Statistical Mechanics - xn--webducation-dbb.com
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©CopyrightKonstantinKLikharev2019
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ISBN978-0-7503-1419-0(ebook)ISBN978-0-7503-1420-6(print)ISBN978-0-7503-1421-3(mobi)
DOI10.1088/2053-2563/aaf504
Version:20190701
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ContentsPrefacetotheEAPSeriesPrefacetoSMProblemswithSolutionsAcknowledgmentsNotation
1 ReviewofthermodynamicsReferences
2 PrinciplesofphysicalstatisticsReferences
3 Idealandnot-so-idealgasesReferences
4 Phasetransitions
5 FluctuationsReferences
6 ElementsofkineticsReferences
Appendices
A Selectedmathematicalformulas
B Selectedphysicalconstants
Bibliography
PrefacetotheEAPSeries
EssentialAdvancedPhysicsEssentialAdvancedPhysics(EAP)isaseriesoflecturenotesandproblemswithsolutions,consistingofthefollowingfourparts1:
PartCM:ClassicalMechanics(aone-semestercourse),PartEM:ClassicalElectrodynamics(twosemesters),PartQM:QuantumMechanics(twosemesters),andPartSM:StatisticalMechanics(onesemester).
Eachpart includestwovolumes:LectureNotesandProblemswithSolutions,andanadditional
fileTestProblemswithSolutions.
Distinguishingfeaturesofthisseriesâinbriefcondensedlecturenotes(âŒ250pppersemester)âmuchshorterthanmosttextbooksemphasisonsimpleexplanationsofthemainnotionsandphenomenaofphysicsafocusonproblemsolution;extensivesetsofproblemswithdetailedmodelsolutionsadditionalfileswithtestproblems,freelyavailabletoqualifieduniversityinstructorsextensivecross-referencingbetweenallpartsoftheseries,whichsharestyleandnotation
LevelandprerequisitesThe goal of this series is to bring the reader to a general physics knowledge level necessary forprofessional work in the field, regardless on whether the work is theoretical or experimental,fundamentalorapplied.Fromtheformalpointofview,thislevel(augmentedbyafewspecialtopiccourses in a particular field of concentration, and of course by an extensive thesis researchexperience)satisfiesthetypicalPhDdegreerequirements.Selectedpartsoftheseriesmaybealsovaluableforgraduatestudentsandresearchersofotherdisciplines,includingastronomy,chemistry,mechanicalengineering,electrical,computerandelectronicengineering,andmaterialscience.TheentrylevelisanotchlowerthanthatexpectedfromaphysicsgraduatefromanaverageUS
college. Inaddition tophysics, theseriesassumes thereaderâs familiaritywithbasiccalculusandvectoralgebra, tosuchanextentthat themeaningof the formulas listed inappendixA,âSelectedmathematicalformulasâ(reproducedattheendofeachvolume),isabsolutelyclear.
OriginsandmotivationTheseriesisaby-productoftheso-calledâcorephysicscoursesâItaughtatStonyBrookUniversityfrom1991to2013.Mymaineffortwastoassistthedevelopmentofstudentsâproblem-solvingskills,ratherthantheiridlememorizationofformulas.(Withacertainexaggeration,mylectureswerenotmuchmorethanintroductionstoproblemsolution.)Thefocusonthismainobjective,undertherigidtime restrictions imposed by the SBU curriculum, had some negatives. First, the list of coveredtheoreticalmethodshadtobelimitedtothosenecessaryforthesolutionoftheproblemsIhadtimeto discuss. Second, I had no time to cover some core fields of physicsâmost painfully generalrelativity2andquantumfieldtheory,beyondafewquantumelectrodynamicselementsattheendofPartQM.Themainmotivationforputtingmylecturenotesandproblemsonpaper,andtheirdistributionto
students,wasmy desperation to find textbooks and problem collections I could use,with a clearconscience, formy purposes. The available graduate textbooks, including the famousTheoreticalPhysics series by Landau andLifshitz, did notmatch theminimalistic goal ofmy courses,mostlybecause they are far too long, and using them would mean hopping from one topic to another,pickingupa chapterhereanda section there, at ahigh riskof losing thenecessarybackgroundmaterialand logicalconnectionsbetween thecoursecomponentsâand thestudentsâ interestwiththem. In addition, many textbooks lack even brief discussions of several traditional and moderntopicsthatIbelievearenecessarypartsofeveryprofessionalphysicistâseducation3.On theproblemside,mostavailablecollectionsarenotbasedonparticular textbooks,and the
problem solutions in them either do not refer to any backgroundmaterial at all, or refer to theincludedshortsetsofformulas,whichcanhardlybeusedforsystematiclearning.Also,thesolutionsarefrequentlytooshorttobeuseful,andlackdiscussionsoftheresultsâphysics.
StyleInanefforttocomplywiththeOccamâsRazorprinciple4,andbeatMalekâslaw5,Ihavemadeeveryefforttomakethediscussionofeachtopicasclearasthetime/space(andmyability:-)permitted,andassimpleasthesubjectallowed.Thisefforthasresultedinrathersuccinctlecturenotes,whichmaybethoroughlyreadbyastudentduringthesemester.Despitethisbriefness,theintroductionofeverynewphysicalnotion/effectandofeverynoveltheoreticalapproachisalwaysaccompaniedbyanapplicationexampleortwo.Theadditionalexercises/problemslistedattheendofeachchapterwerecarefullyselected6,so
thattheirsolutionscouldbetterillustrateandenhancethelecturematerial.Informalclasses,theseproblemsmaybeusedforhomework,whileindividuallearnersarestronglyencouragedtosolveasmanyof themaspracticallypossible.The fewproblemsthatrequireeither longercalculations,ormorecreativeapproaches(orboth),aremarkedbyasterisks.Incontrastwiththelecturenotes,themodelsolutionsoftheproblems(publishedinaseparate
volume for eachpart of the series) aremoredetailed than inmost collections. In some instancestheydescribeseveralalternativeapproachestotheproblem,andfrequentlyincludediscussionsofthe resultsâ physics, thus augmenting the lecture notes. Additional files with sets of shorterproblems (also with model solutions) more suitable for tests/exams, are available for qualifieduniversityinstructorsfromthepublisher,freeofcharge.
DisclaimerandencouragementThe prospective reader/instructor has to recognize the limited scope of this series (hence thequalifierEssentialinitstitle),andinparticularthelackofdiscussionofseveraltechniquesusedincurrenttheoreticalphysicsresearch.Ontheotherhand,Ibelievethattheseriesgivesareasonableintroduction to the hard core of physicsâwhich many other sciences lack. With this hard coreknowledge, todayâs student will always feel at home in physics, even in the often-unavoidablesituations when research topics have to be changed at a career midpoint (when learning fromscratch is terribly difficultâbelieve me :-). In addition, I have made every attempt to reveal theremarkablelogicwithwhichthebasicnotionsandideasofphysicssubfieldsmergeintoawonderfulsingleconstruct.MoststudentsItaughtlikedusingmymaterials,soIfancytheymaybeusefultoothersaswellâ
hencethispublication,forwhichalltextshavebeencarefullyreviewed.
1Note that the (very ambiguous) termmechanics is used in these titles in its broadest sense. The acronymEM stems from another popular name for classicalelectrodynamicscourses:ElectricityandMagnetism.2Foranintroductiontothissubject,IcanrecommendeitherabriefreviewbySCarroll,SpacetimeandGeometry(2003,NewYork:Addison-Wesley)oralongertextbyAZee,EinsteinGravityinaNutshell(2013,PrincetonUniversityPress).3To list just a few: the statics and dynamics of elastic and fluid continua, the basics of physical kinetics, turbulence and deterministic chaos, the physics ofcomputation,theenergyrelaxationanddephasinginopenquantumsystems,thereduced/RWAequationsinclassicalandquantummechanics,thephysicsofelectronsand holes in semiconductors, optical fiber electrodynamics, macroscopic quantum effects in BoseâEinstein condensates, Bloch oscillations and LandauâZenertunneling,cavityquantumelectrodynamics,anddensityfunctionaltheory(DFT).Allthesetopicsarediscussed,ifonlybriefly,inmylecturenotes.4EntianonsuntmultiplicandapraeternecessitateâLatinforâDonotusemoreentitiesthannecessaryâ.5âAnysimpleideawillbewordedinthemostcomplicatedwayâ.6Manyoftheproblemsareoriginal,butitwouldbesillytoavoidsomeoldgoodproblemideas,withlong-lostauthorship,whichwanderfromonetextbook/collectiontoanotheronewithoutreferences.Theassignmentsandmodelsolutionsofallsuchproblemshavebeenre-workedcarefullytofitmylecturematerialandstyle.
PrefacetoSMProblemswithSolutionsThis volume of the EAP series containsmodel solutions of the problems formulated in volume 7,Statistical Mechanics: Lecture Notes. For readerâs convenience, the problem assignments arereproducedinthisvolumeaswell,althoughtheaccompanyingfiguresarefrequentlymoredetailed,extended to explain the solutions. The appendices A (Selected mathematical formulas) and B(Selectedphysicalconstants),commonforallpartsoftheseries,arealsoincludedinthisvolume.
Since the whole series is strongly focused on the development of problem solution skills, themodelsolutionsareratherdetailed,andinsomecases(particularlyinthemoredifficultproblems,markedbyasterisks)extendand/orenhancethelecturematerial.
Numberingofformulaswithineachsolutionislocal,byasterisks;referencestoformulasinothersolutions are clearly indicated. The solutions also have numerous references to formulas in thelecturenotesofthis(SM)partoftheEAPseries,andoccasionallythoseinotherpartsoftheseries.Inthelattercase,theacronymofthepartisincludedintothereference.
A file with 25 additional problems, which allow shorter solutions and hence are suitable forexams (also with model solutions) is available to university instructors from the publisher byrequest.
Theauthortriedhardtoeliminateallerrorsinthesolutions,buttheyhavenotpassedarigorousreviewbyqualifiedothers,andarepresentedherewithoutwarranty.
AcknowledgmentsIamextremelygratefultomyfacultycolleaguesandotherreadersofthepreliminary(circa2013)versionofthisseries,whoprovidedfeedbackoncertainsections;heretheyarelistedinalphabeticalorder7:AAbanov,PAllen,DAverin,SBerkovich,P-TdeBoer,MFernandez-Serra,RFHernandez,AKorotkov,VSemenov,FSheldon,andXWang.(Obviously,thesekindpeoplearenotresponsibleforanyremainingdeficiencies.)
Alargepartofmyscientificbackgroundandexperience,reflectedinthesematerials,camefrommy education, and then research, in theDepartment of Physics ofMoscowStateUniversity from1960 to 1990. The Department of Physics and Astronomy of Stony Brook University provided acomfortableandfriendlyenvironmentformyworkduringthefollowing25+years.
Lastbutnotleast,IwouldliketothankmywifeLioudmilaforallherlove,care,andpatienceâwithoutthese,thiswritingprojectwouldhavebeenimpossible.
I know very well that my materials are still far from perfection. In particular, my choice ofcoveredtopics(alwaysverysubjective)maycertainlybequestioned.Also, it isalmostcertainthatdespiteallmyefforts,notalltyposhavebeenweededout.Thisiswhyallremarks(howevercandid)and suggestions from readers will be greatly appreciated. All significant contributions will begratefullyacknowledgedinfutureeditions.
KonstantinKLikharevStonyBrook,NY
7IamverysorryfornotkeepingproperrecordsfromthebeginningofmylecturesatStonyBrook,soIcannotlistallthenumerousstudentsandTAswhohavekindlyattractedmyattentiontotyposinearlierversionsofthesenotes.Needlesstosay,Iamverygratefultoallofthemaswell.
Notation
Abbreviations Fonts Symbols
c.c.complexconjugate F, scalarvariables8 .timedifferentiationoperator(d/dt)
h.c.Hermitianconjugate F, vectorvariables âspatialdifferentiationvector(del)
scalaroperators âapproximatelyequalto
vectoroperators âŒofthesameorderas
Fmatrix âproportionalto
FjjâČmatrixelement âĄequaltobydefinition(orevidently)
â scalar(âdot-â)product
Ăvector(âcross-â)product__timeaveraging
âšâ©statisticalaveraging
[,]commutator
{,}anticommutator
PrimesignsTheprimesigns(âČ,âł,etc)areusedtodistinguishsimilarvariablesorindices(suchasjandjâČinthematrixelementabove),ratherthantodenotederivatives.
PartsoftheseriesPartCM:ClassicalMechanicsââPartEM:ClassicalElectrodynamicsPartQM:QuantumMechanicsââPartSM:StatisticalMechanics
AppendicesAppendixA:SelectedmathematicalformulasAppendixB:Selectedphysicalconstants
FormulasTheabbreviationEq.maymeananydisplayedformula:eithertheequality,orinequality,orequation,etc.
8Thesameletter,typesetindifferentfonts,typicallydenotesdifferentvariables.
(*)
IOPPublishing
StatisticalMechanicsProblemswithsolutionsKonstantinKLikharev
Chapter1
Reviewofthermodynamics
Problem1.1.Twobodies,with temperature-independentheatcapacitiesC1andC2, anddifferentinitialtemperaturesT1andT2,areplacedintoaweakthermalcontact.Calculatethechangeofthetotalentropyofthesystembeforeitreachesthethermalequilibrium.
Solution:Duetothethermalcontactweakness,eachbodyisclosetointernalthermalequilibriumatall times,so thatwecanuseEq. (1.19)of the lecturenotes todescribe thechangeof itsentropyduringthetransferofanelementaryheatdQjtoit:
dSj=dQjTjâČ,withj=1,2.(Hereandbelowtheprimesignmarksintermediate,instanttemperaturesofthebodies,inordertodistinguish them from the initial values specified in the assignment.) On the other hand, by thedefinitionoftheheatcapacity,forthesamedQjwemaywrite
dQj=CjdTjâČ.
Combining these two relations, and integrating the result through the whole temperatureequilibrationprocess,weget
ÎS=â«inifindS1+dS2=C1â«T1TfindT1âČT1âČ+C2â«T2TfindT2âČT2âČ=C1lnTfinT1+C2lnTfinT2,whereTfinisthefinal,commontemperatureofthesystem.Thistemperaturemaybecalculatedfromtheenergyconservationlaw:
dQ1+dQ2=0,i.e.C1dT1âČ+C2dT2âČ=0.Theintegrationofthelastrelationthroughthewholeprocessyields
C1(TfinâT1)+C2(TfinâT2)=0.Fromhere,weget
Tfin=C1T1+C2T2C1+C2,sothat,finally,Eq.(*)yields
ÎS=C1lnC1T1+C2T2(C1+C2)T1+C2lnC1T1+C2T2(C1+C2)T2.
An analysis of this formula (see, e.g. the figure below) shows that ifC1,2 > 0, the change ofentropyispositiveforanyparametersofthesystemâbesidesthetrivialcaseT1=T2,whenthereisnoheatflowatall,andhenceÎS=0.
Problem1.2.Agasportionhasthefollowingproperties:
(i)itsheatcapacityCV=aTb,and(ii)thework necessaryforitsisothermalcompressionfromV2toV1equalscTln(V2/V1),
wherea, b, and c are some constants. Find the equation of state of the gas, and calculate thetemperaturedependenceofitsentropySandthermodynamicpotentialsE,H,F,GandΩ.
Solution:UsingEq.(1.1)ofthe lecturenotes,andthenthecondition(ii)ofproblemâsassignment,weget
sothattheequationofstatecoincideswiththatofanidealgas(seeEq.(1.44)ofthelecturenotes)withN=c.HencewecanuseEqs.(1.45)â(1.50)ofthenotes,withthissubstitution,tofinalizethesolution.Inparticular,comparingEq.(1.50)andthecondition(i)oftheassignment,weobtain
d2fdT2=â1NTCV=âacTbâ1.Integratingthisequalitytwice,weget
dfdT=âabcTb+d,f=âab(b+1)cTb+1+dT+g,i.e.fâTdfdT=a(b+1)cTb+1+g,wheredandgaresomenewconstants.Now,Eqs.(1.45)â(1.49)give
S=NlnVNâdfdt=clnVc+abTbâcd,F=âNTlnVN+Nf(T)=âcTlnVcâab(b+1)Tb+1+cdT+g,E=NfâTdfdT=ab+1Tb+1+cg,HâĄE+PV=ab+1Tb+1+cg+T,GâĄF+PV=âcTlnVc
âab(b+1)Tb+1+c(d+1)T+g,Ω=âPV=âcT.
Note that all thermodynamic potentials (but Ω) are still determined up to some arbitraryconstants.
Problem1.3.AclosedvolumewithanidealclassicalgasofsimilarmoleculesisseparatedwithapartitioninsuchawaythatthenumberNofmoleculesinbothpartsisthesame,buttheirvolumesaredifferent.Thegas is initially in thermalequilibrium,and itspressure inonepart isP1, and inanother,P2. Calculate the change of entropy resulting from a fast removal of the partition, andanalyzetheresult.
Solution:Before the removalof thepartition, the totalentropy (asanextensiveparameter) is thesumofentropiesoftheparts,Sini=S1+S2,sothatEq.(1.46)ofthelecturenotesyields
Sini=NlnV1NâdfdT+NlnV2NâdfdT=NlnT2P1P2â2dfdT,where the last form is obtained by using the equation of state,V/N =T/P, for each part of thevolume. At a fast gas expansion, we may neglect the thermal exchange of the gas with itsenvironment. Also, at a fast removal of the partition (say, sideways), the gas cannot perform anymechanicalworkonit.Asaresultofthesetwofactors,thegasâenergyisconserved.AccordingtoEq. (1.47) of the lecture notes, this means that the gas temperature T is conserved as well. Inaddition, the total number of molecules (2N) is also conserved. Because of that, we may use Eq.(1.44)tocalculatethefinalpressureofthegas,afterthepartitionâsremoval,as
Pfin=2NTV1+V2=2P1P2P1+P2.NowapplyingEq.(1.46)again,withthesameT,wecancalculatethefinalentropyas
Sfin=2NlnV1+V22NâdfdT=2NlnTPfinâdfdT=2Nln(P1+P2)T2P1P2âdfdTâĄNln(P1+P2)2T24P12P22â2dfdT.
Hence,itschangeduringtheexpansion,ÎSâĄSfinâSini=Nln(P1+P2)24P1P2,
doesnotdependontemperatureexplicitly.Asasanitycheck,ourresultshowsthatifP1=P2,theentropydoesnotchange.Thisisnatural,
(*)
(**)
because the partitionâs removal from a uniform gas of similar molecules has no macroscopicconsequences.Foranyotherrelationof the initialpressures, this irreversibleprocess results inagrowthoftheentropy.
Problem1.4.AnidealclassicalgasofNparticlesisinitiallyconfinedtovolumeV,andisinthermalequilibriumwithaheatbathoftemperatureT.ThenthegasisallowedtoexpandtovolumeVâČ>Vinonethefollowingways:
(i)Theexpansionisslow,sothatduetothesustainedthermalcontactwiththeheatbath,thegastemperatureremainsequaltoTallthetimes.(ii)Thepartitionseparating thevolumesVand(VâČâV) is removed very fast, allowing thegas toexpandrapidly.
Foreachprocess,calculatetheeventualchangesofpressure,temperature,energy,andentropy
ofthegas,resultingfromitsexpansion.
Solutions:
(i)Thefirstprocessisisothermal,attemperatureT,sothatforthefinalpressurePâČtheequationofstate(1.44)gives
P=NTV,PâČ=NTVâČ,i.e.ÎPâĄPâČâP=NT1VâČâ1V.Evidently,thereisnochangeoftemperature,sothattheenergyofthegas,which,accordingtoEq.(1.47)ofthelecturenotes,isafunctionoftemperaturealone,doesnotchangeeither.Sinceattheexpansion,thegasdoesperformanonvanishingmechanicalwork(say,onthepistonthatmoderatestheexpansionspeedtokeeptheprocessisothermal):
thisenergylosshastobeexactlycompensatedbytheheat ,transferredfromtheheatbath.Thisequalityallowsus tocalculate thechangeof theentropyduring theprocess,usingEq.(1.20)withT=const:
ÎS=ÎQT=NlnVâČV>0.(ThesameexpressionfollowsfromEq.(1.46)ofthelecturenotes,withconstantdf/dTâwhich isafunctionofTalone.)
(ii)Thesecond,fastexpansionisirreversible,withouttimeforanyheattransfer,sothatÎQ=0,andwithoutperforminganymechanicalwork, .(Atafreeexpansion,thereisnopistontomove.)Hence,accordingtoEq.(1.18)ofthelecturenotes,theinternalenergyEofthegascannotchange:ÎE = 0. Now using Eq. (1.47) again, we may conclude that the gas temperature cannot changeeither,ÎT=0.1Ontheotherhand,accordingtoEqs.(1.44)and(1.46),thegaspressureandentropyaredeterminedbythecurrentstateofthegasratherthanbythewayithasbeenreached,sothattheirchangesaredescribedbythesamerelations(*)and(**).
Note,however,thatincontrasttothefirst,slowandreversibleprocess,atwhichthenetentropy
ofthegasandtheheatbathdoesnotchange,thesecond,fastprocessisirreversible,withthenetentropyrisingbytheÎSgivenbyEq.(**).Notealsothatsincethegastemperaturedoesnotchangeineitherofthesecases,alltheaboveresultsarevalidregardlesswhethertheheatcapacityofthegasdependsonT.
Problem 1.5. For an ideal classical gas with temperature-independent specific heat, derive therelationbetweenPandVattheadiabaticexpansion/compression.
Solution:AccordingtoEq.(1.50)ofthelecturenotes,d2fdT2=âcVT.
wherecVâĄCV/Nisthespecificheat,namelytheheatcapacityperunitparticle.IfCVistemperature-independent,soiscV,sothatintegratingbothsidesoftheaboveequationovertemperature,weget
dfdT=âcVlnT+a,whereaisanothertemperature-independentconstant.Aswasdiscussedinsection1.3ofthelecturenotes,atanadiabaticprocesstheentropyshouldbeconstant,andhenceEq.(1.46)yields
lnVNâdfdTâĄlnVN+cVlnTâaâĄlnVNTcVâa=const;here and below âconstâ means various amounts remaining constant during the adiabaticexpansion/compression.Thisrelationyieldsthefollowinglawoftemperaturechangewithvolume:
VNTcV=const.
Now using the equation of state (1.44), rewritten as T = PV/N, we finally get the requiredrelation,
PcVVNcV+1=const.Itistraditionallyrepresentedintheform
PVγ=const,where the constantγ⥠(cV+1)/cV, according to Eq. (1.51), is the specific heat (and hence heatcapacity)ratio:
ÎłâĄcV+1cVâĄCV+NCV=CPCV.
PleasenoteagainthattheseresultsareonlyvalidifCV,andhenceCP=CV+N,aretemperature-independent.
Problem 1.6. Calculate the speed and the wave impedance of acoustic waves propagating in anidealclassicalgaswithtemperature-independentspecificheat, inthe limitswhenthepropagationmaybetreatedas:
(i)theisothermalprocess,and(ii)theadiabaticprocess.
Whichoftheselimitsisachievedathigherwavefrequencies?
Solution: As classical mechanics shows2, the speed and wave impedance of a longitudinalacousticwaveinafluid(i.e.amediumwithvanishingshearmodulusΌ)are
whereÏ is the volumic mass density of the fluid:ÏâĄM/V=mN/V (wherem is the mass of oneparticle),andKisitsbulkmodulus(reciprocalcompressivity),whichmaybedefinedas
KâĄâVâPâVX,whereXistheparameterkeptconstantatfluidâsexpansion/compression.Intypicalliquids,Kisveryhigh,anddoesnotdependmuchonwhatXis;however,ingasesthedifferenceissubstantial.
(i)AccordingtoEq.(1.44)ofthelecturenotes,attheisothermalprocess(X=T=const),P=NT/V,sothat
wherenâĄN/Vistheparticledensity.Notethat doesnotdependonthestaticcompressionofthegas.Also,aswillbediscussedinchapter2,this exactlycoincideswiththermsvelocityofthegasparticlesinanydirection.
(ii)Aswasdiscussedinthemodelsolutionofthepreviousproblem,attheadiabaticprocess(X=S),pressuredependsonvolumedifferently:P=fVâÎł,whereÎłâĄCP/CV=(cV+1)/cV,andthefactor fdoesnotdependonV,sothatthedifferentiationyields
Since,bydefinition,Îł>1,theseresultsshowthattheacousticwavevelocityandimpedancein
theadiabaticcasearealwayslargerthanthoseintheisothermalcase.Practically,thelattercaseisimplemented only at extremely low frequencies (where the waveâs period is long enough to givetemperature enough time to equilibrate over the size of the system), so that at the usual soundfrequencies,andatambientconditions,onlytheadiabaticresultisrealistic.
Problem1.7.Aswillbediscussedinsection3.5ofthelecturenotes,theso-calledâhardballâmodelsofclassicalparticleinteractionyieldthefollowingequationofstateofagasofsuchparticles:
P=TÏn,wheren=N/Vistheparticledensity,andthefunctionÏ(n)isgenerallydifferentfromthat(Ïideal(n)= n) of the ideal gas, but still independent of temperature. For such a gas, with temperature-independentcV,calculate:
(i)theenergyofthegas,and(ii)itspressureasafunctionofnattheadiabaticcompression.
Solutions:
(i)Firstofall,letusnotethatatN=const,dnâĄdNV=âNdVV2,sothatdV=âV2NdnâĄâNdnn2.
Now, justaswasdone in section1.4of the lecturenotes for the idealgas,wecan startwith thecalculationofthefreeenergy:
F=ââ«PdVN,T=const=âTâ«ÏndVN,T=const=TNΊn+NfT,whereΊnâĄâ«Ïndnn2,andproceedtothecalculationoftheentropyandtheinternalenergy:
S=ââFâTN,V=âNΊn+dfdT,E=F+TS=NfâTdfdT.(ii)NotethatthecalculatedrelationbetweenEandf(T)isabsolutelythesameasfortheidealgasâseeEq.(1.47)ofthelecturenotes.Asaresult,cVisalsoexpressedbythesameEq.(1.50),giving
d2fdT2=âcVT.Since,accordingtotheassignment,cV istemperature-independent,theintegrationofthisequalityovertemperatureyields
dfdT=âcVlnT+const.Now,justaswasdoneinthesolutionofproblem1.5,therequirementoftheentropyâsconstancyattheadiabaticcompression(atconstantN)yields
Ίn+dfdt=ΊnâcVlnTâĄlnexpΊnTcV=const,i.e.TcV=constĂexpΊn,and plugging T expressed from the given equation of state, T = P/Ï(n), we get the requiredexpression:
P=constĂÏnexpΊn1/cVâĄconstĂÏnexpâ«Ïndnn21/cV.
(*)
(**)
(***)
(****)
Asasanitycheck,foranidealgas,Ï(n)=n,Ί(n)=lnn,exp{Ί(n)}=n,andtheaboveresultis
reducedtoP=constĂn(n)1/cVân(1+1/cV)=(N/V)(cV+1)/cV,
i.e.totheresultofproblem1.5:
Problem1.8.Foranarbitrary thermodynamicsystemwitha fixednumberofparticles,prove thefollowingMaxwellrelations(mentionedinsection1.4ofthelecturenotes):
i:âSâVT=âPâTV,ii:âVâSP=âTâPS,iii:âSâPT=ââVâTP,iv:âPâSV=ââTâVS,andalsothefollowingrelation:
âEâVT=TâPâTVâP.Solution:ThemixedpartialsecondderivativeofthefreeenergyF(T,V)mayberepresentedintwoequivalentforms:
ââVâFâTVT=ââTâFâVTV.ButaccordingtoEqs.(1.35)ofthelecturenotes,theinternalderivativeontheleft-handsideofthisequalityisjust(âS),whilethatintheright-handsideisjust(âP),thusprovingEq.(i).
The remaining three Maxwell relations may be proved absolutely similarly, applying similarargumentstothepartialderivativesofthefollowingthermodynamicpotentials:
(ii)H(P,S)âseeEqs.(1.31);(iii)G(P,T)âseeEqs.(1.39);and(iv)E(S,V)âseeEqs.(1.9)and(1.15).
NowletusdividealltermsofEq.(1.17),
dE=TdSâPdV,bydV,fortheparticularcasewhenallthesesmallchangesareperformedatconstanttemperature.Theresultis
âEâVT=TâSâVTâP.NowusingEq.(i),wegetthelastprovidedequalityproved.
Note that there are quite a few other similar thermodynamic relations for E and otherthermodynamicpotentials,whichmaybeprovedsimilarly3.
Problem1.9.Expresstheheatcapacitydifference,CPâCV,viatheequationofstateP=P(V,T)ofthesystem.
Solution:Subtractingthetwoexpressionsderivedintheveryendofsection1.3ofthelecturenotes,weget
CPâCV=TâSâTPââSâTV,sothatweonlyneedtoexpresstheright-handsideofthisrelationviatheequationofstate.
TheentropySofagivensystem(inparticular,withagivennumberNofparticles)iscompletelydefined by its volume V and temperature T, and hence may be considered a function of twoindependentarguments,VandT.Henceitsfulldifferentialmaybeexpressedas
dS=âSâVTdV+âSâTVdT.Ontheotherhand,thesameargumentsVandTuniquelydeterminepressurePviatheequationofstate.HencewemayalternativelyconsidertheentropyasafunctionofPandT,andrepresentthesamedifferentialinanotherform,
dS=âSâPTdP+âSâTPdT.
ThethreedifferentialsdV,dP,anddTinEqs.(**)and(***)arenotfullyindependent,butrelatedbytheequationofstateP=P(V,T),giving
dP=âPâVTdV+âPâTVdT.PluggingthisdPintoEq.(***),andthenrequiringdSgivenbytheresultingexpressiontobeequaltothatgivenbyEq.(**),weget
âSâVTdV+âSâTVdT=âSâPTâPâVTdV+âPâTVdT+âSâTPdT.This equality has to be satisfied at arbitrary (sufficiently small) changes dV and dT of the twoindependentargumentsVandT.Thisrequirementyieldsthefollowingtwoequalitiesforthepartialderivatives:
âSâVT=âSâPTâPâVT,âSâTV=âSâPTâPâTV+âSâTP.Eliminating the partial derivative (âS/âP)T from the system of these two relations, we get thefollowingexpressionforthedifferencestandinginthesquarebracketsinEq.(*)
âSâTPââSâTV=ââSâVTâP/âTVâP/âVT.NowusingtheMaxwellrelationwhoseproofwasthefirsttaskofthepreviousproblem,
âSâVT=âPâTV,toeliminatetheentropyfromtheright-handsidecompletely,wefinallyget
CPâCV=âTâP/âTV2âP/âVT.
Asasanitycheck,fortheidealclassicalgas,whoseequationofstateisgivenbyEq.(1.44)ofthelecturenotes,P=NT/V,andhence
âPâTV=NV,âPâVT=âNTV2,
(*)
(**)
(*)
Eq.(****)isreducedtothesameresult,CPâCV=N,
whichwasobtainedinthelecturenotesinadifferentwayâseeEq.(1.51).More generally, the derivative (âP/âV)T has to be negative for the mechanical stability of the
system4, so thatEq. (****) confirms the inequalityCPâCV > 0, which was already mentioned insection1.3ofthelecturenotes.NotealsothataccordingtoEq.(****),formaterialswithverylowbulkcompressibilityâ(âV/âP)/V,5suchasmostsolidsand liquids, thedifferencebetweentwoheatcapacitiesismuchlowerthanCVandCPassuch, justifyingthenotionof âheatcapacityCâwithoutspecifyingtheconditionsofitsmeasurementâaswasdone,forexample,inproblem1.1,andwillberepeatedlydoneonotheroccasionsinthiscourse.
FinallynotethatifwerepresentedtheequationofstateinthealternativeformV=V(P,T), andthenactedabsolutelysimilarly,wewouldgetanequivalentexpression6:
CPâCV=TâV/âTP2âV/âPT,butconceptuallytheequalityP=P(V,T),andhenceEq.(****),haveamoredirectphysicalsense.
Problem1.10.Provethattheisothermalcompressibility,definedasÎșTâĄâ1VâVâPT,N,
inasingle-phasesystemmaybeexpressedintwodifferentways:ÎșT=V2N2â2PâÎŒ2T=VN2âNâÎŒT,V.
Solution:CombiningEq.(1.60)ofthelecturenotesforthegrandcanonicalpotential,Ω=âPV,andEq.(1.61)foritsfulldifferential,forthecaseofconstanttemperature(dT=0)weget
dâPV=âPdVâNdÎŒ.Afterthedifferentiationintheleft-handside,andthecancellationofâPdV,wegetsimply
VdP=NdΌ,forT=const.Thisrelation7meansthatwemaywrite
âPâÎŒT=NV,regardlessofwhetherthevolumeVofthesystem,orthenumberNofparticlesinit(ormaybesomecombinationofthetwo)isfixed.
NowletususeEq.(*)totransformthesecondderivativeparticipatinginthefirstequalitytobeproved:
â2PâÎŒ2TâĄââÎŒâPâÎŒTT=ââÎŒNVT.Intheparticularcasewhenthenumberofparticlesisfixed,wemaycontinueas
â2PâÎŒ2T=NââÎŒ1VT,N=âNV2âVâÎŒT,N.Ina systemwith fixedTandN, the state of the system, inparticular bothV andÎŒ, areuniquelydefinedbypressureP,sothatwemaycontinueevenfurtheras
â2PâÎŒ2T=âNV2âVâPT,NâPâÎŒT,N=âNV2âVâPT,NNVâĄâ1VâVâPT,NN2V2,whereatthesecondstep,Eq.(*)wasusedagain.Butthelastexpression,besidesthelastfraction,bydefinitionistheisothermalcompressibility,thusgivingusthefirstrelationwehadtoprove:
ÎșT=V2N2â2PâÎŒ2T.
NowletustransformthisexpressionbyusingEq.(**)again:ÎșT=V2N2ââÎŒNVT.
Performingthedifferentiationforthecasewhenthevolumeratherthanthenumberofparticlesisfixed,wegetthesecondrelationinquestion:
ÎșT=VN2âNâÎŒT,V.
Thisformulaisuseful,inparticular,foraconvenientrepresentationofthestatisticalfluctuationsofthenumberofparticlesinthesystemwithfixedT,V,andÎŒâseechapter5below.
Problem 1.11. A reversible process, performed with a fixed portion of an ideal gas, may berepresentedonthe[P,V]planewiththestraight lineshowninthefigurebelow.Findthepointatwhichtheheatflowinto/outofthegaschangesitsdirection.
Solution:AccordingtothebasicEqs.(1.25)ofthelecturenotes,theelementaryheatdQtransferredtothegasduringanelementarychangedVofvolumeis
dQ=dE+PdV.AccordingtoEq.(1.47),theinternalenergyEofanidealgasdependsonlyonitstemperatureT,sothatwemaycontinueasfollows,usingEq.(1.22)withCV=NcV,and,atthenextstep,theequationofstate(1.44):8
dQ=cVNdT+PdV=cVdPV+PdVâĄcV+1PdV+cVVdP.
InordertoeliminatedP,weneedtousethefunctionP(V)forthisparticularprocess.Astraightlineonthe[P,V]planealwaysrepresentsalinearfunction,
P=aâbV,whereaandbareconstants.CalculatingthemfromthevaluesofPandV intwoboundarypointsshowninthefigureabove,weget
P=P05â4VV0,sothatdP=â4P0V0dV.PluggingtheseexpressionsintoEq.(*),weget
dQ=(cV+1)P05â4VV0dVâcVV4P0V0dVâĄP05(cV+1)â4(2cV+1)VV0dV.Thisdifferentialturnstozero(andhencetheheatflowintothegaschangesforthatoutofthegas)at
V=5(cV+1)4(2cV+1)V0,andhenceP=P05â4VV0=5cV2cV+1P0.
Twocomments.First,notethat thiscalculationdidnotrequiretheassumptionof temperature-independentspecificheatâafactabitcounter-intuitiveforaprocessinwhichthegastemperatureisevidentlychanged.
Second,thecalculatedpoint{P,V}isnotnecessarilyreachedatthepartoftheprocessshowninthefigureabove;thatrequirestheratioV/V0tobebetweenÂœand1,i.e.
12⩜5(cV+1)4(2cV+1)⩜1.
Thefirstoftheserelationsissatisfiedforanypositive(i.e.realistic)valuesofcV,whilethesecondonerequirescVâ©Ÿ1/3,and,aswillbediscussedinsection3.1ofthelecturenotes,isalsosatisfiedinmostmodelsofidealclassicalgasesâsee,e.g.Eq.(3.31).
Problem 1.12. Two bodies have equal, temperature-independent heat capacitiesC, but differentinitialtemperatures:T1andT2.Calculatethelargestmechanicalworkobtainablefromthissystem,usingaheatengine.
Solution:ThelargestworkmaybeextractedbyusingthebodiesasthehotandcoldheatbathsofaCarnotheatengine,witheachcycletakingjustasmallportionofheat,dQHâȘCTH,fromthehotterbody, and passing a smaller amount of heat, dQL < dQH, to the colder body (while turning thedifferencedQHâdQLintothemechanicalwork ).Eachenginecyclecoolstheformerbodyandheatsupthelatterbodyjustabit:
dTH=âdQHC,dTL=dQLC.Sinceeachsuchchangeissmall,wemayalsouseEq.(1.66)ofthelecturenotes,
dQHdQL=THTL,which has been derived for constant TH and TL. Combining these relations, we see that thetemperaturechangesobeythefollowingrule:
dTHdTL=âTHTL.
Integratingthisrelation,rewrittenasdTHTH+dTLTL=0,i.e.d(lnTH+lnTL)âĄdln(THTL)=0,
weseethatthroughtheprocesstheproductTHTLremainsconstant,sothatusingtheinitialvaluesoftemperatureT1andT2,weget
THTL=T1T2=const.
Thisformulameans, inparticular,thattheprocesstendstoTHâTLâTfin=(T1T2)1/2,andthatthisfinaltemperatureisdifferentfromtheTâČfin=(TH+TL)/2thatwewouldgetatthedirectthermalcontactofthebodies(withnomechanicalworkdoneatall). Inordertofindthetotalworkinourcase,wemayintegratetheCarnotâsrelation(1.68)throughthewholeprocess:
TheresultshowsthatonlyinthelimitT1/T2ââ,doestheworktendtoCT1, i.e. to the full initialheatcontentsofthehotterbody.
Another, shorter (but also more formal and hence less transparent) way to derive the sameresults is to note that the heat does not leave the (two bodies + engine) system, so that themechanicalworkhastobeequaltothesumofthechangesofthethermalenergiesofthebodies:
andthencalculateTfinfromtheconditionthatthetotalentropyofthesystemstaysconstant(asitshouldatareversibleprocesssuchastheCarnotcycle):ÎS=ÎS1+ÎS2=â«T1TfindQ1T+â«T2TfindQ2T=Câ«T1TfindTT+Câ«T2TfindTT=ClnTfinT1+lnTfinT2âĄClnTfin2T1T2=0,givingthesameTfin=(T1T2)1/2,andhencethesamefinalresultfor .
Problem 1.13. Express the efficiency η of a heat engine that uses the so-called Joule cycle,consistingoftwoadiabaticandtwoisobaricprocesses(seethefigurebelow),viatheminimumand
(**)
maximumvaluesofpressure,andcomparetheresultwithηCarnot.Assumeanidealclassicalworkinggaswithtemperature-independentCPandCV.
Solution:Letusnumbertheprocessjunctionpointsasshowninthefigureabove.Thework( )performedbythegasatanyadiabaticprocess(suchas1â2and3â4)equalsâÎE,becauseÎQ=0.Sinceforanidealclassicalgasofafixednumberofparticles,theinternalenergyisafunctionoftemperaturealone(seeEq.(1.47)ofthelecturenotes),wemaycalculateitschangejustas
ÎE=â«dETdTdTâĄâ«CVdT,despitethefact that thevolumechangesat theadiabaticprocess.So, ifCV isconstant,ÎE is justCVÎT.Next,theworkatanyisobaricprocess,withP=const(suchas2â3and4â1inthefigureabove)issimplyPÎV.Asaresult,thetotalmechanicalworkperformedduringthecycleis
Afterpluggingintheequationofstate,PV=NT,andusingEq.(1.51)intheformCV=CPâN,thisexpressionbecomes
TheheatintakeQHfromthehotbathtakesplaceonlyattheisobaricprocess2â3,andisequal
toCP(T3âT2),sothattheengineefficiency
In order to express this result via the given values P1 andP2, we may combine the result of
problem 1.5 (PVÎł = const), and the equation of state (PV = NT) to get the relation betweentemperatureandpressureatanyadiabaticprocess:
T=constĂP(Îłâ1)/Îł,withÎłâĄCPCVâĄcV+1cV,i.e.Îłâ1Îł=1cV+1.ApplyingthisresulttotwoadiabaticprocessesoftheJoulecycle(3â4and1â3),andusingtherelationsP1=P4=PminandP2=P3=Pmax(evidentfromthefigureabove),weget
T1=T2PminPmaxÎłâ1/Îł,T4=T3PminPmaxÎłâ1/Îł.Nowpluggingtheserelationsintotheright-handsideofEq.(*),weseethatthedifferences(T3âT2)inthenumeratoranddenominatorcancel,givingusaverysimplefinalresult,
η=1âPminPmaxÎłâ1/ÎłâĄ1âPminPmax1/cV+1.
InordertocomparethisformulawithEq.(1.68)fortheCarnotcycle,itisbettertousetheaboverelationbetweenPandTattheadiabaticprocessagaintorecastEq.(**)inthetemperatureform:
η=1âT1T2=1âT4T3.Ofthefournumberedtemperaturepointsofthecycle(seethefigureabove),T3 isthelargestone(Tmax),whileT1 is the lowestone (Tmin),so that the fractions in thoseequalitiesarealways largerthanTmin/Tmax,andhence
η⩜1âTminTmax=ηCarnot,asitshouldbeforanycycle.
Problem1.14.CalculatetheefficiencyofaheatengineusingtheOttocycle9,whichconsistsoftwoadiabaticand two isochoric (constant-volume) reversibleprocessesâsee the figurebelow.Explorehow the efficiency depends on the ratio r ⥠Vmax/Vmin, and compare it with the Carnot cycleâsefficiency.Assumeanidealclassicalworkinggaswithtemperature-independentheatcapacity.
Solution:ThesolutionisverysimilartothatofthepreviousproblemfortheJoulecycle.Numberingtheprocesschangepointsasshowninthefigureabove,duetothespecificheatconstancy,forthe
(**)
isochoric processes with no mechanical work is performed by the working gas, andhencewithdQ=dE=CVdT,wemaywrite
QH=CV(T3âT2),QL=CV(T4âT1),so that for thecycleâs efficiencyasa functionofprocess junction temperatures,weget the sameexpressionasfortheJoulecycle:
NowpluggingtherelationP=NT/V,followingfromtheequationofstateofanidealgas,intothe
resultofproblem1.5fortheadiabaticprocess,PVÎł=const,wegetTV(Îłâ1)=const.Applyingthisrelationtotheprocesses3â4and1â2(withthesamevolumeratio),wegetsimilarresultsfortheirtemperatureratios:
T2T1=V1V2Îłâ1=rV0V0Îłâ1=rÎłâ1,T3T4=V4V3Îłâ1=rV0V0Îłâ1=rÎłâ1.UsingthererelationstoeliminateT3andT2fromEq.(*),wegetthefollowingfinalresult:
η=1âT4âT1T4rÎłâ1âT1rÎłâ1=1â1rÎłâ1.
SinceÎł>1byitsdefinition,i.e.Îłâ1>0,andr>1(seethefigureabove),thedenominatorinthelastformoftheresultisalwayspositiveandlargerthanone,sothattheefficiencyisbetween0and1âasitshouldbe.Inparticular,atrâ1(averyânarrowâcycle)thedenominatortendsto1aswell,sothatηâ0.Thisisnatural,becausetheusefulwork,proportionaltocycleâsareaonthe[P,V]plane,tendstozerointhislimit.Ontheotherhand,asrgrows,sodoesthedenominator,sothatηâ1.
In order to understand whether such efficiency increase can make it higher than that of theCarnotcyclewiththesameminimalandmaximaltemperatures,wemayuseEqs.(**)torecastourresultintwootherforms:
η=1âT1T2=1âT4T3.Sincetemperaturedropsattheadiabaticexpansion,thefigureaboveshowsthatT2<T3=TmaxandT1=Tmin<T4.Asaresult,eitherofthesetwoformulasforηshowsthat,foranyr,theOttocycleâsefficiencyisalwayslowerthanηCarnot=1âTmax/Tmin.
Problem1.15.Aheatengineâscycleconsistsoftwoisothermal(T=const)andtwoisochoric(V=const)reversibleprocessesâseethefigurebelow10.
(i)AssumingthattheworkinggasisanidealclassicalgasofNparticles,calculatethemechanicalworkperformedbytheengineduringonecycle.(ii)Arethespecifiedconditionssufficienttocalculatetheengineâsefficiency?
Solutions:
(i)Inthiscycle,mechanicalworkisperformedonlyattheisothermalprocesses,inwhichP=NT/V=const/V,sothatthetotalwork
(ii)Inordertocalculatetheefficiency ,wewouldneedtoknowalsotheheatQH takenfromthehotbath.Theheatconsistsoftheisothermal-stagepartÎQT,whichmaybeexpressedbythe first of Eqs. (1.65) of the lecture notes, and an isochoric stage part ÎQV. Let us assume thatduringtheisochoricheating(fromTLtoTH)theworkinggasisbroughtincontactonlywiththehotbath(whichisasmartthingtodotoavoidthedirecttransferofheatbetweentwoheatbaths);thenÎQV=ÎE.ThenwemayuseEqs.(1.46)and(1.47)towrite
QH=ÎQT+ÎE=TH(S2âS1)+E(T)TLTH=NTHlnV2V1+Nf(T)âTdf(T)dTTLTH.
Hence,thecalculationoftheefficiencyηwouldrequire,besidestheaboveassumptions,toknowthefunctionf(T)thatcharacterizestheinternaldegreesoffreedomofthegas,oralternatively,theheatcapacityCV(T)âseeEq.(1.50),whichhavenotbeengivenintheassignment.Theonlyevidentfact is thatwithout the second term in the right-handpart of the last relation, i.e. atÎQV=0,ηwouldbeequaltoηCarnot,butinthepresenceofthisterm(whichisnevernegative),theactualQHishigher,i.e.thecycleâsefficiencyislower.Forexample,fortheidealclassicalgaswithoutthermally-activatedinternaldegreesoffreedom,wemayborrowEq.(3.19),tobederivedinsection3.1ofthe
lecturenotes,togetQH=NTHlnV2V1+32N(THâTL),
sothat
Problem1.16.TheDieselcycle(anapproximatemodeloftheDieselinternalcombustionengineâsoperation)consistsoftwoadiabaticprocesses,oneisochoricprocess,andoneisobaricprocessâseethefigurebelow.Assuminganidealworkinggaswithtemperature-independentCVandCP,expresstheefficiencyηoftheheatengineusingthiscycleviathegastemperaturesinitstransitionalstates,correspondingtothecornersofthecyclediagram.
Solution:Numbering the transitionalstatesasshown in the figureabove,and taking intoaccountthat the mechanical work ( ) performed by the gas during an elementary adiabatic processequalsâdE=âCVdT(see,e.g.themodelsolutionofproblem1.13),wemaycalculatethetotalworkdoneduringthecycleas
wherePmaxâĄP2=P3.Nowusingtheequationofstateoftheidealgas,PV=NT,totransformthesecondtermontheright-handsideas
Pmax(V3âV2)âĄP3V3âP2V2=NT3âNT2=N(T3âT2),andusingEq.(1.51)ofthelecturenotestoreplaceCVwith(CPâN)inthefirstterm,weget
Thereasonwhy the lastexpression for thework isconvenient forourpurposes is that its first
termevidentlyequals theheatQHobtainedbytheworkinggas fromthehotbathduringtheonlystage(2â3)atwhichtheyareincontact11,sothat
(ThelastformofthisexpressionusesthecommonnotationÎłâĄCP/CVâsee,e.g.themodelsolutionofproblem1.5.)
Just for thereaderâsreference: inengineering literature, it iscustomarytousetheequationofstatePV=NTandtherelationbetweenvolumeandtemperatureattheadiabaticprocess,TVÎłâ1=const(seethesolutionofproblem1.13)torecastthisresultintoadifferentform:
η=1â1ÎłrÎłâ1αγâ1αâ1,whereαâĄV3/V2>1iscalledthecut-offratio(characterizingthefuelcombustionstage2â3),andrâĄV1/V2>αisthecompressionratio(characterizingthebackstrokeofthepiston).
References[1]LandauLandLifshitzE1980StatisticalPhysics,Part13rdedn(Pergamon)[2]deWaeleA2011J.LowTemp.Phys.164179
1Notethatthisresultisonlyvalidforanidealgas,whileforrealgases(discussedinchapters3and4ofthelecturenotes),thisprocess,whichisaparticularcaseoftheso-calledJouleâThomsonexpansion,mayleadtoeitherheatingor,muchmoretypically,coolingâsee,e.g.problem4.3.2See,e.g.PartCMsection7.7,inparticularEq.(7.114),andEq.(7.120)withÎŒ=0.3See,e.g.Eqs.(16.6)â(16.8)in[1].4Thiscondition,virtuallyevidentfromfigure1.4,willbefurtherdiscussedinsection4.1ofthelecturenotes.5Thecompressibilityisjustthereciprocalbulkmodulus,1/Kâsee,e.g.PartCMsection7.3.6Notethatthederivativeinthenumeratorofthisexpressionisproportionaltothesystemâsthermalexpansioncoefficient,andthatinthedenominator,toitsisothermalcompressibilityâseethenextproblem.7Itmaybealsoobtained,fordT=0,fromEq.(1.53c),afterusingEq.(1.56),G=ÎŒN.8Notethatforanadiabaticprocess,withdQ=0,Eq.(*)immediatelyyieldsthedifferentialformoftheresultobtainedinproblem1.5:dP/P+ÎłdV/V=0.9Thisname stems from the fact that the cycle is an approximatemodel of operationof the internal-combustion (âpetrolâ) engine,whichwas improved andmadepracticablebyNOttoin1876âthoughitsideahadbeenconceivedearlier(in1860)byĂLenoir.10ThereversedcycleofthistypeisareasonableapproximationfortheoperationofStirlingandGifford-McMahon(GM)refrigerators,broadlyusedforcryocoolingâforarecentreviewsee,e.g.[2].11InapracticalDieselengine,thisisthestageoffuelcombustioninsideengineâscylinder,andtheroleofthehotbathisplayedbythehotgasformedastheresult.
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IOPPublishing
StatisticalMechanicsProblemswithsolutionsKonstantinKLikharev
Chapter2
Principlesofphysicalstatistics
Problem2.1.Afamousexampleofthemacroscopicirreversibilitywassuggestedin1907byPEhrenfest.Twodogsshare2Nâ«1fleas.Eachfleamayjumpontoanotherdog,andtherateÎofsuchevents(i.e.theprobabilityofjumpingperunittime)doesnotdependeitherontime,oronthelocationofotherfleas.Findthetimeevolutionoftheaveragenumberoffleasonadog,andoftheflea-relatedpartofthetotaldogsâentropy(atarbitraryinitialconditions),andprovethattheentropycanonlygrow1.
Solution:Duetotheconservationofthetotalfleanumber2N,wemayrepresentthenumberoffleasoneachdog,averagedoverthestatisticalensembleoftwo-dogpairs(butnotovertime!),asN1,2=N±n.ConsideratimeintervaldtsosmallthatduringitthefleanumbersN1,2donotchangesignificantly.Subtractingtheâfleaflowsâ,wegetthefollowingexpressionforelementarychangeof,say,N1:
dN1âĄd(N+n)âĄdn=N2ÎdtâN1ÎdtâĄ(Nân)Îdtâ(N+n)ÎdtâĄâ2nÎdt,i.e.thefollowingdifferentialequationforthefunctionn(t):
dndt=â2nÎdt.Thisequationmaybeeasilysolvedforarbitraryinitialconditions:
n(t)=n(0)expâ2Ît.So,aswecouldexpect,regardlessoftheinitialdistributionofthefleas,eventuallyn(t)â0,i.e.theaveragenumberoffleasoneachdogbecomesthesameâatypicalirreversibleprocess.
Tocalculatetheentropy,wemayapplyEq.(2.29)ofthelecturenotestotwodifferentpositionsofaflea,withprobabilitiesW1,2=(N±n)/2N,sothattheaverageentropyperfleais
S2N=âW1lnW1âW2lnW2=âN+n2NlnN+n2NâNân2NlnNân2N,andtheentropyofthewholesystem(i.e.ofthesetof2Nfleasonbothdogs)is2
S=âN+nlnN+nâ(Nân)lnNân+const.
Inordertoanalyzetheentropyâsevolution,wemaydifferentiateSovertime,withNconstant,dSdt=dSdndndt=lnNânN+ndndt,
anduseEq.(*)torewritethisresultasdSdt=â2nÎlnNânN+n.
WithinthemeaningfulintervalN1,2â[0,2N],i.e.nâ[âN,+N],thelastlogarithmisnegativeatn>0andpositiveifn<
0,sothatatÎ>0,theright-handsideofthelastrelationisanon-negativefunctionofn,whichapproaches0atnâ0.Hencetheexponentialreductionoftheaveragefleaimbalance2nisaccompaniedbyagrowthoftheentropy(i.e.ofdisorder).
Problem2.2.Usethemicrocanonicaldistributiontocalculatethermodynamicproperties(includingtheentropy,allrelevantthermodynamicpotentials,andtheheatcapacity),ofatwo-levelsysteminthermodynamicequilibriumwithitsenvironment,attemperatureTthatiscomparablewiththeenergygapÎ.Foreachvariable,sketchitstemperaturedependence,andfindasymptoticvalues(ortrends)inthelow-temperatureandhigh-temperaturelimits.
Hint:Thetwo-levelsystemisdefinedasanysystemwithjusttwodifferentstationarystates,whoseenergies(say,E0andE1)are separated by a gapÎâĄE1âE0. Itsmost popular (but by nomeans the only!) example is a spin-Âœ particle, e.g. anelectron,inanexternalmagneticfield3.
Solution:ConsideramicrocanonicalensembleconsistingofmanysimilarsetsofNâ«1non-interacting,distinguishable4two-levelsystems,taking(justforthenotationsimplicity)theloweststateenergyE0fortheenergyorigin,sothatE0=0andE1=Î.Justasinthecaseofquantumoscillatorsanalyzedinsection2.2ofthelecturenotes,withanincreaseofEN,thenumberÎŁNofstateswiththetotalenergyofthesetbelowthisvalueENisincreasedbydiscretestepsatEN=NâČÎ(NâČ=1,2,âŠ,N).The height ÎÎŁN of such a step is equal to the number of ways to distribute NâČ indistinguishable energy increments(âexcitationsâ, or âquantaâ) Î amongN distinct systems. This number is equal to the number of ways to selectNâČ similarobjects(incombinatorics,traditionallycalledâballsâ)ofthetotalnumberofN,inanarbitraryorder,andhenceitisjustthebinomialcoefficient5
ÎÎŁN=NâČCNâĄN!NâČ!(NâNâČ)!.
TakingtheenergyspreadÎEof themicrocanonicalensembleequal toÎ(which is legitimate ifN,NâČ,and(NâNâČ) aremuchlargerthan1,i.e.whenÎÎŁNâ«1andÎEâȘE),fortheaverageentropypersystemweget
S=limâŁN,NâČââlnÎÎŁNN=limâŁN,NâČââ1NlnN!âlnNâČ!âln(NâNâČ)!.TheapplicationoftheStirlingformula(initssimplestform,givenbyEq.(2.27)ofthelecturenotes)reducesthisrelationto6
S=ânlnnâ(1ân)ln(1ân),wherenâĄNâČ/N=EN/NÎâ©œ1istheaveragenumberofenergyquantaÎpertwo-levelsystem,sothattheaverageenergypertwo-levelsystemisE=EN/N=nÎ.7
Nowwecanusethedefinitionoftemperature,givenbyEq.(1.9),tofind1TâĄdSdE=1ÎdSdn=1Îln1nâ1âĄ1ÎlnÎEâ1.
SolvingthisequationforE,wegettheequilibriumvalueoftheaverageenergy:E=ÎeÎ/T+1.
Pluggingthisresultforn=E/ÎbackintoEq.(*)yieldstheequilibriumvalueoftheentropy:S=ln1+eâÎ/T+ÎT1eÎ/T+1.
Nowthatweknowtheentropyasafunctionoftemperature,weareonthethermodynamicsautopilotâseechapter1:
FâĄEâTS=âTln1+eâÎ/T,
(*)
CâĄâEâTV=Î2T2eÎ/T1+eÎ/T2âĄ(Î/2T)cosh(Î/2T)2.(Asintheharmonic-oscillatorproblemdiscussedinsection2.2,thenotionofvolume,andhenceofsuchvariableasP,isnotdefinedforthissystem,sothattherearenodifferencesbetweenCVandCP,EandH,andFandG.)
Thefigureaboveshowsthetemperaturebehaviorofthecalculatedthermodynamicvariables.Inthelow-temperaturelimit
(TâȘÎ),allofthemapproachzero.(ForEandF,thislevelisconditional,butfortheheatcapacity,itismeasurable.)Ontheotherhand,inthehightemperaturelimit(ÎâȘT),thebehaviorofeachvariableisdifferent:EâÎ/2=const,8FââTln2âââ,Sâln2â0.693=const,whileCâ0.Itisinterestingthattheheatcapacityisvanishinginbothlow-temperatureandhigh-temperaturelimits,andhasamaximum(Cmaxâ0.45)atafinitetemperature(Tâ0.43Î).AneasyinterpretationofthisbehaviorwillbecomeavailableaftertheenergyleveloccupancieshavebeencalculatedusingtheGibbsdistributionâseethenextproblem.
Problem2.3.SolvethepreviousproblemusingtheGibbsdistribution.Also,calculatetheprobabilitiesoftheenergyleveloccupation,andgivephysicalinterpretationsofyourresults,inbothtemperaturelimits.
Solution:LetusapplytheGibbsdistributiontoacanonicalensembleconsistingofmanytwo-levelsystems.Eachsystemhasjusttwoenergystates,E0=0andE1=Î,sothattheprobabilitiesthatasystemoccupiesthemobeytheGibbsdistribution(2.58),
Wm=1ZexpâEmT,form=0,1,withthestatisticalsum(2.59),
ZâĄâm=0,1expâEmT=1+expâÎT,sothat
W0=1ZexpâE0T=11+eâÎ/T,W1=1ZexpâE1T=eâÎ/T1+eâÎ/TâĄ1eÎ/T+1.
The figurebelowshows theseprobabilities as functionsof temperature.AtTâ0, the system is almost certainly in itsground state,W0 â 1, while the probability to find the system on the upper energy level is exponentially low:W1 âexp{âÎ/T} â 0. Conversely, at high temperatures, Tâ« Î, both probabilities are virtually equal,W0 âW1 â Âœ. Thisâequilibrationâoftheenergy levelpopulation istypical forhigh-temperaturebehaviorofsystemswithanyfinitenumberofquantizedenergylevelsâbutnotforquantumoscillatorsorrotators,whoseenergylevelâladdersâareinfinite9.
PluggingEq.(*)intothemainrelation(2.63)connectingtheGibbsdistributionwiththermodynamics,wereadilygetthe
freeenergy(persystem)F=Tln1Z=âTln1+eâÎ/T.
This formula coincides with the result obtained from the microcanonical distributionâsee the solution of the previousproblem.NowwecanuseEq.(1.35),S=â(âF/âT),tofindtheentropy,andthenEq.(1.33),rewrittenasE=F+TS,tofindtheaverageenergyE.Alternatively,theenergymaybefoundusingEq.(2.61a):
E=âm=0,1WmEm=W0E0+W1E1=W1Î=ÎeÎ/T+1.(NotethattheparameternâĄNâČ/N=E/NÎused inthemodelsolutionof thepreviousproblem, isnothingmorethanW1.)NowwecanuseEq.(1.24),C=âE/âT,tofindtheheatcapacitypersystem.Alltheseresultscoincidewiththoseobtainedfromthemicrocanonicalensembleâseethemodelsolutionofthepreviousproblemforformulasandplots.
TheaboveresultsforW0andW1enableaneasyinterpretationofthetemperaturebehaviorofthevariables.Inparticular,atlowtemperatures(TâȘÎ) thesystem iseffectivelyconfined to the lowest levelwithzeroenergy,so that theaverageEtends to zero.Also, the systemâs statechoice in this limit is virtuallydefinite, so there is very littledisorder, andentropyapproaches zeroaswell. In theopposite limit, atTâ«Î, i.e.atW1âW0âÂœ, the average energynaturally tends to theaverage between those of the two equally occupied levels. Also, in this limit the choice between twopossible states of aparticular system is completely random; hence the entropy tends to the value ln 2, which corresponds to one lost bit ofinformationaboutthischoice.
Finally,byitsdefinition,theheatcapacityofasystemmaybesubstantialonlyifasmallvariationoftemperaturecausesanoticeableredistributionoftheenergylevelprobabilitiesâandhenceoftheaverageenergy.Astheformulasandplotsaboveshow, in our current problem such redistribution happens only at T ⌠Î; hence the peak of the function C(T) at theseintermediatetemperatures.
Problem2.4.Calculate the low-fieldmagneticsusceptibilityÏofaquantumspin-ÂœparticlewithgyromagneticratioÎł,inthermalequilibriumwithenvironmentat temperatureT, neglecting its orbitalmotion.Compare the resultwith that for a
(**)
(***)
classicalspontaneousmagneticdipolemofafixedmagnitude ,freetochangeitsdirectioninspace.
Hint:Thelow-fieldmagneticsusceptibilityofasingleparticleisdefined10as
wherethez-axisisalignedwiththedirectionoftheexternalmagneticfield .
Solution: According to quantum mechanics11, the interaction of a magnetic dipole with an external magnetic field isdescribedbytheHamiltonianoperator12
Ifthedipolemagneticmoment ofaparticleisentirelyduetoitsspin,thenitsvectoroperatorisrelatedtothatofthespinas ,whereÎłisthegyromagneticratio.Foraspin-Âœparticle,SË=(â/2)ÏË,whereâisthePlanckâsconstant,andÏË isthe vector operator whose Cartesian components, in the standard z-basis, are represented by 2 Ă 2 Pauli matrices, inparticular
Ïz=100â1.Asaresult,pointingthez-axisalongthedirectionofthefield ,wemayrepresentthemagneticmomentâscomponent bythediagonalmatrix
wherethescalarconstant
maybeinterpretedasthemagnitudeofthedipolemomentthatmaybedirectedeitheralongoragainsttheexternalfield.ThecorrespondingeigenvaluesoftheHamiltonian(*)aretheeigenenergies,EâandEâ,separatedbytheenergygap
Butthisisexactlythesystemthatwasdiscussedintwopreviousproblems,sothatwemayusetheirresults.Inparticular,theaveragez-componentofthemagneticmomentis
This function is shownwith thebluecurve in the figurebelow13. In thehigh-field (low-temperature) limit it
describesthemagneticmomentâssaturationasitshighestpossiblevalue ,correspondingtothedefiniteorientationofthedipolealongthefield(âspinpolarizationâ).
Forour task,however,weneed theopposite, low-field limit,where the function tanhmaybewell approximatedby its
argument,sothatassumingthatthedipoleislocatedinfreespace,i.e.(intheSIunits) ,weget
SuchÏmâ1/Tdependenceoftheparamagnetic(positive)magneticsusceptibilityiscalledtheCurielaw; inthiscourse, itslimitationsandextensionswillbediscussedinthecontextoftheIsingmodelofphasetransitionsâseesections4.4â4.5.
Now let us consider the classicalmodel outlined in the assignment14, inwhich the orientation of themagnetic dipolevector ,ofafixedmagnitude ,isarbitrary.ThesystemâsisotropyimpliesthatpossibledipoleorientationsareuniformlydistributedoverallthefullsolidangleΩ=4Ï.15ThisiswhytheGibbsdistribution(2.58),appliedtoacanonicalensembleofsuchdipoles,mayberecastintothatfortheprobabilitydensitywâĄdW/dΩ:
w=1ZexpâET,withZ=âź4ÏexpâETdΩ,andtheaverage maybecalculatedas
Themagneticdipoleâsenergyinanexternalmagneticfield isjusttheclassicalversionofEq.(*),16
sothattheaboveformulasyield
Inthesphericalcoordinates,withthepolaraxisdirectedalongthemagneticfield,wehave ,sothatsincedΩ=sinΞdΞdÏ,bothintegralsovertheazimuthalangleÏareequalto2Ïandcancel,andweget
Introducingaconvenientdimensionlessvariable ,sothat ,and ,wemayreducethisformulatoaratiooftwosimpleintegrals,ofwhichone(inthedenominator)iselementary,whiletheotheronemaybereadilyworkedoutbyparts:
Theredcurveinthefigureaboveshowsthefielddependenceofthis .Inthehigh-fieldlimit ,thefirstterm
intheparenthesestendstosgn ,whilethesecondoneisnegligible,sothatthespiniscompletelypolarized: .Ontheotherhand,intheopposite,low-fieldlimitwemayusetheTaylorexpansionofthefunctioncothΟatΟâ0,truncatedtotwoleadingterms,cothΟâĄcoshΟ/sinhΟâ(1+Ο2/2!)/(Ο+Ο3/3!)â1/Ο+Ο/3,toreduceEq.(***)to
sothatthelow-fieldsusceptibilityis
ThecomparisonofEqs. (**)and (***) shows that the fielddependenceof theaveragemagneticmomentofa spin-œ is
qualitativelysimilarto,butquantitativelydifferentfromthatintheclassicalmagneticdipoleâcf.theblueandredlinesinthefigure above. In particular, in terms of (which gives the momentâs saturation value in both models) the low-fieldsusceptibilityofspin-Âœparticlesisthreetimeshigher.
Onemoreremark:analternativewaytocalculate (forbothmodels)istousetheanalogybetweentheusualpair{P,V}ofthegeneralizedcoordinates,participatinginEq.(1.1)ofthelecturenotes,andhenceallformulasofchapter1,andthepair .Indeed,theexpression usedaboveforthepotentialenergyofadipolemeansthattheelementaryworkofafixedexternalmagneticfield onthemagneticmomentis .17ComparingitwithEq.(1.1), , we see that for the average properties of a particle in the magnetic field , we may use all thethermodynamicequalitiesdiscussed inchapter1,with the replacements . Inparticular, the secondofEqs.(1.39)becomes
whereG is the Gibbs energy per particle. However, since in our approach the product (i.e. the analog of theproductPV),whichconstitutesthedifferencebetweenthethermodynamicpotentialsGandF(see,e.g.thelastofEqs.(1.37)of the lecturenotes), isalready taken intoaccount in theexpression forE,18wemay identifyGwithF,andcalculate thisthermodynamicpotentialusingEq.(2.63):
G=F=âTlnZ.
Itisstraightforwardtocheckthatforbothpartsofourcurrentproblem,thisapproachyieldsthesameresults(*)and(**)âseealsothemodelsolutionofthenextproblem.
Problem 2.5. Calculate the low-fieldmagnetic susceptibility of a particle with an arbitrary spin s, neglecting its orbitalmotion.Comparetheresultwiththesolutionofthepreviousproblem.
Hint:Quantummechanics19 tells us that the Cartesian component of themagnetic moment of such a particle, in thedirectionoftheappliedfield,has(2s+1)stationaryvalues:
whereÎłisthegyromagneticratiooftheparticle,sisitsspin,andâisthePlanckâsconstant.
Solution:Letusconsideracanonicalensembleofsuchparticles.Theenergyoftheparticleinanexternalmagneticfieldofmagnitude is ,sothatthestatisticalsumis
Sincethespinsofaparticlemaybeonlyeitherintegerorhalf-integer,2sisalwaysaninteger,sothatthelastsumisjustthewell-knownfinitegeometricprogression20,andwegetaverysimpleresult:
Nowwecouldcalculatetheaverage justaswasdoneinthemodelsolutionofthepreviousproblem,butforpractice,let
ususethealternativeapproachthatwasdiscussed,butnotusedintheendofthatsolution:
ForthestatisticalsumgivenbyEq.(*),
Forourpurposes,weneedonly the low-field limitof thisexpression,atbâ0, sowemayapproximate thesinh functionsusingonlytwoleadingtermsoftheirTaylorexpansion:
sinhΟâΟ+Ο33!âĄÎŸ1+Ο26.Thisapproximationyields
sothat
andhencethelow-fieldatomicsusceptibilityis
(*)
With thenotation , compatiblewith thoseaccepted inbothpartsof thepreviousproblem, this result
reads
showingagradualtransitionfromtheresultsforthespin-Âœmodelconsideredinthefirstpartofthatproblem,tothosefortheclassicalHeisenbergmodelanalyzedinitssecondpart,atsisincreasedfromÂœtoâ.
Problem2.6.*Analyzethepossibilityofusingasystemofnon-interactingspin-Âœparticles,placedintoastrong,controllableexternalmagneticfield,forrefrigeration.
Solution:Combiningtheresultsofproblem2.2withtherelation (seethesolutionofproblem2.4),fortheaverageentropyperspinweget
Notethattheentropydependsononlyonedimensionlesscombinationofparameters, ,sothatanincreaseofthe
appliedfieldjuststretchestheplotofthefunctionS(T)alongthehorizontalaxisâseethefigurebelow.TheseplotsimplythefollowingpossiblewaytoorganizetheCarnotcoolingcycleusingthespinsystemasaworkingâgasâ.(Typically,thespinsarethoseofatoms inasolidsaltsampleâseebelowâwhich iscalledeithermore formally, therefrigerant,or inthetechnicalslang,thesaltpill.)
Starting,forexample,frompoint1(negligiblemagneticfield,spinsareinbothpossibleeigenstateswithequalprobability,
i.e.completelydisordered,sothattheentropyperspinis largest,S/N=ln2), thefield isslowly increasedtosomevalue,while keeping the refrigerant in contactwith the âhot bathâ of temperatureTH.21 Since the entropy is being
decreased(physically,becausealmostallspinscondenseontothelowestenergylevel,thusdecreasingthespindisordertoalmostzero),heatâQH=THÎS>0isbeingtransferredtothehotbath.Then(atpoint2)therefrigerantisbeingthermallyinsulated frombothbaths, and then the external field is decreased. The entropy of the refrigerant cannot change in thisadiabaticprocess,sothattheproduct (whoseuniquefunctionSis)cannotchangeeither.Thismeansthatrefrigerantâstemperature drops proportionally to the field22. At point 3, when T decreases to the temperature TL of the âcold bathâ(practically, the object being cooled), the refrigerant is brought into thermal contactwith that bath, and then the fieldâsdecrease is continued isothermally until point 4, inwhich the energy level splitting is negligible, so that the spin energylevelsareequallypopulatedagain,andtheentropyperspinapproachesitsmaximumvalueln2.ThecycleisnowcompletedadiabaticallyusingaslightfieldincreaseuntilthespinsystemtemperaturerisestoTHagain23.
Practicalcyclesofsuchâadiabaticmagneticrefrigerationâsomewhatdifferfrom,andhencehavelowerCOPcoolingthantheCarnotcycledescribedabove,mostlybecauseofthetechnicaldifficultiesoffastchangingthethermalcontactsbetweentherefrigerant and the heat bathsâtypically letting in and pumping out small portions of gaseous helium. Themost popularmodificationofthecycleisskippingitsisothermalpart(3â4),byallowingaslowheatingoftherefrigeranttogetherwiththecooledobjectinafixedmagneticfield,duetounavoidableunintentionalheatleaksâsee,forexample,dashedarrowsinthe figure above. (In carefully designed systems, suchheat-upmay last for up to aweek; in such cases, engineers speakaboutsingle-shotcooling.)
Anotherdifferenceofexperimentalimplementationsofthistechniquefromthesimplestschemedescribedaboveisthatinsomeusedmaterials24,theappliedmagneticfieldsplitsenergylevelsofatomsintoM>2ratherthanjusttwosublevels25,making the maximum entropy per atom (ln M) larger than ln 2, and hence decreasing the necessary amount of therefrigerant.
Problem2.7.TherudimentaryâzipperâmodelofDNAreplicationisachainofNlinksthatmaybeeitheropenorclosedâseethe figurebelow.Openinga link increases the systemâsenergybyÎ>0;anda linkmaychange its state (eitheropenorclose)onlyifalllinkstotheleftofitarealreadyopen,whilethoseontherightofit,arealreadyclosed.Calculatetheaveragenumberofopenlinksatthermalequilibrium,andanalyzeitstemperaturedependence,especiallyforthecaseNâ«1.
Solution:Accordingtothemodel, thechainmayhaveonly(N+1)differentstates,eachwithsomenumbernof left linksopen and other links closed (see the figure above), so that the total link-related energy is En = nÎ. Hence the Gibbsdistribution,givenbyEqs.(2.58)and(2.59)ofthelecturenotes,givesthefollowingprobabilityofthestatewithnopenlinks:
Wn=1ZexpânÎT,withZ=ân=0NexpânÎT.FromhereandthegeneralEq.(2.7),theaveragenumberoftheopenlinksmaybecalculatedas
n=ân=1NnWn=ân=1NnexpânÎT/ân=0NexpânÎT.
Thesuminthedenominatoristhewell-knownfinitegeometricprogression26:ân=0NexpânÎTâĄân=0Nλn=1âλN+11âλ,whereλâĄeâÎ/Tâ©œ1,
whilethatinthenumeratormaybecalculatedviaitsderivativeovertheparameterλ.Indeed,ân=1NnexpânÎTâĄân=1Nnλn=λââλân=0Nλn,
sothatusingEq.(*)wegetân=1NnexpânÎT=λââλ1âλN+11âλ=λ1âλN+1âN+1λN1âλ1âλ2,
(**)
(*)
(**)
and,finally,n=λ1âλâN+1λN+11âλN+1âĄ1eÎ/Tâ1âN+1e(N+1)Î/Tâ1.
Asthefigurebelowshows,thisresultisnotquitetrivial,especiallyatNâ«1.Letusstartfromtheobvious:iftemperature
is low,TâȘ Î, the probability of having even one (the leftmost) link open is exponentially low. Indeed, in this limit bothexponentsparticipatinginEq.(**),exp{Î/T}andexp{(N+1)Î/T},aremuch larger than1.Moreover, foranyN>1, thesecondexponentismuchlargerthanthefirstone.Asaresult,despitetheadditionalfrontmultiplier(N+1),thesecondterminEq.(**)isnegligibleincomparisonwiththefirstone,andtheformulaisreducedto
nâeâÎ/TâȘ1,forTâȘÎ,independentlyofN.27
Theoppositelimitisalsoreadilypredictable.IfTismuchlargerthanbothÎand(N+1)Î,bothexponentsexp{Î/T}and
exp{(N + 1)Î/T} approach 1, and the denominators in both terms of Eq. (**) become smallâapproximately equal to,respectively,Î/Tand(N+1)Î/T,sothatthemagnitudesofbothtermsbecomelarge.Duetotheadditionalfactor(N+1)inthenumeratorofthesecondterm,thesetermsmostlycanceleachother,withtheremainingbalance,
nâN2,forTâ«Î,(N+1)Î,duetothethirdtermsintheTaylorexpansions
expÎTâ1+ÎT+12ÎT2,expN+1ÎTâ1+N+1ÎT+12N+12ÎT2.Thephysicsofthishigh-temperaturelimitisprettysimple:atveryhightemperatures,theenergygainÎisnegligible,andeachlinkhasanequalchancetobeopenorclosed.
Lessobviousisonemoresimplebehavioroflongchains(Nâ«1)inabroadrangeofintermediatetemperatures:nâTÎ,forÎâȘTâȘN+1T
âsee theslopeddashedstraight line in the figureabove. (Mathematically, it follows fromEq. (**)when its first termhasalready reached its high-temperature limit,T/Î, while the second term is still in its low-temperature limit, and hence isnegligible.)ThephysicalinterpretationofthissimpleformulaisthatthethermalagitationwiththecharacteristicenergyTâ«Îissufficienttoopen,onaverage,T/Îleftlinksofthechain,butnotmorethanthat.
Problem 2.8. Use the microcanonical distribution to calculate the average entropy, energy, and pressure of a classicalparticleofmassm,withnointernaldegreesoffreedom,freetomoveinvolumeV,attemperatureT.
Hint:Trytomakeamoreaccuratecalculationthanhasbeendoneinsection2.2forthesystemofNharmonicoscillators.ForthatyouwillneedtoknowthevolumeVdofand-dimensionalhypersphereoftheunitradius.Toavoidbeingtoocruel,Iamgivingittoyou:
Vd=Ïd/2/Îd2+1,whereÎ(Ο)isthegamma-function28.
Solution:LetusconsideramicrocanonicalensembleofmanysetsofNâ«1distinctparticles29.Anevidentgeneralizationofthequantumstatecountingrule(see,e.g.Eq.(2.82)ofthelecturenotes),withk=p/â,showsthatthenumberofdifferentquantumstatesoftheparticleset,withthetotalenergybelowcertainvalueEN,is
ÎŁN=1(2Ïâ)3Nâ«pj2/2m<EN,1â©œjâ©œ3Nd3Nqd3Np=VN(2Ïâ)3NpE3NV3N,wherepE ⥠(2mEN)1/2 is the momentum of a particle with the energy EN, i.e. the radius of the hypersphere in the 3N-dimensionalmomentumspace,containingthestateswearecounting.UsingtheformulaforVd,providedintheHint,withd=3N,weget
ÎŁN=VN(2Ïâ)3N(2mEN)3N/2Ï3N/2Î(3N/2+1),sothat
g(EN)âĄdÎŁdEN=3N2VNÎ(3N/2+1)m2Ïâ23N/2EN3N/2â1,SNâĄlng(EN)+const=NlnV+3N2â1lnEN+3N2lnm2Ïâ2+lnNâlnÎ3N2+1+const.
InthelimitNââwemayapplytheStirlingformulatoln[Î(3N/2+1)]âln[(3N/2)!],andgetSNâNlnV+3N2lnEN+3N2lnm2Ïâ2â3N2ln3N2â1+const=NlnVm2Ïâ23/22EN3N3/2+32N+const.
Nowwecanusethedefinition(1.9)oftemperature,
1TâĄâSNâENV=3N2EN,togetEN=(3N/2)T,i.e.theaverageenergyperparticle30
EâĄENN=32T.NowexpressingENviaT,theentropySperparticlemayberecastasafunctionofTandV:
SâĄSNN=lnVm2Ïâ23/2T3/2+32+const,andthususedtocalculatefreeenergy(perparticle)asafunctionofthesetwoarguments:
FâĄEâTS=âTlnVmT2Ïâ23/2=âTlnV+f(T),where
f(T)âĄâTlnmT2Ïâ23/2.
(*)
Theresult(*)isexactlyEq.(1.45)ofthelecturenotes(derivedtherefromtheequationofstatePV=NT)fortheparticular
caseN=1.Thisiswhyallotherthermodynamicrelationsfortheparticle,withthisspecificformoff(T),coincidewithEqs.(1.44)â(1.51)ofthelecturenotes,againwithN=1.However,Eq.(**)forthefunctionf(T)isnew,specificforaparticlewithno internal degrees of freedom; its generalization is discussed in section 3.1 of the lecture notesâsee, in particular Eq.(3.16b).
Problem2.9.SolvethepreviousproblemstartingfromtheGibbsdistribution.
Solution:CombiningEqs.(2.59)and(2.82)ofthelecturenotes,wegetZâĄâneâEn/TâV(2Ïâ)3â«eâE(p)/Td3p=V(2Ïâ)3â«eâp2/2mTd3p=V(2Ïâ)3â«ââ+âeâpx2/2mTdpxâ«ââ+âeâpy2/2mTdpyâ«ââ+âe
âpz2/2mTdpz=V(2Ïâ)32ÏmT3/2,sothat
F=Tln1Z=âTlnVmT2Ïâ23/2,i.e.wehavearrived(muchfaster)atthesameresultasusingthemicrocanonicaldistributioninthepreviousproblem.
Insection3.1ofthelecturenotes,thiscalculationwillbegeneralizedtoaclassicalgasofNparticles,withanontrivialdifferenceoftheso-calledâcorrectBoltzmanncountingâ,whichdoesnotcontributetotheequationofstate,butaffectstheentropyofthegas.
Problem 2.10. Calculate the average energy, entropy, free energy, and the equation of state of a classical 2D particle(withoutinternaldegreesoffreedom),freetomovewithinareaA,attemperatureT,startingfrom:
(i)themicrocanonicaldistribution,and(ii)theGibbsdistribution.
Hint:Fortheequationofstate,maketheappropriatemodificationofthenotionofpressure.
Solutions:
(i)Rewritingthesolutionofproblem2.8,withtheappropriatereplacementofvolumeVwithareaA,andthechangeofphasespacedimensionalityfrom6Nto4N,weget
ÎŁN=1(2Ïâ)2Nâ«pj2/2m<E,0<j<2Nd2Nqd2Np=AN(2Ïâ)2NpE2NV2N=AN(2Ïâ)2N(2mEN)NÏNÎ(N+1),g(EN)=dÎŁNdEN=NANN!m2Ïâ2NENNâ1,
SN=lng(EN)+const=NlnA+Nâ1lnEN+Nlnm2Ïâ2+lnNâlnN!+const,SNâNââNlnAm2Ïâ2ENN+const,1TâĄâSNâENA=NEN,
tothattheaverageenergyEâĄEN/NperparticleequalsT(inaccordancewiththeequipartitiontheorem),andSâĄSNN=lnAmT2Ïâ2+const.
Fromhere,FâĄEâTS=âTlnAmT2Ïâ2=âTlnA+f(T),withf(T)=âTlnmT2Ïâ2.
Inour2Dsystem,theusualconjugatepairofvariables{âP,V}hastobereplacedwiththepair{âÏ,A},whereâÏisthe
surface âanti-tensionâ, i.e. the average pressure force exerted by the particle per unit length of the border contour. As aresult,thesecondofEqs.(1.35)ofthelecturenoteshastobereplacedwith
Ï=ââFâAT,togetherwithEq.(*)givingusessentiallythesameequationofstateasinthe3Dcase:
ÏA=T.(ii)ApplyingtheGibbsdistributiontoasingleclassicalparticle,wehave
ZâĄâneâEn/TâA(2Ïâ)2â«eâp2/2mTd2p=A(2Ïâ)2â«ââ+âeâpx2/2mTdpxâ«ââ+âeâpy2/2mTdpy=A(2Ïâ)22ÏmT,sothat
F=Tln1Z=âTlnAmT2Ïâ2,i.e.thesameformulaasobtainedbythefirstmethod,thusenablingthethermodynamicsautopilottore-calculateallotherresults,includingtheequationofstate.
Problem2.11.Aquantumparticleofmassmisconfinedtofreemotionalonga1Dsegmentoflengtha.Usinganyapproachyou like, calculate the average force the particle exerts on the âwallsâ (ends) of such â1D potential wellâ in thermalequilibrium,andanalyzeitstemperaturedependence,focusingonthelow-temperatureandhigh-temperaturelimits.
Hint:YoumayconsidertheseriesÎ(Ο)âĄân=1âexp{âΟn2}aknownfunctionofΟ.31
Solution:Thewell-knowneigenenergiesofthisproblemare32
En=pn22m=(âkn)22m=E1n2,whereE1âĄÏ2â22ma2,andn=1,2,âŠHencethestatisticalsumoftheGibbsdistributionforthesystemis
ÎâĄân=1âeâEn/T=ân=1âeâE1n2/TâĄÎE1T,whereÎ(Ο)isthefunctionmentionedintheHint,sothatthefreeenergyis
F=Tln1Z=âTlnÎE1T.
Sincetheelementaryexternalwork ofslowlymovingwallsonour1Dsystemmayberepresentedas ,whereistheaverageforceexertedontheparticlebythewalls,theusualcanonicalpairofmechanicalvariables{âP,V}hastobereplacedwiththepair .HencethesecondofEqs.(1.35)ofthelecturenoteshastobereplacedwith
Combiningtheaboveformulas,weget
AlogâlogplotofthefunctionÎ(Ο)anditsasymptotesareshownontheleftpanelofthefigurebelow,whileitsrightpanel
showstheresultingtemperaturedependenceoftheforce ,anditshigh-temperatureasymptote.
(*)
(**)
AtΟââ,theseriesdefiningthefunctionÎ(Ο)isdominatedbyitsfirstterm,sothat
Thisisexactlythe(temperature-independent)resultwewouldgetfromapurelyquantum-mechanicalanalysisofthegroundstateoftheparticle.
Ontheotherhand,atΟâ0theseriesisconvergingveryslowlyandmaybeapproximatedbyaGaussianintegral33:ÎΟ=ân=1âeâΟn2ââ«0âeâΟn2dn=Ï1/22Ο1/2ââ,atΟâ0.
Asaresult,inthis(classical)limitweget
The last resultmaybealsoobtained fromelementaryclassical arguments: according to theequipartition theorem, the
average(kinetic)energyofafree1Dparticle,p2/2m= 2/2,isequaltoT/2,sothatitsrmsmomentumis(mT)1/2,andthermsvelocityis(T/m)1/2.Sinceeachreflectionfromthewalltransferstoittwicethemomentumoftheincidentparticle,andthetimeintervalÎtbetweenparticleâscollisionswiththesamewallistwicethewelllengthadividedbyparticleâsvelocity,weget
Notethataccordingtothesolutionsofproblems2.8â2.11,theequationofstateofafreeclassicalparticleisessentiallythe
sameforanydimensionality.
Problem2.12.*Rotationalpropertiesofdiatomicmolecules(suchasN2,CO,etc)maybereasonablywelldescribedbytheâdumbbellâmodel:twopointparticles,ofmassesm1andm2,withafixeddistancedbetweenthem.Ignoringthetranslationalmotionofthemoleculeasthewhole,usethismodeltocalculateitsheatcapacity,andspellouttheresultinthelimitsoflowandhigh temperatures.Discusswhether your solution is valid for the so-calledhomonuclearmolecules, consisting of twosimilaratoms,suchasH2,O2,N2,etc.
Solution:Asweknowfromclassicalmechanics34,themotionofatwo-particlesystemmaybeconsideredasasuperpositionofthetranslationmotionoftheircenterofmassasapointparticleofmassM=m1+m2,locatedatpointR=(m1r1+m2r2)/M,andthemutualrotationofparticles1and2aboutthispoint,equivalenttotherotationofasingleparticlewiththeso-calledreducedmass
aboutanimmobilepoint.Suchareductionofrotationtothatofasingleparticleisvalidinquantummechanicsaswell35.Inourcaseoffixeddistanced,thismeansthattherotationalpropertiesofthemoleculeareequivalenttothoseoftheso-calledsphericalrotatorâaparticlefreetomoveonthesurfaceofaspherewithradiusd.
Accordingtoquantummechanics36,eigenfunctionsofsucharotatorarethesphericalharmonics,indexedbytwointegerquantumnumbers:l=0,1,âŠ,andm,withpossiblevalueswithinthelimitsâlâ©œmâ©œ+l.Intheabsenceofanexternalfieldaffectingtherotation,thecorrespondingeigenenergiesdependonlyontheâorbitalâquantumnumberl:
El=â22Il(l+1),where is the particleâs moment of inertia. Hence the lth energy level is (2l + 1)-degenerate, with differenteigenfunctionscorrespondingtodifferentvaluesoftheâmagneticâquantumnumberm.ThismeansthatthestatisticalsumoftheGibbsdistributionis
Z=âl=0â(2l+1)expââ22ITl(l+1).Theaverageenergymaybefoundas37
E=1Zâl=0â(2l+1)Elexpââ22ITl(l+1)=â22I1Zâl=0â(2l+1)l(l+1)expââ22ITl(l+1),andtheheatcapacityasC(T)=âE/âT.
InthegeneralcasethesumsinEqs.(*)and(**)maybecalculatedonlynumerically.TheresultingfunctionC(T)=dE/dThasaweakmaximum,Cmaxâ1.1attemperatureTâ0.8(â2/2I)âseethefigurebelow.(Thephysicaloriginofthismaximumissimilartothatintwo-levelsystemsâseethediscussioninthemodelsolutionofproblem2.3.However,noticethatinourcurrentcasethemaximumismuchweaker,becausetheenergyspectrumoftherotatorisinfinite,sothattheprobabilityre-distributionamongitsvaluescontinuesevenathightemperaturesâseebelow.)
(***)
Inthehigh-andlow-temperaturelimitstheresultsmaybesimplified.Intheformer(classical)limit,Tâ«â2/2I,thesum(*)
isconvergingatlâ«1,andhencemaybewellapproximatedwithanintegral:Zââ«l=0â2lexpââ22ITl2dl=â«l=0âexpââ22ITl2d(l2)=2ITâ2â«0âeâΟdΟ=2ITâ2,
whilethattheaverageenergy(**)isEâ1Zâ«l=0âEl2lexpââ22ITl2dl=â22ITâ«l=0ââ22Il2expââ22ITl2d(l2)=Tâ«0âΟeâΟdΟ=T,
givingtheheatcapacityCâ1âseethefigureaboveagain.Thisresultisnatural,becauseinaninertialreferenceframe,theclassicalrotatorâsenergymaybeexpressedas
withthelinearmomentumvectorphavingtwoCartesiancomponentsp1,2âinanytwodirectionsperpendiculartoeachotherandthesphereâsradius.Accordingtotheequipartitiontheoremdiscussedinsection2.2(andvalidinthisclassicallimit),theaverageenergyofeachofthesetwoâhalf-degreesoffreedomâisT/2.
In the opposite, low-temperature (i.e. quantum) limit T âȘ â2/2I, the terms of the statistical sum (*) drop fast(exponentially)withl,andwemaykeeponlytwofirsttermsâwithl=0andl=1:
Zâ1+3expââ2ITâĄ1+3expââ2ÎČI,sothatlnZâ3expââ2ÎČI,whereÎČâĄ1/T.Fromheretheaverageenergy(**)is38
E=âlnZââÎČâ3â2Iexpââ2ÎČIâĄ3â2Iexpââ2IT,andtheheatcapacity
C=âEâTâ3â2IT2expââ2ITâȘ1,forTâȘâ2I.
HenceatTâ0theheatcapacityisexponentiallysmallâthepropertycommonforallsystemswithafinitegapbetweentheground-stateenergyandthelowestexcitedstate(s).Notealsothatthedegeneracyofexcitedstatesofthesystemdoesaffectitsthermodynamicproperties,inparticularbeingresponsibleforthenumericalfactorinthelastresult(3=2l+1forl=1).
Nownotethatthissolutionhastoberevisedinthecasewhentwoatomsofthemoleculeareindistinguishablefromeachother. For that, they have to be not only chemically similar (i.e. the molecule has to be homonuclear rather thanheteronuclear),butalsotheirinternaldegreesoffreedom,includingelectronic,vibrational,andnuclear-spinones,tobeinexactlythesame(forexample,ground)quantumstate.Inthiscase,thewavefunctionofthesystemhastobesymmetricwithrespecttotheatomsâswap(âpermutationâ)39.Butsuchaswapisequivalenttothereplacementrââr,while thesphericalharmonicswithoddvaluesoflareantisymmetricwithrespecttosuchreplacement40.Asaresult,onlythestateswithevenvaluesl=2p(withp=0,1,2,âŠ)arepermitted,andwehavetoredotheabovecalculationskeepingonlycontributionsfromthesestates:
Zs=âp=0â(4p+1)expââ2ITp(2p+1).
Inthelow-temperaturelimit,thisformulayieldsZsâ1+5expâ3â2IT,lnZsâ5expâ3ÎČâ2I,Esâ15â2Iexpâ3â2IT,Csâ45â2IT2expâ3â2ITâȘ1,forTâȘâ2I,
i.e. the heat capacity is much lower than that given by Eq. (***). (At T â 0, the change of the exponent is muchmoreimportantthanthatofthepre-exponentialfactor.)Superficially,itmaylooklikethequantumbanonthepopulationofeachotherlevel(withoddvaluesofl)shouldaffecteventhehigh-temperatureresults.However,thisisnotso;indeed,becauseofthisleveldepletionthestatisticalsumbecomestwicelower,
ZsâITâ2,forâ2IâȘT,but since this constant factor changes lnZs only by an additive constant, it does not affect the average energy and heatcapacity:EsâT,Csâ1.
Anevenmoreinterestingcaseispresentedbyhomonuclearmoleculeswhosequantumstatemaybeeithersymmetricorantisymmetricwithrespecttotheatomsâswap,dependingonthenuclearspinstate.Ahistoricallyimportantexampleofsuchamolecule isgivenbyN2,withthenuclearspinofeachnitrogenatom(orratherof itsprevailing14Nisotope)equalto1.Accordingtotherulesofspinaddition41,thesystemoftwosuchspinsmaybeinanyof6symmetricstates(withthenetspinequal to either 0 or 2) and 3 asymmetric states (with the net spin equal to 1). Since the electronic ground state of thenitrogenmoleculeissymmetricwithrespecttotheatomsâswap,theformerstates(asinthepreviouscase)permitonlyevenl,whilethelatterstatespermitonlyoddvaluesl=2p+1.42Intheseasymmetricstates,Eq.(*)shouldbereplacedwith
Za=âp=0â(4p+3)expââ2ITp+1(2p+1),inthelow-temperaturelimitgivingevenlowerheatcapacity:
Zaâ3expââ2IT+7expâ6â2ITâĄ3expâÎČâ2I1+73expâ5ÎČâ2I,lnZaâln3âÎČâ2I+73expâ5ÎČâ2I,Eaââ2I+353â2Iexpâ5â2IT,Caâ1753â2IT2expâ5â2ITâȘ1,forTâȘâ2I.
(Inthehigh-temperaturelimit,EandCareagainnotaffectedbythequantumsymmetryeffects.)Since theenergydifferencebetween thenuclearspinstates isnegligibleon thescaleof temperaturesTâŒâ2/Iweare
considering,itdoesnotaffecttheprobabilityofstatepopulationinthermalequilibrium.AtTâȘâ2/IthetotalstatisticalsumZ=Zs+Zaisdominatedbythatofthesymmetricstates(namely,bytheground-statetermZsâ1),sothatthemoleculesarepredominantly in that state, and C â Cs. On the other hand, at high temperatures Tâ« â2/I, Zs â Za, and the ratio ofprobabilities for a molecule to be in the symmetric/antisymmetric state is determined by the relative number of thecorrespondingnuclearspinstates:
WsWa=63âĄ2.
SinceinthislimitCsâCaâ1,thiscompositionoftheequilibriumensembleofthemoleculesdoesnotaffectitsaverageheatcapacity.However,itdoesaffecttherelativeprobabilityofquantumtransitionsfromdifferentrotationalstatestohigherexcited states. In the late 1920s, i.e. before the experimental discovery of neutrons in 1932, measurements of suchprobabilitiesoftheN2molecules(carriedoutbyLOrnstein)havehelpedtoestablishthefactthatthespinofthenucleus14Nis indeedequal to1,andhencetodiscardthethen-plausiblemodel in that thenucleuswouldconsistof14protonsand7
(*)
(**)
electrons,givingittheobservedmassmâ14mpandnetelectricchargeQ=7e.(Inthatmodel,theground-statevalueofthenuclearspinwouldbesemi-integerratherthaninteger.)
Problem2.13.Calculatetheheatcapacityofaheteronucleardiatomicmolecule,usingthesimplemodeldescribedinthepreviousproblem,butnowassumingthattherotationisconfinedtooneplane43.
Solution:Repeating theargumentsgiven in themodelsolutionof thepreviousproblem, thesystemâsHamiltonianmaybereducedtothatof theso-calledplanarrotatorâaparticlewiththereducedmass , freetomoveonacircleof theradiusequaltod.TheHamiltonianconsistsonlyof1Dcomponentofthekineticenergy,
whereLËz=dpË is the angular momentumâs component normal to the motion plane. Quantummechanics says44 that theeigenvaluesLz of this systemare equal tomâ,wherem is an integer (the âmagnetic quantum numberâ). As a result, therotatorâsenergylevelsaredescribedbytherelation
This spectrum is similar to that studied inproblem2.11.Note,however, that incontrast to thatproblem, the rotatorâs
groundstatecorrespondstom=0(andhastheenergyE0=0),whileallitsexcitedenergylevels(withmâ 0)aredoublydegenerate,correspondingtotwopossiblesignsofm,becausetherotatorâseigenfunctions,
Ïm(Ï)=12Ï1/2eimÏ,correspondingtothesesigns,aredifferent.Asaresult,thestatisticalsumofthesystemis
Z=eâE0/T+2âm=1âeâEm/TâĄ1+2ÎE1TâĄ1+2Î(ÎČE1),withÎČâĄ1T,whereÎ(Ο)isthesamefunction,
ÎΟâĄâm=1âeâΟm2,aswasdiscussed(andplotted)inthemodelsolutionofproblem2.11.Fromthere,andEq.(2.61b)ofthelecturenotes, foraverageenergyoftheparticleweget
E=â(lnZ)ââÎČ=âE1ddΟln1+2ÎΟΟ=E1/T,sothatitsheatcapacity
CâĄâEâT=Ο2d2dΟ2ln1+2Î(Ο)Ο=E1/T.
Inthelow-temperaturelimit(TâȘE1),thelargestcontributiontoCisprovidedbythefirsttermofthesumÎ(Ο):
Zâ1+2eâÎČE1,lnZâ2eâÎČE1,Eâ2E1eâÎČE1âĄ2E1eâE1/T,Câ2E1T2eâE1/T,so that the heat capacity is exponentially low. In the opposite (essentially, classical) limit of high temperatures, Î(Ο) isreducedtoastandardGaussianintegral,
ÎΟââ«0âeâΟm2dm=Ï1/22Ο1/2,sothat
ZâÏÎČE11/2,lnZâ12lnÏÎČE1,EâT2,Câ12.Thisresultisnatural,becauseatTâ«E1thesystemisessentiallyclassical,withtheHamiltonianfunctionbeingaquadraticfunctionofoneâhalf-degreeoffreedomâ,sothattheclassicalequipartitiontheorempredictsthatE=T/2.
Betweenthesetwolimits,theheatcapacityasafunctionoftemperaturehasamaximumatTâŒE1(seethefigureabove),
whose origin is similar to that of two-level systemsâsee the discussion in themodel solutions of problem 2.3 and of thepreviousproblem.
Problem2.14.Aclassical,rigid,stronglyelongatedbody(suchasathinneedle),isfreetorotateaboutitscenterofmass,andisinthermalequilibriumwithitsenvironment.AretheangularvelocityvectorÏandtheangularmomentumvectorL,ontheaverage,directedalongtheelongationaxisofthebody,ornormaltoit?
Solution:Accordingtoclassicalmechanics45,theenergyofrotationofanyrigidbodymayberepresentedasE=âj=13IjÏj22,
whereIjaretheprincipalmomentsofinertia,andÏjaretheCartesiancomponentsoftheangularvelocityvectorÏalongthecorrespondingprincipalaxes.EachÏjmaybeconsideredasageneralizedvelocity, i.e.a âhalf-degreeof freedomâ,givingaquadratic contribution to the energy (*). Hence, according to the equipartition theorem, the statistical average of eachquadraticcomponentofEisequaltoT/2,sothat
I1Ï12=I2Ï22=I3Ï32.
Inastronglyelongatedbody,oneofthemomentsIj (say, I3),correspondingtotherotationalongtheelongationaxis, ismuchsmallerthantwootherones(I1,2),sothat
Ï32=I1I3Ï12=I2I3Ï22â«Ï1,22.Thismeansthattheaxisoftherandomthermalrotationofthebodyis,onaverage,veryclosetotheelongationaxis.
On theotherhand,whenrewritten for theCartesiancomponentsLj= IjÏjof theangularmomentumvectorL, Eq. (**)takestheform
L12I1=L22I2=L32I3,sothatthesecomponentsaretheoppositerelation:
L32=I3I1L12=I3I2L22âȘL1,22,i.e.thevectorLis,onaverage,directedalmostnormallytotheelongationaxis.
Note that these results should be generalized to the low-temperature (quantum) case with caution, because classical
(*)
(**)
mechanicsdoesnothavethenotionofparticleindistinguishability,andtheaboveformulasimplythattheturnofthebodybyanyangleisdistinguishable.Thisisnottrue,forexample,fordiatomicmolecules,whoserotationabouttheaxisconnectingtheatomicnucleiis(atallrealistictemperatures)purelyquantum;thisisthereasonwhythesolutionsofproblems2.12and2.13donotincludethecorrespondingenergy/Hamiltoniancomponent.
Problem2.15.Twosimilarclassicalelectricdipoles,ofafixedmagnituded,areseparatedbyafixeddistancer.Assumingthateachdipolemomentdmaytakeanyspatialdirection,andthatthesystemisinthermalequilibrium,writethegeneralexpressions for its statistical sum Z, average interaction energy E, heat capacity C, and entropy S, and calculate themexplicitlyinthehigh-temperaturelimit.
Solution:Accordingtothebasicelectrostatics46,theenergyofinteractionoftwoindependentdipolesisU=14ÏΔ0d1â d2r2â3râ d1râ d2r5âĄ14ÏΔ0d1xâ d2x+d1yâ d2yâ2d1zâ d2zr3,
whereinthelastexpression(andbelow)thez-axisisdirectedalongthevectorr,i.e.alongthelineconnectingthedipoles.PluggingintothelastformofthisrelationtheexpressionsofCartesiancomponentsofbothdipolemomentsviathepolarandazimuthalanglesoftheirorientation,
djx=dsinΞjcosÏj,djy=dsinΞjsinÏj,djz=dcosΞj,wherej=1,2,wemayrewritetheinteractionenergyasU=af,withaâĄd24ÏΔ0r3,fâĄsinΞ1cosÏ1sinΞ2cosÏ2+sinΞ1sinÏ1sinΞ2sinÏ2â2cosΞ1cosΞ2âĄsinΞ1sinΞ2cosÏ1âÏ2â2cosΞ1cosΞ2.
AtTâȘa,whenthermaleffectsareverysmall, thesystemshouldstayveryclosetooneof itspotentialenergyminima.
AccordingtoEq.(*),therearetwoofthem47;inbothcasesthedipolemomentsdarealignedwitheachother,andthelineconnectingthem:
Umin=afmin,fmin=â2,atΞ1=Ξ2=0,Ï.(Inanyofthesepositions,theanglesÏ1andÏ2areuncertain,anddonotaffectf.)Inthislimit,
EâUmin=afminâĄâ214ÏΔ0d2r3.Thenegativesignoftheenergy,andthegrowthofitsmagnitudeatrâ0showthatthedipoles,inequilibrium,attracteachother.
Sinceeachdipoleisfreetotakeanydirection,possiblestatesofitsorientationareuniformlydistributedoverthefullsolidangleΩj=4Ï.Asaresult,theprobabilitydensitywâĄdW/dΩ1dΩ2tofindthesystematacertainpoint{Ξ1,Ï1;Ξ2,Ï2}maybecalculatedusingtheGibbsdistributionintheform
w=1ZexpâUTâĄ1ZeâÎČU,whereÎČâĄ1/Tisthereciprocaltemperature,andZisthestatisticalsum:
Z=âź4ÏdΩ1âź4ÏdΩ2eâÎČU.
AsweknowfromEq.(2.61b)ofthelecturenotes,theaverageinteractionenergymaybecalculatedfromZasE=âź4ÏdΩ1âź4ÏdΩ2UwâĄ1Zâź4ÏdΩ1âź4ÏdΩ2UeâÎČUâĄ1ZâZââÎČâĄââlnZâÎČ.
From theseZ andE, two other variables of our interestmaybe readily calculatedusing the general relations (1.22) and(2.62):
CâĄâEâT,S=ET+lnZ.
Thus,weneedtocalculateonlyoneintegral,namelyZ=â«0ÏsinΞ1dΞ1â«02ÏdÏ1â«0ÏsinΞ2dΞ2â«02ÏdÏ2eâÎČU.
Duetothe2Ï-periodicityofthefunctionundertheintegralwithrespecttobothargumentsÏj,theintegralwouldnotchangeifwereplacetheintegrationinterval[0,2Ï]foroneoftheseangles,sayÏ1,withany2Ï-longinterval,forexample[Ï2,Ï2+2Ï].Nowinthisintegral,tobeworkedoutatfixedÏ2,wemaywritedÏ1=dÏ,whereÏâĄÏ1âÏ2.Since,accordingtoEq.(*),thefunctionundertheintegraldependsonlyonÏbutnotonÏ2,wemayfirsttaketheintegraloverÏ2,givingthefactor2Ï,sothatthegeneralexpressionforZisreducedto
Z=2Ïâ«0ÏsinΞ1dΞ1â«0ÏsinΞ2dΞ2â«02ÏdÏeâÎČaf,withf=sinΞ1sinΞ2cosÏâ2cosΞ1cosΞ2.
SinceâŁfâŁâŒ1,inthehigh-temperaturelimit,Tâ«a,i.e.ÎČaâȘ1,theargumentoftheexponentinthisexpressionissmallforanydipoleorientations,andwemayexpanditintotheTaylorseriesinthisparameter,keepingonlythreeleadingterms:
eâÎČafâ1âÎČaf+12ÎČaf2.TheintegrationofthefirsttermaloneyieldsaÎČ-independentcontribution,(4Ï)2,toZ.Thesecondterm,proportionaltof,hastwoparts.TheintegralofthepartproportionaltocosÏvanishesbecauseoftheintegraloverÏ,andthatoftheremainingpartisaproductoftwosimilarintegralsofthetype
â«0ÏsinΞjdΞjcosΞj=â«â1+1cosΞjdcosΞj=â«â1+1ΟdΟ=0.This is exactlywhyweneeded to keep the last, quadratic term in theaboveTaylor expansion: it doesgive the largestÎČ-dependentcontributiontoZ.Indeed,inthisapproximation
Zâ4Ï2=ÏÎČa2â«0ÏsinΞ1dΞ1â«0ÏsinΞ2dΞ2Ăâ«02ÏdÏsinΞ1sinΞ2cosÏâ2cosΞ1cosΞ22.Openingtheparentheses,weseethatthemixedterm,proportionaltocosÏ,givesavanishingcontributiontotheintegraloverÏ,sothatwemaycontinueasfollows:
Zâ4Ï2=ÏÎČa2â«0ÏsinΞ1dΞ1â«0ÏsinΞ2dΞ2Ăâ«02ÏdÏsin2Ξ1sin2Ξ2cos2Ï+4cos2Ξ1cos2Ξ2âĄÏÎČa2â«0ÏsinΞ1dΞ1â«0ÏsinΞ2dΞ2sin2Ξ1sin2Ξ2+8cos2Ξ1cos2Ξ2âĄÏÎČa2â«0ÏdcosΞ1Ăâ«0ÏdcosΞ21âcos2Ξ11âcos2Ξ2+8cos2Ξ1cos2ΞIntroducingthevariablesΟjâĄcosΞjagain,wegetZâ4Ï2ÏÎČa2=â«â1+1dΟ1â«â1+1dΟ2(1âΟ12)(1âΟ22)+8Ο12Ο22âĄâ«â1+1dΟ1â«â1+1dΟ2(1âΟ12âΟ22+9Ο12Ο22)âĄ4â4â«â1+1Ο2dΟ
+9â«â1+1Ο2dΟ2=4â4â 23+9â 232âĄ163,sothat,finally,
Z=4Ï2+163Ï2ÎČ2a2.
Thishigh-temperatureapproximationisvalidonlyifÎČaâȘ1,sothatwiththeaccuracyO(ÎČ2a2),lnZ=ln4Ï21+14Ï2163Ï2ÎČa2âln4Ï2+14Ï2163Ï2ÎČ2a2âĄ2ln4Ï+13ÎČ2a2.
Withthatresult,wegetE=ââlnZâÎČââ23a2ÎČâĄâ2a23T.
NowwemayusethisexpressiontocalculateC=âEâT=2a23T2,S=ET+lnZ=â2a23T2+2ln4Ï+13ÎČ2a2âĄconstâa23T2,
whereinthelastexpression,theconstantmeansatermindependentofthedipole-dipoleinteraction.Theseresultsshowthatinthehigh-temperaturelimit,alleffectsofdipoleinteractionarerelativelysmall(proportionalto
a2âȘT2).Thisisnatural,becausetheprobabilitydensityw isnearlyuniformlydistributedoveralldipoleorientations,thusvirtuallyaveragingouttheinteractionenergy.Inparticular,Eq.(**)fortheaverageinteractionenergyshowsthatwhilethedipolesstillattracteachothereveninthis limit,theattractionismuchweakerthanatlowtemperatures,anddropsmuchfasterwithdistance:
E=â23Td24ÏΔ0r32ââ1r6.
Note that such distance dependence is typical for the one ofmolecular (âvan derWaalsâ) forcesâthe so-calledLondon
(*)
(**)
(***)
(*)
(**)
dispersion force, which dominates the long-range interaction of electroneutral atoms and molecules48. This similarity isnatural,becausetheLondonforceisalsoduetothestatistically-averagedinteractionofelectricdipoles.However,incontrastto the fixed-magnitude, free-orientationdipolemodelanalyzed in thisproblem, theLondondispersion forcebetweenmostmolecules (having no spontaneous electric dipolemoments) is due to theweakmutual induction of randomly fluctuatingdipoles. These fluctuations have not only the classical, but also a quantum contribution, so that the force has the samedependenceonrevenatTâ0,whilebecomingtemperatureâindependentinthislimit49.
Problem2.16.Aclassical1Dparticleofmassm,residinginthepotentialwellUx=αxγ,withγ>0,
isinthermalequilibriumwithitenvironment,attemperatureT.CalculatetheaveragevaluesofitspotentialenergyUandthefullenergyEusingtwoapproaches:
(i)directlyfromtheGibbsdistribution,and(ii)usingthevirialtheoremofclassicalmechanics50.
Solutions:
(i)ThecontinuousversionoftheGibbsdistribution(2.58)forsuchaparticlemaybewrittenaswx,p=1ZexpâEx,pT,withZ=â«ââ+âdxâ«ââ+âdpexpâEx,pT,
whereEisparticleâsfullenergy:Ex,p=p22m+Ux,
sothattheexponentparticipatinginthepartitionfunctionmayberepresentedasaproduct:expâEx,pT=expâp22mTĂexpâUxT.
Duetosuchfactoring,theintegralsoverpintheaboveexpressionforZ,andinEq.(2.11)ofthelecturenotes,appliedtoU,U=1Zâ«ââ+âdxâ«ââ+âdpUxexpâEx,pT,
areexactlythesameandcancel51.Also,duetothesymmetryofthefunctionU(x),bothintegralsoverxmaybelimitedtox>0:
U=â«0+âUxexpâUxTdx/â«0+âexpâUxTdx.
Theintegralinthedenominator,forourparticularformofthefunctionU(x),maybeworkedoutbyitsreductiontotheusualdefinitionofthegamma-function52bythefollowingvariablereplacement:ΟâĄÎ±xÎł/T(givingx=(TΟ/α)1/Îł,andhencedx=(T/α)1/ÎłÎŸ1/Îłâ1dΟ/Îł):
â«0+âexpâUxTdx=â«0+âexpâαxÎłTdx=1ÎłTα1Îłâ«0+âeâΟΟ1Îłâ1dΟ=1ÎłTα1ÎłÎ1ÎłâĄ1Îł1αÎČ1ÎłÎ1Îł,whereÎČâĄ1/Tisthereciprocaltemperature.NowtheintegralinthenumeratorofEq.(*)maybecalculatedbydifferentiationoftheaboveexpressionovertheparameterÎČ:
â«0+âUxexpâUxTdx=âââÎČâ«0+âexpâÎČUxdx=âââÎČ1Îł1αÎČ1ÎłÎ1Îł=1αÎČ2Îł21αÎČ1Îłâ1Î1Îł,so that Eq. (*), after the cancellation of common factors (including the gamma-function), yields a very simple result,independentofthecoefficientα,i.e.ofthepotentialâsstrength:
U=1ÎČÎłâĄTÎł.
Forthekineticenergy,thesamefactoringofthepartitionfunctionyieldsthesameresultasforafreeparticle(seeEq.(2.48)again):
p22m=â«0+âp22mexpâp22mTdp/â«0+âexpâp22mTdp=T2,sothattheaverageenergy,
E=p22m+U=T12+1Îł.(ii)Appliedtoasingleparticle,thevirialtheoremreads
p22mÂŻ=âFâ rÂŻ2,relating the time averages of its kinetic energy and the scalar force-by-position product. Since we may represent thethermalizationoftheparticleastheeventualresultofitsveryweakinteractionwithaheatbath,notperturbingeachmotionperiodnoticeably,thestatisticalaveragesofbothsidesarealsoequal:
p22mÂŻ=âFâ rÂŻ2.
Accordingtotheequipartitiontheorem,fora1Dparticlemovingalongaxisx,theleft-handsideofthisrelationinthermalequilibriumequalsT/2,whilethescalarproductontheright-handsideisjustFx=(ââU/âx)x,sothatweget
xâUâxÂŻ=T.Forour(time-independent)potential,
xâUxâx=xddxαxÎł=αγxsgnxxÎłâ1âĄÎ±ÎłxÎłâĄÎłUx,sothattheequipartitiontheoremyields53
ÎłUÂŻ=T,i.e.UÂŻ=TÎł,EâĄp22mÂŻ+UÂŻ=T2+TÎłâĄT12+1Îł,i.e.gives,inamuchsimplerway,thesameresultsaswereobtainedbythefirst,directapproachâseeEqs.(**)and(***).
FortheparticularcaseÎł=2,i.e.forthequadraticconfiningpotentialU=αâŁx2âŁâĄÎ±x2,whentheparticleisjustaharmonicoscillator,thisresultreturnsustoEq.(2.48)ofthelecturenotes.However,itshowsthatgenerally,âšEâ©â T;forexample,forverysoftconfiningpotentials(Îłâ0),theaverageenergyismuchlargerthanT.Thisfactshedsanadditionallightonwhythegeneralnotionoftemperaturehastobedefineddifferentlythantheaverageenergyperparticle,andgivesagoodpretexttohaveonemore,thoughtfullookatEq.(1.9).
Problem 2.17. For a thermally-equilibrium ensemble of slightly anharmonic classical 1D oscillators, with massm andpotentialenergy
Uq=Îș2x2+αx3,withsmallcoefficientα,calculateâšxâ©inthefirstapproximationinlowtemperatureT.
Solution:AccordingtothebasicEq.(2.11)ofthelecturenotes,forthis1Dsystemx=â«ââ+âdxâ«ââ+âdpxwx,p,
wherew(x,p)istheprobabilitydensity.Inthermalequilibrium(i.e.forthecanonicalensemble)thedensitymaybecalculatedfromthecontinuousversionoftheGibbsdistribution(2.58):
wx,p=1ZexpâEx,pT,withZ=â«ââ+âdxâ«ââ+âdpexpâEx,pT.whereEisoscillatorâsfullenergy:
Ex,p=p22m+Ux,sothatexpâEx,pT=expâp22mTĂexpâUxT.Duetothisfactoring,theintegralsoverpintheexpressionsforâšxâ©andZareexactlythesameandcancel,sothat(similarlytoEq.(*)inthesolutionofthepreviousproblem)
x=â«ââ+âxexpâUxTdx/â«ââ+âexpâUxTdx.
Accordingtotheequipartitiontheorem(2.48),inaclassicalharmonicoscillator,thecharacteristicscaleofxis(T/Îș)1/2,sothat,inthefirstapproximationintemperature,wemaytreatthedimensionlesscombinationαx3/TâŒÎ»âĄÎ±T1/2/Îș3/2âȘ1asasmallparameterandexpandtheexponentsinEq.(*)as
expâUxTâĄexpâÎșx22Texpâαx3TâexpâÎșx22T1âαx3TâĄexpâÎșx22Tâαx3TexpâÎșx22T.Thesecondtermofthisexpansion,oddinx,giveszerocontributionintotheintegralinthedenominatorofEq.(*),sothatitis,inthisapproximation,thesameasfortheharmonicoscillator:
â«ââ+âexpâUxTdxââ«ââ+âexpâÎșx22Tdx=2TÎș1/2â«ââ+âexpâΟ2dΟ=2ÏTÎș1/2,whereatthelaststepthestandardGaussianintegral54hasbeenused.However,intheintegralinthenumerator,itisthefirsttermoftheexpansion(**)thatvanishes,whilethesecondoneyields
x=âαTâ«ââ+âx4expâÎșx22Tdx/2ÏTÎș1/2âĄâαTÎș2ÏT1/22TÎș5/2â«ââ+âΟ4expâΟ2dΟ.ThisisalsoatableGaussianintegral55,equalto(3/4)Ï1/2,sothat,finally,
x=âαTÎș2ÏT1/22TÎș5/234Ï1/2âĄâ3αTÎș2âĄâ3λTÎș1/2.ThisformulaisstrictlyvalidonlyifλâȘ1,i.e.ifâŁâšxâ©âŁâȘ(T/Îș)1/2âŒâšx2â©1/2.
This result conceptually explains the thermal expansion of solids, because at interatomic interactions, the effectivecoefficientα is typically negativeâsee, e.g. figure 3.7 of the lecture notes. However, for a quantitative comparison withexperiment,thetheoryhastobedulygeneralizedtophononmodesâsee,e.g.thediscussioninsection2.6(ii).
Problem 2.18.* A small conductor (in this context, usually called the single-electron island) is placed between twoconductingelectrodes,withvoltageV appliedbetween them.Thegapbetweenoneof the electrodes and the island is sonarrowthatelectronsmaytunnelquantum-mechanicallythroughthisgap(theâweaktunneljunctionâ)âseethefigurebelow.CalculatetheaveragechargeoftheislandasafunctionofV.
Hint:Thequantum-mechanicaltunnelingofanelectronthroughaweakjunction56betweenmacroscopicconductors,anditssubsequentenergyrelaxationinsidetheconductor,maybeconsideredasasingleinelastic(energy-dissipating)event,sothattheonlyenergyrelevantforthethermalequilibriumofthesystemisitselectrostaticpotentialenergy.
Solution:ThecalculationoftherelevantelectrostaticenergyofthissystemUasafunctionofthenetchargeQ=âneoftheisland(wheren,thedifferencebetweenthetotalnumberofelectronsandprotonsintheisland,maytakeonlyintegervalues)wasthesubjectofPartEMproblem2.27.Theresultis
Un=Q+Qext22CÎŁ+constâĄQextâne22CÎŁ+const,whereQextâĄCVistheeffectivepolarizationchargeoftheisland(whosevalues,incontractwithQâĄâne,arenotlimitedtothemultiplesofthefundamentalcharge57),andCÎŁâĄC+C0isthetotalcapacitanceoftheisland.ApplyingtothissystemtheGibbsdistribution(2.58)â(2.59),weget
Q=1Zân=ââ+ââneexpâQextâne22CÎŁT,withZ=ân=ââ+âexpâQextâne22CÎŁT,sothatintroducingthedimensionlessexternalchargenextâĄQext/eâĄCV/eandthenormalizedtemperatureÏâĄT/(e2/CÎŁ),weget58
Qe=âân=ââ+ânexp{â(nextân)2/2Ï}ân=ââ+âexp{â(nextân)2/2Ï}.
Thisresultisplottedinthefigureaboveforseveralvaluesofthenormalizedtemperature.AtTâȘe2/CÎŁ,thedependenceof
âšQâ©onQext(i.e.ontheappliedvoltageV)followsthevertical-stepâCoulombstaircaseâpatternâseethemodelsolutionofPartEMproblem2.27.However,non-zerotemperaturesevenaslowasâŒ0.3e2/CÎŁresult inanalmostcompletesmearingofthepattern.Because of the similar smearing,most single-electrondevices (suchas single-electron transistors, single-electrontraps,etc59)alsorequiretemperaturestobelowerthanâŒ0.03e2/CÎŁfortheirproperoperation.Thisrequirementspresentsoneoftwomajorchallenges60forthedevelopmentofdigitalsingle-electronics,becauseforoperationatroomtemperature(TâŒ25meV)itdemandse2/CÎŁtobeoftheorderof1eV,correspondingtosingle-electronislandsofjustafewnanometersinsize.
Problem2.19.AnLCcircuit(seethefigurebelow)isinthermodynamicequilibriumwithitsenvironment.CalculatethermsfluctuationÎŽVâĄâšV2â©1/2ofthevoltageacrossit,foranarbitraryratioT/âÏ,whereÏ=(LC)â1/2istheresonancefrequencyofthisâtankcircuitâ.
Solution:ThoughtheexpressionfortheclassicalenergyoftheLCcircuitiswellknownfromundergraduatephysics,wewanttheresultthatwoulddescribeitsquantumpropertiesaswell,soletusderiveitsHamiltoniancarefully,firstusingthebasicnotionsofclassicalanalyticalmechanics61.Atnegligibleenergylosses,thecircuitmaybedescribedbyaLagrangianfunction,whichisthedifferenceofthekineticandpotentialenergiesofthesystem.Forexample,wemaywrite
(*)
(**)
(***)
(****)
wheretheelectricchargeQofthecapacitorandthecurrentI=Qthroughtheinductivecoilmaybeconsidered,respectively,asageneralizedcoordinateandthegeneralizedvelocityofthesystem.Thecorrespondinggeneralizedmomentumis
(Physically,Ί=LIisthetotalmagneticfluxintheinductivecoil.)HencetheHamiltonianfunctionofthesystemis
Now we may perform the transfer to quantum mechanics just by the replacement of I and Q with the operators
representingtheseobservables62.TheresultingHamiltonian,
isevidentlysimilartothatofamechanicalharmonicoscillatorâsee,e.g.Eq.(2.46)ofthelecturenotes,withthefollowingreplacements:qâQ,pâpQ=Ί,mâL,Îșâ1/C(givinginparticularÏ2=Îș/mâ1/LCâawellknownresult).With thesereplacements,Eq.(2.78)ofthelecturenotesyields
Q2=âÏC2cothâÏ2T.ButthevoltageweareinterestedinisjustV=Q/C,sothat
V2=Q2C2=âÏ2CcothâÏ2T,andthermsfluctuationÎŽVisjustthesquarerootofthisexpression.Thephysicsofthisresult,anditsimplications,willbediscussedindetailinchapter5ofthelecturenotes.
Problem2.20.DeriveEq.(2.92)ofthelecturenotesfromsimplisticarguments,representingtheblackbodyradiationasanidealgasofphotons,treatedasultra-relativisticparticles.Whatdosimilarargumentsgiveforanidealgasofclassical,non-relativisticparticles?
Solution:LetusconsiderarectilinearcavityofvolumeV=LxĂLyĂLz,containingNphotonselasticallyreflectedfromthewalls,butotherwiseempty.Eachreflectionfromthewallthatisperpendiculartoaxisx,transfersthemomentumÎpx=2pxtoit.Forafreeultra-relativisticparticle,thefullmomentummagnitudepandtheenergyΔarerelatedjustasΔ=cp,wherecisthespeedoflight63,sothatthetransferredmomentummaybeexpressedas
Îpx=2px=2ΔccosΞ,whereΞistheincidenceangleâseethefigurebelow.Sincephotonsinfreespacemovewithvelocityc,thereflectedphotonwillreturntothesamewallagain(afterbeingreflectedbytheoppositewall)aftertheballisticflighttime
sothatthe(time-)averageforceexertedbythisparticularphotononthewallis
correspondingtoitsaveragepressure
NowsummingupthepressurecontributionsbyallNphotons,weget
P=EVcos2ΞwhereE is thesumofallΔ, i.e. the full energyof thephotongas,andcos2Ξ is averagedover the statistical ensembleofrandomdirectionsofphotonpropagation:
cos2Ξ=14Ïâź4ÏdΩcos2Ξ=14Ïâ«02ÏdÏâ«Îž=ÏΞ=0cos2Ξd(cosΞ)=14Ï2Ïâ«â1+1Ο2dΟ=13.Asaresult,wegetEq.(2.92)ofthelecturenotes:
P=13EV.
Let us make a similar calculation for a non-relativistic gas, with just a few (but significant!) changes. Indeed, themomentumofsuchaparticleisp=mv,sothatinsteadofEq.(*)wemaywrite
Usingthefirstofequalities(**)forthetimeinterval, ,weget
Again, summing the pressure contributions by all particles, while assuming the gasâ isotropy (so that64wegetaresult,
whichdiffersfromEq.(***)byafactorof2âaswasalreadydiscussedinsection2.6ofthelecturenotes.NownotethatwhiletheabovederivationsofEqs.(***)and(****)hadrequiredtheassumptionoftheisotropyofparticle
flight directions, they did not require the gas to be in thermal equilibrium. If this condition is added, we may use theequipartitiontheorem(2.48),validfornon-relativisticparticlesonly,torecastEq.(****)as
thusrecoveringtheequationofstateoftheidealclassicalgas,Eq.(1.44)âwhichwillbederivedinadifferent,moregeneralwayinsection3.1.
Problem 2.21. Calculate the enthalpy, the entropy, and the Gibbs energy of blackbody electromagnetic radiation with
(*)
(**)
(***)
temperatureT,andthenusetheseresultstofindthelawoftemperatureandpressuredropatanadiabaticexpansion.
Solution:PluggingEq. (2.88)of the lecturenotes for theenergy,Eq. (2.91) for the freeenergy,andEq. (2.92) for thePVproductoftheblackbodyradiation,
E=Ï215â3c3VT4,F=âE3,PV=E3,intothegeneralthermodynamicrelations(1.27),(1.33),and(1.37),wereadilyget
HâĄE+PV=43E=4Ï245â3c3VT4,S=EâFT=43ET=4Ï245â3c3VT3,GâĄF+PV=0.
Actually,thelastresultwasalreadygiven(inadifferentform)inthelecturenotesâseeEq.(2.93).Itssimplestphysicalinterpretationisthatthethermally-equilibriumradiationmaybeconsideredasagasofultra-relativistic,masslessparticles(photons),whichmaybecreatedâfromnothingâ,i.e.maybeformallyconsideredascomingfrom(andgoingto)someexternalsourcewithavanishingchemicalpotentialÎŒ=G/N.
Aswasdiscussedinsection1.3ofthelecturenotes(andmentionedseveraltimesafterthat),atanadiabaticexpansionofasystem,itsentropyhastostayconstant,sothattheaboveexpressionforSyields
VT3=const,i.e.Tâ1V1/3,i.e. at an isotropic expansion, the temperature is inversely proportional to the linear size of the region occupied by theradiation.Next,accordingtoEq.(*),theradiationâspressure
P=E3V=Ï245â3c3T4isindependentofV,sothatpluggingthereciprocalrelation,TâP1/4,intoEq.(**),weget
Pâ1V4/3.
Thisrelationmayberewrittenintheformsimilarfortheadiabaticexpansionoftheâusualâ(non-relativistic)idealgas65:PVÎł=const,
ifwe takeÎł=4/3.Note,however, that inourcurrentcaseof thephoton (andanyultra-relativistic)gas, thecoefficientÎłcannot be interpreted as the CP/CV ratio, because, according to Eq. (1.23) of the lecture notes, the notion of CP isundeterminedifthepressureisauniquefunctionofT,asitisinthiscaseâseeEq.(***)again.
Problem 2.22. Aswasmentioned in section 2.6(i) of the lecture notes, the relation between the temperaturesTâ of thevisibleSunâssurfaceandthat(To)oftheEarthâssurfacefollowsfromthebalanceofthethermalradiationtheyemit.Provethat this relation indeed follows,withgoodprecision, froma simplemodel inwhich the surfaces radiateasperfectblackbodieswithconstant,averagetemperatures.
Hint:Youmaypickuptheexperimentalvaluesyouneedfromany(reliable)source.
Solution: According to the Stefan radiation law (seeEq. (2.89) of the lecture notes), the full radiation power of the Sun,withinthismodel,is
whereRâisSunâsradius.AttheEarthâsdistance,ro,fromtheSun,thisradiationisuniformlydistributedoverthesphericalsurfacewitharea
A=4Ïro2,sothattheEarth,visiblefromtheSunasaplanerounddiskoftheradiusRoâȘro,picksuppower
IfthepowerradiatedbytheEarthâssurfacetospaceisexpressedsimilarlytoEq.(*):
thenfromtheradiationbalance ,wegettheresultTo=TâRâ2ro1/2,
independentofboth theEarthradiusRoandtheStefanâBoltzmannconstantÏâandhenceof thePlanckconstantand thespeedoflightâseeEq.(2.89b)ofthelecturenotes.
Plugging in the experimental valuesTââ5778K,Râ â 0.6958Ă 106 km, and ro â 149.6Ă 106 km, for the averagetemperatureofEarthâssurfacewegetanumber,Toâ278.6K(i.e.âŒ6°C),whichdiffersfromtheexperimentalvalueof288Kby justâŒ3%.Thissurprisinglygood fit isdueto thehighemissivity (see thenextproblem),ΔâŒ99%,ofboth theSunâsphotosphereandtheEarthoceanscoveringmostofourplanetâssurface,atthemostrelevantfrequencies.
Problem2.23.Ifasurfaceofabodyisnotperfectlyradiation-absorbing(âblackâ),thepowerofitsthermalradiationdiffersfromthevaluegivenbytheStefanlaw(2.89a)byafactorΔ<1,calledtheemissivity:
Provethatsuchsurfacereflects(1âΔ)partoftheincidentradiation.
Solution:Considersuchsurfaceinathermalequilibrium,attemperatureT,withalargesurroundingvolume.Accordingtothediscussioninsection2.6(i)ofthelecturenotes,thepowerincidentonthesurfaceis
Butinthermalequilibrium,thetotalpowerflowingfromthesurface,
whereristhereflectioncoefficientwearetryingtofind,hastoequal ,becauseatthermalequilibrium,thereshouldbenonetheat flow into the interior of thebodyâif it is not transparent. This balance, using the expression for given in theassignment,immediatelyyields
i.e.provesthestatedresultforthereflectioncoefficient:r=1âΔ.
This fundamental relation is sometimes called theKirchhoff lawof thermal radiation. (Such a long name is probably
justifiedtoavoidconfusionwiththefamoustwolaws,orârulesâ66,governinglumpedelectriccircuits,formulatedbythesameGKirchhoff.)Noteagainthatbyitsderivation,thelawisnotvalidfortransparentobjects.
Notealsothatinthisproblem,Δwastreatedasanangle-averagedparameter,hidingpossibledependenceoftheso-calleddirectionalemissivityΔΩontheradiationdirection.
Problem2.24.Iftwoblacksurfaces,facingeachother,havedifferenttemperatures(seethefigurebelow),thenaccordingtotheStefanradiationlaw(2.89),thereisanetflowofthermalradiation,fromawarmersurfacetothecolderone:
(**)
Formanyapplications,notablyincludingmostlowtemperatureexperiments,thisflowisdetrimental.OnewaytosuppressitistoreducetheemissivityΔ(foritsdefinition,seethepreviousproblem)ofbothsurfacesâsaybycoveringthemwithshinymetallic films.Analternativeway toward thesamegoal is toplace,between thesurfaces,a thin layer (usuallycalled thethermalshield),withalowemissivityofbothsurfacesâseethedashedlineinthefigureabove.Assumingthattheemissivityisthesameinbothcases,findoutwhichwayismoreefficient.
Solution:AnyemissivityΔ<1 reduces the thermal radiationpowerofa surface to thevalue , lower than thevalue followingfromtheStefanlaw.Thetotalpowerflowingfromsuchasurfacemaybecalculatedasasumofits own thermal radiation, , and , where is the power incident on this surface from outside, where r is thereflectivityofthesurface.Butaswasprovedinthepreviousproblem,thereflectivityr isrelatedtotheemissivityΔofthesamesurface(atthesamefrequency)bytheverysimpleKirchhoffradiationlaw:
r=1âΔ.
Applying these arguments to the power flowing from surface 1 to surface 2, and its counterpart , for the firstapproachdiscussedintheassignment(seethefigurebelow),wemaywritetwosimilarrelations:
Solvingthissimplesystemoftwolinearequations,weget
sothatthenetpowerflowfromsurface1tosurface2is
Asasanitycheck, in the limitΔâ0 (perfectly reflectingsurfaces) thenetpower tends tozero,while forΔ=1 (perfectlyabsorbingsurfaces)wegettheblackbodyformulacitedintheassignment.
Forthesecondmethoddescribedintheassignment(seethefigureabove),thepowersradiatedbythesurfaces1and2,
consideredperfectlyblack,followtheunmodifiedStefanlaw,butweneedtowriteequationssimilartoEqs.(*)forthepowersflowingfromeachsurfaceofthethermalshield,assigningtoitsome(sofar,unknown)temperatureTs:
Since the shield is in thermal equilibrium, and is not connecteddirectly to any external objects, the net radiationpower,flowingfromtherighttotheleft,shouldbethesameontheleftandontherightoftheshield:
Solvingthissystemofthesefourequations,weget,inparticular,
ComparingthelastresultwithEq.(**),weseethatbothmethodsareveryefficient,withthesecondmethodyieldinga
somewhatbetterresult,i.e.alowernetpowerflow.However,fortypicalwell-reflectingsurfaces,withΔoftheorderofonepercent,thedifferenceisminor.Still,thethermalshieldmethodismoreconvenientpractically,andmaybefurtherimprovedbyusingseveralshields,thermallyisolatedfromeachotherandfromthebothbodies.(Calculating forasystemwithnsimilarshields,usingsuchapproach,isasimpleadditionalexercise,highlyrecommendedtothereader.)
Problem2.25.Twoparallel,wellconductingplatesofareaAareseparatedbyafree-spacegapofaconstantthicknesstâȘA1/2. Calculate the energy of the thermally-induced electromagnetic field inside the gap at thermal equilibrium withtemperatureTintherange
âcA1/2âȘTâȘâct.Doesthefieldpushtheplatesapart?
Solution:Electrodynamics67tellsusthatwithinabroadfrequencyrange,cA1/2âȘÏâȘct,inwhichtâȘλâĄ2ÏkâĄ2ÏcÏâȘA1/2,
such a system supports transverse (âTEMâ) electromagnetic waves of speed c. Hence if temperature is within the rangespecifiedintheassignment,wemayjustreproducethecalculationscarriedinsection2.6(i)ofthelecturenotes,replacingthe3DdensityofstatesinEqs.(2.82)and(2.83)withthe2Ddensity,andthedegeneracyfactorg=2withg=1:
dN=gA2Ï2d2k=A2Ï22Ïkdk=A2Ïc2ÏdÏ.
Withthisreplacement,Eq.(2.84)becomes
(*)
(*)
(**)
uÏâĄEAdNdÏ=âÏ22Ïc21eâÏ/Tâ1,sothatthetotalenergyoftheradiationinthegapis68
E=Aâ«0âuÏdÏ=Aâ2Ïc2â«0âÏ2dÏeâÏ/Tâ1âĄAâ2Ïc2Tâ3â«0âΟ2dΟeΟâ1=Aâ2Ïc2Tâ3Î3ζ3â0.383AT3c2â2.
NotethattheenergyscalesasT3,ratherthanasT4intheStefanlaw,becauseofadifferentdimensionalityofthesystem.(AtTbecomeslargerthanâŒâc/t,whennon-TEMmodesmaybeexcitedinthegap69,wemayexpectagradualcrossovertotheStephanlaw.)
The sufficient condition of applicability of this result is given by the double inequality given in the assignment. Note,however, thatamoderateattenuationof thewaves (say,due toa finiteconductivityof theplanematerial),with thewavedecay lengthmuch larger than t,butmuchsmaller thanA1/2, sustains theTEMwaves,butmakes them insensitive to theplateboundary70.Inthiscase,ourresultmaybevalidevenattemperaturesbelowâc/A1/2.
Finally,sincethecalculatedaverageenergyEofthespontaneousradiation,andhenceitsfreeenergyF,areindependentofthegapthicknesst,itdoesnotapplyanynormalpressuretotheconductingplates.This(perhaps,counterintuitive)resultisduetothefactthatthewavevectork(andhencethephotonmomentump=âk)ofthesewavesareparalleltotheplatesurfaces71.(Notethatanaccountoftheground-stateenergyoftheTEMmodeswouldnotchangethesituation.)Moreover,since the pressure of unavoidable 3D (non-TEM) electromagneticwaves outside of the plates is not compensated, at thefrequenciesÏ<Ïc/t,bytheTEMwavesbetweenthem,theplatesareeffectivelyattractedtoeachother,evenatTâ0âtheso-calledCasimireffect72.
Ontheotherhand,ourcalculationshowsthattheenergyoftheTEMwavesinsidethegapisproportionaltothesystemareaAâjustasinthe3DcaseitisproportionaltothecontainingvolumeVâseeEq.(2.88)ofthelecturenotes.Asaresult,thethermally-inducedTEMradiationbetweentheplatesdoesapplyanoutwardstresstosystemâsedges,tryingtostretchtheplates.
Problem 2.26. Use the Debye theory to estimate the specific heat of aluminum at room temperature (say, 300 K), andexpresstheresultinthefollowingpopularunits:
(i)eV/Kperatom,(ii)J/Kpermole,and(iii)J/Kpergram.
Comparethelastnumberwiththeexperimentalvalue(fromareliablebookoronlinesource).
Solution:Aswasmentionedinsection2.6(ii)ofthelecturenotes,theDebyetemperatureofaluminumiscloseto430K,sothatattheroomtemperature,T/TDâ300/430â0.70.UsingEqs.(2.97)and(2.98),orjustreadingthevaluefromoneoftheplotsinfigure2.11,weget73
CnVâĄCNâ2.71,withasub-1%accuracy.(Thepossiblesmallerrorresultsfromalimited,a-few-KaccuracyoftheexperimentalvalueofTD.)
Theresult(*)isvalidifthespecificheatisdefinedbyEq.(1.22),CâĄâQ/âT,withtemperatureTexpressedinenergyunits,sayjoules.Withtemperatureexpressedinkelvins,TKâĄT/kB,thespecificheatbecomes
CinJ/K=âQâTK=âQâT/kB=kBâQâT=kBC,sothatEq.(*)yields
CNinJ/Kâ atomâ2.71kBâ2.71Ă1.38Ă10â23JKâ atomâ3.74Ă10â23JKâ atom.
Nowletusexpressthisestimateintherequestedunits:
(i)CNineV/Kâ atomâCNinJ/Kâ atom1eâ3.74Ă10â231.60Ă10â19eVKâ atomâ0.233meVKâ atom,(ii)CNinJ/Kâ moleâCNinJ/Kâ atomNAâ3.74Ă10â23Ă6.02Ă1023JKâ moleâ22.5JKâ mole,(iii)CNinJ/Kâ gâCNinJ/Kâ mole1ÎŒâ22.527.0JKâ gâ0.833JKâ g,
whereÎŒ â 26.98 â 27.0 is the average atomic weight of the natural aluminum (dominated by the 27Al isotope), i.e. theaveragemass(ingrams)ofitsmole.
For the last number, a linear interpolation of the experimental valuesgiven in the classical tablesbyKaye andLaby74(0.880JKâ1·gâ1for273Kand0.937JKâ1·gâ1for373K)totheroomtemperatureof300Kgives0.895JKâ1·gâ1.Anotherrespectablesource75givesjustaâŒ1%highervalue,0.904JKâ1·gâ1,forthesame300K.TheWikipediaâsarticleâAluminiumâ(with themetalâs name in British English) lists, without an explicit reference and temperature specification, the number24.20JKâ1·moleâ1,equivalentto0.897JKâ1·gâ1,i.e.justin-betweenthesetwoabovepoints.So,theaverageexperimentalvalue is by âŒ7.5% higher than our estimate. This deviation shows a limited accuracy of the Debye theory, but still isastonishinglysmallforsuchasimplemodel.
Problem2.27.Low-temperaturespecificheatofsomesolidshasaconsiderablecontributionfromthermalexcitationofspinwaves,whosedispersionlawscalesasÏâk2atÏâ0.76Neglectinganisotropy,calculatethetemperaturedependenceofthiscontributiontoCVatlowtemperaturesanddiscussconditionsofitsexperimentalobservation.
Hint:Justasthephotonsandphonons,discussedinsection2.6ofthelecturenotes,thequantumexcitationsofspinwaves(calledmagnons)maybeconsideredasnon-interactingbosonicquasiparticleswithzerochemicalpotential,whosestatisticsobeysEq.(2.72).
Solution:Actingexactlyas insection2.6, for isotropicwavesofanykindwemaycalculatethenumberofdifferentmodeswithinasmallintervaldÏoffrequenciesas
dN=gV2Ï34Ïk2dk=gV2Ï34Ïk2dkdÏdÏ.IfthewaveexcitationstatisticsobeysEq.(2.72),wemaycalculatethecorrespondingenergyas
dE=âÏeâÏ/Tâ1dN=gV2Ï3âÏeâÏ/Tâ14Ïk2dk=gV2Ï3âÏeâÏ/Tâ14Ïk2dkdÏdÏ,sothatthetotalenergyoftheexcitationsattemperatureTis
E=4ÏgV2Ï3â«0ââÏeâÏ/Tâ1k2dkdÏdÏ,givingthefollowingcontributiontotheheatcapacity:
CVâĄâEâT=4ÏgV2Ï3â«0âââTâÏeâÏ/Tâ1k2dkdÏdÏ=4ÏgV2Ï3â«0ââÏT2eâÏ/TeâÏ/Tâ12k2dkdÏdÏ.
ForâÏâ«T,thesecondfractionunderthelastintegraltendstoexp{ââÏ/T},sothatforanyplausibledispersionrelationÏ(k), the integral converges at frequencies Ïmax ⌠T/â. Hence, for sufficiently low temperatures, we may use the low-frequencyapproximationforthedispersionlawk(Ï);inthespecificcaseofspinwaves,wemaytakek=αÏ1/2(whereαisaconstant),sothatk2=α2Ï,anddk/dÏ=(α/2)Ïâ1/2.Withthesesubstitutions,Eq.(*)becomes
CV=4ÏgV2Ï3α3â«0ââÏT2eâÏ/TeâÏ/Tâ12Ï1/2dÏâĄ4ÏgV2Ï312α2Tâ3/2â«0âeΟeΟâ12Ο5/2dΟ,whereΟâĄâÏ/T.Thelast integral is justadimensionlessconstant,anddoesnotaffectthetemperaturedependenceoftheheatcapacity:
CV=constĂT3/2.
AsEq. (2.99) of the lecturenotes shows, thephonon contribution toCV at low temperatures is proportional toT3, i.e.drops, atTâ0, faster thanEq. (**) predicts.On the otherhand, aswill be shown in section3.3 of the lecturenotes, inconductorsthe free-electroncontributiontospecificheat isproportional toT, i.e.decreasesslower thanthatbymagnons.Hence,thereisachancetoobservethefullspecificheatbeingproportionaltoT3/2ininsulatorswithorderingofatomicspinsatlowtemperatures.(Thespinwavesarecollectivedeviationsofthemagneticmomentsfromsuchanorderedstate.)Indeed,aclassicalexampleofamaterialwithsuchbehaviorofCViseuropiumoxide(EuO);theverysubstantial(âŒ7ÎŒB)spontaneousmagneticmomentsofitsEuatomsleadtotheirferromagneticorderingbelowtheCurietemperatureTCâ69K.
(*)
(**)
(***)
(***)
(**)
(***)
Problem2.28.Deriveageneralexpressionforthespecificheatofaverylong,straightchainofsimilarparticlesofmassm,confinedtomoveonlyinthedirectionofthechain,andelasticallyconnectedwitheffectivespringconstantsÎșâseethefigurebelow.Spellouttheresultinthelimitsofverylowandveryhightemperatures.
Hint:Youmayliketousethefollowingintegral77:â«0+âΟ2dΟsinh2Ο=Ï26.
Solution:According toclassicalmechanics78, small longitudinalmotions in this systemmaybe representedasa sumofNindependentstandingwaveswithfrequencies
Ïk=Ïmaxsinkd2,withÏmax=2Îșm1/2,wherekareequidistantwavenumbersseparatedbyintervalsÎk=Ï/l=Ï/Nd,Nisthenumberofparticlesinthechain,anddisthespatialperiodofthesystem,sothatl=Ndisthetotallengthofthechain.Inbothclassicalandquantummechanics,eachofthesestandingwavesmaybetreatedasanindependentharmonicoscillator,sothataccordingtoEq.(2.75)ofthelecturenotes,itsheatcapacityis
Ck=âÏk/2Tsinh(âÏk/2T)2,andthetotalheatcapacityofthechainis
C=âk=1NâÏk/2Tsinh(âÏk/2T)2,wherethesummationhastobelimitedtoNphysicallydistinguishablewavemodes.SincethemodesseparatedbyÏ/d-longintervalsofkareidentical,thesummationinEq.(***)maybecarriedoveronesuchinterval,forexample[0,+Ï/d].
IfNâ«1(astheproblemâsassignmentimplies),thesummationmaybereplacedbyintegration:C=â«k=0k=Ï/dâÏk/2Tsinh(âÏk/2T)2dkÎk=NÏâ«0ÏâÏk/2Tsinh(âÏk/2T)2dkd,
sothattherequestedgeneralexpressionfortheheatcapacityperparticleisCN=1Ïâ«0ÏâÏk/2Tsinh(âÏk/2T)2dkd,
withÏkgivenbyEq.(*).Sincethefunctionundertheintegraldropsveryfast(exponentially)assoonasâÏkbecomessubstantiallylargerthanT,at
temperaturesmuchlowerthanâ(Îș/m)1/2,i.e.thanâÏmax,theintegraliscutoffatfrequenciesmuchsmallerthanÏmax.Forallthesefrequencies,Eq.(*)maybesimplified79,
Ïk=Ïmaxkd2,sothatd(kd)=2ÏmaxdÏk,forÏkâ©Ÿ0,andEq.(***)isreducedto
CN=2ÏÏmaxâ«0ââÏk/2Tsinh(âÏk/2T)2dÏkâĄ4TÏâÏ0â«0âΟ2dΟsinh2Ο.ThisistheintegralmentionedintheHint,sothatwefinallyget
CN=4TÏâÏmaxÏ26=2Ï3TâÏmaxâȘ1,forTâȘâÏmax.NotethatthisspecificheatincreaseswithTmuchfasterthaninsimilar3DsystemsâcfEq.(2.99)ofthelecturenotes80.
In the opposite limit, when temperature is much higher than âÏmax, Eq. (**) yieldsCk = 1 for each ofN elementaryoscillatorsofthesystem,sothat
CN=1,forTâ«âÏmax,inagreementwiththeclassicalequipartitiontheorem,whichinthislimitmaybeappliedtoeachoscillationmode.
Problem2.29.Calculatethermsthermalfluctuationofthemiddlepointofauniformguitarstringoflengthl,stretchedbyforce ,attemperatureT.Evaluateyourresultforl=0.7m, ,androomtemperature.
Hint:Youmayliketousethefollowingseries:1+132+152+âŠâĄâm=0â12m+12=Ï28.
Solution:Classicalmechanicstellsus81 thatsmall transversedisplacementq(z,t) of a thin (flexible) string,directedalongaxisz,hasthefollowingenergyperunitlength:
andobeystheusualwaveequation
whereÎŒ is stringâs linear density (i.e. its mass per unit length), and its tension (equal to the stretching force). Thisequation,togetherwiththeboundaryconditionsatthestringâsends,
q(0,t)=q(l,t)=0,issatisfiedbythesumofvariable-separatedterms,eachdescribingastanding-wavemode:
q(z,t)=ân=1âqn(z,t),withqn=Zn(z)Tn(t).HereZn(z)arethesinusoidalstanding-waveprofilesthatcomplywiththeboundaryconditions(**),
Zn=sinÏnzl,withn=1,2,3,âŠ.andeachfunctionTn(t)obeys thesameordinarydifferentialequationas theusualharmonicoscillator,butwith themode-specificfrequencyÏn:
withthewell-knownsolutionTn=AncosÏnt+Ïn.
Thefactthateachofthesestandingwavesobeysitsindividualequationofmotion,i.e.isuncoupledfromotheroscillators,
meansthatitsenergyEnisconserved.Since,withtime,theenergyisperiodicallyandfullyâre-pumpedâbetweenitspotentialandkineticforms,wemaycalculateit,forexample,asthemaximumvalueofthepotentialenergy,givenbythesecondtermofEq.(*):
Nowwemayuseeithertheequipartitiontheorem(2.48)orEq.(2.80)inthethermallimit(Tâ«âÏn),towrite
Letusconsiderthemiddle-pointâsfluctuations,
ql2,t=ân=1âsinÏn2Tn(t)=ân=1âAncosÏnt+ÏnsinÏn2,andcalculatetheirvarianceofasfollows:
q2=ân=1âAncosÏnt+ÏnsinÏn22=ân,nâČ=1âAnAnâČcosÏnt+ÏncosÏnâČt+ÏnâČsinÏn2sinÏnâČ2.
(*)
(**)
(***)
(*)
(**)
Duetotherandomicityofmodephases,theaveragesofallcross-termsvanish,sothat
NowusingtheseriesprovidedintheHint,wefinallyget
Notethattheusedseriesconvergesveryfast(Ï2/8â1+0.234),sothatstringfluctuationsarevirtuallydominatedbythe
contribution from the fundamentalmodewithm = 0, i.e.n = 1. Another interesting fact is that the calculated varianceformally does not depend on ÎŒ, though in practice the string mass affects the result, because usually the tension isadjustedtoobtainthedesirablevalueofthefundamentalmodefrequency, âseeEq.(***).
Nowplugginginthe(quitepracticable)parametersgivenintheassignment,inparticularT=kBTKâ(1.38Ă10â23Ă300)J, we get ÎŽq â 0.85 Ă 10â12 m. On the human scale, this is not too much, but still can be readily measured even withinexpensive lab equipmentâfor example, a capacitive sensor followed by a low-noise electronic amplifier. (Sensors ofgravitationalwaveobservatories,suchasthenow-famousLIGO,canmeasuredisplacementsaboutsevenordersofmagnitudesmaller,thoughatfrequencieslowerthanthetypicalguitartune.)
Problem2.30.UsethegeneralEq.(2.123)ofthelecturenotestore-derivetheFermiâDiracdistribution(2.115)forasysteminequilibrium.
Solution:Asdiscussedinsection2.8ofthelecturenotes,Eq.(2.123),Sk=âNklnNkâ1âNkln1âNk,
expressestheentropyrelatedtothekthquantumstateofFermiparticlesasafunctionofitsaverageoccupancyâšNkâ©,andisvalidevenoutofequilibrium.Nowwemayrequirethetotalentropy,
S=âkSk,ofasystemofNparticleswiththetotalenergyE,consideredasafunctionofallâšNkâ©,toreachitsmaximuminthethermalandchemicalequilibrium,withtwoconstraints:
âkNk=N=const,âkNkΔk=E=const.
Aswasdiscussedinsection2.2,therequirementrequiresallconditionalderivativestovanish:âSâNkcondâĄâSâNk+âkâČâ kâSâNkâČâNkâČâNk=0.
Thesystemofsuchequations forallk,aswellasthefirstof theconditions(*),maybesatisfiedbytakingâS/ââšNkâ©=λ=const.However, such a formwould leave the second of the conditions (*) unsatisfied, so thatwemay try to look for thesolutioninamoregeneralform:
âSâNk=λ+λâČΔk,with the Lagrange multiplies λ and λâČ independent of k. Plugging this form into the expression (**) for the conditionalderivative,weget
âSâNkcond=λ+λâČΔk+âkâČâ kλ+λâČΔkâČâNkâČâNk=λââNkâkâČNkâČ+λâČââNkâkâČΔkâČNkâČ,sothatwiththeadditionalconditions(*),theconditionalderivativesindeedvanish,thussatisfyingthesystemofequations(**).
Intheparticularcaseofthefermionicentropy,Eq.(***)givesâlnNk+ln(1âNk)=λ+λâČΔk.
Solvingthisequationfortheaverageoccupancy,wegetNk=1expλ+λâČΔk+1.
InordertoexpresstheLagrangemultipliersλandλâČviathestandardthermodynamicnotions, letususeourintermediateresult(***)tospelloutthechangeofSatanarbitrarysmall,reversiblevariationofthesystemâsparameters,whichchangestheaverageoccupanciesâšNkâ©,butkeepssystemâsvolume(andhencetheparticlesâenergyspectrumΔk)intact:
dS=âkâSâNkdNk=âkλ+λâČΔkdNkâĄÎ»âkdNk+λâČâkΔkdNk.ComparingthisexpressionwiththecorrespondingchangesofNandE,
dN=âkdNk,dE=âkΔkdNk,weseethattheyarerelatedas
dS=λdN+λâČdE.Nowcomparingthisexpressionwiththethermodynamicsrelation(1.52)withdV=0,
dE=TdS+ÎŒdN,i.e.dS=âÎŒTdN+dET,wegetλ=âÎŒ/T,λâČ=1/T,sothattheaboveresultforâšNkâ©indeedcoincideswithEq.(2.115).
Absolutely similarly, the general bosonic Eq. (2.126) may be used for an alternative calculation of the BoseâEinsteindistribution(2.118)validinequilibrium.
Problem 2.31. Each of two similar particles, not interacting directly,may be in any of two quantum states,with single-particleenergiesΔequalto0andÎ.WritedownthestatisticalsumZofthesystem,anduseittocalculateitsaveragetotalenergyEattemperatureT,forthecaseswhentheparticlesare:
(i)distinguishable;(ii)indistinguishablefermions;(iii)indistinguishablebosons.
AnalyzeandinterpretthetemperaturedependenceofâšEâ©foreachcase,assumingthatÎ>0.
Solutions:Letusdescribethepossiblestatesofthesystembyasetoftwoarrowsthatdenotethesingle-particlestates:âforthestateofenergy0,andâforthestateofenergyÎ.Thenthepossiblestatesandtheirtotalenergiesare:
ââ:E=0;ââ:E=Î;ââ:E=Î;ââ:E=2Î.(i)Fordistinguishableparticles,allthesestatesarealsodistinguishableandpossible,sothat
ZâĄâmexpâEmT=1+2eâÎ/T+eâ2Î/T.(Asausefuldetour,notethatthisexpressionmayberewrittenas
Z=1+eâÎ/T2,andderived,inthisform,fromthefollowinggeneralargument(whichwillbeusedinsection3.1ofthelecturenotesforagasofNparticles):sinceforasystemof2particlestheenergyE=Δ1+Δ2,ifallthestatesaredifferent,wemaywrite
ZâĄâmexpâEmT=âmexpâΔ1+Δ2mTâĄâmexpâΔ1mTexpâΔ2mT.Fordistinguishableparticles,thepossiblestatesofeachparticleareindependent,andwemaywrite
Z=âm1,m2expâΔ1m1TexpâΔ2m2TâĄâm1expâΔ1m1Tâm2expâΔ2m2T.
(***)
Buteachofthesepartialsumsisjustthepartialstatisticalsumofasingleparticle,sothatwegetaverysimpleresult:Z=Z1Z2,
whichisreducedtoanevensimplerform,Z=Z12,ifthepartialsumsareequal,foroursimplesystemimmediatelygivingEq.(**),andthuscompletingthedetour.)
Nowpluggingthisresult,intheformZ=1+eâÎČÎ2,whereÎČâĄ1T,
intoEq.(2.61b)ofthelecturenotes,E=ââlnZâÎČâĄâ1ZâZâÎČ,
wereadilygetE=2ÎeÎČÎ+1âĄ2ÎeÎ/T+1.
According to this expression, the average energy tends to 2Îexp{âÎ/T}â 0 atT/Î â 0, and to Î at T/Î â â. Both
asymptotic behaviors arenatural, because atTâȘ Î both particles,with an almost 100%probability, reside on the lowerenergylevel,whileatTâ«Îtheyhaveanequalprobabilitytobeonthelowerandthehigherenergylevels,eachgivinganaveragecontributionofÎ/2tothetotalenergyofthesystem.
(ii) In the case of indistinguishable fermions, the first and the last states of the list (*) are impossible due to the Pauliprinciple,whiletheremainingtwocombinations,ââandââ,arepossibleonlyasentangledcomponentsofoneasymmetricstate,withenergyE=Î.(Inthestandardquantumshorthandnotation,thenormalizedket-vectorofthestateis82
a=expiÏ2âââââ,whereÏisanarbitraryrealphaseshift.)ThismeanthatZisreducedtojustoneterm:
Z=expâÎTâĄexpâÎČÎ,i.e.lnZ=âÎČÎ,sothatEq.(***)immediatelyyieldsanaturalresult,
E=Î,foranytemperature.
Notethatduetotheentanglednatureofthesingletstate,itisunfairtoprescribethisenergytoanyparticularparticle.
(iii) In the case of bosons, all states (*) are possible, but again, the middle two combinations exist only as entangledcomponentsofonequantumstateânowasymmetricone,
s=expiÏ2ââ+ââ,butalsowiththesameenergyE=Î.Asaresult,thestatisticalsumis
Z=1+eâÎ/T+eâ2Î/TâĄ1+eâÎČÎ+eâ2ÎČÎ,andEq.(***)yields
E=ÎeâÎČÎ1+2eâÎČÎ1+eâÎČÎ+eâ2ÎČÎâĄÎeâÎ/T1+2eâÎ/T1+eâÎ/T+eâ2Î/TâÎeâÎ/Tâ0,atT/Îâ0,Î,atT/Îââ.
Besidesthemissingfactor2inthelow-temperaturelimit,theseasymptotes,andtheirinterpretations,arethesameasinthecase(i)ofdistinguishableparticles.
Problem2.32.CalculatethechemicalpotentialofasystemofNâ«1indistinguishable,independentfermions,keptatafixedtemperatureT,ifeachparticlehastwonon-degenerateenergylevels,separatedbygapÎ.
Solution:Ifwedealtwiththegrandcanonicalensemble,i.e. ifthechemicalpotentialÎŒwasexactlyfixed,wecouldreadilycalculatetheaveragenumbersofparticlesatthelowerlevel(whoseenergyΔ0wemaytakefor0),andonthehigherlevel(ofenergyΔ1=Î),byapplyingtheFermiâDiracdistribution(2.115)toeachlevel:
N0=Ne(Δ0âÎŒ)/T+1âĄNeâÎŒ/T+1,N1=Ne(Δ1âÎŒ)/T+1âĄNe(ÎâÎŒ)/T+1.
Nowwemayusethetrickdiscussedinsection2.8ofthelecturenotes.If thetotalnumberofparticles isso largethatâšN0,1â©â«1,therelativefluctuationsofthesenumbersarenegligiblysmall,andwemayusetheaboveformulasevenforthecanonical(Gibbs)ensemble,inwhosemembersystemsthetotalnumberN=N0+N1ofparticlesisexactlyfixed,usingthiscondition83:
NeâÎŒ/T+1+Ne(ÎâÎŒ)/T+1=N,i.e.1eâÎŒ/T+1+1e(ÎâÎŒ)/T+1=1,to calculate the average value of the chemical potential, whose relative fluctuations, atNâ« 1, are very small. The lastequationmaybeeasilysolved,giving
ÎŒ=Î/2.Notethatthisresultmaybeusedasatoymodeloftheelectron/holestatisticsinundoped(âintrinsicâ)semiconductors,tobediscussedinsection6.4âcf.figure6.6andEq.(6.60).
References[1]WhiteGandMeesonP2002ExperimentalTechniquesinLow-TemperaturePhysics4thedn(OxfordSciencePublications)[2]LikharevK1999Proc.IEEE87606[3]KayeGandLabyT1995TablesofPhysicalandChemicalConstants16thedn(Longman)[4]HultgrenRetal1973SelectedValuesofThermodynamicPropertiesoftheElements(ASM)
1Thisisessentiallyasimpler(andfunnier)versionoftheparticlescatteringmodelusedbyLBoltzmanntoprovehisfamousH-theorem(1872).Besidesthehistoricsignificanceofthattheorem,themodelusedinit(seesection6.2ofthelecturenotes)isascartoonish,andhencenotmoregeneral.2AnotherwaytogetthesameresultforSistousethemicrocanonicaldistribution(2.24),S=lnM,withMbeingthenumberofdifferentwaystoplaceN1=N+nindistinguishablefleasononedog,andhenceN2=Nânontheotherone:
M=N1C2NâĄ(2N)!N1!(2NâN1)!âĄ(2N)!(N+n)!(Nân)!,(whereEq.(A.5)hasbeenused),andthenapplytheStirlingformula.3See,e.g.PartQMsections4.6and5.1,forexampleEq.(4.167).4Say,bytheirfixedspatialpositions.5See,e.g.Eq.(A.5).6Wewillrunintothesimilarexpressionagaininsection2.8ofthelecturenotes,discussingtheFermiâDiracparticlesoutofequilibriumâseeEq.(2.123).7NotethatthisderivationofEq.(*)hasfollowedthederivationofmoregeneralEq.(2.29)inthelecturenotes,sothatalternatively,wemightjustusethatformulawithM=2,W1=nandW2=1âW1âĄ1ân.8Thisfactmeansthatfortwo-levelsystemstheequipartitiontheorem(2.48)isnotvalidevenathightemperatures.Onemaysaythatincontrasttotheharmonicoscillatorsorrotatorswiththeirinfiniteenergyspectra,two-level(andmoregenerally,anyfinite-energy-level)systemsareneverclassical!9See,e.g.PartQMsections5.4and5.6.10Thisâatomicâ(orâmolecularâ)susceptibilityshouldbedistinguishedfromtheâvolumicâsusceptibility ,where isthemagnetization,i.e.themagneticmomentofaunitvolumeofasystemâsee,e.g.PartEMEq.(5.111).ForauniformmediumwithnâĄN/Vnon-interactingdipolesperunitvolume,Ïm=nÏ.11See,e.g.PartQMEqs.(4.105),(4.115),(4.116),and(4.163).12LetmehopethattheoperatorâhatâabovetheletterHclearlydistinguishesitsusefromthatfortheenthalpyânotusedinthissolution.13AccordingtoEq.(**),âšmzâ©isanodd(asymmetric)functionof ,sothattheplotisonlyshownfor .14Thisisanultimatelysimple(no-interaction)versionoftheso-calledclassicalHeisenbergmodel,whichfollowsfromquantummechanicsforparticleswithverylargespins,sâ«1.15This(virtuallyself-evident)assumptionisconfirmedbythequantumtheoriesofbothorbitalandspinangularmomenta,intheirclassicallimitsâsee,e.g.PartQMsections3.6and5.6.16See,e.g.PartEMEq.(5.100).Notethatthisformulaisvalidonlyifthemagnitudem0ofthemagneticmomentisindependentoftheappliedfield.17SeeEq.(1.3b)forthecaseofjustoneparticle.18Intermsofthediscussioninsection1.4,thismeansthatweareusingthefirstoptionforthedescriptionofthesystemofparticlesintheexternalfield.19See,e.g.PartQMsection5.7,inparticularEq.(5.197).20See,e.g.Eq.(A.11a).21Foratypicalapplicationofthistechnique,withTHcorrespondingtoâŒ4K,thetermâhotbathâisprettyawkward,andengineerspreferthetermâcoolingsourceâ(whichisofcoursewrongfromthepointofviewofphysics).
22Thisstageofadiabaticdemagnetizationrendersitsnametothewholerefrigerationtechnique,whichisalternativelycalledthemagneticrefrigeration.ItwassuggestedindependentlybyPDebyein1926andWGiauquein1927,andimplementedexperimentallybyseveralresearchgroupsintheearly1930s,enablingthemtoreachtemperatureswellbelow1Kinthelaboratoryforthefirsttime.23Comparingthiscyclewiththoseshowninfigure1.9bofthelecturenotes,oneshouldtakeintoaccountthattheSandTaxisarenowswapped,sothattheclockwiserotationofthepointrepresentingthesystemâsstatecorrespondstoarefrigeratorratherthantoaheatengine.24Theinitiallyusedmaterialswereparamagneticsalts,suchasMg3N2.ThecurrentmaterialsofchoiceincludesuchalloysasGd5(Si2Ge2)andPrNi5; theyallowtoreachtemperaturesbelow10â3Kusingmodestappliedfieldsofafewtesla.Formoreontheadiabaticrefrigerationsee,e.g.sections8.2â8.5in[1].25AquantitativediscussionofthisZeemaneffectmaybefound,e.g.inPartQMsection6.4.26See,e.g.Eq.(A.11a).27Asthefigureaboveshows,actuallythisapproximationworksprettywellallthewayuptoTâÎ.28Foritsdefinitionandmainproperties,see,e.g.Eqs.(A.33)â(A.36).29Pleasenotethatdespitethefactthatasingleclassicalparticlehasanessentiallycontinuousenergyspectrum,theapplicationofthismethodtoN=1wouldgivesubstantialerrors(inparticular,E=T/2insteadofthecorrectE=3T/2)âexplainwhy.30Thisresultpassesasanitycheck:forafreeparticle,withthreeâhalf-degreesoffreedomâ,thisequalitycorrespondstotheequipartitiontheorem.31Itmaybereducedtotheso-calledelliptictheta-functionΞ3(z,Ï)foraparticularcasez=0âsee,e.g.section16.27intheAbramowitzâStegunhandbookcitedinsectionA.16(ii).However,youdonotneedthat(oranyother)handbooktosolvethisproblem.32See,e.g.PartQMsection1.4.33See,e.g.Eq.(A.36b).34See,e.g.PartCMsection3.4.35See,e.g.themodelsolutionofPartQMproblem8.11.36See,e.g.PartQMsections3.6and5.6.37ThesameresultmaybeobtainedfromEq.(2.61b)ofthelecturenotes:E=â(lnZ)/â(âÎČ),whereÎČâĄ1/T.38Alternatively,thisapproximateexpressionforEmaybecalculatedasâšEâ©âE0W0+3E1W1=3E1exp{âE1/T}.39See,e.g.PartQMsection8.1.Notethatthissymmetrymaybevaliddespitetheasymmetryofthewavefunctionwithrespecttotheswapofanypairofmoleculeâselectrons(asfermions)âsee,e.g.themodelsolutionofPartQMproblem8.13foradetaileddiscussionofthispoint.40See,e.g.PartQMEqs.(3.168)and(3.171),and/orfigure3.20ofthatpart.41See,e.g.PartQMsection8.1andthemodelsolutionsofproblems8.12â8.13ofthatpart.42Amazingly,thismeansthatthegroundstateofthemolecule,withp=0andl=1,correspondstoanonvanishingrotationofthemolecule,withtheenergyEl=E1=â2/I.Evenmorecounter-intuitive,thisrotationisimposedonthemoleculebyitsnuclearspinsystem,whosecouplingwithotherdegreesoffreedomofthemoleculeismuchweakerthantherotationenergy!43Thisisanapproximatebutreasonablemodeloftheconstraintsimposedonsmallatomicgroups(e.g.ligands)bytheiratomicenvironmentinsidesomelargemolecules.44See,e.g.PartQMsection3.5.45See,e.g.PartCMsection4.2,inparticularEq.(4.25).46See,e.g.PartEMEq.(3.16),withthenotationreplacementp1,2âd1,2.47Ifanexplanationofthispointisneeded,see,e.g.themodelsolutionofPartEMproblem3.5.48Notethatthetraditionalform,1/r12,ofthesecondterminthevanderWaalsformula,describingmolecular/atomicrepulsionatsmalldistances,doesnothaveasimilarlyquantitativephysicalfoundationâseesection4.1ofthelecturenotes.49See,e.g.thesolutionsofPartQMproblems3.16,5.15,and6.18.50See,e.g.PartCMproblem1.12.51Asimilarcancellationwillleadus,insection3.1ofthelecturenotes,toamoregeneralresult,theso-calledBoltzmanndistributionâseeEq.(3.26).52See,e.g.Eq.(A.34a).53SinceinthisHamiltoniansystemthetotalenergyEisconserved,thetimeaveragingsignoveritmaybedropped.54See,e.g.Eq.(A.36b).55See,e.g.Eq.(A.36d)56In thisparticularcontext, theadjective âweakâdenotesa junctionwith the tunneling transparencyso low that the tunnelingelectronâswavefunction looses itsquantum-mechanicalcoherencebefore theelectronhasachancetotunnelback.Inatypicaljunctionofamacroscopicareathisconditionisfulfilledifitseffectiveresistanceismuchhigherthanthequantumunitofresistance(see,e.g.PartQMsection3.2),RQâĄÏâ/2e2â6.5kΩ.57Ifthisimportantconceptualpointisnotclear,pleasereviewitsdiscussioninPartEMsection2.6.58LetmeemphasizeagainthatthevalidityofthisGibbsdistributionisnotaffectedbythefactthatelectronsobeytheFermiâDiracstatistics,becauseherewedealwiththeenergyUofthewholesystem,ratherthanoneofits(indistinguishable)components.59Forareviewsee,e.g.[2].60Thesecondmajorchallengeistherandomnessoftheso-calledbackground(orâoffsetâ)chargesâseethejustcitedreviewpaper.61See,e.g.PartCMsections2.1and2.3.62See,e.g.PartCMsection10.1and/orPartQMsection1.2.63Ifthisisnotevident,pleasesee,e.g.PartEMsection9.3,inparticularEq.(9.78).64HereΔisgenerallythekineticenergyoftheparticle,andmaybeassociatedwithitstotalenergy(besidesaconstant)onlyifitsinternaldegreesoffreedomareintheirgroundstateâseesection3.1formorediscussionofthispoint.65See,e.g.thesolutionofproblem1.5.66See,e.g.PartEMsections4.1and6.6.67See,e.g.PartEMsection7.6.IntheTEMwaves,theelectricfield isnormaltotheplatesurfaces,whilethemagneticfield isnormaltoboth andthewavevectork,i.e.paralleltotheplatesurfaces.68Forthelasttwostepsofthiscalculation,wemayuseEq.(A.35b)withs=3,andthenEqs.(A.10b)and(A.34c).69Accordingtoelectrodynamics,allsuchmodeshavefrequenciesexceedingthecriticalvalueÏc=Ïc/t.70See,e.g.PartEMsection7.10.71Ontheelectromagneticfieldlanguage,duetotheuniversalrelation betweentheelectricandmagneticfieldmagnitudesintheTEMwaves,theattractionforce duetotheelectricfield,normaltotheplatesurfaces,isexactlycompensatedbytherepulsiveforce duetothemagneticfield,paralleltothesurfacesâsee,e.g.PartEM section9.8, inparticularEqs.(9.240)and(9.242).72Foritsdiscussion,see,e.g.PartQMsection9.1.73Aswasmentionedinsection2.6,forsolids(likethealuminumatroomtemperature),withtheirverysmallexpansioncoefficient,thedifferencebetweenCVandCPisnegligible,sothattheindexâVâinEq.(2.97)maybedropped.74[3].75[4].76Notethatthesamedispersionlawistypicalforbendingwavesinthinelasticrodsâsee,e.g.PartCMsection7.8.77Itmaybereduced,viaintegrationbyparts,tothetableintegralEq.(A.35d)withn=1.78See,e.g.PartCMsection6.3,inparticularEq.(6.30).79Thismeansthat,atsuchtemperatures,onlyacousticwavesgiveasubstantialcontributiontoheatcapacity.80Experimentalobservationsofsuchtemperaturedependencehavehelpedtorevealthe1Dcharacterofatomicmotioninsomeorganicmaterials.81See,e.g.PartCMsection6.3andthesolutionofproblem6.10ofthatpart.82See,e.g.PartQMsection8.1,inparticularEq.(8.11).83Thiswayofcalculationofthe(average)chemicalpotentialwillberepeatedlyusedinchapters3and6ofthiscourseforothersystems.
IOPPublishing
StatisticalMechanicsProblemswithsolutionsKonstantinKLikharev
Chapter3
Idealandnot-so-idealgases
Problem 3.1. Use the Maxwell distribution for an alternative (statistical) calculation of themechanicalworkperformedbytheSzilardenginediscussedinsection2.3ofthelecturenotes.
Hint:Youmayassumethesimplestgeometryoftheengineâseefigure2.4.
Solution:Letusassumethatinitiallythepartitionwithadoorinitwasinthemiddleofthecylinder(in the figure below, at x = l/2), and that the information provided by the Maxwell demon hasallowed us to close the door when the molecule was in the left part of the cylinder. Then therepeatedhitsbythemoleculeprovidetheforcepushingthepartitiontotheright.Themomentumtransferred fromtheparticle to thepartitionatasingleelastichit is2px,wherepx is theparticlemomentumâscomponentnormaltothepartition.Time-averagingthe2ndNewtonlaw(orratheritsx-component),
overthetimeintervalÏbetweenthehits,weseethatthetimeaverageoftheforce actingonthepartitionisequaltotheratio2px/Ï.Atanarbitrarypositionxofthepartition(withl/2â©œxâ©œl),theintervalÏequals ,where isthex-componentofparticleâsvelocity.Thus,
Inordertoaveragethisresultoverastatisticalensembleofsimilarexperiments,wemayusethe
Maxwelldistribution(oralternativelytheequipartitiontheorem),giving ,sothat
Nowwecancalculatetheworkdonebythemoleculeataslow(reversible)motionofthepartitionfromxini=l/2toxfin=l,ataconstanttemperature:
This isexactly theworkthatwascalculated insection2.3 fromthethermodynamicrelationdQ=TdS.
OnemorewaytogetthesameresultistousetheequationofstateoftheidealclassicalgaswithN=1molecule,PV=T, andcalculate the sameworkby integrationoveran isothermal two-foldexpansionofthecontainingvolumeV:
Problem3.2.UsetheMaxwelldistributiontocalculatethedragcoefficient ,whereistheforceexertedbyanidealclassicalgasonapistonmovingwithalowvelocityu,inthesimplestgeometry shown in the figurebelow, assuming that collisions of gasparticleswith thepiston areelastic.
(**)
(*)
(**)
Solution:Thisproblemisessentiallyarefinementofthepreviousone,becausenowwehavetotakeintoaccountanonvanishingvelocityuofthepiston.Consideraparticlehittingthepiston,withtheinitial horizontal velocity (in the cylinderâs reference frame)âsee the figure above. In thepistonâsreferenceframe,thiscomponentequals .Sincethecollisioniselastic,inthepistonâsframethecomponentâsmodulusisconserved,sothatafterthecollisionitis ,andin the cylinderâs frame it is . Hence themomentum transferred to thepistonis .
ThismomentumshouldbeattributedtoanappropriatetimeintervalÎtaroundthehitmomentt,for example the sum of the intervals and between that moment and theinstancesofthepreviousandthenextreflectionsofthisparticlefromtheopposite(immobile)endofthecylinderâseethefigureabove:
Hencethetime-averageforceexertedonthepistonbyoneparticleis
(As a sanity check, atu = 0 this expression is reduced to the one derived in the solution of thepreviousproblem.)
Sinceaparticlemoving initially in theoppositedirection,but the same , exerts on thepistonthesimilaraverageforce,wecangeneralizethisformulaas
Nowthis forceshouldbeaveragedoverthe1DMaxwelldistributionofthevelocitycomponent ,followingfromEqs.(3.5)â(3.8)ofthelecturenotes:
TheresultingaverageforceexertedbyNmoleculesonthepistonis
whereAisthepistonarea,V=Axisthecurrentvolumeoccupiedbythegas(seethefigureabove),and
Thelargest,firsttermintheexpression(*)istheusualpressureforceoftheidealclassicalgas,
while the second one is the linear approximation for the drag (viscous-friction) force ,alwaysdirectedagainstbodyâsvelocity,andhencedissipatingitsenergy1.
Theparticularexpression(**)forthedragcoefficientηwillbeusedinchapter5ofthelecturenotestoillustratethefundamentalrelationbetweenfluctuationsanddissipation.
Problem 3.3. Derive the equation of state of the ideal classical gas from the grand canonicaldistribution.
Solution:AccordingtoEq.(2.109)ofthelecturenotes,inthegrandcanonicalensemble,i.e.inthestatisticalensemblewithfixedtemperatureTandchemicalpotentialΌ,butvariablenumberNoftheparticles(seefigure2.13anditsdiscussion),thegrandthermodynamicpotentialΩequals
Ω=âTlnâm,NexpÎŒNâEm,NTâĄâTlnâNeÎŒN/TâmexpâEm,NT.Butaccording toEq. (2.59), the lastsum is just thepartition functionZN of thecanonical (Gibbs)distribution, for a particular value ofN. As was discussed in detail in section 3.1, for the idealclassicalgasofindistinguishableparticlesthisfunctionisgivenbyEq.(3.15):
ZN=1N!gVmT2Ïâ23/2N,whereNmayvaryfrom0toâ,sothatthelastformofEq.(*)reducesto
Ω=âTlnâN=0âeÎŒN/T1N!gVmT2Ïâ23/2NâĄâTlnâN=0â1N!eÎŒ/TgVmT2Ïâ23/2N.AccordingtotheTaylorexpansionoftheexponentialfunctionattheorigin,
eΟ=âk=0â1k!dkeΟdΟkΟ=0Οk=âk=0â1k!Οk,thesuminthelastformofEq.(**)isjusttheexponentoftheexpressioninthesquarebrackets,and
(***)
(*)
wegetΩ=âTeÎŒ/TgVmT2Ïâ23/2.
Nowwecancalculatetheaveragenumberofparticlesinthegas,usingthelastofEqs.(1.62):2
N=ââΩâÎŒT,V=âââÎŒâTeÎŒ/TgVmT2Ïâ23/2T,V=eÎŒ/TgVmT2Ïâ23/2.PluggingthisrelationbackintoEq.(***),wegetaverysimpleresult,
Ω=âTN,sothatthethermodynamicrelation(1.60),Ω=âPV,immediatelyyieldstheequationofstateofthegas,
PV=NT.
In the thermodynamic limitN ââ,when the difference between âšNâ© andN is negligible, thisexpressioncoincideswithEq.(3.18),whichwasderivedinsection3.1ofthelecturenotesfromtheGibbs(canonical)distribution,validforstatisticalensembleswithfixedTandN,ratherthanÎŒ.
Problem3.4.ProvethatEq.(3.22)ofthelecturenotes,ÎS=N1lnV1+V2V1+N2lnV1+V2V2,
derivedforthechangeofentropyatmixingoftwoidealclassicalgasesofcompletelydistinguishableparticles(thatinitiallyhadequaldensitiesN/VandtemperaturesT),isalsovalidifparticlesineachoftheinitialvolumesareidenticaltoeachother,butdifferentfromthoseinthecounterpartvolume.Assumethatmassesandinternaldegeneracyfactorsgofalltheparticlesareequal.
Solution:ForeachofthegasesbeforemixingwemayuseEq.(3.20)ofthelecturenotes,S1,2=N1,2lnV1,2N1,2âdf(T)dT,
resultingfromEq.(3.12)withthecorrectBoltzmanncounting.(Duetotheequalityofmandgofallparticles,theirfunctionsf(T)arethesame,anddonotneedtheindices.)Forthegasaftermixing,wehavetomodifythecountinginthefollowingway:
Z=1N1!N2!âkexpâΔkTN1+N2,inordertoaccount for the internal indistinguishabilityofparticlesofeachsort.Nowcarryingoutthecalculationssimilartothosedoneinsection3.1,fortheentropyofthemixedgasweget
S=N1lnV1+V2N1+N2lnV1+V2N2âN1+N2df(T)dT.Fromhere,weseethatthemixingentropy,
ÎSâĄSâS1+S2=N1lnV1+V2V1+N2lnV1+V2V2,is indeed described by the same expression as for the completely distinguishable particles. Thisresultisnatural,becausethemixingdoesnotchangetheinternaldisorder(i.e.theentropy)ofeachcomponentofthegas.
Problem3.5.AroundcylinderofradiusRandlengthL,containinganidealclassicalgasofNâ«1particlesofmassmeach,isrotatedaboutitssymmetryaxiswithangularvelocityÏ.Assumingthatthegasasthewholerotateswiththecylinder,andisinthermalequilibriumattemperatureT,
(i)calculatethegaspressuredistributionalongitsradius,andanalyzeitstemperaturedependence,and(ii)neglectingtheinternaldegreesoffreedomoftheparticles,calculatethetotalenergyofthegasanditsheatcapacityinthehigh-andlow-temperaturelimits.
Solutions:
(i) From classicalmechanics3 we know that in the non-inertial reference frame connected to thecylinder (inwhich the gas as thewhole rests)we have to add, to all real forces, the centrifugalâinertialforceâ
whereÏistheparticleradius-vectorâscomponentintheplanenormaltotherotationaxis.ThisforcemayberepresentedasââU,whereUistheeffectivepotentialenergy
U(r)=âmÏ2Ï22+const.Withthisspecification,Eq.(3.26)ofthelecturenotesyieldsthefollowingparticledensityperunitvolume:
n(r)=n0expmÏ2Ï22T.Theconstantn0 (physically, theparticledensityatÏ=0, i.e.onthecylinderâsaxis)maybefoundfromtheconditionthatthetotalnumberofparticlesinthecylinderisequaltothegivenN:
â«Vn(r)d3râĄ2ÏLn0â«0RexpmÏ2Ï22TÏdÏâĄn02ÏLTmÏ2expmÏ2R22Tâ1=N.Thisconditionyields
n0=NLmÏ22ÏT1expmÏ2R2/2Tâ1,sothatn(r)=NLmÏ22ÏTexpmÏ2Ï2/2TexpmÏ2R2/2Tâ1.Nowapplyingtheequationofstateofanidealgaslocally,intheformP(r)=n(r)T,weget4
P(r)=n(r)T=NLmÏ22ÏexpmÏ2Ï2/2TexpmÏ2R2/2Tâ1.
IfthetemperatureTisrelativelyhigh(and/ortherotationisrelativelyslow),thentheexponentsin this expression may be Taylor-expanded, with only leading terms kept, giving only a minorcorrectiontotheusualresultP=NT/V:
PâNTÏR2L1+mÏ2Ï2âR2/22TâĄNTV1+mÏ2Ï2âR2/22TâNTV,atmÏ2R2âȘT.
(**)
(***)
(*)
(**)
Intheopposite,low-temperature(i.e.high-Ï)limit,TâȘmÏ2R2,theexponentinthedenominatorofEq.(*)ismuchlargerthan1,andtheformulaisreducedtothefollowingrelation:
PâNÏ22ÏLexpÏ2âR2Ï02,whereÏ0âĄ2TmÏ21/2âȘR,showingthatallthegasiscompressedintoalayer,ofthicknessâŒÏ0âȘR,atthecylinderâswall.
(ii) In the lab reference frame, theparticleâsvelocityv is thevectorsumof thevelocityvrelof itsthermalmotionintherotatingreferenceframe,andtheaveragelocalrotationvelocityvrot=ÏĂr,sothatitstotalkineticenergyis
Since the thermal velocity vrel is random, with the isotropic distribution of its directions, thestatisticalaverageofthesecondtermvanishes,whiletheaverageofthefirstterm(accordingtotheequipartitiontheorem)is3T/2.Asaresult,thetotalenergyofthegas5maybecalculatedasE=3T2N+â«Vn(r)mÏ2Ï22d3r=3T2N+mÏ222ÏLn0â«0RexpmÏ2Ï22TÏ2ÏdÏ=3T2N+2ÏLT2mÏ2n0mÏ2R22T
â1expmÏ2R22T+1=TN32+mÏ2R2/2Tâ1expmÏ2R2/2T+1expmÏ2R2/2Tâ1.
Inthehigh-temperaturelimit,Tâ«mÏ2R2,theTaylorexpansionoftheexponents,withonlytwoleading terms kept, yields the usual thermal energy (3/2)NT, plus a relatively small correctiondescribing the kinetic energy of rotation of the gas (which, in this limit, is distributed virtuallyuniformlyoverthecylinderâsbulk)asthewhole:
EâTN32+mÏ2R24TâĄ32NT+NmÏ2R24=32NT+I1Ï22,whereI1âĄMR22,andMâĄmNisthetotalmassofthegas,sothatI1istheusualmomentofinertiaofauniformroundcylinderofmassM.Sincethisrotationcorrectionistemperature-independent,itdoesnotaffecttheheatcapacityofthegas,CâĄdE/dT=3N/2.
Intheopposite, low-temperature limit, theexponents inthenumeratoranddenominator inthelast form of Eq. (***) aremuch larger than 1. As a result, they cancel, andwe get a result verysimilarinform:
EâTN32+mÏ2R22Tâ1âĄmÏ2R22N+NT2=I2Ï22+NT2,whereI2âĄMR2.However, here the first, temperature-independent term, which describes the kinetic energy ofrotationof thegasasawhole (nowcompressed to thecylinderâswall,andhencehavinga largermomentof inertia, I2=2 I1), ismuch larger than the second, temperature-dependent term.Notethat,counter-intuitively,thesecondtermisthreetimessmallerthanthatofafreegas,sothattheheatcapacityof thesystem isalso three times lower:C=N/2.As the full result (***) shows, thisreductionisduetotheincrease,withgrowingtemperature,ofthethicknessÏ0ofthegaslayerâseeEq. (**).This increase,pushing thegas,on theaverage,closer to thecylinderâsaxis, reduces thekineticenergyofitsrotation,andthusslowsthegrowthofthefullenergyofthegas.
Problem3.6.Nâ«1classical,non-interacting,indistinguishableparticlesofmassmareconfinedinaparabolic,spherically-symmetric3DpotentialwellU(r)=Îșr2/2.Usetwodifferentapproachestocalculate all major thermodynamic characteristics of the system, in thermal equilibrium attemperatureT, including itsheatcapacity.Whichof theresultsshouldbechanged if theparticlesaredistinguishable,andhow?
Hint:Suggestareplacementofthenotionsofvolumeandpressure,appropriateforthissystem.
Solution:Firstofall, letuscalculatethecharacteristics thatdonotrequirethenotionsofvolumeandpressure,startingfromthestatisticalsum.Thesummaybecalculatedin(atleast)twodifferentways.
Approach1 is to rely on systemâs classicity from the verybeginning.Thismakes allCartesian
coordinates andmomenta independent arguments, and allows us to useEq. (3.24) of the lecturenotes,whichmayberewrittenasaproduct:
w(r,p)=constĂexpâp22mTĂexpâU(r)T.Sincethefirstoftheseoperandsistheprobabilitydensityofafreeparticle,wemaygeneralizeEq.(3.14)(validforindistinguishableparticles)asfollows:
Z=1N!zpâ«expâU(r)Td3rN,where themomentum factor zp is the same as for a free-particle gas, andmay be calculated byGaussianintegration,exactlyasthiswasdoneatthederivationofEq.(3.15):
zp=g2Ïâ3â«ââ+âexpâpj22mTdpj3=gmT2Ïâ23/2.ThecoordinatefactorinEq.(*),forourquadraticpotential,isalsoaproductofthreesimilar,simpleGaussianintegrals,whichmaybecalculatedsimilarly:
â«expâU(r)Td3r=â«expâÎșr12+r22+r32Td3râĄâ«ââ+âexpâÎșrj22Tdrj3=2ÏTÎș3/2,sothat,finally,
Z=1N!gmâ2Îș3/2T3N.
NowwemayuseEq.(2.63),assumingthatNâ«1andapplyingtheStirlingformulatosimplifyln(N!),tocalculatethefreeenergy:
F=âTlnZ=âNTlnNÎș3/2+NfT,withfTâĄâTlngmT2â23/2+1.ThereasonwhyNisgroupedwithÎș3/2underthefirstlogarithm(justasitisgroupedwith1/Vattheusual,rigidconfinementwithinvolumeVâseeEq.(3.16a)ofthelecturenotes) isthatourcurrent
(***)
(****)
soft-well system does not have any clearly defined volume, and the only parameter thatcharacterizestheconfinementistheeffectivespringconstantÎș.ThecomparisonwithEq.(3.16)ofthe lecturenotes shows that apossibleanalogof the fixedvolumeV in this case is the followingfixedparameter:
despite its different dimensionality. In this notation, the expression for the free energy becomesformallysimilartoEq.(3.16a)fortherigidlyconfinedgas,
sothattheremainingcalculationsaresimilaruntilweneedtospelloutthefunctionf(T)âwhichisdifferent.Inparticular,usingEq.(1.35)tocalculatethesystemâsentropy,atthefixedparameter ,weget
sothat,accordingtoEq.(1.47),theinternalenergyofthesystemisE=F+TS=NfTâTdfTdT=3NT.
Mercifully,EisagainstrictlyproportionaltoN,sowemayreadilycalculatetheaverageenergyperparticle, E/N = 3T, and the heat capacity per particle, also at fixed âvolumeâ , i.e. a fixedpotentialâsprofile:
This simple result is in the agreementwith the equipartition theorem (2.48), because in contrastwiththerigidlyconfinedgas,eachofthreedegreesoffreedomoftheparticleprovidesitwithnotonlyaquadratickineticenergy,butalsoaquadraticpotentialenergy.Finally, fromtheanalogyofEq. (***) and Eq. (3.16) for the âusualâ (rigidly confined) ideal classical gas, it is clear that ifweintroduce the corresponding analog of pressure using the second of Eqs. (1.35), the resultingequationofstatewouldbealsothesame:
At this point we have to notice that an effective volume of the wellâs part occupied by the
particlesmaybedefinedbytherelationVefâĄNn0=1n0â«n(r)d3r,
wheren(r)isparticleâsdensityatthepointr.Since,accordingtoEq.(3.26),thedensityn(r)isequalton(0)exp{âU(r)/T},thenecessaryintegralhasbeenessentiallycalculatedabove,giving
In contrast to , the effective volume Vef has the usual physical dimensionality (m3), but it
dependsontemperature,andhencecouldnotbeemployedinthebasicrelationsofthermodynamicswithoutmodification.
Approach2.LetusnoticethatEq.(**)mayberewrittenas
Z=1N!gTâÏ3N,whereÏâĄ(Îș/m)1/2hasthephysicalsenseoftheeigenfrequencyofoscillationsofasingleparticleofthegas inourharmonicpotential. Thisunsolicitedappearanceof theoscillation frequency showsthatthisexpressionforZmaybealsoobtaineddifferently.Inthis,morequantum-mechanics-basedapproach, letus first treateachgasparticleasa3Dharmonicoscillator, anduse thewell-knownquantum-mechanicalresultforitsenergyspectrum6:
Δk=âÏn1+n2+n3+32,withnj=0,1,2,âŠ,sothat,withtheinconsequentialtemperature-independentshiftoftheparticleenergyreferencetotheground-stateenergy(3/2)âÏ,Eq.(3.12)ofthelecturenotesbecomes
Z=1N!gân1,n2,n3=0âexpââÏTn1+n2+n3N=1N!gânj=0âexpââÏTnj3N.Thissumisjustageometricprogression,andmaybecalculatedexactly(seeEqs.(2.67)and(2.68)ofthelecturenotes),butforourcurrentpurposeswemayusejustitsclassicallimit,validatâÏâȘT:
Zâ1N!gâ«0âexpââÏTnjdnj3N=1N!gTâÏ3N,thusreturningustoEq.(**)andallthefollowingresultsobtainedbyapproach1.Moreover,wemayobtain some of these results directly from Eqs. (2.73)â(2.75) in the classical limit, with propermultiplicationsby3torecognizethe3Dnatureofourcurrentproblem.
Finally, the above results show that just as in the âusualâ gas, the distinguishability of theparticles,which kills the âcorrectBoltzmann countingâ factor 1/N! inEq. (**), does not affect theequationofstate,itsenergyandheatcapacity,butdoesaffectthefreeenergyandentropyâjustasintherigidlyconfinedgas,aswasdiscussedinsection3.1ofthelecturenotes.
Problem 3.7. In the simplest model of thermodynamic equilibrium between the liquid and gasphasesofthesamemolecules,temperatureandpressuredonotaffectthemoleculeâscondensationenergy Î. Calculate the concentration and pressure of such saturated vapor, assuming that it
(*)
(*)
(**)
(***)
(****)
behavesasanidealgasofclassicalparticles.
Solution:Inthismodel,fromthepointofviewofthegas/vapor,theliquidisanunlimitedsourceofmoleculeswithenergy(âÎ), i.e.anenvironmentwithaconstantchemicalpotentialâjustasinthegrandcanonicalensembleâseesection2.7of the lecturenotes.Referring themoleculeenergy tothatofoneinrestinthegaseousphase,wemaywriteÎŒ=âÎ.Hencewemayapplytothevaporallformulasderived from thegrand canonical distribution; in particular, for non-interacting classicalparticles we may use Eq. (3.32) of the lecture notes. For the molecule density n âĄN/V in thegaseousphase,itimmediatelyyields
n=gmT2Ïâ23/2expâÎT.
Since the gas itself remains classical, it obeys the Maxwell distribution (3.5) and hence theequationofstate(3.18)initslastform:
P=nT.Note,however,thatsincenowthenumberofgasparticles isnot fixed(theymaygoto,andcomefromtheliquidasnecessaryforthechemicalandthermalequilibrium),inthismodelthepressureisafunctionoftemperatureonly,regardlessofthevolume:
P=PT=nT=gm2Ïâ23/2T5/2expâÎT.
WiththevalueÎâ0.71Ă10â19Jâ0.44eV(correspondingtoâŒ5130K),7thisformulagivesasurprisinglyreasonableapproximationforthetemperaturedependenceofthepartialpressureofthesaturated water vapor, for all temperatures not too close to waterâs critical point of 647 Kâseechapter4.Aswillbediscussedinthesamechapter,suchnearlyâexponentialdependence(includingthe Arrhenius factor exp{âÎ/T}), is a common feature of most phenomenological models of realgasesâincluding the famous van der Waals model, though this is frequently not immediatelyapparentfromtheequationofstate.
Problem3.8.AnidealclassicalgasofNâ«1particlesisplacedintoacontainerofvolumeVandwallsurfaceareaA.Theparticlesmaycondenseoncontainerwalls,releasingenergyÎperparticle,andforminganideal2Dgas.Calculatetheequilibriumnumberofcondensedparticlesandthegaspressure,anddiscusstheirtemperaturedependences.
Solution: Since the surface condensate and the volume gasmay exchange particles atwill, theirchemicalpotentialsÎŒhavetobeequal.InordertocalculateÎŒofthegaseousphase,wemayuseEq.(3.32)of the lecturenotes,withN replacedwith (NâN0),whereN0 is thenumberof condensedparticles8:
ÎŒvolume=TlnNâN0NVT,whereNVTâĄgVVmT2Ïâ23/2.
Ontheotherhand,thesurfacecondensateofnon-interactingparticlesisanideal2Dgas,andweneedtoreproducethecalculationsofsection3.1usingthe2Ddensityofstates.UsingtheevidentmodificationofEq.(3.13),9
âkâŠâgA2Ïâ2â«âŠd2p,foranisotropicsurface(withd2p=2Ïpdp,anddΔ=d(p2/2m)âĄpdp/m)weget
dN2=gA2Ïâ2d3p=gA2Ïâ22Ïpdp,g2(Δ)âĄdN2dΔ=gA2Ïâ22Ïpdppdp/mâĄgmA2Ïâ2.(Notethatincontrasttothe3DdensityofstatesgivenbyEq.(3.43)ofthelecturenotes,g3(Δ)âΔ1/2,thefunctiong2(Δ)isactuallyaconstant.)Onemorenecessarymodificationisthatinaccordancewiththeproblemassignment,allenergiesofparticlesonthesurface,includingÎŒ,shouldbeshifteddownbytheenergyÎ.Theresultis10
ÎŒsurface=âÎ+TlnN0NST,whereNSTâĄgSAmT2Ïâ2.(ThedegeneracygSofsurfacestatesmaybedifferentfromthat,gV,ofthebulkstates.)
Equating these two expressions for Ό to describe the chemical equilibrium between the twophases,wegetthefollowingequationforthecalculationofN0:
TlnNâN0NVT=âÎ+TlnN0NST,whichreadilyyields
N0N=11+ÎșT/Î1/2expâÎ/Tâ©œ1,whereÎșâĄgVVgSAmÎ2Ïâ21/2.Hencethenumberofparticlesinthegaseousphaseis
NâN0=NÎșT/Î1/2expâÎ/T1+ÎșT/Î1/2expâÎ/T,andsincetheyobeytheideal-gasequationofstate,theirpressureis
P=NâN0VT=ÎșT/Î1/2expâÎ/T1+ÎșT/Î1/2expâÎ/TNTVâĄ11+Îșâ1Î/T1/2expÎ/TNTV.
This result, plotted in the figure below for several values of the dimensionless parameter Îș,shows that the temperature dependence of P is very much different at temperatures below andabovethevalueTcthatshouldbecalculatedfromthetranscendentalequation
Tc=ÎlnÎșTc/Î1/2,and (becauseof the slownessof the logarithm functionat largevaluesof its argument) is alwayssomewhatlower,butstilloftheorderofÎ.11Inparticular,iftemperatureiswellbelowTc,virtuallyallparticlesarecondensedatthesurface:N0âN,andthepressureprovidedbythefewparticlesremaininginthegaseousphaseisexponentiallylow:
P=NTVÎșTÎ1/2expâÎTâȘNTV,atTâȘTc.ThisfunctionP(T),whichalsoincludestheArrheniusfactorexp{âÎ/T},iscloseto,butstilldifferent
(*)
(**)
(***)
from the one obtained in the solution of the previous problemâwhich is closer to experiment fortypicalliquids.
Ontheotherhand,thecurrentmodeloftheliquid/vaporequilibriumismorerealistic inthat it
doesdescribethecondensateâsfullevaporationattemperatureswellaboveTc,whenN0âȘN,andthepressureinthegaseousphaseobeystheequationofstateoftheusualidealgas,P=NT/V.
Note also that in contrast to genuine phase transitions, to be discussed in chapter 4, thetemperature dependences of all variables at T ⌠Tc are smooth even at N â â. Such smoothtemperature borderlines are frequently called crossovers; they are typical for systems whoseparticles(orotherelementarycomponents)donotinteract12.
Problem3.9.TheinnersurfacesofthewallsofaclosedcontainerofvolumeV, filledwithNâ«1particles,haveNSâ«1similartraps(smallpotentialwells).Eachtrapcanholdonlyoneparticle,atpotentialenergyâÎ<0.Assumingthatthegasoftheparticlesinthevolumeisidealandclassical,derivetheequationforthechemicalpotentialÎŒofthesysteminequilibrium,anduseittocalculatethepotentialandthegaspressureinthelimitsofsmallandlargevaluesoftheratioN/NS.
Solution:Thetotalnumberofparticles,N,isthesumofsomenumberN0oftheparticlescondensedonthesurface(localizedatthesurfacetraps),andthenumber(NâN0)of theparticlesthermallyactivated into the gas phase. Due to the given conditions that the gas is classical, and that thenumber(NâN0)oftheseparticlesislarge,theformernumbermaybecalculated(justaswasdoneintwopreviousproblems)fromEq.(3.32)ofthelecturenotes,
NâN0=NVTexpÎŒT,whereNVTâĄgVVâ3mT2Ï3/2â«1.
InordertocalculateN0,wemayconsiderastatisticalensembleofsingletraps,andapplytoitthegrandcanonicaldistribution,with (1+gS)different states: oneempty-trap state, of a certain(inconsequential)energyΔ0,andgSdifferentpossiblestateswithonetrappedmolecule,eachwiththeenergy(Δ0âÎ).Asaresult,Eqs. (2.106)and(2.107)yieldthefollowingprobabilitiesof thesestates:
W0=1ZGexpâΔ0T,W1=1ZGexpÎŒ+ÎâΔ0T,with
ZG=expâΔ0T+gSexpÎŒ+ÎâΔ0T,sothattheaveragenumberoffilledtraps(withanystateofthetrappedmolecule)is
N0=NSgSW1=NSgSexpÎŒ+Î/T1+gSexpÎŒ+Î/T,whereinthethermodynamicequilibrium,ÎŒandThavetobethesameasinEq.(*).Combiningtheseexpressions,weget
NVTexpÎŒT+NSgSexpÎŒ+Î/T1+gSexpÎŒ+Î/T=N.
Unfortunately for this model of the condensate/gas equilibrium (which is more realistic thanthose discussed in two previous problems), the transcendental equation for ÎŒ defies an exactanalyticalsolutionforarbitraryparameters.However, itmaybereadilysolved inthe limitsof lowandhighvaluesoftheN/NSratio.Indeed,sinceN0cannotbelargerthanNS, inthe limitNâ«NSmostparticleshavetobeinthegasphase,sothatinthe0thapproximation,thesecondterminEq.(**) may be ignored, and this equation is reduced to Eq. (3.32) of the ideal classical gas ofNparticles,giving
ÎŒââTlnNNVT,PâNTVâT,forNâ«NS.WemayalsousethisvalueofÎŒtocalculatethe(relativelysmall)numberofcondensedparticles:
N0âNSgSâ1NVT/NexpâÎ/T+1âȘN.AccordingtoEq.(*)andEq.(3.35)ofthe lecturenotes,theratioNV(T)/N iscloseto(T/T0)3/2,andhastobemuchlargerthan1tokeepthegasclassical.However,sincetheexponentissuchasteep
(****)
(**)
function,theratioN0/NSdependsmostlyonthecondensationenergyÎ:ifitismuchlargerthanthecrossovervalue13
ÎcâĄTlnNVTgSN,thefirstterminthedenominatorofEq.(***)isnegligiblysmall,andN0âNS.Intheoppositelimit,ÎâȘÎc,theratioN0/NSisexponentiallysmall.NowEq.(***)maybeusedwithEq.(**)tocalculatethesmall,firstordercorrectionstoÎŒandP,etc.
Intheoppositelimitofarelativelylargenumberoftraps,NâȘNS,Eq.(**)maybesatisfiedonlyifitssecondtermismuchsmallerthanNS,i.e.atexp{â(Î+ÎŒ)/T}â«gSâŒ1,sothattheequationisreducedto
NVTexpÎŒT+NSgSexpÎ+ÎŒT=N,andmaybereadilysolvedforÎŒ:
expÎŒT=NNVT+NSgSexpÎ/T,i.e.ÎŒ=âTlnNVTN+NSgSNexpÎT,givingtheresultfunctionallyalmostsimilartoEq.(***)ofthepreviousproblem:
NâN0=N1+NSgS/NVTexpÎ/T,P=NâN0TV=NTV11+NSgS/NVTexpÎ/T,justwithadifferentpre-exponentialcoefficientinthedenominator.
Again, the ratioNV(T)/N is of the order of (T/T0)3/2, and has to be large for the gas to stayclassical. However, since in our limit the ratioNS/N is also large, and exp{Î/T} is a very steepfunction, the gas pressure dependsmostly on the condensation energy Î. If the energy ismuchlargerthanthecrossovervalueÎcgivenbyEq.(****), thenumberofparticles inthegasphase isexponentiallysmall,
NâN0=NVTNNSgSexpâÎTâT3/2expâÎT,(becausevirtuallyallparticlesarecondensedonthesurfacetraps),andsoisitspressure:
P=NTVNVTNSgSexpâÎTâNT5/2expâÎT.
As in the results of two previous problems, the pressure in this limit includes the Arrheniusactivationfactorexp{âÎ/T}.Notealsoaverynaturaltrend,Pâ1/NS,thoughtheconditionNSâ«Nusedforthederivationofthisresultdoesnotallowusingittofollowuptheno-traplimitNSâ0.IntheoppositelimitofalowcondensationenergyÎâȘÎc,ourresultisagainreducedtothepressureofanidealclassicalgasofNparticles,P=NT/V.This isnatural,because in this limitvirtuallyallparticlesarethermallyactivatedintothegasphase.
To summarize the above analysis, the particle condensation on the surface affects the gaspropertiessubstantiallyonlyifthenumberNSoftrapsisoftheorderoftheN(orhigher),andthecondensationenergyislargerthanthetemperature-dependentcrossovervalue(****).
Notethatthesolvedproblemmapsverycloselyonthatofstatisticsofchargecarriersindopedsemiconductorsâseesection6.4ofthelecturenotes,andtheassociatedproblems.
Problem3.10.Calculate themagnetic response (thePauli paramagnetism) of a degenerate idealgasofspin-Âœparticlestoaweakexternalmagneticfield,duetoapartialspinalignmentwiththefield.
Solution: According to the basic quantum mechanics14, a charged, spin-œ particle, placed intomagneticfield ,mayhaveonlytwostationaryspinstates,withenergies
where istheeffectivemagneticmomentassociatedwiththespin15.Hence,wemayrepresenttheFermi gas of non-interacting particles as two independent gases with different particle spinorientations,withshiftedparticleenergies
(Letmehope that the usage of different fonts leaves very little chance of confusionbetween theparticleâsmassmanditsmagneticmoment ).
If , the number of particles in each partial gas is the same, but after the field has beenapplied,thespinsofsomeparticlesfliptoalignwiththefieldtoreducetheirenergies,i.e.transferfromonepartialgas toanother.Sucha transfer continuesuntil thechemicalpotentialsÎŒofbothpartial gases become equalâsee the schematic figure below. As this scheme shows, at relativelysmall fields and temperatures ( ), the number of transferred particles, in the linearapproximationinfield,is
(***)
(****)
whereg3(Δ)isthedensityofstatesgivenbyEq.(3.43)ofthelecturenotes,andthefrontfactorœisduetothefactthatg3(Δ) isproportionaltothespindegeneracyg=2s+1 (in thecaseofspin-œparticles,g=2),i.e.countsparticlesofbothspindirections,whileinEq.(**)weneedtocountonlyoneofthem.
Superficially, itmay lookthat thisresult isvalidonlyatextremely lowtemperatures, ,when the thermal smearing of the Fermi surface (see figure 3.2a of the lecture notes and itsdiscussion) ismuchsmaller thantheenergyshiftcausedbytheappliedfieldâaspictured, for thesakeofsimplicity,inthefigureabove.However,thisisnotso.Indeed,wemightderiveEq.(**)bysubtractingtwoEqs.(3.65)ofthelecturenotes,withtheappropriatesubstitutiong(ÎŒ)â(Âœ)g3(ÎŒ),written for two values ofÎŒ that differ by , andkeepingonly the leading term, linear in .SinceEq.(3.65),byitsderivation,isvalidforanychemicalpotentialshiftsandtemperaturesmuchlowerthanΔF,soisEq.(**).
Each flipped spin changes the total magnetic moment of the gas by , so that the netmagnetization(themagneticmomentofaunitvolume)becomes
withthevector paralleltothevector ,andhencedescribingaparamagneticresponseofthegasto the applied field. As the figure above shows, in this linear approximation in , wemay take
, so that the magnetization is proportional to the field, and hence may becharacterizedbypositivemagneticsusceptibility16
The result in this form is convenient for applications, because it is to someextent stablewith
respecttodeviationsoftheFermisurfacefromthesphericalshape,typicalforconductionelectronsin most metals17. If these deviations are negligible, and the Fermi surface is indeed virtuallyspherical(asitis,e.g.inalkalimetals),wemayuseEq.(3.55b)ofthelecturenotestorecastitinanotherform:
wherenâĄN/V is theparticle density.Comparing this expressionwith that for a similar but non-degenerategas,followingfromthesolutionofproblem2.4,
we see that cooling of an ideal Fermi gas results in the Curie-law growth of its paramagneticsusceptibilityuntilitsaturatesatTâŒÎ”F,i.e.attheonsetofthegasdegeneracy.
Fortheparticularcaseofelectrons,wemayusetheexpression .TogetherwiththegeneralrelationΔF=â2kF2/2me,andEq.(3.54)intheformn=2(4Ï/3)kF3/(2Ï)3,itenablesustorewriteourresultinamorecommonform
Ïm=ÎŒ04Ï2e2kFme.
In order to facilitate a comparison of Eq. (***) for the Pauli paramagnetism with the Landaudiamagnetism resulting from the orbital motion of particles of the same degenerate Fermi gas(whose calculation is the subject of the next problem), let us also derive this result in a slightlydifferentway.AccordingtoEq.(*),thereversalofthemagneticmoment ofoneparticle,fromthedirection against the field to the direction along the field changes its energy by .Hence the transfer of ÎNâ« 1 particles, shown by the arrow in the figure above, caused by agraduallygrowingfield,resultsinthefollowingmagneticenergychange18:
But, according to the general electrodynamics (or rather magnetostatics)19, the magnetization-
(**)
relatedpartofthetotalmagneticfieldenergy inthevolumeVofalinear,weakly-polarizablemedium(withÏmâȘ1)is
ComparingthisexpressionwithEq.(****),wearriveatEq.(***)again.
Problem3.11.Calculatethemagneticresponse(theLandaudiamagnetism)ofadegenerateidealgasofelectricallychargedfermionstoaweakexternalmagneticfield,duetotheirorbitalmotion.
Solution:Accordingtoquantummechanics20,externalmagneticfieldcausesthequantizationofthemotionofafreeparticleofmassmandelectricchargeq,intheplanenormaltothefield,sothatthespectrumofthecorrespondingcomponentΔâ„ofitsorbitalenergyformsasetofLandaulevels:
The orbital degeneracy of each level, in a plane of areaA, is g2/âÏc, where g2 =Am/Ïâ2 is theenergy-independent2Ddensityofstatesintheabsenceofthefield(seethesolutionofproblem3.8),so that themagnetic fielddoesnot affect the statedensity averagedover theLandau levels.Themagneticfieldalsodoesnotaffectthemotionoftheparticlealongitsdirection,includingitsenergyΔâ=pâ
2/2m,andthecorresponding1Ddensityoforbitalstates,
g1ΔââĄdN1dΔâ=L2ÏâdpâdΔâ=L2ÏâddΔâ2mΔâ1/2=LÏâ2mΔâ1/2,whereListhelengthofthesysteminthedirectionofthefield,sothatitsvolumeisLA.Asaresult,thetotalenergyoftheparticlemaybecalculatedas
Δ=Δn+Δâ=âÏcn+12+pâ22m.IfNâ«1fermions,confinedinthevolumeV,donotdirectlyinteract,theaboveexpressionsenablethecalculationofthemagneticresponseofthesystem,using(atleast:-)threedifferentapproaches.
Thefirst,mostgeneralapproach21,validforarbitraryfieldsandtemperatures,istosumuptheexpressions (2.114) for all quantum states of the system to calculate the total grand canonicalpotentialΩofthesystemasafunctionofÎŒ,T,and .ThisexpressionnaturallyincludesasumoverallLandaulevels(i.e.overtheindexn),whichmaybeexplicitlycalculatedinthelimit usingtheEulerâMaclaurin formula22. The resulting expression forΩmay then be used to calculate themagneticresponseusingthesecondofEqs.(1.62),andtheanalogybetweenthecanonicalpairsofvariables{âP,V}and ,discussedinsection1.1.Theadvantageofthisapproachisthatits full form (before following the limit ) is valid for arbitrary temperatures and fields, inparticularintherangeâÏcâŒÎ”F,whereitsresultsdescribetheso-calleddeHaasâvanAlpheneffectâperiodicoscillationsofthemagnetizationasafunctionof ,astheLandaulevels(*)sequentiallycrosstheFermisurface23.
For our limited purposes of analysis of the degenerate Fermi gas, withT/ΔF â 0, this generalapproach is, however, somewhat excessive, because in this limit the susceptibility Ïm tends to afinitevalue,whichmaybemorereadilycalculatedbytakingT=0fromtheverybeginning.Inthiscase, theabove formulas, and the level fillingdiagramshown in the figurebelow,maybe readilyusedtocalculatethetotalenergyEofthesystemasafunctionofthemagneticfieldandtheFermienergyΔF, i.e. the largest value of the single-particle energy (**).24 From this function, we maycalculatethedifference ,anduseittofindÏm,exactlyaswasdoneattheend of the solution of the previous problem. Such an approach is rather straightforward, and ishighlyrecommendedtothereaderasanadditionalexercise.
However,hereIwouldliketoshowanevenshorter,veryelegantalternativecalculation,whichre-usesthesolutionofthepreviousproblem25.LetusconsiderthesystemsofLandaulevelsinfieldsand âsee the figure below. Sincewe are only pursuing the limit ,wemay consider thelevelsplittingonascaleâÏcthatismuchsmallerthanT(whichisinturnmuchlowerthanΔF),sothatthefullnumberofstatesonadjacentLandaulevels isvirtuallythesame.Thediagramshowsthatatthefieldincreasefrom to ,theenergyofstatesoneachotherlevel,andhenceofahalfofallstatesofthesystem,increasestheenergybyâÏc/4,whileanotherhalfgoesdownbythesameamount.Butthisisexactlytheeffectanalyzedinthepreviousproblem(seeinparticularthefigureinitssolution),withthereplacement
(*)
(**)
(***)
HencetheexpressionfortheenergychangeÎEduetotheresultingre-distributionofparticlesovertheenergylevelswithΔclosetoΔF,derivedinitssolution,nowbecomes
Nowreproducingthisresultfortheseriesofsequentialtwo-foldincreasesofthefield,weget
Thelastsumisjustthegeometricseries26,equalto4/3,sothatwefinallyget
Butthisexpression,forq=âeandm=me,differsonlybythefactor(â1/3)fromEq.(****)ofthe
solution of the previous problem, if in the latter formulawe take =eâ/2meâaswe should forelectrons.Henceforthis(practicallymostimportant)casewemaywrite
Ïmorbital=â13Ïmspin,i.e.forthefreeelectrongas,theLandaudiamagnetismisexactlythreetimesweakerthanthePauliparamagnetism27. Note, however, that this ratio is sensitive to several effects common to realmetals; forexample, it isaffectedby thedifferencebetweentheeffectivemassofelectrons28 andme.Notealsothatforatomswithuncompensatedspins,thespinparamagnetismismuchstrongerthantheorbitaldiamagnetismâsee,e.g.PartEMproblem5.14andPartQMproblem6.14.
Problem3.12.*ExploretheThomasâFermimodelofaheavyatom,withnuclearchargeQ=Zeâ«e,inwhichtheelectronsaretreatedasadegenerateFermigas,interactingwitheachotheronlyviatheir contribution to the common electrostatic potential Ï(r). In particular, derive the ordinarydifferential equation obeyedby the radial distribution of the potential, anduse it to estimate theeffectiveradiusoftheatom29.
Solution:DuetotheconditionZâ«1,wemayexpectthecharacteristicradiusrTF(Z)oftheatom(i.e.of the electron cloud surrounding the point-like nucleus) to bemuch larger than the radius r0 ofsingle-electron motion in the Coulomb field of bare nucleus of charge Q = Ze, defined by theequalityofthescalesofthequantum-kineticandpotentialenergiesofanelectroninsuchhydrogen-likeâatomâ(actually,ion):
â2mer02=Ze24ÏΔ0r0,givingr0=â2me/Ze24ÏΔ0âĄrBZ,whererB istheBohrradius30.Thisassumption,rTFâ«r0,willbeconfirmedbyoursolution.Duetothis relation,whichmeans thatelectronâselectrostaticpotentialenergyU(r)=âeÏ(r) changes inspaceveryslowly,wemaycalculatetheelectrondensityn(r)âĄdN/d3rinasmalllocalvolume(withr0âȘdrâȘrTF)byneglectingthegradientofU(r),i.e.consideringtheelectronsasfreeparticleswithenergy31
Δ=p22meâeÏ(r),wherethesecondtermistreated,ateachpoint,asalocalconstant.Asaresult,wemayapplytothissmalllocalvolumeofthisdegenerategastheanalysiscarriedoutinthebeginningofsection3.3ofthelecturenotes,inparticular,Eq.(3.54)withthespindegeneracyg=2,towrite
â22me3Ï2n(r)2/3=pF2(r)2me.
Next,ifweacceptthefreeelectronenergyatrââasthereference,thechemicalpotentialÎŒofan atomâs electrons in this model has to be zero, because they have to be in the chemicalequilibriumwithfreeelectronsintheenvironment32.(OnemaysaythattheionizationenergyoftheThomasâFermiatomequalszero.)Hence,fornegligibletemperatures33,thelargestvalueofthetotalenergyΔhastoequalzeroforanyr,sothat,accordingtoEq.(*),themaximumvalue,pF
2/2me, ofthe local kinetic energy p2/2me has to be equal to âqÏ(r) ⥠eÏ(r). Together with Eq. (**), thisequalityyields
n(r)=13Ï22meeÏ(r)â23/2.
The second relation between the functions n(r) and Ï(r) is given by the Poisson equation of
(****)
electrostatics34,â2Ï(r)=âÏ(r)Δ0âĄâeZÎŽ(r)ân(r)Δ0,
wherethespelled-uptwopartsof theelectricchargedensityÏ(r)representthepoint-likepositivechargeQ=Zeofthenucleusattheorigin,andthespace-distributednegativechargeoftheelectroncloud,withthedensityâen(r).Pluggingn(r) fromEq.(***),andspellingouttheLaplaceoperatorforourspherically-symmetricproblem35,wegetthefollowingThomasâFermiequationfortheradialdistributionoftheelectrostaticpotentialÏ:
1r2ddrr2dÏdr=e3Ï2Δ02meeÏâ23/2,forr>0.
Thisordinarydifferential equationhas tobe solvedwith the followingboundaryconditions.AstheabovePoissonequationshows,atrâ0,thepotentialhastoapproachthatoftheatomicnucleus:
ÏrâQ4ÏΔ0râĄZe4ÏΔ0r,atrâ0.Ontheotherhand,duetoatomâsneutrality,atlargedistancesitselectrostaticpotentialshouldnotonlytendtozero,butalsodothisfasterthanthatofanynonvanishingnetcharge36:
rÏrâ0,atrââ.
It isconvenient (andcommon) torecast thisboundaryproblemby introducingadimensionlessdistanceΟfromtheorigin,definedas
ΟâĄrrTFZ,withrTFZâĄbrBZ1/3=br0Z2/3â«r0,andbâĄ123Ï42/3â0.8853,andalsoadimensionlessfunctionÏ(Ο),definedbythefollowingequality:
ÏrâĄZe4ÏΔ0rÏΟ.With these definitions, the boundary problem becomes free of any parameters, and in particularindependentoftheatomicnumberZ:
d2Ïd2Ο=Ï3/2Ο1/2,withÏ(Ο)â1,atΟâ0,0,atΟââ.
Unfortunately,thisnonlineardifferentialequationmaybesolvedonlynumerically,butthisisnotabigloss:thesolutionshowsthatastheargumentΟisincreased,thefunctionÏ(Ο)goesdownfromunity at Ο = 0 to zero at Ο â â monotonically (and very uneventfully), at distances Ο ⌠1. (Forexample,Ï(1)â0.4.)Thisiswhy,evenwithouttheexactsolution,wemayconcludethatEq.(****)gives a fair scale of the effective atomâs size. This relation shows that the effective radius rTF(Z)decreaseswiththeatomicnumberZveryslowly,asrB/Z1/3,andhence,atZâ«1,ismuchlargerthanr0âĄrB/Z,confirming,inparticular,ourinitialassumption.Thisresultisingoodagreement(atZâ«1)withthosegivenbymoreaccuratemodelsâinparticularthosedescribingthequantizedenergyspectraoftheatoms.
Problem3.13.*Use theThomasâFermimodel,explored in thepreviousproblem, tocalculate thetotalbindingenergyofaheavyatom.Comparetheresultwiththatforthesimplermodel,inwhichtheCoulombelectronâelectroninteractionofelectronsiscompletelyignored.
Solution:Thebindingenergyoftheatommaybefoundas
where istheworknecessarytodecreasetheatomicnumberfromZâČto(ZâČâ1). Inorder tocalculate , let us note that the process of decreasing the atomic number by one may bedecomposedintotwosteps:takingoneelectronoutoftheelectroncloud,andoneproton,ofcharge+e, out of the nucleus. Since the chemical potential of the electrons in the ThomasâFermimodelequalsexactlyzero,thefirststepofthisprocessrequiresnowork,whilethesecondsteprequireswork ,whereÏe(r)isthepartofpotentialÏ(r)thatisduetoelectronsonly37.Usingtheaboverelations,withthereplacementZâZâČ,weget
Ïer=ÏrâZâČe4ÏΔ0râĄZâČe4ÏΔ0rÏΟâZâČe4ÏΔ0râĄZâČe4ÏΔ0ÏΟâ1r.SinceÏ(0)â1=0byconstruction,atrâ0thelastfractiontendsto(dÏ/dr)r=0,andweget
DuetothepropertiesoftheuniversalfunctionÏ(Ο),discussedinthemodelsolutionofthepreviousproblem,wemayexpectthederivativedÏ/dΟtobenegative,withamodulusoftheorderof1atΟ=0.Indeed,thenumericalsolutionoftheboundaryproblemforthisfunctionyields
âdÏdΟΟ=0â1.588,so that for anyZâČ. As a result, the total binding energyEb (*) is positive aswell. (Thismeans that theatomâscomponents,after theyhavebeenbrought farapart,haveahigherenergythantheinitialatom.)DuetotheconditionZâ«1,thesum(*)maybecalculatedasanintegral:
ButthelastfractionisjusttheHartreeenergyEH,38sothatwefinallyget
(**)
(***)
(****)
Eb=37bâdÏdΟΟ=0Z7/3EHâ0.7688Z7/3EHâ«EH,forZâ«1.Notetheverynon-trivialscalingoftheenergywiththeatomicnumberZ.
Nowletusconsiderasimplermodel39, inwhichtheelectronâelectron interaction iscompletelyignored,sothatÏ(r)istheunscreenedpotentialofthenucleus,
Ï(r)=Ze4ÏΔ0r,forallr.HerewestillmayuseEqs.(**)ofthemodelsolutionofthepreviousproblem,
â22me3Ï2n(r)2/3=pF2(r)2me,but since the chemical potential ÎŒ in this case is not known in advance, the local value of themaximummomentum,pF(r),shouldbefoundfromEq.(*)ofthepreviousproblem,withΔ=ÎŒ:40
pF22me=ÎŒ+eÏ(r)âĄÎŒ+Ze24ÏΔ0r.In order to have the electrons localized near the nucleus, ÎŒ cannot be positive (relative to theelectronenergyatrââ),sothatpF,andhencetheelectrondensityhavetoturntozeroatsomefiniteradiusref,definedas
Ze24ÏΔ0refâĄâΌ⩟0,andplayingtheroleofatomâsradius.Withthisnotation,Eqs.(**)and(***)yield
n(r)=13Ï2mee2Z2ÏΔ0â21râ1ref3/2.
Now we may calculate ref (and hence the chemical potential) by requiring that the atom beneutral,i.e.thenumberofelectronstobeequaltoZ:41
â«râ©œRn(r)d3r=Z.Carryingouttheintegration,weget
â«râ©œRn(r)d3r=4Ïâ«0Rr2drn(r)=43Ïmee2Z2ÏΔ0â23/2â«0R1râ1ref3/2r2dr=43Ïmee2Zref2ÏΔ0â23/2â«011âΟ3/2Ο1/2dΟ.
ThisdimensionlessintegralmayberecastintoasumofelementaryintegralswiththesubstitutionΟâĄsin2αandthenworkedoutusingEqs.(A.18d)and(A.19).TheresultisÏ/16,sothattheelectroncountingyields
43Ïmee2Zref2ÏΔ0â23/2Ï16=Z,giving
ref=181/34ÏΔ0â2e2me1Z1/3âĄ181/3rBZ1/3â2.620rBZ1/3.
So, this simple model gives the same order of magnitude of rTF as the ThomasâFermi model,thoughwithasignificantlylargernumericalcoefficient42.
Now let us calculate the binding energy (*) within this simple model. In order to avoid thecalculationoftheelectronpotentialÏe(0)feltbythenuclearcharges,thepartialwork maybecalculateddifferently than for theThomasâFermimodel.Namely, letuscalculate theradiusref(ZâČ)andthechemicalpotentialÎŒ(ZâČ)ofanion,withZâČelectrons,butthenuclearchargeQstillequaltoZe.Reviewingtheabovecalculations,weseethat thismaybedonemerelybyreplacingZ on theright-handsideofEq.(****)withZâČ:
43Ïmee2ZrefZâČ2ÏΔ0â23/2Ï16=ZâČ,givingrefZâČ=181/3ZâČ2/3ZrB.Nowwemaycalculatethechemicalpotentialas
âÎŒZâČ=Ze24ÏΔ0refZâČ=1181/3Ze24ÏΔ0rBZZâČ2/3âĄEH181/3Z2ZâČ2/3.Thework necessaryfortheremovalofanadditionalelectronfromtheiontoinfinityisâÎŒ(ZâČ),sothat,replacingthesum(*)withthecorrespondingintegral,weget
Eb=ââ«0ZÎŒZâČdZâČ=EH181/3Z2â«0ZdZâČZâČ2/3=3181/3Z7/3EHâ1.145Z7/3EH.
Verynaturally,thisvalueishigherthanthatcalculatedintheThomasâFermimodel,becauseinthesimplemodeleachelectronisattractedtothenucleusbytheCoulombfieldunscreenedbyotherelectrons,makingtheirinteractionstronger.Note,however,thatthedifferenceisnottoolargeâjustabout50%.ThisrelativeinsignificanceoftheCoulombinteractionofelectronsinatomsechoesthatin the degenerate electron gas in metalsâsee table 3.1 and its discussion in section 3.3 of thelecturenotes.
Problem 3.14. Calculate the characteristic ThomasâFermi length λTF of weak electric fieldâsscreening by conduction electrons in a metal, modeling their ensemble as an ideal, degenerate,isotropicFermigas.
Hint:AssumethatλTFismuchlargerthantheBohrradiusrB.
Solution:Aswasargued in themodelsolutionofproblem3.12, therelationλTFâ«rBallowsus toconsider,intheelectronâsenergy
Δ=p22me+U(r)âĄp22meâeÏ(r),thepotential-energytermasa(local)constant.Asaresult,theconductionelectrons(i.e.fermionsatTâȘΔF), at point r, fill all stateswith kinetic energiesp2/2me satisfying the condition p < pF(r),where
pF2(r)2meâeÏ(r)=ÎŒ.Sincethefielddoesnotpenetratedeepintotheconductorâsbulk,thechemicalpotentialÎŒhastobeequaltoitsfield-unperturbedvalueΔF.43Asaresult,thelocalvaluepF(r)oftheFermimomentum,andhencethelocalelectrondensityn(r),calculatedfromEq.(3.54)ofthelecturenotes(withthespin degeneracy g = 2s + 1 = 2, and N/V = n), become functions of the local value of the
(*)
(**)
(***)
(*)
(**)
electrostaticpotential:â22m3Ïn(r)2/3=ΔF+eÏ(r).
The second relation between the functions n(r) and Ï(r) may be obtained from the Poisson
equationofelectrostatics44,â2Ï(r)=en(r)ân0ÎșΔ0,
whereÎșisthedielectricconstantoftheconductorâsionlattice,andn0istheequilibriumdensityoftheelectronsintheabsenceofthefield,i.e.inanelectricallyneutralconductor.Letusconsiderthesimplest geometry, when the applied electric field is normal to the plane surface of theconductor45.ThenbothnandÏarefunctionsofjustoneCartesiancoordinate(say,x)normaltothesurface,andthePoissonequationisreducedto
d2Ïxdx2=enxân0ÎșΔ0.
SinceEq. (*) is nonlinear, the systemof differential equations (*) and (**) generally cannotbesolvedanalytically.However,iftheexternalfieldisrelativelyweak,
eventhelargestchangeofÏ itcauses(onconductorâssurface) isstillmuchlessthanΔF/e. InthiscasewemaylinearizeEq.(*)withrespecttosmalldensityvariation
nËxâĄnxân0âĄnxânâ,wherexââcorrespondstoconductorâsbulk.Thelinearizationyields
ddnâ22m3Ïn2/3n=n0nËxâĄâ22m3Ï2/323n0â1/3nËxâĄ2ΔF3n0nËx=eÏx,giving
nËx=g3ΔFeÏx,withg3ΔFâĄ32n0ΔF.(Inthisg3(ΔF),wemayreadilyrecognizethevolume-normalizeddensityofstates,(dN/dΔ)/V,at theFemisurfaceofafreeFermigasâseeEq.(3.55b)ofthelecturenotes.)Pluggingthisexpressionintotheright-handsideofEq.(**),wemayrewritetheresultingequationas
d2Ïxdx2=ÏxλTF2,whereλTFâĄÎșΔ0e2g3ΔF1/2âĄ2ÎșΔ0ΔF3e2n01/2â1n01/6.
Thislineardifferentialequation,solvedwiththeboundaryconditions
where x = 0 at the conductorâs surface, describes screeningâan exponential decrease of theelectrostaticpotential,andoftheelectricfield,bothbeingproportionaltoexp{âx/λTF}.Hence,theaboveexpressionforλTFgivesthesolutionoftheposedproblem46.
Ingoodmetals,thedensityofstatesontheFermisurfaceisoftheorderof1022eVâ1cmâ3âĄ1028
eVâ1mâ3,andthedielectricconstantÎșrangesapproximatelyfrom3to10,sothatλTFisoftheorderofafewtenthsofananometer,i.e.isonlymarginallylargerthanrBâ0.05nm,makingtheThomasâFermitheoryofscreeningapplicableonlysemi-quantitatively.However,thetheoryisquantitativelyvalidforsomedegeneratesemiconductors,withsomewhatsmallern0andhencelargerλTF,butstillwithΔFâ«T.
Problem3.15.Foradegenerateideal3DFermigasofNparticles,confinedinarigid-wallboxofvolumeV,calculatethetemperaturedependencesofitspressurePandtheheatcapacitydifference(CPâCV), in the leading approximation inTâȘ ΔF. Compare the results with those for the idealclassicalgas.
Hint:Youmayliketousethesolutionofproblem1.9.
Solution: According to the universal (statistics-independent) relation for any non-relativistic idealgas,givenbyEq.(3.48)ofthelecturenotes,
PV=23E,wemayusethetemperaturedependenceofEgivenbyEq.(3.68),andthenEq.(3.55b)forthe3Ddensityofstates,towrite
PTâ23VE0+Ï26g3ΔFT2âP0+Ï26NVT2ΔF,forTâȘΔF.A comparison of this result with Eq. (1.44), P =NT/V, shows that the pressure (*) grows withtemperaturemuchslowerthanthatoftheclassicalgasofthesamedensityN/V.Thisisverynatural,becauseatlowtemperatures,onlyaminorfractionâŒT/ΔFoftheoccupiedparticlestatesareinsuchaclosevicinitytotheFermisurfacethattheymaybethermallyexcitedâseefigure3.2ofthelecturenotes.
Proceedingtothedifference(CPâCV),perhapstheeasiestwaytocalculateistouseEq.(****)ofthemodelsolutionofproblem1.9:
CPâCV=âTâP/âTV2âP/âVTâĄVTKâPâTV2,whereKâĄâV(âP/âV)T is thebulkmodulus(reciprocalcompressibility)of thegas.UsingEq. (*) tocalculatetheneededderivative,
âPâTV=Ï23NVTΔF,weseethatatT/ΔFâȘ1,Eq.(**)givesustheleadingterm(âT3)ofthetemperaturedependenceofthedifference(CPâCV)evenifweignorethe(weak)temperaturedependenceofthebulkmodulus.Hencewecanuseitszero-temperaturevaluegivenbyEq.(3.58)ofthelecturenotes,
(*)
K=23NVΔF,tofinallyget
CPâCV=VTÏ23NVTΔF2/23NVΔFâĄÏ46TΔF3N.
ThusthedifferencebetweentwoheatcapacitiesofadegenerateFermigasismuchsmallerthanthe capacities as such:CV,CP ⌠(T/ΔF)N (see Eq. (3.70) of the lecture notes), and also than thedifferenceCPâCV=Noftheidealclassicalgas.
Problem3.16.HowwouldtheFermistatisticsofanidealgasaffectthebarometricformula(3.28)?
Solution:Letusstartwithcalculatingtheparticledensityn(r), i.e.thenumberofallparticlesinaunitvolumenearcertainpointr,regardlessoftheirkineticenergy.Thismaybedone,forexample,usingEq. (3.39) of the lecture notes (with theupper sign,which corresponds to theFermiâDiracstatistics),forasmallvolumed3ratthepointr,includingthepotentialenergyU(r)ofaparticleintoitsfullenergyΔ:
n(r)âĄdN(r)d3r=g2Ïâ3â«1expp2/2m+U(r)âÎŒ/T+1d3p.Forhigh temperatures,whenÎŒ is stronglynegative (see figure3.1 of the lecturenotes),wemayignorethelastterm(the1)inthedenominatorofthefraction,getting
n(r)=g2Ïâ3â«expâp22m+U(r)âÎŒ/Td3p=eâU(r)/TeÎŒ/Tg2Ïâ3â«expâp22mTd3p.This integral may be readily worked out in Cartesian coordinates (see section 3.1), but evenregardlessofit,ifthetemperaturethroughoutthegasisconstant,theintegraldoesnotdependofr,sothattheaboveexpressionalreadygivestheexplicitdependencen(r),
n(r)=n0expâU(r)T,wherer = 0 is the point accepted for the potential energy reference:U(0) = 0. Now using theequation of state of the ideal classical gas in the local form P(r) = n(r)T, for the isothermalatmosphere,withU(r)=mgh,wegettheclassicalbarometricformula(3.28):
Ph=P0expâhh0,withh0âĄTmg.
These resultshavealreadybeenobtained in section3.1of the lecturenotes from theMaxwelldistribution; theadvantageofourcurrentapproach is that itmaybegeneralizedtodescribenon-classicalstatisticaleffectsaswell.Inparticular,foradegenerateFermigas(TâȘÎŒ),thefractioninEq.(*) isastepfunction,equalto1 ifp2/2m+U(r)<ÎŒ,andzerootherwise,so that thisrelationyields
n(r)=g(2Ïâ)3â«p2/2m<ÎŒâU(r)d3p=g2Ïâ34Ï3pF3(r),wherepF2(r)2mâĄÎ”F(r)âĄÎŒâU(r).ThefirstexpressionexactlycoincideswithEq.(3.54)ofthelecturenotes,derivedforU=0,sothattheâonlyâroleoftheexternalpotential istocontrolthelocalFermienergyΔF(r).47This iswhywemayusethereadyEq.(3.57)toexpressthegaspressureviaΔF(r)andn(r)andthenviaU(r):
P(r)=25ΔF(r)n(r)=25ΔF(r)g2Ïâ34Ï32mΔF(r)3/2=25g2Ïâ34Ï32m3/2ÎŒâU(r)5/2,sothatforU(r)=mghwehaveabarometricformulaverymuchdifferentfromtheclassicalone:
P(h)=P(0)1âmghÎŒ5/2.
Theformulashows, inparticular,thatthepressure(anddensity)ofthegasturntozeroath=hmax=Ό/mg, because the stateswith energies aboveΌ are unpopulated. (The value ofΌ may befixed,forexample,bythegivennumberofparticles,
NA=â«0ânhdh,perareaAofthegaslayer.)
Problem 3.17. Derive general expressions for the energy E and the chemical potential ÎŒ of auniform Fermi gas of N â« 1 non-interacting, indistinguishable, ultra-relativistic particles48.CalculateE,andalsothegaspressurePexplicitlyinthedegenerategaslimitTâ0.Inparticular,isEq.(3.48)ofthelecturenotes,PV=(2/3)E,validinthiscase?
Solution:TheenergyΔofafree,ultra-relativisticparticle(withΔâ«mc2,wheremisitsrestmass)isrelated to itsmomentump asΔ(p)=cp.49 In this case,Eqs. (3.39) of the lecturenotes (with theuppersign,correspondingtotheFermiâDiracstatistics)yields
N=gV2Ïâ3â«0â4Ïp2dpecpâÎŒ/T+1,whileEq.(3.52),alsowiththeuppersign,becomes
E=gV2Ïâ3â«0âcp4Ïp2dpecpâÎŒ/T+1.As was repeatedly discussed in the lecture notes, in the usual caseNâ« 1, these two relations,exactlyvalidforagrandcanonicalensemblewithgivenÎŒ,maybealsousedastwoequationsforthecalculationofÎŒandEinacanonicalensemblewithgivenN.
The calculations are much simplified in the degenerate limit T â 0, when the FermiâDiracdistribution tends to a step function (see figure3.2 and its discussion),when the above relationstakeasimpleformandtheintegralsinthemmaybereadilyworkedout:
NâgV2Ïâ3â«0pF4Ïp2dp=gV2Ïâ34ÏpF33,EâgV2Ïâ3â«0pFcp4Ïp2dp=gV2Ïâ34ÏcpF44,wherepF is theFermimomentumdefinedby theequalityΔ(pF)âĄcpF=ÎŒâĄÎ”F. The first of theserelations,identicaltoEq.(3.54)forthenon-relativisticgas,yields
pF=2Ïâ34ÏgNV1/3,sothatÎŒ=cpF=6Ï2g1/3câN1/3V1/3,sothatpluggingitintotheexpressionforE,weget
E=gVÏc2Ïâ3pF4=346Ï2g1/3câN4/3V1/3.
(*)
(**)
Now,usingthefactthataccordingtothefundamentalEq.(1.33),FâEatTâ0,wemaycalculatethegaspressurefromEq.(1.35)(withtheimpliedconditionN=const)as
P=ââFâVT=0,N=ââEâVN=146Ï2g1/3câNV4/3.
ComparingtheaboveexpressionsforPandE,weseethatP=13EV.
ThisformulacoincideswithEq.(2.92b)ofthelecturenotesfortheelectromagneticfield(which
maybeconsideredasagasofultra-relativisticBoseparticles,photons),butdiffersbyafactorof2fromEq. (3.48) fornon-relativisticparticles.Aswewillbeshown in thesolutionofnextproblem,thisrelationisvalidforanytemperature.
Problem3.18.UseEq.(3.49)ofthelecturenotestocalculatethepressureofanidealgasofultra-relativistic, indistinguishablequantumparticles, foranarbitrary temperature,asa functionof thetotalenergyEofthegas,anditsvolumeV.Comparetheresultwiththecorrespondingrelationsfortheelectromagneticblackbodyradiation,andanidealgasofnon-relativisticparticles.
Solution:AccordingtoEq.(3.49),thegrandthermodynamicpotentialofasingle-particlestate,withenergyΔ,is
ΩΔ=âTln1±expÎŒâΔT,where theuppersigncorresponds to fermions,andthe lowerone, tobosons.Thepotentialof thewholegasmaybecalculatedas
Ω=â«0âgΔΩΔdΔ,whereg(Δ)isthedensityofquantumstates:g(Δ)âĄdN(Δ)/dΔ,withN(Δ)beingthenumberofquantumstates with energies below Δ. For an isotropic 3D gas, the general rule (3.4) of quantum statecounting50,withtheaccountoftheparticleâsspindegeneracyg,yields
NΔ=gV2Ïâ3â«pâČâ©œpd3pâČ=gV2Ïâ34Ï3p3,wherepisthemagnitudeoftheparticleâsmomentumcorrespondingtotheenergyΔ.Aswasshowninsection3.2of the lecturenotes, fornon-relativisticparticles,withΔ=p2/2m, theseexpressionsimmediatelyleadtoEq.(3.48),
P=2E3V.
Accordingtotherelativitytheory51,forultra-relativisticparticleswemaytakeΔ=pc,sothatp=Δ/c,andweget
NΔ=gV2Ïâ34Ï3Δ3c3,gΔâĄdNΔdΔ=gV2Ïâ34Ï33Δ2c3âĄgV2Ï2Δ2âc3.Asaresult,weobtain
Ω=âTgV2Ï2âc3â«0âΔ2ln1±expÎŒâΔTdΔ.ThisintegralissimilartotheoneinEq.(2.90),anditsintegrationbypartsgivesasimilarresult:
Ω=âgV6Ï2âc3â«0âΔ3expΔâÎŒ/kBT±1dΔâĄâgV6Ï2âc3â«0âΔ3fΔdΔ,wheref(Δ)is,respectively,eithertheFermiâDiracortheBoseâEinsteindistribution:
fΔâĄ1expΔâÎŒ/kBT±1.ButthelastexpressionforΩisjust(â1/3)ofthegasâenergy
E=â«0âgΔΔfΔdΔ=gV2Ï2âc3â«0âΔ3fΔdΔ,sothatΩ=âE/3.Since,accordingtoEq.(1.60),foranarbitrarysystemΩ=âPV,weimmediatelyget
P=E3V.(In the previous problem this result was proved, by simpler means, for the particular case offermionsatTâ0;seealsothesolutionofproblem2.20.)
Eq.(**)coincideswithEq.(2.92)fortheelectromagneticblackbodyradiation(whichmaybealsoconsideredasagasofultra-relativisticparticlesâphotons),butdiffers,byafactorofÂœ,fromEq.(*)foranidealgasofnon-relativisticparticles.(Strictlyspeaking,EinEq.(*)doesnotincludetherest-energy contribution, Nmc2, while E in Eq. (**) does, but in our ultra-relativistic case thiscontributionisnegligiblysmallincomparisonwiththetotalenergyofthegasparticles.)
Problem3.19.*Calculatethespeedofsoundinanidealgasofultra-relativisticfermionsofdensityn,atnegligibletemperature.
Solution:Asweknowfromclassicalmechanics52,thesoundpropagationvelocitymaybecalculatedas
Usingthesolutionofproblem3.17,intheformE=ÏgV2Ïâ3cpF4,P=E3V=Ï3g2Ïâ3cpF4âĄ14cpFnâ1V4/3,
wherenâĄNV=4Ï3g2Ïâ3pF3
istheparticle-numberdensityofthegas,andpFâ1/V1/3isitsFermimomentum,wereadilyget
K=43P=13cpFn.
ThecalculationofthemassdensityÏofthegasisabitmoretricky.Indeed,itwouldbewrongtotakeitequaltonm,wheremistherestmassoftheparticle,becauseforultra-relativisticparticles,withpâ«mc, thismassdoesnotaffectanypropertyof thegas.RatherÏ has tobe calculated asnmef,wheremefameasureofthecollectivegasâinertiaattheapplicationofasmallexternalforceFtoeachofitsparticles:
(Indeed,thisisexactlythedefinitionusedatthederivationofEq.(*)inclassicalmechanics.)Inordertocalculatemef,letustakeintoaccountthatintherelativitytheory,the2ndNewtonlaw
remainsvalidintheform53
providedthatpistherelativisticmomentum.ForthedegenerateFermigas,whichintheabsenceoftheforceoccupiesallstatesinsidetheFermispherepâ©œpF,thismeansthatundertheeffectofforceduringashorttimeintervaldt,thewholesphereshiftsbyasmallinterval inthedirectionoftheforceâalongthez-axis in the figurebelow.Thismeans thateven thougheachparticle stillmoveswiththesamespeed,âŁvâŁ=c,asbeforetheforceapplication,thenumberofparticlesattheFermisurface,withthez-componentofvinthedirectionoftheforce,increases,whilethatoftheparticlesmovingintheoppositedirection,decreases.Wemaydescribethischangeas
where the first fraction is the density dN/d3p of states in the momentum space, Ξ is the anglebetweenthevectorspFand (seethefigureabove),anddΩp isanelementarysolidangle inthemomentumspace.Theparticleâsvelocity ineachstatehasthez-componentequaltoccosΞ(otherCartesiancomponentsofvareaveragedout),sothattheaveragevelocitychangeundertheeffectoftheimpulse maybecalculatedas
Afteraneasyintegration,âź4Ïcos2ΞdΩp=2Ïâ«0Ïcos2ΞsinΞdΞ=2Ïâ«â1+1cos2ΞdcosΞ=2Ï23=4Ï3,
wegetaverysimpleresult:
(Asasanitycheck,thesimilarcalculationforthenon-relativisticFermigas,withthereplacement,immediatelyyieldsmef=mandÏ=nm.)
WiththeseKandÏ,Eq.(*)yieldsthesoundvelocity
Itisremarkablethatthisresultdoesnotdependonthedensityn(andalsothespindegeneracy
g)of theparticles. (Actually, this is correct for the ideal classicalgasaswellâsee the solutionofproblem1.6.)
Problem 3.20. Calculate basic thermodynamic characteristics, including all relevantthermodynamicpotentials,specificheat,andthesurfacetension,forauniform,non-relativistic2DelectrongaswithgivenarealdensitynâĄN/A:
(i)atT=0,and(ii)atlowbutnonvanishingtemperatures(inthelowestsubstantialorderinT/ΔFâȘ1),
neglectingtheCoulombinteractioneffects54.
Solution:Inthestatedconditions,theelectrongasmaybetreatedasanidealFermigas,sothattheaverage value of any thermodynamic variable f(Δ) is the sum of its values in all quantum states,
(*)
(**)
(***)
weighedwiththeFermidistribution.Inthe2Dcase,f=âkfkNk=2A2Ïâ2â«0âf(Δ)NΔ2Ïpdp=â«0âf(Δ)NΔg2(Δ)dΔ,
whereΔ=p2/2m is thekinetic energyof a singleparticle (so thatdΔ=pdp/m), the factorg=2describesthedoublespindegeneracyofeachorbitalstate,âšN(Δ)â©istheFermiâDiracdistribution,
NΔ=1expΔâÎŒ/T+1,andg2(Δ) is the 2Ddensity of states.Aswas discussed in themodel solution of problem3.8, thedensityturnsouttobeenergy-independent:
g2(Δ)=mAÏâ2.
LetusapplyEq.(*)tothekeyvariables.Asusual,thechemicalpotentialΌ,participatingintheFermi distribution, may be found from the formal calculation of the (actually, given) number ofparticlesN.Takingf(Δ)=1(asappropriateforparticlecounting),weget
N=â«0âNΔg2(Δ)dΔ=mAÏâ2â«0âdΔexp(ΔâÎŒ)/T+1=mAÏâ2Tâ«âÎŒ/TâdΟeΟ+1.Thisisatableintegral55,giving
N=mAÏâ2Tln1+expÎŒTâĄmAÎŒÏâ2+mAÏâ2ln1+expâÎŒT.However,forothervariables,forexample,thetotalenergy
E(T)=â«0âΔNΔg2(Δ)dΔ=mAÏâ2â«0âΔdΔexp(ΔâÎŒ)/T+1,integralscannotbeworkedoutanalytically(moreexactly,expressedviathefunctionsdiscussedinthis series) for arbitrary temperatures. Let us proceed to the limiting cases specified in theassignment.
(i)T=0.InthiscasethesecondterminthelastformofEq.(**)vanishes,andityieldsN=(mA/Ïâ2)ÎŒ,sothat
ÎŒ(T=0)âĄÎ”F=Ïâ2mNA.Next,sinceatT=0,âšN(Δ)â©isastepfunction(seeEq.(3.53)andfigure3.2ainthelecturenotes),Eq.(***)gives
E(0)=mAÏâ2â«0ΔFΔdΔ=mAÏâ2ΔF22=mAÏâ212Ïâ2mNA2âĄ12Ïâ2mN2A,sothattheaverageenergyperparticle
ΔâĄE(0)N=ΔF2.
Concerningotherthermodynamicpotentials,bydefinitionF(0)=E(0)andH(0)=G(0).ThelatterpotentialmaybefoundusingEq.(1.56):
G(0)=ÎŒ(T=0)N=ΔFN=2E(0),sothatthegrandpotential,definedbythefirstofEqs.(1.60),isΩ(0)âĄF(0)âG(0)=âE(0).56Inthe3Dcase,ΩisequaltoâPVâseeEq.(1.60)again.Aswasrepeatedlydiscussedabove,inthe2DcasethevolumeâsanalogistheareaA,sothattheanalogofpressurePâĄâ(âF/âV)T,Nisthesurfaceâanti-tensionâ57ÏâĄâ(âF/âA)T,N,sothatΩ=âÏA.ForT=0,weget
Ï(0)=âΩ(0)A=E(0)A=12Ïâ2mN2A2.Finally,inordertocalculatethespecificheat,weneedtoconsiderthecaseoffinitetemperatures.
(ii)TâȘΔF, i.e.TâȘÎŒ. In this limit the second term inEq. (**) gives only an exponentially smallcorrectiontoÎŒâanon-analyticalfunctionofT,whichisnotcapturedbytheSommerfeldexpansion(3.59),sothatwemaystilluseforitthesamerelationasatT=0:
ÎŒ(TâȘΔF)âΔF=Ïâ2mNA=const.Fromhere
G(T)âΔFN=G(0)=2E(0).
For the energy, we may use the Sommerfeldâs expansion (3.59), which does not depend onsystemâsdimensionality:
I(T)âĄâ«0âÏΔNΔdΔââ«0ÎŒÏ(Δ)dΔ+Ï26T2dÏ(ÎŒ)dÎŒ,Taking Ï(Δ) = (mA/Ïâ2)Δ (with dÏ(ÎŒ)/dÎŒ =mA/Ïâ2 = const), we may calculate the temperaturedependenceofenergy:
E(T)âE(0)+Ï26T2mAÏâ2=E(0)1+Ï23TΔF2,sothatthespecificheatcA(theheatcapacityperparticleatfixedarea)issmallincomparisonwiththatoftheclassicalgas,andlinearintemperature:
cA=1NâEâTA,N=Ï23TmAÏâ2=Ï23TΔFâȘ1.Justasinthe3Dcasediscussedinthemodelsolutionofproblem3.15,thedifferencebetweenthelow-temperature values of the specific heats at constant area (cA) and at constant tension (cÏ) isproportional to (T/ΔF)3, and is much smaller than the (virtually equal) cA and cÏ, which areproportionaltoT/ΔF.
NowletuscalculatetheeffectoftemperatureonthegrandpotentialΩ:Ω(T)=âTâkln1+expÎŒâΔkT=âTmAÏâ2â«0âln1+expÎŒâΔTdΔ.
Integratingbyparts,wegetΩ(T)=âmAÏâ2â«0âNΔΔdΔ=âE(T)=Ω(0)1+Ï23TΔF2.
Thisresultenablesareadilycompletionofallcalculations:Ï(T)=âΩ(T)A=Ï(0)1+Ï23TΔF2,F(T)=G(T)+Ω(T)=F(0)1âÏ23TΔF2,S(T)=E(T)
(*)
(*)
(**)
âF(T)T=Ï23mAÏâ2T,H(T)=E(T)+Ï(T)A=H(0)1+Ï23TΔF2.
Problem 3.21. Calculate the effective latent heat Îef ⥠âN(âQ/âN0)N,V of evaporation of thespatially-uniform BoseâEinstein condensate as a function of temperature T. Here Q is the heatabsorbedbythe(condensate+gas)systemofNâ«1particlesasawhole,whileN0isthenumberofparticlesinthecondensatealone.
Solution:Foraslow(reversible)process,wecanusethefundamentalEq.(1.17)ofthelecturenotes,intheformdE=dQâPdV,torepresenttheeffectivelatentheatas
Îef=âNâEâN0N,V.ForthenumberN0ofcondensedparticlesinaspatially-uniformsystem,wehadEq.(3.74a):
N0=N1âTTc3/2,whereTcisthecriticaltemperature.SolvingthisequationforT,weget
TTc=1âN0N2/3.Ontheotherhand,forthesystemenergybelowTcwehadEqs.(3.75)and(3.78):
E(T)=E(Tc)TTc5/2=32ζ(5/2)ζ(3/2)NTcTTc5/2.Combiningtheseformulas,wegetaconvenientdirectrelationbetweenEandN0,
E=32ζ(5/2)ζ(3/2)NTc1âN0N5/3,whosedifferentiationyields
Îef=âNâEâN0N,V=52ζ(5/2)ζ(3/2)NTc1âN0N2/3.NowwecanuseEq.(*)againtorewritethisexpressionas
Îef=52ζ(5/2)ζ(3/2)NTâ1.284NT.
Thisisaprettyinterestingresult:theeffectivelatentheatofthesystemdoesnotdependonTc,andhenceonthesystemâsvolumeâwhileitsenergyEdoes.Inchapter4,wewillseethatthisÎefcoincideswiththelatentheatinitsusualdefinition.
Problem 3.22.* For an ideal, spatially-uniform Bose gas, calculate the law of the chemicalpotentialâs disappearance at T â Tc, and use the result to prove that the heat capacity CV is acontinuousfunctionoftemperatureatthecriticalpointT=Tc.
Solution:LetusconsiderasmallincreasedT=TâTcâȘTcoftemperatureaboveitscriticalvalue.SincewekeepthenumberNofparticlesconstant,thedirecteffectofdTonthepre-integralfactorinEq. (3.44)of the lecturenoteshas tobecompensatedby thechangeof the integraldue to theappearanceofasmallpositiveparametera2âĄâÎŒ/TâȘ1:
dN=gVm3/22Ï2â332Tc1/2dTI(0)+Tc3/2IaâI0=0,where
IaâĄâ«0âΟ1/2dΟexpΟ+a2â1=2â«0âÏ2dÏexpÏ2+a2â1,withÏâĄÎŸ1/2,sothatthe(small)differenceofintegrals,participatinginEq.(*),is
IaâI0=2â«0âÏ2dÏ1expÏ2+a2â1â1expÏ2â1.Sinceataâ0,themaincontributiontothisexpressioncomesfromtheregionofsmallÏ,wemayexpandtheexpressioninbracketsintheTaylorseriesinbotha2andÏ2,getting
IaâI0â2â«0âÏ2dÏ1Ï2+a2â1Ï2=â2a2â«0âdÏÏ2+a2.This is a table integral58, equal to Ï/2a, so that I(a) â I(0) = âÏa, and with the value I(0) =Î(3/2)ζ(3/2)(seeEq.(3.71)ofthelecturenotes),Eq.(*)yields
âÎŒTâĄa2=3I(0)2ÏdTTc2=3Î(3/2)ζ(3/2)2ÏdTTc2â1.222dTTc2âȘ1.So, thesmallchemicalpotential,appearingaboveTc, isproportional to the temperaturedeviationsquared.This is fullyconsistentwiththeplotshownwiththeblue line in figure3.1of the lecturenotes.(Onthelogâlogplotoffigure3.3a,thisrelationislessobvious.)
Fortheheatcapacitycalculation,itisconvenienttouseEq.(3.52),withtheappropriatenegativesign,tointroducethenotionoftheenergyE0(T)thatthegaswouldhaveatT⩟Tc ifthechemicalpotentialretainedthevalueΌ=0:
E0âĄgVm3/2T5/22Ï2â3â«0âΟ3/2eΟâ1dΟ=E(Tc)TTc5/2.Bythisdefinition,thedifferencebetweentheactualenergyEandE0atthesametemperatureT>Tcisonlyduetothechangeofthechemicalpotential,sothatatdTâȘTc,wemaywrite
EâE0=âEâÎŒTÎŒ.NowwemayuseEq.(3.48),thenEq.(1.60),andthenthelastofEqs.(1.62)towrite
âEâÎŒT=32â(PV)âÎŒT=â32âΩâÎŒT=32N,wheretheconstancyofvolumeisalsoimplied.CombinedwithEq.(**)forÎŒ,theseformulasyield
EâE0=â32NT3Î(3/2)ζ(3/2)2ÏdTTc2,for0â©œdTâȘTc.
NownotethatsincetheabovedefinitionofE0coincideswithEq.(3.78),whichisvalidforactualenergyatTâ©œTc,thetemperaturederivativeofthedifferenceEâE0(atconstantvolume),takenatT=Tc,givesthedifferenceofthelimitingvaluesoftheheatcapacity:
CVT=Tc+0âCVT=Tcâ0=limTâTcddTEâE0=limTâTcâ33Î(3/2)ζ(3/2)2Ï2NTcdTâ0.
(*)
(**)
Hence, the heat capacity is indeed a continuous function of temperature. However, itstemperaturederivativeisnot:
dCVdTT=Tc+0âdCVdTT=Tcâ0=â33Î(3/2)ζ(3/2)2Ï2NTcââ3.665NTc.Thisâcuspâisclearlyvisibleinthe(numericallycalculated)figure3.5ofthelecturenotes.
Problem3.23.Inchapter1ofthelecturenotes,severalthermodynamicrelationsinvolvingentropyhavebeendiscussed,includingthefirstofEqs.(1.39):
S=ââG/âTP.Ifwe combine this expressionwith the fundamental relation (1.56),G=ÎŒN, it looks like for theBoseâEinstein condensate, whose chemical potential ÎŒ equals zero at temperatures below thecriticalpointTc, theentropy shouldvanishaswell.On theotherhand,dividingbothpartsofEq.(1.19)bydT,andassumingthatatthistemperaturechangethevolumeiskeptconstant,weget
CV=TâS/âTV.(Thisequalitywasalsomentionedinchapter1.)IfCVisknownasafunctionoftemperature,thelastrelationmaybeintegratedoverTtocalculateS:
S=â«V=constCV(T)TdT+const.AccordingtoEq.(3.80),thespecificheatfortheBoseâEinsteincondensateisproportionaltoT3/2,sothattheintegrationgivesanonvanishingentropySâT3/2.Resolvethisapparentcontradiction,andcalculatethegenuineentropyatT=Tc.
Solution:Eq.(1.39)ofthelecturenoteshasbeenderivedforuniformsystemswithafixednumberofparticles,and(asEq.(1.53c)shows)inthegeneralcaseshouldberewrittenas
S=ââGâTP,N.Weknow,however,thatbelowTc,theBoseâEinsteincondensateisessentiallyatwo-phasesystem,inwhichthedisorderedgascoexistswiththecompletelyordered(andhenceentropy-free)condensate,andthechangeoftemperatureresultsinthechangeofthenumberofthegaseous-phaseparticles,
NâČâĄNâN0=NTTc3/2,evenifthetotalnumberNofparticlesinthesystemdoesnotchange.Moreover,anytemperaturevariationalsoleadstoavariationofpressureP(T)aswellâseeEq.(3.79),sothatitcannotbekeptconstant.Asaresult,theargumentsbasedonEq.(1.39)arenotvalid,andwemayindeedcalculatetheentropy(at0â©œTâ©œTc)usingEq.(3.80):
STâS0=â«0TCV(T)TdTV=const=52ETcTc5/2â«0TT3/2TdT=53ETcTc5/2T3/2,sothattheentropyofthesystem59isindeedproportionaltoT3/2,and(ifwereasonably,thoughitisnotcompulsory,takeS(0)=0)atTâTcitreachesthevalue
STc=53ETcTcâ1.284N.
Itisstraightforward(andhighlyrecommendedtothereaderasanadditionalexercise)tocheckthat the entropy of the gas approaches the same value at the approach to Tc from the highertemperature side,whereÎŒâ 0, and allN Bose particles are in the gas phase, so thatSmaybeindeedcalculatedmerelybydifferentiationâeitherfromthefirstofEqs.(1.39)withG=ÎŒN,orevensimpler,fromthefirstofEqs.(1.62),
S=ââΩâTV,ÎŒ,withthepotentialΩgivenbythefirstformofEq.(3.51),withtheappropriate(lower)sign.
Problem3.24.ThestandardanalysisoftheBoseâEinsteincondensation,outlinedinsection3.4ofthelecturenotes,mayseemtoignoretheenergyquantizationoftheparticlesconfinedinvolumeV.Use the particular case of a cubic confining volume V = a Ă a Ă a with rigid walls to analyzewhether the main conclusions of the standard theory, in particular Eq. (3.71) for the criticaltemperatureofthesystemofNâ«1particles,areaffectedbysuchquantization.
Solution: An elementary quantum-mechanical analysis of a single particle placed in such a box60yieldsthefollowingenergyspectrum:
Δnx,ny,nz=Δ0nx2+ny2+nz2,whereΔ0âĄÏ2â22ma2=Ï2â22mV2/3,with quantum numbersnx, etc, taking all positive integer values starting from 1. Let us use theBoseâEinsteindistribution(2.118)tocalculatetheaveragenumberofparticlesinthecasewhentheboxisinthermalandchemicalequilibriumwiththeenvironmentwithtemperatureTandchemicalpotentialÎŒ:
N=gânx,ny,nz=1âNnx,ny,nz=gânx,ny,nz=1âexpΔnx,ny,nzâÎŒTâ1â1=gânx,ny,nz=1âexpΔ0nx2+ny2+nz2âÎŒTâ1â1,
whereg=2s+1isthespindegeneracy.(Forelectrons,s=œ,sothatg=2.)Generally,asweknowfromsection2.8ofthelecturenotes,thisexpressionisonlyvalidforthe
grandcanonicalensemble,inwhichthenumberNofparticlesintheboxisnotfixed.However,aswas repeatedly discussed in the lecture notes, in the canonical ensemble of systems with thenumberNofparticlesintheboxisfixedbutverylarge,wemayuseit,withthereplacementâšNâ©âN,forthecalculationoftherelationbetweentheaveragevaluesofNandÎŒ,neglectingtheirsmallfluctuations,whose relative rms values scale as 1/N1/2âȘ 1. In particular, in accordancewith thediscussioninsection3.4,inordertocalculateTc,wehavetotakeÎŒequaltothelowestvalueofthesingle-particleenergyΔ(inthiscase,thegroundstateenergyis3Δ0ânotexactlyzero!),i.e.solvethefollowingequation:
N=gânx,ny,nz=1âexpΔ0Tcnx2+ny2+nz2â3â1â1.
Suchasumconvergesassoonasthemagnitudeoftheargumentundertheexponentbecomes
(*)
(**)
muchlargerthan1,i.e.atnâĄ(nx2+ny2+nz2)1/2âŒnmaxâĄ(Tc/Δ0)1/2.Sincethefirst,mostsignificanttermsofthesumareoftheorderof1,thesumitselfmaybeestimatedasn3
max,sothat,bytheorderofmagnitude,Eq.(**)gives
NâŒgnmax3âĄTcΔ03/2,i.e.TcâŒÎ”0Ng2/3.But sinceNâ« 1 andgâŒ1, thismeans thatTc ismuch larger thanΔ0,which is the scale of thedistance between the adjacent energy (which differs by a unit change of one of the quantumnumbers).HenceatTâŒTc,manylowerlevelsarepopulated,sothatinthesumsinEq.(**),theterm(â3)maybeneglected,andthesumasawholemaybeapproximatedbyanintegral.Asaresult,theequationforTctakestheform
N=gâ«0âdnxâ«0âdnyâ«0âdnzexpΔ0Tcnx2+ny2+nz2â1â1âĄg8â«expΔ0Tcn2â1â1d3n.wherenâĄ{nx,ny,nz}, so thatn2=nx2+ny2+nz2, and the factor (1/8) before the last integralreflectstheconditionnx,ny,nz>0.Nowusingthesphericalcoordinatesinthespaceofvectorsn,weget
N=g84Ïâ«0âexpΔ0Tcn2â1â1n2dnâĄÏg2â«0ân2dnexpΔ0n2/Tcâ1.Butthisequation,withtheintegrationvariablereplacementΟâĄÎ”0n2/Tc,andwiththeaccountofEq.(*) forΔ0, exactly coincideswithEq. (3.73) forTc, and naturally, yields the same result (3.71). Inhindsight,thisisnotsurprising,becausetheargumentsusedinthissolutionessentiallyreproduce,foraparticularsystem,thereasoningleadingtothegeneralquantumstatecountingrule(3.13).
Nevertheless,thesolutionofthisproblemwasnotinvain:itclearlyshowswhyEq.(3.71)isvalidonlyforspatially-uniformsystems,i.e.iftheparticleconfinementisratherstiff,inthesensethatthewallvolumetowhichthewavefunctionsoftheparticlespartlypenetrateismuchsmallerthanthevolumeVoftheirfreemotion.Intheoppositecaseofsoftconfinement,forexampleatthebottomofa quadratic-parabolic potential well, the value of Tc is rather differentâsee Eq. (3.74b) and itsdiscussion in the lecture notes, and the next problem (to which this solution gives a perfectpreparation).
Problem 3.25.* N â« 1 non-interacting bosons are confined in a soft, spherically-symmetricpotentialwellU(r)=mÏ2r2/2.DevelopthetheoryoftheBoseâEinsteincondensationinthissystem;in particular, prove Eq. (3.74b) of the lecture notes, and calculate the critical temperature Tc*.Looking at the solution, what is the most straightforward way to detect the condensation inexperiment?
Solution: A well-known quantum-mechanical analysis61 of a single particleâs motion in such apotentialwell(frequentlycalledthe3Dharmonicoscillator)hasthefollowingenergyspectrum:
Δnx,ny,nz=âÏnx+ny+nz+32,withthequantumnumbersnx,ny,andnz taking independent,non-negative integervaluesstartingfrom 0. Just aswas done in the solution of the previous problem,wemay use theBoseâEinsteindistribution (2.118) to calculate theaveragenumberofparticles in thecasewhen thegas is inathermalandchemicalequilibriumwiththeenvironmentwithtemperatureTandchemicalpotentialÎŒ:
N=gânx,ny,nz=0âexpΔnx,ny,nzâÎŒTâ1â1=gânx,ny,nz=0âexpâÏnx+ny+nz+3/2âÎŒTâ1â1,wheregisthespindegeneracyofeachâorbitalâstate.Usingthestandardargumentsforthetransferfromthegrandcanonicaltothecanonicalensemble,quantitativelycorrectinthelimitNâ«1,andtakingthechemicalpotentialÎŒequal to thegroundstateenergyΔg (inourcurrentcase,equal toΔ0,0,0=(3/2)âÏ),wegetthefollowingequationforthecriticaltemperatureTc*:
N=gânx,ny,nz=0âexpâÏTc*nx+ny+nzâ1â1.
Suchasumconvergesassoonasthemagnitudeoftheargumentundertheexponentbecomesmuchlargerthan1,i.e.atnâĄnx+ny+nzâŒnmaxâĄTc*/âÏ.Sincethefirst,mostsignificanttermsofthesumareoftheorderof1,thesumasawholemaybeestimatedasnmax3,sothatEq.(*)givesthefollowingestimate:
NâŒgnmax3âĄTc*âÏ3,i.e.Tc*âŒâÏNg1/3.ButsinceNâ«1andgâŒ1,so that (N/g)1/3 ismuch larger than1aswell, thismeans thatTc* ismuchlargerthanâÏ,whichisthescaleofthedistancebetweentheadjacentenergylevels(whichdifferbyaunitchangeofoneofthequantumnumbers).HenceatTâŒTc*,many lower levelsarepopulated,sothatthesum(*)maybewellapproximatedbyanintegral.
Asaresultofthisapproximation,theequationforthecriticaltemperaturetakestheformN=gâ«0âdnxâ«0âdnyâ«0âdnzexpâÏTc*nx+ny+nzâ1â1=gâ«nx,ny,nzâ©Ÿ0expâÏTc*nx+ny+nzâ1â1dÎŁ,
wheredÎŁâĄdnxdnydnz is an elementary volumeof the statenumber space{nx,ny,nz}.Since thefunction under the integral depends only on one linear combination, n ⥠nx + ny + nz, of theCartesiancoordinatesofthisspace,itisbeneficialtoselectthedifferentialdÎŁintheform62
dÎŁ=dn36âĄn22dn(seethefigurebelow),sothatour3Dintegralreducestoa1Done:
N=gâ«0âexpâÏTc*nâ1â1n2dn2.
(***)
(****)
(*)
WiththeintegrationvariablereplacementΟâĄ(âÏ/Tc*)n,thisequationtakestheformN=g2Tc*âÏ3â«0âΟ2dΟeΟâ1.
Thistableintegral63equalsÎ(3)ζ(3)âĄ2ζ(3)â2.404,sothat,finally,wegetN=ζ3gTc*âÏ3,i.e.Tc*=âÏNgζ31/3â0.9405âÏNg1/3,
infullagreementwithourinitialestimate(**).AcomparisonofthisexpressionwithEqs.(3.35)and(3.71)ofthelecturenotes(whicharevalid
for the rigid confinement case) shows that the dependence of the critical temperature on thenumberofparticlesatthesoftconfinementismuchweaker:Tc*âN1/3vsTcâN2/3.Thisisnatural,becausetheeffectiveradiusRoftheconfinedgascloud,whichmaybeestimatedfromtherelation
UâŒmÏ2R22âŒT2,andhenceitseffectivevolumeVefâŒR3,nowgrowswithtemperature,andhence(atTâŒTc*)withN.
Thedifferencebetweenthetwoconfinementtypesalsomanifestsitselfinadifferentdependenceof the condensed particle number N0 on temperature at T â©œ Tc*. Indeed, using the sameargumentationaswasusedforthespatially-uniformsystemdiscussedinsection3.4ofthelecturenotes64,wemaygetthisdependencefromEq.(***)byreplacementsNâ(NâN0)andTc*âT:
NâN0=g2TâÏ3â«0âΟ2dΟeΟâ1,atTâ©œTc*.Nowcombining thisexpressionwithEq. (***),wegetEq. (3.74b)of the lecturenotes (whichwasgiventherewithoutaproof):
N0=N1âTTc*3,forTâ©œTc*.
ThereaderisencouragedtoexploreotherdifferencesbetweenBECfeaturesatthesoftandrigidconfinement, but I will limit this solution to answering the last question of the assignment.According to Eq. (****), the optically visible area of all the gas cloud above Tc*, and of itsuncondensedfractionbelowthecriticaltemperature,isoftheorderof
AâŒR2âŒTmÏ2âŒTc*mÏ2âŒâmÏNg1/3.However,allparticlesof thecondensed fractionof thegas,atT<Tc*,are in theirgroundstate,withenergyΔgâĄÎ”0,0,0=(3/2)âÏ,sothattheradiusRcoftheircloudshouldbeestimatednotfromEq.(****),butfromtherelation
UcâŒmÏ2Rc22âŒÎ”g=3âÏ2,givingthevisiblearea
AcâŒRc2âŒ3âmÏâŒ3AN/g1/3âȘA,forNâ«1.
As a result, the most direct manifestation of the BoseâEinstein condensation at the softconfinementistheappearance,atT<Tc*,ofasmall,denseâblobâontheopticalimageofthegas,on thebackgroundof a largergas cloud.Some spectacular imagesof this appearancehavebeenpublished;afewofthemareavailableonline65.
Problem3.26.Calculatethechemicalpotentialofanideal,uniform2Dgasofspin-0Boseparticlesasafunctionofitsarealdensityn(thenumberofparticlesperunitarea),andfindoutwhethersuchagascancondenseatlowtemperatures.Reviewyourresultforthecaseofalarge(Nâ«1)butfinitenumberofparticles.
Solution:Aswasalreadydiscussedinthesolutionsofproblems3.8and3.20,thedensityg2(Δ)of2DquantumstatesisindependentoftheparticleenergyΔ:
g2(Δ)=dNstatesdE=gA2Ïâ2d2pdp2/2m=gA2Ïâ22Ïpdppdp/m=gmA2Ïâ2,whereg=2s+1isthespindegeneracy,inthecaseofspin-0particlesequalto1.HencethenumberofparticlesinsideareaAmaybecalculatedas
N=mA2Ïâ2â«0âdΔeΔâÎŒ/Tâ1=mA2Ïâ2Tâ«âÎŒ/TâdΟeΟâ1,whereΟâĄ(ΔâÎŒ)/T.Thisisatableintegral66,giving
N=mA2Ïâ2Tln11âexpÎŒ/T,sothatthechemicalpotentialis
ÎŒ=Tln1âexpâ2Ïâ2nmT,withnâĄNA.
Since theexponential function in thisexpression isbetween0and1 foranygasdensitynand
(*)
(**)
temperatureT,theargumentofthelogarithmisalwaysbelow1,sothatthelogarithmisnegative,andhence thechemicalpotential isnegative forallT>0.Hence theBoseâEinsteincondensation(whichrequiresÎŒ=0)isimpossibleinauniform2Dgas.Thisfactmightbeevidentalreadyfromtheverybeginning,becauseatÎŒ=0,theintegral(*)divergesatitslowerlimit.Thisargumentdoesnothold in 3D,where the different dependence of the density of states on the particle energy,g3(Δ),providesanextra factorofΔ1/2 in thenumeratorof the functionunder the integral,preventing itsdivergenceatÎŒ=0,andmakingtheBoseâEinsteincondensationpossible.
Note,however,thatthisconclusion(Tc=0)isstrictlyvalidonlyinthelimitNââ(andhenceAââ),becausethedivergenceoftheintegral(*)atthelowerlimitatÎŒ=0isveryweak(logarithmic),andmaybecutoffbyvirtuallyanyfactor.Inparticular,alargebutfiniteareaAofthegas-confiningbox keeps the particle energy quantized on a small scale ΔâŒÏ2â2/2mA, corresponding to ΟminâŒÏ2â2/2mATcâȘ1.Withthismodification,Eq.(*)givesforTcthefollowingtranscendentalequation
N=AmTc2Ïâ2ln2mATcÏ2â2,where theargumentof the logarithm isapproximate. (Itdoesnotmakemuchdifference,becausethis argument is very large, and the log function of a large argument is very insensitive to itschange.)Theapproximate(withtheso-calledlogarithmicaccuracy)solutionofthisequationis
Tcâ2Ïâ2nm/lnN,sothatTcindeedtendstozeroatNââ(atafixeddensitynâĄN/A),butextremelyslowly.
Inthiscontext,notethattheverynotionoftheBoseâEinsteincondensation(and,moregenerally,ofanyphasetransition)makesfullsenseonlyinthelimitNââ.Indeed,asthesolutionsofthelastthreeproblemsindicateveryclearly,ifthenumberofbosonsinasystemisfinite,thereductionoftemperature leads âmerelyâ to theirgradualaccumulationon the lowest,groundenergy level.Thewholeideaofthephasetransition,withacertaincriticaltemperatureTc,isthatatNâ«1,mostofthisaccumulationhappensinaverynarrowtemperatureintervalnearsometemperature,calledTc.AtafiniteN,thisintervalisalwaysnonvanishing,andgraduallybroadenswiththereductionofN.
Problem3.27.CantheBoseâEinsteincondensationhappenina2DsystemofNâ«1non-interactingbosonsplacedintoasoft,axially-symmetricpotentialwell,whosepotentialmaybeapproximatedasU(r) =mÏ2Ï2/2, where Ï2 ⥠x2 + y2, and {x, y} are the Cartesian coordinates in the particleconfinementplane?Ifyes,calculatethecriticaltemperatureofthecondensation.
Solution:Withthenaturalchangefrom3Dto2D,Eq.(*)ofthemodelsolutionofproblem3.25forthecriticaltemperatureTc*ofthegasbecomes
N=gânx,ny=0âexpâÏTc*nx+nyâ1â1,whereg is thespindegeneracy.Suchasumconvergesassoonasthemagnitudeof theargumentundertheexponentbecomesmuchlargerthan1,i.e.atnâĄnx+nyâŒnmaxâĄTc*/âÏ.Sincethefirst,significanttermsofthesumareoftheorderof1,thesumitselfmaybeestimatedasn2
max,sothatEq.(*)givesthefollowingestimate:
NâŒgnmax2âĄgTc*âÏ2,i.e.Tc*âŒâÏNg1/2.
In order to calculateTc* exactly,wemay again use the strong inequalityNâ« 1 to justify thetransitionfromthesummation(*)tointegration,getting
N=gâ«0âdnxâ«0âdnyexpâÏTc*nx+nyâ1â1=gâ«nx,nyâ©Ÿ0expâÏTc*nx+nyâ1â1dÎŁ,wheredΣ⥠dnxdny is an elementary area on the plane of quantum numbers {nx,ny}. Since thefunctionunderthe integraldependsonlyonone linearcombination,nâĄnx+ny,oftheCartesiancoordinatesofthisspace,wemayselectthedifferentialdÎŁintheform67,
dÎŁ=dn22âĄndn(seethefigurebelow),sothatour2Dintegralreducestoa1Done:
N=gâ«0âexpâÏTc*nâ1â1ndn.
WiththeintegrationvariablereplacementΟâĄ(âÏ/Tc*)n,thisequationtakestheformN=gTc*âÏ2â«0âΟdΟeΟâ1.
Thistableintegral68equalsÎ(2)ζ(2)âĄÏ2/6,sothat,finally,wegetN=gÏ26Tc*âÏ2,i.e.Tc*=âÏÏ6Ng1/2,
inagreementwiththeestimate(**).Thisexpressionshowsthatjustasinthesimilar3Dsystem(seeproblem3.25),ifthefrequencyÏ
of the effective 2D harmonic oscillator, formed by each particle in the quadratic potential, isnonvanishing, thecritical temperature isdifferent fromzeroaswell.This factdoesnotcontradictthe solution of the previous problem, because the free (uniform) 2D gas analyzed there may beconsideredastheultimatelimitofthesoftconfinementwithÏâ0andhencewithTc*â0.
Problem 3.28. Use Eqs. (3.115) and (3.120) of the lecture notes to calculate the third virialcoefficientC(T)forthehardballmodelofparticleinteractions.
Solution:AccordingtoEq.(3.120),C(T)=J22J12âJ33J1V2,
where,accordingtoEq.(3.115),
andthelettersrdenotethe interparticledisplacementvectorsshowninfigure3.6bof the lecturenotes(seealsothetwofiguresbelow).
For thehardballmodel, the integral J2 is contributed by only by the spatial region r<2r0, inwhichtwospheres,outliningthehardballs,overlap,andwasalreadycalculatedinthelecturenotesâseethederivationofEq.(3.96):
J2=â8V0V,whereV0=(4Ï/3)r03istheeffectivevolumeoftheparticle.Similarly,theintegralJ3 iscontributedonlybythoseregionsofthe6Dspaced3râČd3râłinwhichallthreespheresoverlap.(Indeed,ifjusttwoofthemoverlap,forexamplewhenrâČ<2r0,butrâł,râŽ>2r0,thenU(râČ,râł)=U(râČ)=â,whileU(râł)=U(râŽ)=0,sothat )Therearetwooptionshere:
(i)Twopairsofthethreespheresoverlap,butthethirdpairdoesnot,forexamplerâČ,râł<2r0,râŽ>2r0,
asshowninthefigurebelow.Inthisparticularregion,whose6DvolumewillbecalledWi,U(râČ,râł)=U(râČ)=U(râł)=â,whileU(râŽ)=0,sothat ,andtheregionâscontributiontoJ3equalsWi/V2.Sincetherearethreesimilarregionslikethis(differingbythechoiceofthepairofspheresthatdonotoverlap),theirtotalcontributiontoJ3is3Wi/V2.
(ii)Allthreespheresoverlap,forexampleasshowninthefigurebelow:râČ,râł,râŽ<2r0.
Here all potentials U are infinite, and , so that this 6D volume, Wii,contributestoJ3withcoefficient2.Asaresult,thetotalintegral
J3=3Wi+2WiiV2,andweneedonlytocalculatetwo6Dvolumes:WiandWii.
Firstofall,wecanexploitthesphericalsymmetryoftheproblemwithrespecttorotationofoneofthedisplacementvectors(say,râČ),providedthatthesecondvectorisrotatedtogetherwithit:WiâĄâ«râČ,râł<2r0râŽ>2r0d3râČd3râł=4Ïâ«02r0râČ2drâČVirâČ,withVi(râČ)âĄâ«râł<2r0râŽ>2r0d3râł,WiiâĄâ«râČ,râł,râŽ<2r0d3r
âČd3râł=4Ïâ«02r0râČ2drâČViirâČ,withVii(râČ)âĄâ«râł,râŽ<2r0d3râł,sothatinternaltheintegralsViandViimaybeworkedoutconsideringthedirectionofthevectorrâČfixedâsay,vertical.Thefigurebelowshowsthegeometricalsenseoftheseintegrals:Vi is just thevolumeshowngray,whileViiisitscomplementtothevolume8V0ofthesphereofradius2r0:Vi+Vii=8V0.
(*)
(**)
Thustheproblemisreducedtoabitofbulky,butelementarygeometry:asshowninthefigure
below,Vii/2isjustthevolumeofthesphericalsectorwiththepolarangleΞ0=cosâ1(râČ/4r0),whichmaybecalculatedasthedifferencebetweenthevolumesofthesphericalcone,
Vsphericalcone=(2r0)332Ïâ«0Ξ0sinΞdΞ=(2r0)332Ï1ârâČ4r0=4V01âΟ,
whereΟâĄrâČ/4r0=cosΞ0,andtheflat-baseconewiththesamepolarangleandtheheighth=râČ/2:Vflat-basecone=13Ah=13Ï(2r0)2ârâČ22râČ2=2V01âΟ2Ο.
Asaresult,wegetVii=2VsphericalconeâVflat-basecone=24V01âΟâ2V01âΟ2Ο=4V02â3Ο+Ο3,Vi=8V0âVii=4V03Ο
âΟ3,3Vi+2Vii=4V04+3ΟâΟ3.Nowwecancompletethecalculationofthe6DintegralJ3,
J3=1V23Wi+2Wii=1V24Ïâ«02r0râČ2drâČ3Vi+2Vii=1V24Ï4r03â«01/2Ο2dΟ16Ï3r034+3ΟâΟ3=1V2212Ï23r064â (1/2)33+3â (1/2)44â(1/2)66=1V2288Ï2r06=162V02V2,
andhencethethirdvirialcoefficientC(T)=J22J12âJ33J1V2=64â54V02=10V02.
Thisexpressionshowsthatinthehardballmodel,thiscoefficientis(quitenaturally)temperature-
independent69.Justforthereaderâsreference,thefourthvirialcoefficient(calculatedanalyticallybyLBoltzmann)isapproximately18.36V0
3,thefifthone(calculatedonlynumerically)iscloseto28.2V0
4,etc,withthenumericalfactorsbeforeV0kâ1growingratherslowlywiththecoefficientnumber
k.
Problem 3.29. Assuming thehardballmodel,with volumeV0 permolecule, for the liquid phase,describehow the resultsofproblem3.7 change if the liquid forms sphericaldropsof radiusRâ«V0
1/3.Brieflydiscusstheimplicationsoftheresultforwatercloudformation.
Hint: Surface effects in macroscopic volumes of liquids may be well described by attributing anadditionalenergyÎł(calledthesurfacetension)tounitsurfacearea70.
Solution: Inthe limitRâ«V01/3,whenthenumberNofmolecules ineachdrop is large, its radius
maybecalculateddisregardingthepeculiaritiesofamoleculeâsshape,fromtherelationN=4Ï3R3V0,givingR=3V04Ï1/3N1/3.
Inthesamelimit,thedropsurfaceareaisA=4ÏR2=4Ï3V04Ï2/3N2/3.
AddingonemoleculetothedropincreasesitsareabyÎAââAâN=4Ï3V04Ï2/323Nâ1/3;
plugging inN from the first of Eqs. (*), we get ÎA=2V0/Râagain, regardless of themoleculeâsshape.
Due to the surface increase, and the resulting increase of the surface energy ÎłA, the totaldifference of molecular energy between the liquid and gaseous phases is not just âÎ as wasassumedinproblem3.7(orasatRââ),butrather
âÎâČ=âÎ+ÎłÎA=âÎ+2ÎłV0R.
Nowrepeatingtheargumentsgiveninthemodelsolutionofproblem3.7,weseethatthesaturatedpressureP(T),calculatedthere,hastobemultipliedbyanadditionalfactor
ÎșR=exp2ÎłV0RT.
Thoughthisresult isquantitativelyvalidonly forRâ«V01/3,qualitatively itworksevenfor few-
moleculedroplets(âclustersâ),enablingasemi-quantitativediscussionofcloudformationdynamics.Asanairmass,withacertainconcentrationnofwatermolecules,risesupintheatmosphereandasa result cools down, the saturated pressure value P(T) as calculated in problem 3.7, which is anearly-exponentialfunctionoftemperature,dropsdowntotheactualpartialpressureofthewatervapor,stillbehavingalmostasanidealgas,PânT.However,sincethemasslacksliquidphase,inthe absence of other condensation centers, the water liquefaction cannot start. Only when thetemperaturedecreasestothelevelwhenthelargestproductÎș(R)P(T),oftheorderofÎș(V0
1/3)P(T),approachesnT,thefirstrandomdropletsform.Since,accordingtoEq.(**),theenergygainÎâČatthecondensationislargestforlargerdrops,theystarttoaccumulatemoremolecules,thusgrowinginsizeandsuppressingtheaveragepressureÎș(R)P(T),sothatsmallerdropletsstarttoevaporate.Intheabsenceofgravity, thiscompetitionwouldresult in the formationofonegigantic liquid âdropâwithnegligiblecurvature,butgravityforceslargerdrops,assoonastheyhavebeenformed,togodownintheformofrain.
References[1]LandauLandLifshitzL1980StatisticalPhysics,Part13rdedn(Pergamon)[2]BatyevE2009PhysicsâUspekhi521245
1See,e.g.thediscussionsofthisforceinPartCMchapters5and8.2NotethatwiththereplacementsâšNâ©âNandÎŒââšÎŒâ©(justifiedforNâ«1),thisexpressioncoincideswithEq.(3.32)ofthelecturenotes,whichwasderivedtherefromtheGibbsdistribution.3See,e.g.PartCMsection4.6.Notethattheother(Coriolis)inertialâforceâduetorotationwithconstantangularvelocity, ,isperpendiculartoparticleâsvelocityv,andhencecannotdoanyworkontheparticle(justbendsitstrajectory),sothatitdoesnotcontributetotheeffectivepotentialenergy.4NotethataccordingtoEq.(*),P(R)/P(0)=exp{mÏ2R2/2T},sothatatmÏ2R2âłT,evenaminordifferenceinthemassmoftheparticlemayleadtoaconsiderabledifferenceofthispressureratio.Thisisexactlytheeffectemployedfortheseparationofisotopes(inparticular,of238Ufrom235U,intheformofthehexafluoridegasUF6)incentrifugesusedinnuclearenrichmenttechnology.5Notethatintheinertiallabframe,thecentrifugalâforceâandtheassociatedeffectivepotentialU(r)donotexist,sothattheonlycontributiontothefullenergyoftheparticleisgivenbyitskineticenergy .6See,e.g.PartQMsection3.5,inparticularEq.(3.124).7Withthewatermoleculemassofm=3.00Ă10â26kg,thisvalueofÎisinagoodagreementwiththewaterâslatentheatofvaporization:Î(2.27MJkgâ1atambientconditions).8As evident from this expression,NV(T) has the physical meaning of the effective number of states available for occupation in the gaseous phase at the giventemperature.Similarnotionswillbeusedatthediscussionofelectronsandholeinsemiconductors,insection6.4ofthelecturenotes.9Ifindoubt,pleaseconsultPartQMsections1.7â1.8,inparticularEq.(1.99).Fortheparticularcaseofelectromagneticwaves,thisexpressionwasalreadyusedinthesolutionofproblem2.25.10SimilarlytoNV,theNAintroducedthiswayhasthephysicalsenseoftheeffectivenumberofthesurfacestatesavailableforoccupationattemperatureT.11Indeed,accordingtoEq.(**),theparameterÎșisoftheorderofV/Arc(Î)whererc(T)isthetemperature-dependentcorrelationlengththatwasdiscussedinsection3.2ofthelecturenotesâseeEq.(3.37).Asfollowsfromtheestimatesmadeduringthatdiscussion,rcismicroscopicevenatverylowtemperatures,whiletheratioV/Aisoftheorderofthelinearsizeofthecontainer,sothatforallâmacroscopicâ(human-scale)containers,Îșisextremelylarge,i.e.thelogarithminEq.(****)islarger,thoughnottoomuchlargerthan1.12Suchsmoothtransitionsarealsocommonforvirtuallyallsystemswithafinitenumberofparticlesâsee,forexample,thesolutionofproblem3.26below,andalsothediscussionofthisissueinsection4.5ofthelecturenotes.13Seethediscussionofthisnotioninthesolutionofpreviousproblem.14See,e.g.PartQMsections4.4â4.6.15Aswasalreadydiscussedinthesolutionofproblem2.4,themagnitude isequaltoÎłâ/2,whereandÎłisthegyromagneticratiooftheparticle.16See,e.g.PartEMEq.(5.111).Notethatsuchsimpletransferfrom to isonlyvalidatâŁÏmâŁâȘ1.17Thesedeviationsaredue,mostly,toelectronsâinteractionswiththecrystallatticeâsee,e.g.PartQMsections2.7and3.4.18Notethatthesimplemultiplicationof bytheÎNgivenbyEq.(**)wouldnotgivethecorrectfactorÂœ,whichreflectstheinducedcharacterofthemagnetization,proportionaltothegrowingfield.(Ifthisissueisnotabsolutelyclear,pleasereviewthederivationofEq.(1.60)inPartEMsection1.3.)19See,e.g.PartEMsections5.5and6.2.20See,e.g.PartQMsection3.2.21Itisfollowed,forexample,insection59of[1].22See,e.g.Eq.(A.15a).23TheseoscillationsarecloselyrelatedtotheShubnikovâdeHaaseffectâtheaccompanyingoscillationsoftheconductivity,anditsextreme2DformâthequantumHalleffect.24ThisintermediateresultalsodescribesthedeHaasâvanAlphenoscillations,whichatT=0areverysharp.25Tothebestofmyknowledge,thistrickwasinventedonlyrecentlyâsee[2].26See,e.g.Eq.(A.11b),withλ=1/4.27NotethatmodifyingEq.(*)toincludethecontributionofparticleâsspin(asisdone,e.g.inthesolutionofPartQMproblem5.44),itispossibletocalculatethenetmagneticresponsearisingfromthePauliparamagnetismandtheLandaudiamagnetisminoneshot.28See,e.g.PartQMsection2.8.29Sincethisproblem,andthenextone,areveryimportantforatomicphysics,andattheirsolutionthethermaleffectsmaybeignored,theyweregiveninchapter8ofPartQMoftheseriesaswell,forthebenefitofreaderswhowouldnottakethisSMcourse.Note,however,thatthesolutionofthesetwoproblemsisstreamlinedbyusingthenotionofthechemicalpotentialÎŒ,whichwasintroducedonlyinthiscourse.30See,e.g.PartQMEq.(1.13).31LetmehopethatthedifferencebetweenelectronâsenergyΔandtheelectrostaticconstantΔ0isabsolutelyclearfromthecontext.32ThereaderwhohasalreadyrunintothesolutionofthisprobleminPartQMofthisseries,mightnoticethatinthatcourse(whichinmysequenceprecedesthisone)Ihadtousesomeawkwardreasoningtomakethispointwithoutusingthenotionofthechemicalpotential.33TheapparentscaleoftemperaturesatthatthisassumptionbecomesinvalidisgivenbytheHartreeenergyEH=(me/â2)(e2/4ÏΔ0)
2â27.2eV(see,e.g.PartQMEq.(1.9)anditsdiscussion),correspondingtoTK=EH/kBâŒ3Ă105Kâaboutthousandtimeshigherthanthestandardroomtemperatureof300K.(Actually,thesolutionofthenextproblemshowsthattherealvaliditythresholdfortemperatureisevenâŒZ4/3â«1timeshigher.)34See,e.g.PartEMEq.(1.41).35See,e.g.Eq.(A.67)withâ/âΞ=â/âÏ=0.36Ausefulsanitycheckoftheself-consistencyoftheThomasâFermimodelisusingtheaboverelationstoprovethatthetotalnumberofelectrons,calculatedas
N=â«n(r)d3râĄ4Ïâ«0ânrr2dr,equalsexactlyZâasimpleexercise,highlyrecommendedtothereader.37Ofcourse,theremovedprotonalsointeracts(andverystrongly)withotherprotonsinthenucleus.However,ourgoalistocalculatetheelectronbindingenergy,i.e.differencebetweenthesumofenergiesoftheâassembledânucleusandindividualelectrons,farapartfromitandeachother,andthatofthewholeâassembledâatom.Atthecalculationofsuchenergy,thechangeoftheintrinsicenergyofthenucleushastobeignored.
38See,e.g.thesolutionofthepreviousproblem.39Veryunfortunately,thismodelissometimescalledâstatisticalââasifitscounterpart,theThomasâFermimodel,isnotstatistical.40Justasinthepreviousproblem,weconsidertheelectrongastobedegenerate,withTâȘÎŒ.41Evidently, in contrast to the ThomasâFermi model, this rudimentary model is not self-consistent, because it impliesÏ(ref) = Ze/4ÏΔ0ref â 0, while the exactelectrostaticpotentialofaneutral,spherically-symmetricalsystemofchargesshouldvanishatitseffectivesurface.42Tobefair,refisthelargestelectrondistancefromthenuclei,andtheaveragedistance,givenby
râĄ1Zâ«rn(r)d3r,isclosertorTF.(Agoodoptionalexercise:calculatethisdistance.)43Thisvalueislowerthantheenergyofelectronsinfreespacebyamaterial-dependentconstantcalledtheworkfunctionÏ,formostmetalsbetween4and5eVâsee,e.g.section6.3ofthelecturenotes,andalsoPartEMsection2.6andPartQMsection1.1.Note,however,thatÏdoesnotparticipatedirectlyinthesolutionofthisproblem.44See,e.g.PartEMEq.(1.41),andsection3.4.45Indeed,ingenuineelectrostatics,i.e.atnocurrentflowingintheconductor,theelectricfieldatthesurfacehastobenormaltoitâsee,e.g.PartEMsection2.1.46Actually,theproblemwassolved,ifonlysemi-quantitatively,inPartEMsection2.1.47Actually, a particular form of this relationwas already used for the analysis of the ThomasâFermimodel of heavy atoms in themodel solutions of problems3.12â3.13.48Thisis,forexample,anapproximatebutreasonablemodelforelectronsinwhitedwarfstars,whoseCoulombinteractionismostlycompensatedbythechargeofnucleioffullyionizedheliumatoms.49See,e.g.PartEMEq.(9.79).50See,e.g.PartQMEq.(1.82).51See,e.g.eitherthesolutionofthepreviousproblem,orPartEMEq.(9.79).52See,e.g.PartCMEq.(7.114).Aswasdiscussedinthemodelsolutionofproblem1.6,generallywehavetodistinguishtheisothermal(T=const)andadiabatic(S=const)compressibility,withtheformeronerelevantonlyatextremelylowfrequencies,butatTâ0, theentropyofan idealgas isconstant,so that thesenotionscoincide.53See,e.g.PartEMsection9.6,inparticular,Eq.(9.144).54Thisconditionmaybeapproachedreasonablywell,forexample,in2Delectrongasesformedinsemiconductorheterostructures(see,e.g.thediscussioninPartQMsection1.8,andthesolutionofproblem3.2ofthatcourse),duetotheelectronfieldâscompensationbybackgroundionizedatoms,anditsscreeningbyhighlydopedsemiconductorbulk.55See,e.g.Eq.(A.31a).56Asareminder,inthe3Dcase,thecoefficientinsuchrelationisdifferent,Ω=â(2/3)EâseeEq.(3.52).57AdiscussionofmechanicaleffectsofthesurfacetensioninliquidsmaybefoundinPartCMsection8.2.58See,e.g.Eq.(A.32a).59Essentially,ofitsuncondensedfractionofNâČparticles.60Ifneeded,see,e.g.PartQMsection1.7,inparticularEq.(1.86).61See,e.g.PartQMsection3.5,inparticularEq.(3.124)withd=3.62Thiscalculationissimilartotheoneinsection2.2ofthelecturenotesâseefigure2.3c,andalsoEq.(2.40)withN=3.63See,e.g.Eq.(A.35b)withs=3,andthenEqs.(A.10b)and(A.34e).64Inparticular,seeEqs.(3.72)and(3.73).65See,e.g.https://en.wikipedia.org/wiki/Bose-Einstein_condensate.66See,e.g.Eq.(A.31b).67Thisisessentiallythesamecalculationthathadbeendoneinsection2.2ofthelecturenotesâseefigure2.3b,andalsoEq.(2.40)withN=2.68See,e.g.Eq.(A.35b)withs=2,andthenEq.(A.10b).69Asareminder,soisthesecondvirialcoefficient,B(T)=â(J2/2J1)V=4V0âseeEq.(3.96)ofthelecturenotes.70See,e.g.PartCMsection8.2.
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IOPPublishing
StatisticalMechanicsProblemswithsolutionsKonstantinKLikharev
Chapter4
Phasetransitions
Problem4.1.ComparethethirdvirialcoefficientC(T)thatfollowsfromthevanderWaalsequation,with itsvalueforthehardballmodelofparticle interactions(whosecalculationwasthesubjectofproblem3.28),andcomment.
Solution: In thehardballapproximation(i.e.witha=0andb=4V0), thevanderWaalsequationreads
P=NTVâ4NV0.Expanding its right-handsideof thedimensionless ratioPV/NT into theTaylor series in the smalldimensionlessparameterNV0/V,weget
PVNT=VVâ4NV0âĄ11â4NV0/V=1+4V0NV+4V0NV2+âŠThus,whiletheresultB(T)=4V0forthesecondvirialcoefficientcomplieswiththatfollowingfromitsdirectcalculation,theresultforthethirstvirialcoefficient,C(T)=16V0
2,issignificantlydifferentfromtheexactone(10V0
2).ThisdifferenceemphasizesthephenomenologicalnatureofthevanderWaalsmodel.
Problem4.2.CalculatetheentropyandtheinternalenergyofthevanderWaalsgas,anddiscusstheresults.
Solution:Actingjustasfortheidealgas(seesection1.4ofthelecturenotes),forfreeenergywegetF=ââ«PdVT=const=ââ«NTVâNbâaN2V2dVT=const=âNTlnVâNbNâaN2V+Nf(T).
Inordertosimplifyfurthercalculations,wemayrecallthatata=b=0,thevanderWaalsmodeldescribestheidealclassicalgas,sothataddingandsubtractingNTln(V/N)to/fromtheright-handsideofEq.(*),wemayrecastitintothefollowingform:
F=NTlnVN+Nf(T)âNTlnVâNbVâaN2VâĄFidealâNTlnVâNbVâaN2V.Now,therestofcalculationissimple:
S=ââFâTV=Sideal+NlnVâNbV,E=F+TS=EidealâaN2V,whereSidealandEidealaregivenbyEqs.(1.46)and(1.47)ofthelecturenotes.
Notethattheconstantb(describingtheshort-rangerepulsionoftheparticles)doesnotgiveanycorrectiontotheinternalenergyofthegas,atfixedN,V,andT.Thiscouldbeexpected:weknowthattheinternalenergyofanidealgasdoesnotdependonvolume,sothatlosingsomenetvolume(Nb)tocollisionsdoesnotaffectiteither.
Problem4.3.Usetwodifferentapproachestocalculatetheso-calledJouleâThomsoncoefficient(âE/âV)T forthevanderWaalsgas,andthechangeoftemperatureofsuchagas,withatemperature-independentCV,atitsfastexpansion.
Solutions:Approach1istouseoneoftheresultsofthepreviousproblem:
E=EidealTâaN2V.Sincetheidealgasâenergydependsontemperatureonly(see,e.g.Eq.(1.47)ofthelecturenotes),thepartialderivativeofEoverV,atfixedT,iscontributedbythesecondtermalone:
âEâVT=aN2V2.
Accordingtothisresult,theJouleâThomsoncoefficientdoesnotdependontheconstantb,i.e.isnot contributed by hard-core interaction between the particles, and depends only on their long-range interaction characterized by parameter a. (As was discussed in section 3.5 of the lecturenotes,formostneutralatomicandmoleculargasesthiscoefficientispositive,duetotheattractivelong-rangeinteractionbetweentheparticles.)
NowletususeEq.(**)toanalyzetherapidgasexpansionintofreespaceâtheso-calledJouleâThomsonprocess. At such an expansion, the gas does not have time to exchange heat with theenvironment,andisgivennochancetoperformanymechanicalwork,sothatitsinternalenergyEhastostayconstant.ApplyingthisconditiontoEq.(*)writtenfortheinitial(index1)andthefinal(index2)pointsofgasexpansion,weget
Eideal(T1)âaN2V1=Eideal(T2)âaN2V2.
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IfCV is temperature-independent, thenEideal(T2) âEideal(T1) =CVÎT, where ÎT âĄT2 âT1 is thetemperaturechange,andweget
ÎT=âaN2CV1V1â1V2.Sinceforexpansion(V2>V1)theexpressionintheparenthesesispositive,foragaswithpositivea,thechangeoftemperatureisnegative.SuchJouleâThomsonexpansionisoneofthemainprocessesusedforgasliquefaction.
Approach 2. Let us start from the thermodynamic relation whose proof was the last task of
problem1.8:âEâVT=TâPâTVâP.
ForthevanderWaalequationofstate,P=NTVâNbâaN2V2,
itimmediatelygivesEq.(**)again.Proceedingtooursecondtask:sincetheenergyofanygas,withafixednumberNofparticles,is
uniquely determined by its temperature and volume, it may be described by a function E(V,T).Differentiatingthisfunctionoveritsindependentarguments,weget
dE=âEâTVdT+âEâVTdV.Bydefinition,thefirstofthesederivativesisjustCV,sothat
dE=CVdT+âEâVTdV.
AccordingtotheabovediscussionoftheJouleâThomsonexpansion,forthisprocesswemaytakedE=0,sothat,foragaswithconstantCV,theabovedifferentialrelationgives
dT=â1CVâEâVTdV,anditsintegralis
ÎTâĄT2âT1=â1CVâ«V1V2âEâVTdV.(Thekeyroleplayedbythepartialderivative(âE/âV)T in thisrelationexplainswhy it iscalledtheJouleâThomsoncoefficient.)NowusingEq.(**),andcarryingoutaneasyintegration,weget
ÎT=âaN2CV1V1â1V2,alsothesameresultasinthefirstapproach.
Problem4.4.CalculatethedifferenceCPâCVforthevanderWaalsgas,andcompareitwiththatforanidealclassicalgas.
Solution:WemayusethegeneralthermodynamicrelationCPâCV=âTâP/âTV2âP/âVT,
whose derivationwas the task of problem 1.9. Calculating the needed partial derivatives for theequationofstateofthevanderWaalsgas,
P=âaN2V2+NTVâNb,weget
âPâTV=NVâNb,âPâTV2=NVâNb2,âPâVT=2aN2V3âNTVâNb2,sothat,finally,
CPâCV=âTNVâNb2/2aN2V3âNTVâNb2âĄN1â2aNVâNb2/V3T.
SincethismodelcangivephysicallymeaningfulvaluesofPonlyforaâ©Ÿ0andV>Nb(see,e.g.figure4.1inthelecturenotes),thelastterminthedenominatorofthisresultcannotbenegative.HenceCP âCVâ©ŸN, with the equality reached only either at a vanishing long-range attractionbetweentheparticles(aâ0),orintheidealgaslimit(Tââ).
Problem4.5.Calculate the temperaturedependenceof thephase-equilibriumpressureP0(T) andthelatentheatÎ(T),forthevanderWaalsmodel,inthelow-temperaturelimitTâȘTc.
Solution:PluggingthevanderWaalsexpressionforP,P=NTVâNbâaN2V2,
into the Maxwell rule given by Eq. (4.11) of the lecture notes, performing the integration, anddividingby(V2âV1),wegetthefollowingrelation:
P0(T)=NTV2âV1lnV2âNbV1âNbâaN2V1V2.SinceP0(T)hastosatisfythesingle-phaseequationofstate(*)atpoints1and2(seefigure4.2ofthelecturenotes),Eqs.(*)and(**)giveusasystemoftwoequationsforfindingV1andV2:
NTV1,2âNbâaN2V1,22=NTV2âV1lnV2âNbV1âNbâaN2V1V2.
Unfortunately,forarbitrarytemperaturesthesetranscendentalequationsarehardtoanalyze,soletusproceed to the limitTâ0 specified in theassignment. In this limit, in order to satisfy theMaxwell ruleAu = Ad (see figure 4.2 again), P0(T) should tend to zero very fast. (See, e.g. thenumericalplotoftheequationofstateforT/Tc=0.5below;forreallylowvaluesofT/Tc,thetrendistoostrongtoshowitonsuchlinearscale.)
Asaresult,thegas-phasevolumeV2ismuchlargerthannotonlytheliquid-phasevolumeV1~
Nb,butalsotheunstable-equilibriumvolumeV0~aN/T~(Tc/T)Vc>Vc.(ThiscrudeestimateforV0maybeobtainedbyrequiringthatatV=V0,bothcontributionstoPinEq.(*)arecomparable;thenumerical plot above confirms this estimate for the particular value Tc/T = 2.) Hence for thedefinitionofV2wemaydropbothcorrectionstotheidealgaslawandtake
V2=NTP0(T).Also,thevolumeV1isclosetothedivergencepointNboftheright-handsideofEq.(*),sothatwecantakeV1=NbinallexpressionsbesidesthedifferenceV1âNb.ThisdifferencemaybeevaluatedfromthevanderWaalsequationwithP=0andV1=Nbinthea-term:
0=NTV1âNbâaN2(Nb)2,givingV1âNb=NTb2a.
Plugging these approximations forV1,2 into Eq. (**), and canceling common factors, we get asimpleequationforP0(T):
1=lnaP0b2âaNbT,whichyieldstheresultgivenbyEq.(4.12)ofthelecturenotes(there,withoutproof):
P0(T)=ab2expâaNbTâ1âĄ27Pcexpâ278TcTâ1âȘPc.(Asasanitycheck,thisresultgives
V2=NTP0(T)âŒTTcVcexp278TcTâ«V0,V1,thusjustifyingtheaboveassumption.)
NotethatthistemperaturedependenceofP0(T) isdominatedby theArrheniusexponent1,withthe activation energy Î = (27/8)Tc per molecule, in accordance with the physical picture ofevaporation as the thermal activation of the molecules from the condensed phase. However, itscomparisonwithEq.(*)ofthemodelsolutionofproblem3.7showsthatthe(phenomenological)vanderWaalsmodelfallsshortofdescribingthepre-exponentialfactorâT5/2,givenbythemicroscopicmodelofvapor/liquidequilibrium,whichisveryreasonableattemperaturesmuchbelowthecriticalpoint.
Now, using the ClapeyronâClausius law (4.17), for the latent heat of evaporation we get atemperature-independentvalue
Î(T)=T(V2âV1)dP0dTâTNTb2a278TcT2âĄ278NTcâĄNÎ,atT/Tcâ0,which is completely consistent with the activation picture of evaporation. Note that thisapproximationisonlyvalidifTâȘÎ,i.e.ifÎâ«NT.
Problem4.6.Performthesametasksas inthepreviousproblemintheopposite limitâinaclosevicinityofthecriticalpointTc.
Solution:AtTâTc(andTâ©œTc),thephaseequilibriumregionissmallandlocatednearcriticalpoint{Pc,Vc,Tc}âseeEq.(4.3)ofthelecturenotes.Letusintroducenormalizeddeviationsfromthepointasfollows:
PluggingtheserelationsintothenormalizedvanderWaalsequation(4.4),inthelimit â0weget
Notethatinthisexpression,onlythetermslinearin arekept,butboththelinearandcubicterms in are retained, because this is necessary to describe the non-monotonic shape of theisotherm, and hence the coexistence of the liquid and gaseous phases atT <Tcâsee the figurebelow.
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Eq. (*) shows that in this limit, the function is asymmetric: . According to theMaxwellrule(11),thismeansthatthephaseequilibriumlineP=P0(T)passesthroughthepoint =0(i.e.V=Vc),sothatEq.(*)yields:
pË0=4tËâ©œ0,i.e.P0(T)=Pcâ4Pc1âTTc<Pc.Becauseofthesameasymmetry,theendpointsV1,2ofthesegmentareontheequaldistancefromVc:
whereaccordingtoEq.(*),
andhence,
NowusingtheClapeyronâClausiusformula(4.17),forthelatentheatweget
Î(T)=T(V2âV1)dP0dT=Tc4VcTcâTTc1/24PcTcâĄ16PcVc1âTTc1/2.This expression may be simplified by noting that according to Eqs. (4.3), PcVc = (1/9)aN/b =(3/8)NTc,sothat
Î(T)=6NTc1âTTc1/2.(Thisapproximationisonlyvalidif0â©œTcâTâȘT,i.e.ifÎâȘNT.)
Ascouldbeexpected,thelatentheatdisappearsatT=Tc,becauseabovethistemperature,thesystemmaybeonlyinone(gaseous)phase.
Problem4.7.CalculatethecriticalvaluesPc,Vc,andTcfortheso-calledRedlichâKwongmodel oftherealgas,withthefollowingequationofstate2:
P+aVV+NbT1/2=NTVâNb,withconstantparametersaandb.
Hint:Bepreparedtosolveacubicequationwithparticular(numerical)coefficients.
Solution: Just as in the van derWaalsmodel case, theRedlichâKwong equation of state gives anexplicitexpressionofpressureasafunctionofvolumeandtemperature:
P=NTVâNbâaVV+NbT1/2âĄTb1Οâ1âaN2b2T1/21Ο1+Ο,wherethelastformusesthedimensionlessparameterΟâĄV/Nb,whichmakesthecalculationslessbulky.Indeed,thisformmakesthecriticalpointconditions,
âPâVT=0,â2PâV2T=0,atV=VcandT=Tc,easytospellout:
N2bTc3/2a1Οâ12â2Ο+1Ο21+Ο2=0,N2bTc3/2a1Οâ13â3Ο2+3Ο+1Ο31+Ο3=0.
Eliminatingtheleftmostfractionfromthesystemofthesetwoequations,wegetasimplecubicequationforthedimensionlesscriticalvolume:
Ο3â3Ο2â3Οâ1=0.As theplots in the figurebelow show, this equationhas just onepositive (i.e. physically-sensible)root3:
Οcâ3.84732,sothatVcâĄÎŸcNbâ3.84732Nb.(This value is to be comparedwithVc = 3Nb for the van derWaalsmodelâsee Eq. (4.3) of thelecturenotes.)
NowpluggingthisvalueintoanyofEqs.(**),wemaycalculatethecriticaltemperature:
Tc=ÏabN22/3,whereÏâĄ(Οcâ1)2(2Οc+1)Οc2(Οc+1)22/3â0.34504,
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whileEq.(*)yieldsthefollowingresultforcriticalpressure:Pc=ζa2b5N41/3,whereζâĄÏΟcâ1â1Οc(Οc+1)Ï1/2â0.029894.
(The results for Pc and Tc are functionally different from Eqs. (4.3) for the van der Waals gas,becauseofthedifferentdefinitionofthecoefficienta.)
Note that in literature, these results are frequently representedbackwards, as expressions forthecoefficientsaandbviathecriticalvaluesoftemperatureandpressure:
a=ζÏ5/2N2Tc5/2Pcâ0.42748N2Tc5/2Pc,b=ζÏTcPcâ0.086640TcPc.
Problem4.8.CalculatethecriticalvaluesPc,Vc,andTcforthephenomenologicalDietericimodel,withthefollowingequationofstate4:
P=NTVâbexpâaNTV,with constant parameters a and b. Compare the value of the dimensionless factorPcVc/NTc withthosegivenbythevanderWaalsandRedlichâKwongmodels.
Solution:Accordingtothediscussioninsection4.1ofthelecturenotes(see,e.g.figure4.1),atthecriticalpointthefollowingtwoconditionsshouldbefulfilled:
âPâVT=0,â2PâV2T=0,atV=VcandT=Tc.LetuscalculatethefirstandsecondderivativesoftheratioP/NT,sofarforarbitraryVandT:
1NTâPâVT=1VâbaNTV2â1Vâb2expâaNTV,1NTâ2PâV2T=1VâbaNTV2â1Vâb2aNTV2+âaNT3Vâ2bVâb2V3+2Vâb3ĂexpâaNTV.
Thesecondexpressionlooksbulky,butweneedtoevaluateitonlyatthecriticalpoint{Vc,Tc},
where the first derivative ofP, andhence the expression in the squarebrackets inEq. (*), equalzero:
1(Vcâb)aNTcV2â1(Vcâb)2=0,givingNTc=a(Vcâb)Vc2.HencethesimilarexpressioninthefirstsquarebracketsinEq.(**)hastoequalzeroaswell,sothattherequirementofhavingthesecondderivativeequaltozeroisreducedtotheconditionthatthesecondsquarebracketinthisexpressionalsovanishes:
âaNTc(3Vcâ2b)(Vcâb)2Vc3+2(Vcâb)3=0,givingNTc=a(3Vcâ2b)(Vcâb)2Vc3.NowrequiringbothobtainedexpressionsforNTctocoincide,andcancelingthecommonfactora(Vcâb)/Vc
2,wegetanelementaryequationforVc:
1=3Vcâ2b2Vc,givingVc=2b.Fromthisresult,andEq.(***),wemayreadilycalculatethecriticaltemperature,
NTc=a(Vcâb)Vc2=a4b,andnowthepressureatthecriticalpointmaybecalculatedfromtheequationofstate:
Pc=NTcVcâbexpâaNTcVc=a4b2eâ2â0.0338ab2.
Note that according to these results, the dimensionless combination PcVc/NTc, which may beconvenientlyusedtocharacterizethedeviationfromtheidealclassicalgas(forwhichPV/NT=1atanypoint),doesnotdependonthefittingparametersaandb,andisjustafixednumber:
PcVcNTcâŁDieterici=2eâ2â0.2707.ThesameistrueforthevanderWaalsmodel(seeEqs.(4.3)ofthelecturenotes),andtheRedlichâKwongmodel(seethepreviousproblem),buttherethenumbersaresomewhatdifferent:
PcVcNTcâŁvanderWaals=38=0.375,PcVcNTcâŁRedlich-Kwong=ΟcζÏâ0.333.
Forcomparison,theexperimentalvalueofthisparameterforwateriscloseto0.23,whileforthediethylether(seeitsdiscussioninsection4.1ofthelecturenotes)itiscloseto0.27.Thisdifferenceshows that it is impossible to design a two-parameter model (with a parameter-independentPcVc/NTc)thatwouldfitallrealfluidsveryclosely.
Problem 4.9. In the crude sketch shown in figure 4.3b of the lecture notes (partly reproducedbelow),thevaluesofderivativesdP/dTofthephasetransitionsliquid-gas(âvaporizationâ)andsolid-gas(âsublimationâ)atthetriplepointaredifferent,with
dPvdTT=Tt<dPsdTT=Tt.
Isthisoccasional?Whatrelationbetweenthesederivativescanbeobtainedfromthermodynamics?
Solution:Theby-product(4.16)oftheClapeyronâClausiusrelationâsderivation(insection4.1ofthelecturenotes)mayberewrittenas
dP0dT=S1âS2V1âV2,
(*)
(**)
wheretheindices1and2numberthephasesseparated,onthephasediagram,bythecriticallineP0(T).Formostmaterials,thegasphasevolume(ofafixednumberofparticles)ismuchlargerthanthoseintheliquidandsolidphases,atthesamepressure,sothatEq.(*),appliedtothevaporizationandsublimationtransitions,maybewellapproximatedas
dPvdTâSgasâSliquidVgas,dPsdTâSgasâSsolidVgas.But the solid phase is more ordered than the liquid phase, so that its entropy is lower, i.e. thedifference(SgasâSsolid)hastobelargerthanthedifference(SgasâSliquid),atthesametemperature.Sincetheonlytemperaturewhereboththesetransitionstakeplace isTt, therelationgivenintheassignmentisindeedvalidforallusualmaterialswithVgasâ«Vliquid,Vsolid.
However, for some materials, notably including the usual water H2O, the difference betweenthesederivativesisverysmall, implyinginparticularthatthewatericeisnotmuchmoreorderedthantheliquidwater.Thisisindeedthecase:nearthetriplepoint(273.16K,i.e.0.01°C),iceisamixture of 16 different âpacking geometriesââessentially, different phases, though one of them(calledIh)isprevalent.
ArelatedpeculiarfeatureofwateratTâTtisthattheiceisslightlylessdensethantheliquidwater(Vliquid<Vsolid),5sothatthesameEq.(*),appliedtotheâfusionâ(orâmeltingâ)transitionwaterâice,yields
dPfdT=SliquidâSsolidVliquidâVsolid<0.ThismeansthattheslopeofthecurvePf(T)forwaterisnegativeatthispoint,i.e.oppositetothatshowninthefigureaboveâwhichistypicalformostcommonmaterials.
Problem 4.10. Use the ClapeyronâClausius formula (4.17) to calculate the latent heat Î of theBoseâEinsteincondensation,andcompare theresultwith thatobtained in thesolutionofproblem3.21.
Solution: According to the discussion in section 3.4 of the lecture notes, an isothermof an ideal,uniform BoseâEinstein gas looks as sketched in the figure below, where the temperaturedependence of the critical volume Vc may be found from the critical transition condition (3.73),takingintoaccountEq.(3.71):
VcT=2Ï2Î3/2ζ3/2Ngâ2mT3/2.
This plot (to be compared with figure 4.2) shows that for this phase transition, the volume V1correspondingtothepureâliquidâ(condensed)phaseisformallyequaltozero,whileV2=Vc(T),sothattheClapeyronâClausiusformulaisreducedto
Î=TVc(T)dP0dT.
As the figureaboveshows, theequilibriumpressureP0(T)corresponds toVâ©œVc(T), i.e. to theBoseâEinsteincondensationregion,andhencemaybecalculatedusingEqs.(3.76)and(3.79):
P0T=ζ5/2ζ3/2NVTc3/2T5/2,sothatdP0dT=52ζ5/2ζ3/2NVTc3/2T3/2,where Tc should now be understood as a function of V. (It may be obtained from the criticalcondition(*)bymovingthesubscriptâcâfromVtoT.6)Theserelationsmaybeusedattheultimate,criticalpointV=Vcaswell,whereTc=T,sothatwefinallyget
Î=TVc(T)52ζ5/2ζ3/2NVcTT3/2T3/2âĄ52ζ5/2ζ3/2NTâ1.284NT.
Thisisexactlythesameresultaswasobtainedusingacompletelydifferentapproach(andevenasomewhatdifferentdefinitionofÎ) inproblem3.21.Thiscoincidenceshows that regardlessof its(meaningful) definition, the latent heat describes the same physics: Î/N is the average energynecessarytoexciteaparticle fromthecondensedphase intothegaseousphase. It isonlynaturalthat this energy tends to zero atTâ0, because in this limit the average energy of thegaseous-phaseparticlesalsotendstozero.
Problem4.11.
(i) Write the effective Hamiltonian for which the usual single-particle stationary SchrödingerequationcoincideswiththeGrossâPitaevskiiequation(4.58).(ii)UsethisGrossâPitaevskiiHamiltonian,withtheparticulartrappingpotentialU(r)=mÏ2r2/2,tocalculate the energy E of N â« 1 trapped particles, assuming the approximate solution Ï âexp{âr2/2r02},asafunctionoftheparameterr0.7(iii)ExplorethefunctionE(r0)forpositiveandnegativevaluesoftheconstantb,andinterprettheresults.(iv)Forsmallb<0,estimatethelargestnumberNofparticlesthatmayformametastableBoseâEinsteincondensate.
Solutions:
(i)TheonlydifferencebetweenEq.(4.58)andtheusual(linear)stationarySchrödingerequation,
(*)
(**)
ââ22mâ2Ï+UrÏ=ΔÏ,withΔ=aÏ,isthetermbâŁÏ2âŁaddedtoU(r),sothattheGrossâPitaevskiiHamiltonianis
HË=ââ22mâ2+bâŁÏâŁ2+Ur.Note that the GrossâPitaevskii Hamiltonian should be used with care, because thisphenomenological construct does not belong to the family of linear operators, for which thestandardformalismofquantummechanicsisstrictlyvalid.
(ii)Forthequadratic-parabolicpotentialspecifiedintheassignment,theHamiltonianbecomesHË=ââ22mâ2+mÏ2r22+bâŁÏâŁ2,
i.e. isdifferent fromtheusualHamiltonianof the3Dharmonicoscillatoronlyby the last term. InordertocalculatetheenergyEcorrespondingtotheassumed(âtrialâ)wavefunctionÏ(r),weneedtocalculatetheexpectationvalueofthecorrespondingoperator,i.e.oftheHamiltonian:
E=H=â«Ï*rHËÏrd3r=ââ22mâ«Ï*râ2Ïrd3r+mÏ22â«âŁÏrâŁ2r2d3r+bâ«âŁÏrâŁ4d3r,withthenormalizationcondition
N=â«Ï*rÏrd3râĄâ«âŁÏrâŁ2d3r.Forourfactorabletrialfunction,
Ïr=Cexpâr22r02âĄCexpâx22r02expây22r02expâz22r02,(wherethenormalizationconstantCmaybealwaystakenreal),andcoordinate-separableoperators,
â2=â2âx2+â2ây2+â2âz2,andr2=x2+y2+z2,perhapsthesimplestwaytoworkoutall the involved integrals is to factorthemintosimilar (andhenceequal)coordinateparts:
â«Ï*râ2Ïrd3r=3C2â«ââ+âexpâx22r02â2âx2expâx22r02dxĂâ«ââ+âexpây2r02dyâ«ââ+âexpâz2r02dzâĄ3C2r0â«ââ+âexpâΟ22â2âΟ2expâΟ22dΟâ«ââ+âexp{âΟ2}dΟ2=3C2r0â«ââ+âexp{âΟ2}
(Ο2â1)dΟâ«ââ+âexp{âΟ2}dΟ2=â3C2r0Ï3/22,â«âŁÏrâŁ2r2d3r=3C2â«ââ+âexpâx2r02x2dxâ«ââ+âexpây2r02dyâ«ââ+âexpâz2r02dzâĄ3C2r05â«
ââ+âexp{âΟ2}Ο2dΟâ«ââ+âexp{âΟ2}dΟ2=3C2r05Ï3/22,â«âŁÏrâŁ4d3r=C4â«ââ+âexpâ2x2r02dxâ«ââ+âexpâ2y2r02dyâ«ââ+âexpâ2z2r02dzâĄC4r03â«
ââ+âexp{â2Ο2}dΟ3=C4r03Ï3/223/2,â«âŁÏrâŁ2d3r=C2â«ââ+âexpâx2r02dxâ«ââ+âexpây2r02dyâ«ââ+âexpâz2r02dzâĄC2r03â«
ââ+âexp{âΟ2}dΟ3=C2r03Ï3/2.whereat the last stepsofall calculations, thewell-knowndimensionlessGaussian integrals8 havebeenused.Asaresult,weget
N=C2r03Ï3/2,givingC2=NÏ3/2r03,E=ââ22m
â3C2r0Ï3/22+mÏ223C2r05Ï3/22+bC4r03Ï3/223/2âĄN3â24mr02+3mÏ2r024+bN2Ï3/2r03.(iii)Iftheproductofthecoefficientb(characterizingtheinteractionbetweentheparticles)bythenumberNofparticlesisnegligible,thentheenergy(*)isjustthesumofNsimilarenergies,
Δ=3â24mr02+3mÏ2r024,of single particles placed into the quadratic-parabolic 3D potential well, i.e. just 3D harmonicoscillators9.Followingthevariationalmethodofquantummechanics10, theground-stateenergyofsuchanoscillatormaybefoundbyminimizingΔ(andhenceE=NΔ)overthefittingparameterr0:
âΔâr02âĄâ3â24mr04+3mÏ24=0,atr0=(r0)opt,giving
(r0)opt=âmÏ1/2,Δ=ΔgâĄ32âÏ.(This isararecase inwhichthevariationalmethodofquantummechanicsgivesexactresults forboththeground-stateenergy,andthecorrespondingwavefunction.)
Iftheconstantbispositive,describingaweakparticlerepulsion,andtheproductbNissmall,itdoesnot affect the functionE(r0), given byEq. (*), quantitativelyâsee the red lines in the figurebelow,whereλisthedimensionlessinteractionparameter
λâĄbmÏ2Ïâ3/2.
(Afurtherincreaseofband/orofthenumberNofparticlesincreasesthecriticaltemperatureTc*ofthecondensationfasterthanatb=0âseethesolutionofproblem3.25.ThereaderischallengedtocombinethesetwosolutionstocalculateTc*asafunctionofNforb>0.)
However,ifthecoefficientbisnegative,describingamutualattractionoftheparticles,thenforanyNâ«1theenergyEbecomesinfinitelynegativeatr0â0âseethebluelinesinthesamefigure.Thistrenddescribesapossiblefreeze-outoftheparticlesintoanâiceblobââtheessentiallyclassicaleffect, very different from the BoseâEinstein condensation, and in practice, preventing itsimplementationforsystemswithalargenumberofparticles.(TheGrossâPitaevskiiequationwithb<0,whichdoesnotdescribetheshort-distancerepulsionoftheparticles,isinsufficienttocalculatethesizeofthisâblobâ.)
(iv)The largestnumberNmaxofattractingparticles forwhich theBoseâEinsteincondensationstillmay be implemented (at T â 0) may be estimated by requiring the function E(r0) to retain aminimum corresponding to the condensation. For that, at N = Nmax this function must have ahorizontalinflectionpoint(r0)inf,where
âEâr0=0andâ2Eâr02=0âseethedashedbluecurveinthefigureabove.WithEq.(*),thesetwoconditionsbecome
â3â22mr03+3mÏ2r02â3bN2Ï3/2r04=0,9â22mr04+3mÏ22+12bN2Ï3/2r05=0,A bit counter-intuitively, this system of two nonlinear equationsmay be readily solved (say, by abrute-forceeliminationofthetermproportionaltobN),giving
(r0)inf=151/4(r0)optâ0.6687(r0)opt,Nmax=255/41âŁÎ»âŁâ0.26751âŁÎ»âŁ.
For typicalparametersofexperimentswithweaklyattractingatoms(ahistoricexample is7Li),thisresultyieldsNmax~103, inasemi-quantitativeagreementwithexperimentaldata.Noteagainthat even atN<Nmax, theminimum of the functionE(r0) at r0 â 0, in which the BoseâEinsteincondensatemay format sufficiently low temperatures, is local rather than global, so that such acondensate is always metastable, with a certain finite lifetime with respect to the âice blobâformation.
Problem4.12.Superconductivitymaybesuppressedbyasufficientlystrongmagneticfield.Inthesimplestcaseofabulk,longcylindricalsampleofatype-Isuperconductor,placedintoanexternalmagneticfield paralleltoitssurface,thissuppressiontakesthesimpleformofasimultaneoustransitionofthewholesamplefromthesuperconductingstatetotheânormalâ(non-superconducting)stateatacertainvalue of the fieldâsmagnitude.Thiscriticalfield gradually decreaseswithtemperaturefromitsmaximumvalue atTâ0tozeroatthecriticaltemperatureTc.Assumingthatthefunction isknown,calculatethelatentheatofthisphasetransitionasafunctionoftemperature,andspelloutitsvaluesatTâ0andT=Tc.
Hint:Inthiscontext,theâbulksampleâmeansasampleofsizemuchlargerthantheintrinsiclengthscalesof the superconductor (suchas theLondonpenetrationdepthÎŽL and the coherence lengthΟ)11.Forsuchbulksamplesoftype-Isuperconductors,magneticpropertiesofthesuperconductingphasemaybewelldescribed justastheperfectdiamagnetism,withthemagnetic inductioninsideit.
Solution:Inthissimplegeometry(only!),propertiesofthesamplecannotaffectthemagneticfieldoutsideit,sothatatanyexternalpoint,includingthoseclosetothesamplesurface, .Thisfield is parallel to the sample surface, and according to the basic electrodynamics12, should becontinuousattheborder.So,insidethesample, , i.e.doesnotdependonwhetherthe sample is superconducting or not. On the other hand, in the ideal-diamagnetismapproximation13,
(**)Since the field is definedby the relation ,where is themagnetization of themedium14,Eq.(*)mayberewrittenas
Aswasdiscussedinsection1.1ofthelecturenotes,theeffectoftheexternalmagneticfieldon
theenergyofamagneticmaterialissimilartotheexternalpressure.Inparticular,thecomparisonofEqs. (1.1)and(1.3a)showsthatwemaydescribetheeffectbyreplacing, inall formulasof thetraditional thermodynamics, the usual mechanical pair {âP, V} of the generalized force andcoordinatewiththesetofmagneticpairs foreachCartesiancomponent j=1,2,3(perunit volume). For our simple geometry, each of these vectors may be described with just onecomponent,paralleltotheappliedfield(andhencetothesampleâssurface),sothatthenecessaryreplacements are just , and , and the ClapeyronâClausius relation (4.17) for thelatentheatperunitvolumebecomes
where the indices ânâ and âsâ denote, respectively, the normal and superconducting phases. NowusingEq.(**),forthelatentheatperunitvolumewefinallyget15
Sincethecriticalfielddropswithtemperature, ,thisexpressionyieldspositivelatent
heat,asitshould.(Asuperconductorneedstobeheatedtomakeitnormal.)Inparticular,Eq.(***)showsthatthelatentheatvanishesbothatT=0andT=Tcâinthelattercasebecause .Thelastfactshowsthatintheabsenceoftheexternalmagneticfield,thethermal-inducedtransferfromthesuperconductingtothenormalstateatT=Tcmaybeconsideredasacontinuousphasetransitionâseesections4.2â4.3ofthelecturenotes.
Problem4.13.Insometextbooks,thediscussionofthermodynamicsofsuperconductivityisstartedwithdisplaying,asself-evident,thefollowingformula:
where Fs and Fn are the free energy values in the superconducting and non-superconducting(ânormalâ) phases, and is the critical value of the external magnetic field. Is this formulacorrect,andifnot,whatqualificationisnecessarytomakeitvalid?Assumethatallconditionsofthesimultaneous field-induced phase transition in the whole sample, spelled out in the previousproblem,aresatisfied.
Solution:Withthereplacements and ,discussedinsection1.1ofthelecturenotes(andinthepreviousproblem),theusualrelationG=F+PVbetweenthefreeenergyandtheGibbsenergy(perunitvolume)becomes16
Aswasdiscussed in themodel solutionof thepreviousproblem, in thebulkcylindricalgeometryinanyphase,while
sothatEq.(*)yields
Next,onthebasisofthesameanalogy ,repeatingallargumentsofsection1.4,wemay
concludethatthethermodynamicequilibriumofthesystem,atfixed andT,correspondstotheminimumoftheGibbsenergyâseeEq.(1.43)anditsdiscussion.Asthemagneticfieldreachesthecriticalvalue ,thedifferenceofGhastovanish,soEq.(**)yieldstheresult,
whichdiffersfromtherelationcitedintheassignmentbyafactorof2.However,letusconsidertherelationbetweenthefreeenergiesFn(0)andFs(0)ofthesephasesin
theabsenceofmagneticfield,atthesametemperatureT.Duetothefreeenergyâsdefinition(seeEq. (1.33)of the lecturenotes),F=EâTS, in thenormal phase it includes the sameadditionalmagneticfieldenergy ascontributestotheinternalenergyE:17
whileinthesuperconductingphase,with ,thereisnosuchadditionalcontribution:FsT=Fs0T.
Pluggingtheserelations, takenat , intoEq. (***),wesee thatone-halfof its right-handsidecancelswiththefieldenergyterm,andweget
(*)
(*)
Hencetherelationcitedintheproblemâsassignmentisvalidonlyforthefield-freevaluesofthe
freeenergy(whileforthevaluesinthefield,Eq.(***)isvalid).Fortunately,thisqualificationismadeinmorecompetenttexts.
Problem 4.14. In section 4.4 of the lecture notes, we have discussed theWeissâmolecular-fieldapproachtotheIsingmodel,inwhichtheaverageâšsjâ©playstheroleoftheorderparameterη.UsetheresultsofthatanalysistocalculatethecoefficientsaandbintheLandauexpansion(4.46)ofthefree energy. List the critical exponents α and ÎČ, defined by Eqs. (4.26) and (4.28), within thisapproach.
Solution:FortheWeissmolecular-fieldtheorywithh=0,Eq.(4.68)ofthelecturenotesreads:Z=2coshhmolTâĄ2cosh2JηdT.
Usingittoconstructthefreeenergy,wehavetoincludethebackgroundtermÎF=JdNη2(seethefirstterminEq.(4.64)ofthelecturenotes)aswell,becauseitalsodependsontheorderparameter.Fromhere,andthefundamentalEq.(2.63),thefreeenergyperparticleis
FN=âTlnZ+ÎFN=âTln2cosh2JdηT+Jdη2.Asasanitycheck,theequilibriumconditionâF/âη=0yieldsthefollowingequation,
η=tanh2JdηT,coincidingwithEq.(4.71),whichwasderiveddirectlyfromstatistics.
ExpandingEq. (*) intotheTaylorseries insmallη,andusingthevalueTc=2Jd of the criticaltemperature,givenbytheWeisstheory(seeEq.(4.72)ofthelecturenotes),weget
FN=âTln2+Tc21âTcTη2+T12TcT4η4+âŠ..ComparingthisresultwiththeLandauâsexpansion(4.46)attemperaturesclosetoTc,
ÎfâĄFηâF0V=âaÏη2+12bη4+âŠ,whereÏâĄ(TcâT)/TcâȘ1,weseethattheirleadingtermscoincideif
a=nTc2,b=nTc6,wheren=N/Visthevolumicdensityoftheâspinsâ.
Regardingthecriticalexponents,sincewehavebeenabletoreducethefreeenergy(*),atTâTc, to that inLandauâsmean-field theory (withh=0andâη=0),wemayuse the resultsof thecalculationscarriedoutforthismodelinsection4.3:
α=0,ÎČ=Âœ.
This calculation illustrates again that the Weissâ molecular-field approximation and Landauâstheorybelongtodifferentlevelsofphysicalphenomenology,andshowshowdangerousitistolabelthembothâmean-fieldtheoriesâ.(Unfortunately,inphysicsthistermisover-used,andoftenrequiresaqualification.)
Problem4.15.ConsideraringofN=3Isingâspinsâ(sk=±1),withsimilarferromagneticcouplingJbetweenallsites,inthermalequilibrium.
(i)Calculatetheorderparameterandthelow-fieldsusceptibilityÏ.(ii)Usethelow-temperaturelimitoftheresultforÏtopredicttheresultforaringwithanarbitraryN,andverifyitbyadirectcalculation(inthislimit).(iii)Discuss therelationbetween the last result, in the limitNââ,andEq. (4.91)of the lecturenotes.
Solutions:
(i)TheenergyofeachstateofthesystemmaybeexpressedbyEq.(4.78)ofthelecturenotes,withN=3:
Em=âJ(s1s2+s2s3+s3s1)âh(s1+s2+s3).Inoneof23=8possiblestatesofthesystem,inthatallspinsarealignedwiththefield,systemâsenergyislowestandequalto(â3Jâ3h).Inonemorestate,withallspinsdirectedagainstthefield,theenergyis(â3J+3h).Inthesixremainingstates,oneofthespinshasadirectionoppositetotheothertwo,sothatthenetcouplingenergyisâJ+2JâĄ+J,whiletheenergyofinteractionwiththefieldis±h,dependingontheorientationofthetwosimilarspins,withthreestatesineachgroup.Hencethesystemâsstatisticalsumis
ZâĄâmexpâEmT=expââ3Jâ3hT+expââ3J+3hT+3expâJ+hT+3expâJâhTâĄ2e3J/Tcosh3h/T+6eâJ/Tcoshh/T.
FromhereandthefundamentalEq.(2.63),thefreeenergyofthesystemisF=âTlnZ=âTln2e3J/Tcosh3h/T+6eâJ/Tcoshh/T.
NowusingEq.(4.90)ofthelecturenotes,wegetη=â1NâFâhT=T3ââhln2e3J/Tcosh3h/T+6eâJ/Tcoshh/T=e3J/Tsinh3h/T+e
âJ/Tsinhh/Te3J/Tcosh3h/T+3eâJ/Tcoshh/T.
Forlowfields,h/Tâ0,bothsinhfunctionstendtothevaluesoftheirarguments,whilebothcoshfunctionstendto1,sothat
ηâe3J/T3h/T+eâJ/Th/Te3J/T+3eâJ/TâĄhT3+eâ4J/T1+3eâ4J/T,i.e.Ï=1T3+eâ4J/T1+3eâ4J/T.
(**)
(***)
Notethatforthissystem(andforanysystemwithafiniteN,foranyreasonablephysicalmodel)
all average variables are continuous functions of temperature, so that we cannot speak about adefinite phase transition temperature, and even about the phase transition as suchâthis is a(useful!)abstractionstrictlyvalidonlyinthelimitNââ.
(ii)Inthelow-temperaturelimit(T/Jâ0),theexponentinthelastexpressionisnegligible,andtheresultisreducedto
Ï=3T,forTâȘJ.Comparing this resultwithEq. (4.77) forasinglespin,wemayguess that for the IsingringofNspins,
Ï=NT,forTâȘJ.Indeed,reviewingtheaboveexactcalculationforN=3,wemayseethatthelow-temperaturelimitcorrespondstoanegligibleeffectofthecontributionfromexp{â4J/T}inEq.(*).ButthisexponentisjusttheGibbsfactordescribingtheeffectofthethermalexcitationofthesystem,whoseenergy(in the vanishing field) is the additional energy of flipping one spin, i.e. increasing the couplingenergyofeachofitstwobondswiththeneighborsby2J,sothatÎE=4J.18Hence,Eq.(**)maybederivedbyignoringsuchexcitationsaltogether,i.e.takingintoaccountonlytwostatesofthesystem(oftheall2Npossible!):bothwithallspinsalignedâeitheralongthefieldoragainstit,withenergiesEâ=âN(J+h)andEâ=âN(Jâh):
ZâexpââNJ+hT+expââNJâhTâĄ2expNJTcoshNhT,atTâȘJ,sothat
lnZâNJT+ln2coshNhT.
But the field-dependentpartof thisexpression isabsolutely similar to thatof a single âspinâ19,withtheonlydifferencethatthefieldeffectisNtimeslarger,sothatinthelow-fieldlimit,
NhâȘTit immediately yields Eq. (**). This result should not be surprising, because firmly aligned spinsbehaveasasingleone,onlywiththemagneticmomentNtimeslarger.
(iii)InthelimitNââ,Eq.(**)yieldsÏââ, i.e.aresultdifferentfromthefinite(ifexponentiallylarge) value (4.91), given by the exact theory described in section 4.5 of the lecture notes. Thereasonforthisdiscrepancyisthatforsuchaninfinitesystem,thelimit(***)cannotbefollowed.Thisparadoxemphasizesagainthatthenotionofanâinfinitesystemâshouldbetakenwiththesamegrainof salt as that of the âphase transitionâ, especially in systems of low dimensionalityâsee also Eq.(4.93)anditsdiscussion.
Problem4.16.Calculatetheaverageenergy,entropy,andheatcapacityofathree-siteringofIsing-type âspinsâ (sk = ±1), with anti-ferromagnetic coupling (of magnitude J) between the sites, inthermalequilibriumattemperatureT,withnoexternalmagneticfield.Findtheasymptoticbehaviorofitsheatcapacityforlowandhightemperatures,andgiveaninterpretationoftheresults.
Solution: The internal energy of the systemmay be represented similarly to that in the previousproblem(seealsoEq.(4.78)ofthelecturenotes),butwiththeoppositesignofthecouplingenergy:
Em=J(s1s2+s2s3+s3s1),withJ>0.Intwoof23=8possiblestatesofthesystem,allspinsarealigned,andthesystemâsenergyequals+3J, while in all other 8 â 2 = 6 states, one of the spins has a direction opposite to its twocounterparts,sothattheenergyequalsJâ2JâĄâJ.Hencethesystemâsstatisticalsumis
Z=2expâ3JT+6expJT,andtheGibbsprobabilitiesofitsstatesare
W1,2=expâ3J/T2expâ3J/T+6expJ/T,andW3â8=expJ/T2expâ3J/T+6expJ/T.
Fromhere,wemayreadilycalculatethefreeenergy,F=âTlnZ=âTln2expâ3JT+6expJT,
theaverageinternalenergy20,E=3JĂ2W1,2+âJĂ6W3â8=âJ1âexp{â4J/T}1+(1/3)exp{â4J/T},
theentropy,S=EâFT=ln2expâ3JT+6expJTâJT1âexp{â4J/T}1+(1/3)exp{â4J/T},
andtheheatcapacity:CâĄâEâT=163JT21exp2J/T+(1/3)exp{â2J/T}2.
Atlowtemperatures,TâȘJ,thesecondexponentinthedenominatorvanishes,sothattheheat
capacityisexponentiallylow,Câ163JT2expâ4JT,atTâ0.
This result is natural, because the system stays mostly in one of its six lowest-energy, âalmost-antiferromagneticâstates(withenergyE=âJ),separated fromthetwohigher-energystates (withenergyE=+3J)bytheenergygapÎ=4J.
IntheoppositelimitTâ«J,bothfactorsexp{±2J/T}approach1,sothatCâ3JT2â0,atTââ.
Suchagradualdecreaseoftheheatcapacitywithtemperatureisalsonatural,becauseatsuchhightemperatures, all eight states of the systemare almost equally populated, and the remaining low
heatcapacityisduetothesmallremainingimbalanceofthesepopulations.
Problem4.17.Usingtheresultsdiscussedinsection4.5ofthelecturenotes,calculatetheaverageenergy, free energy, entropy, and heat capacity (all per spin) as functions of temperature T andexternal fieldh, for the infinite 1D Isingmodel. Sketch the temperature dependence of the heatcapacityforvariousvaluesofratioh/J,andgiveaphysicalinterpretationoftheresult.
Solution:FromEq.(4.88)ofthelecturenotes,rewrittenasZ=λ+N,withλ+=coshÎČh+sinh2ÎČh+expâ4ÎČJ1/2expÎČJ,
whereÎČâĄ1/T,wecanreadilycalculatethefreeenergyperspinsite:FN=âTNlnZ=âlnλ+ÎČ,
the(average)energypersite:EN=1NâlnZâ(âÎČ)=ââ(lnλ+)âÎČ,
andthenusethermodynamicrelationstocalculatetheentropyandtheheatcapacity(alsopersite):SN=FâENT=lnλ++ÎČâ(lnλ+)âÎČ,CN=1NâEâT=ÎČ2â2(lnλ+)âÎČ2.
Thefigurebelowshowsthelogâlogplotofthespecificheatasafunctionoftemperature,forseveralvaluesoftheratioh/J.Atnegligiblemagneticfield(hâȘJ,T),
λ+=2coshJT,FN=âTln2coshJT,EN=âJtanhJT,CN=J/Tcosh(J/T)2.This behavior of the heat capacity21 is absolutely similar to that of the usual two-level system(formallycorrespondingtothe0DIsingmodel)âsee,e.g.themodelsolutionofproblem2.2,withthesubstitutionÎ=2J.Thisfactmaybeinterpretedasthedominanteffect,onthethermodynamicsofthesystem,ofindependentlowest-energyexcitations,namelyoftheBlochwallswithenergyEW=2Jâseetheirdiscussioninsection4.5ofthelecturenotes.
IntheoppositelimitwhentheenergyhofthespininteractionwiththefieldismuchhigherthantheenergyJofcouplingbetweenthem,butstillmaybecomparablewiththethermalenergyscaleT,thegeneralformulaforλ+reducesto
λ+â2coshhT,forJâȘh,T,showingthatallcharacteristicsofthesystembecomedependentonlyontheh/Tratio.(ThistrendisclearlyvisibleinthefigureaboveasarigidhorizontalshiftoftheplotalongtheaxisofT/Jastheratioh/Jisincreasedwellabove1.)Moreover,thisexpressionforλ+,andhencethoseforallotherthermodynamicvariablesincludingtheheatcapacity,aresimilartothoseforthelow-fieldlimit,butwiththereplacementJâh.So,thetemperaturedependenceofCisagainsimilartothatobtainedinthe solution of problem 2.2, but now with the substitution Î = 2h. This behavior is readilyexplainable:atnegligiblecouplingJ,thesystemisjustasetofNindependentIsingâspinsâ(i.e.two-levelsystems),withtheenergydifferenceÎ=2hbetweentheirpossibleorientationsinthefield.
Asthefigureaboveshows,in-betweenthesetwolimits,i.e.athâŒJ,thetemperaturedependenceofthespecificheatisquantitativelydifferent,butqualitativelysimilar:CvanishesbothatTâ0andatTââ,withamaximumCmaxâŒNatTâŒh.
Problem4.18.Usethemolecular-fieldtheorytocalculatethecriticaltemperatureandthelow-fieldsusceptibility of a d-dimensional cubic lattice of spins, described by the so-called classicalHeisenbergmodel22:
Em=âJâk,kâČskâ skâČââkhâ sk.Here, in contrast to the (otherwise, very similar) Ising model (4.23), the spin of each site isdescribedbyaclassical3Dvectorsk={sxk,syk,szk}oftheunitlength:sk2=1.
Solution: Let us align the z-axis with the direction of the externalmagnetic fieldh, and use theshorthandszkâĄsk;thentheenergyofthemthstatemayberewrittenas
Em=âJâk,kâČ(skskâČ+sxksxkâČ+syksykâČ)âhâksk.
In themolecular-field theory,eachCartesiancomponentof thespinshouldberepresented ina
(*)
(**)
formsimilartoEq.(4.62)ofthelecturenotes.However,duetothesymmetryoftheproblemwithrespecttoreflectionsxââxandyâây,theaverageâšsâ©âĄÎ·maybedifferentfromzeroonlyforthez-component.Hence the two last terms in theparentheses includeonly thesquaresof fluctuationterms:
Em=âJâk,kâČ(η+sËk)(η+sËkâČ)âJâk,kâČ(sËxksËxkâČ+sËyksËykâČ)âhâksk,withâŁsËâŁâȘ1.Multiplying the parentheses under the first sum, and neglecting all the terms quadratic in smallfluctuations,(seethediscussionoftransferfromEq.(4.63)toEq.(4.64)ofthelecturenotes),wegetanexpressionformallysimilartoEq.(4.64)fortheIsingmodel,
EmâNJdη2âhefâksk,describing a set of N independent classical âspinsâ sk placed into the effective (external +âmolecularâ)field(4.65):
hefâĄh+2Jdη.However,incontrastwiththeIsingmodel,nowskmaytakeanyvaluesfromâ1to+1.
Thissituationwasoneofthesubjectsofproblem2.4,andwecanuseitssolution.Inourcurrentnotation,itsresultfortheorderparameterreads
ηâĄs=cothhefTâThef.Thisfunctionisqualitatively,butnotquantitativelysimilartoEq.(4.69),η=tanh(hef/T),fortheIsingmodel;mostimportantly,ithasathree-foldlowerslopeattheorigin:
âηâhefâŁhef=0=13T.Thisdifference immediatelymapsonthephasetransitiontemperatureTc. Indeed,combiningEqs.(*)and(**),andlinearizingtheresultinsmallh,hef,andη(exactlyaswasdoneinsection4.4ofthelecturenotesfortheIsingmodel),forthelow-fieldsusceptibilitywegetthesameCurieâWeisslaw(4.76),
ÏâĄâηâhâŁh=0=1TâTc,butwithathreetimeslowercriticaltemperature:
Tc=2Jd3.
Thisreductionisanaturalresultofthespin-to-fieldinteractionweakeningduetotheavailabilityofintermediatevalues,â1<sk<+1,forthefield-alignedspincomponents.Inturn,thisavailabilityistheimmediateresultoftakingintoaccounttwootherCartesiancomponentsofthevectorsintheHeisenbergmodel.
1Again,itisamazinghowwellisthisexponentiallawhiddeninsidethevanderWaalsequationofstate!2Thisequationofstate,suggestedin1949,describesmostrealgasesbetterthannotonlytheoriginalvanderWaalsmodel,butalsoothertwo-parameteralternatives,suchastheBerthelot,modified-Berthelot,andDietericimodels,thoughsomeapproximationswithmorefittingparameters(suchastheSoaveâRedlichâKwongmodel)workevenbetter.3Actually,using thebulky(âquasi-anayticalâ)TartagliaâCardanoformulas, it ispossible toshowthat thisvalue is just (2â +2â +1).However, forparameter-freeequations,thedemonstratedmethodofnumericalsolutionisfaster,moregeneral,andperfectlysuitableformostapplications.4ThismodeliscurrentlylesspopularthantheRedlichâKwongmodel(alsowithtwofittingparameters),whoseanalysiswasthetaskofthepreviousproblem.5Thisisofcourseveryfortunate:ificewasmoredensethanwater,thenmostlakes,rivers,andevenseaswouldfreezetothebottominwinters,andlifeontheEarthmightbepossibleonlyinthetropics.6Since,accordingtothatformula,Tc
3/2â1/V,thefunctionP0(T)givenbythefirstofEqs.(**)actuallydoesnotdependofVâasatshouldbeâseethefigureabove.7Thistaskisessentiallythefirststepofthevariationalmethodofquantummechanicsâsee,e.g.PartQMsection2.9.8See,e.g.Eqs.(A.36b)and(A.36c).9See,e.g.PartQMsection3.6.10See,e.g.PartQMsection2.9.11Adiscussionoftheseparameters,aswellasofthedifferencebetweenthetype-Iandtype-IIsuperconductivity,maybefoundinPartEMsections6.4â6.5.However,thosedetailsarenotneededforthesolutionofthisproblem.12See,e.g.PartEMEq.(5.117).13Noteadifferentdescriptionoftheelectrodynamicsofsuperconductors,inwhichthesurfacesupercurrentsaredescribedexplicitlyâsee,e.g.PartEM section6.4.However,forbulk,type-Isuperconductorsthisalternative(andmuchmoreinvolved)analysisyieldsthesamefinalresultfor .14Ifyouneedareminder,youmayhavealookatPartEMEq.(5.108)anditsdiscussion.15NotethataccordingtoEq.(4.14)ofthelecturenotes,thefactorafterTalsogivesthedifferenceofentropiesinthesuperconductingstateandthenormalstate.16Noteagainthat(aswasdiscussedinsection1.4ofthelecturenotes),thisrelationisonlytrueiftheeffectofthefield isnotincludedintheenergyofeachparticleofthemedium,asisdone,forexample,intheIsing-typeproblemsâsee,e.g.thelasttermsinEqs.(4.21)and(4.23).InthelattercasethereisnodifferencebetweenthethermodynamicpotentialsGandFâunlesstheusualpressureP(oranyothergeneralizedforcebut )interferes.17Iamsurethereaderknowsthisformulaâbutifnot,pleaseseePartEMsection5.3and6.2,inparticularEq.(5.57).18NotethatforanopenIsingchain,thelowestexcitationhasthetwicelowerenergyâseefigure4.11anditsdiscussioninsection4.5.However,thisdifferencedoesnotaffectthevalidityofEq.(**).19See,e.g.Eq.(4.68)ofthelecturenotes,fortheparticularcasehef=h.20Aswasrepeatedlydiscussedinthiscourse,analternativewaytocalculateE,whichdoesnotrequirethepreliminarycalculationofthestatesâprobabilities,istouseEq.(2.61b)ofthelecturenotes:E=ââ(lnZ)/âÎČ.21TheformulasforF/NandE/Nare functionallydifferent fromthose in thesolutionofproblem2.2onlybecauseof theshiftof their reference levelbyâJ. (Thistemperature-independentshiftdoesnotaffecttheheatcapacityCâĄâE/âT.)22Thisclassicalmodelisformallysimilartothegeneralizationofthegenuine(quantum)Heisenbergmodel(21)toarbitraryspins,andservesasitsinfinite-spinlimit.
IOPPublishing
StatisticalMechanicsProblemswithsolutionsKonstantinKLikharev
Chapter5
Fluctuations
Problem 5.1. Treating the first 30 digits of number Ï = 3.1415⊠as a statistical ensemble ofintegersk(equalto3,1,4,1,5,âŠ),calculatetheaverageâškâ©,andthermsfluctuationÎŽk.Comparethe results with those for the ensemble of completely random decimal integers 0, 1, 2,âŠ,9, andcomment.
Solution:Thehigh-precisionvalueofÏmaybeeither foundonmanyWeb sites, e.g. inWikipedia(http://en.wikipedia.org/wiki/Pi),orgeneratedbyanycalculator(say,thatonyoursmartphone)âforexample,as4·tanâ1(1).Forourpurposes,weneedjustM=30firstdigits:
Ï=3.14159265358979323846264338327âŠFromhere, the calculation can be done eitherwith a computer, bywriting and running a simplescript, using the number rounding routine readily available in virtually all numerical libraries toreformatthenumberintotheintegerarraykm=3,1,4,âŠ,orjustbybruteforce,usingacalculator.For30digits,therequiredtimeiscomparable(andinsignificant:-).
Theresultsare:kâĄ1Mâm=1Mkmâ4.70;
âškË2â©âĄ1Mâm=1MkËm2âĄ1Mâm=1Mkmâk2=1Mâm=1Mkm2âk2â6.08,sothat
ÎŽkâĄâškË2â©1/2â2.47.
Foraninfinitesetofrandomdecimalintegers(withequalprobabilitiesWn=1/NtotakeanyofN=10possiblevalueskn=nâČâĄnâ1=0,1,2,âŠ9),theexpectationvalueofknis1
k=ân=1NknWn=1Nân=1Nkn=1NânâČ=0Nâ1nâČ=1N(Nâ1)N2âĄNâ12=4.5.Usingthisresult,weget2
âškË2â©=1Nân=1Nkn2âk2=1Nâk=0Nâ1k2âk2=1NNâ1N2Nâ16âNâ122âĄN2â112=8.25,sothat
ÎŽkâĄâškË2â©1/2=12N2â131/2â2.87,i.e.virtuallythesameresultsasforthedigitsofnumberÏ.
A good sanity check here is that the difference between the calculated averages of these two(verysimilar)statisticalensemblesismuchsmallerthantheirrmsuncertainties.Itisalsoimportantto understand that the relative statistical uncertainty of the random integer set would benonvanishing,andsubstantial,eveniftheirnumber(N)wasinfinite:
ÎŽkk=N+13Nâ11/2âNââ=13â0.577.
Problem 5.2. Calculate the variance of fluctuations of a magnetic moment placed into anexternalmagneticfield ,withinthesametwomodelsasinproblem2.4:3
(i)aspin-ÂœwithagyromagneticratioÎł,and(ii)aclassicalmagneticmoment ,ofafixedmagnitude ,butwithfreeorientation,
bothinthermalequilibriumattemperatureT.Discussandcomparetheresults.
Hint:MindallthreeCartesiancomponentsofthevector .
Solutions:
(i)Aswasdiscussedinthemodelsolutionofproblems2.2â2.4,inthiscasethestationaryvaluesofthe magnetic moment are , where and nz is the fieldâs direction, and thecorrespondingeigenenergiesare
sothattheirprobabilitiesW±inthecanonicalensembleare
W±=1ZexpâhT=expâh/Texp+h/T+expâh/T,withW++Wâ=1,andtheaveragemagneticmomentisdirectedalongthefield: ,with
Theaveragesquareof maybecalculatedsimilarly:
NowwecanusethegeneralEq.(5.4b)of the lecturenotes tocalculate thevarianceofmomentâsfluctuations:
Note that this resultmay be also obtained in a differentway, usingEq. (5.37a) of the lecture
notes, and the analogy between two canonical pairs of variables: {âP, V} for a system undermechanicalpressure,and forasinglemagneticmomentâorratheritscomponentinthedirectionofthefield4.ThisanalogyimmediatelyyieldsEq.(*)again:
Inthelimitofrelativelyhightemperatures,i.e.relativelylowfields(hâȘT),thedenominatorof
thisexpressiontends to1,so that themomentâsvariance is largest,approaching ,while in theopposite,low-temperature(high-field)limit,thefluctuationsareexponentiallysmall:
ThisisanaturalresultforasystemwiththeenergygapÎ=2h,separatingthegroundstateofthesystemfromits(only)excitedstate.
Now let us consider two other components of themagneticmoment. Their averages evidentlyequalzeroduetotheaxialsymmetryofthesystem:
andthesamesymmetrymaybeusedtowrite ,sothatsince ,
Here we should avoid the error of taking equal to . Indeed, in quantum mechanics theexpectationvalueof the spinâs square is âšS2â©=â2s(s+1), so that for spin-Âœ,with s=Âœ, âšS2â©=(3/4)â2. (Since this isequality isvalid foranyquantumstateof thesystem, it isalsovalid for theaverageoveranystatisticalensembleaswell,includingthatinthermodynamicequilibrium.)Hence,for ,wehave
sothat
âthesameresultasfor intheabsenceofthefield.So, the fluctuations of the lateral components of the magnetic moment are temperature-
independent,andphysicallyarecausedjustbytheirquantumuncertainty.(WemaysaythatevenatT=0thesefluctuationsareultimatelylarge,sothatthermalexcitationsofthespinatT>0cannotincreasethem.)
(ii)Aswasdiscussedinthemodelsolutionofproblem2.4,inthismodeltheprobabilitydistributioniscontinuous,withtheangulardensity
(withthesamenotation, ,forthenormalizedfield),givingthefollowingaverages:
Wemaynowcalculate theaveragesquaresofallCartesiancomponentsof thevector inthe
sameway.However, due to the axial symmetry of theproblem, those of thex- andy-componentshavetobeequal,sothatitisconvenienttocalculatethembothinoneshot:
Theseintegralsmaybeworkedoutexactlyasinthesolutionofproblem2.4,byintroducingthenewvariableÎŸâĄ (h/T)cosΞ, so that cosΞ= (T/h)Ο, sinΞdΞâĄâd(cosΞ)=â(T/h)dΟ, and sin2Ξ=1â(T/h)2Ο2:
â«0ÏexphcosΞTsinΞdΞ=âThâ«âh/T+h/TeΟdΟ=âTh[eΟ]âh/T+h/T=â2ThsinhhT,â«0Ïsin2ΞexphcosΞTsinΞdΞ=âThâ«âh/T+h/T1âTh2Ο2eΟdΟâĄâThâ«âh/T+h/TeΟdΟ+Th3â«
âh/T+h/TΟ2eΟdΟ.The firstof these two integrals isexactly thesameaswas (easily :-)workedoutabove,while thesecondonerequirestwosequentialintegrationsbyparts,giving
â«âh/T+h/TΟ2eΟdΟ=(Ο2â2Ο+2)eΟâh/T+h/T=2hT2+2sinhhTâ4hTcoshhT.Asaresult,weget
Finally,sincethemagnitude ofthevector isthesameinallitspossiblestates,wemaywrite
so that the calculation of the square of the remaining, field-aligned component of the magneticmomentmaybeperformedverysimply:
andthegeneralrelation(5.4b)yields
Aswasalreadydiscussedinthemodelsolutionofproblem2.4,inthehigh-temperature(low-field)
limithâȘT,theexpressionintheparenthesesparticipatinginEqs.(**)and(***)approachesh/3TâȘ1,sothatourresultsarereducedtoaverysimple,field-independentexpression
Thisisverynatural,becauseintheabsenceofthefield,thesystemisfullyisotropic.However, in the opposite, low-temperature (high-field) limit, when the same expression in the
parenthesestendsto1âT/h,Eqs.(**)and(***)giveverydifferentresultsforthefield-alignedandfield-normalcomponents:
Thephysicalreasonforthisdifferenceisthatsmalldeviationsofthemomentvectorfromthefield-aligneddirectionnzgiveacontributiontotheenergy,whichisquadraticin and :
sothattheirvarianceshaveto(anddo!)satisfytheequipartitiontheorem(2.48):
On the other hand, the deviations of the z-component of the momentum from its valuecorrespondingtotheexactalignment,inthislimitaremuchsmaller:
asaresult,theirvarianceisofahigherorderinthesmallparameterT/h.Nowcomparingtheresultsforthetwomodelsofaspininmagneticfield,weseethattheyare
rather different. Most significantly, in the low-temperature limit, the classical Heisenberg model(which isasymptoticallycorrect for largevaluesofspin,sâ«1)doesnotexhibit theexponentiallysmallfluctuationsof ,whichare typical fors=Âœ (andany finitevalueof spin),becauseof the
(*)
availabilityofintermediatestateswith0<Ξ<Ï,fillingthegapbetweentwoextremevalues(E=±h)oftheenergy.
Problem 5.3. For a field-free, two-site Ising systemwith energy valuesEm = âJs1s2, in thermalequilibrium at temperature T, calculate the variance of energy fluctuations. Explore the low-temperatureandhigh-temperaturelimitsoftheresult.
Solution:Thissystemhastwodoubly-degeneratevaluesofitsenergyE:Em=âJ,fors1=s2,+J,fors1=âs2.
HenceitsstatisticalsumisZ=2expJT+2expâJTâĄ4coshJT,
theaverageofthesystemâsenergyisE=1ZâmEmexpâEmT=14cosh(J/T)2JexpâJTâ2JexpJTâĄâJtanhJT,
andthatofE2isâšE2â©=1ZâmEm2expâEmT=14cosh(J/T)2J2expâJT+2J2expJTâĄJ2,
sothattheenergyfluctuationvarianceâšEË2â©=âšE2â©âE2=J21âtanh2JTâĄJ2cosh2(J/T).
Alternatively,thesameresultmaybeobtainedusingEq.(5.19)ofthelecturenotes,asââšEâ©/â(âÎČ).At low temperatures, TâȘ J, the function cosh(J/T) is very large, so that the fluctuations are
exponentiallysmall,duetothegap2Jintheenergyspectrumofthesystem.Ontheotherhand,inthe opposite limit of high temperatures, cosh(J/T) â 1, and the variance approaches thetemperature-independentvalueJ2.
Problem 5.4. For a uniform three-site Ising ring with ferromagnetic coupling (and no externalfield),calculatethecorrelationcoefficientsKsâĄâšskskâČâ©forbothk=kâČandkâ kâČ.
Solution:InallIsingmodels,eachâspinâskmaytakeonlythevalues±1,sothatitssquareequals1in any state of the system, and hence âšsk2â© equals 1 for any parameters. However, the mutualcorrelationcoefficientsâšskskâČâ©,5withkâČâ k,requirecalculation.
Aswasalreadydiscussed in themodelsolutionofproblem4.15, theenergyofaringâsstate isgivenbyEq.(4.78)ofthelecturenotes(withN=3andh=0):
Em=âJ(s1s2+s2s3+s3s1),withJ>0.Thesystemevidentlyhas23=8differentstates.Intwoofthemtheâspinsâareallaligned(ineitheroftwopossibledirections),sothatforeachofthemEm=â3J.Inallremainingsixstates,onespinhas a direction different from its two counterparts, so thatEm = âJ(+1 â 1 â 1) = +J. Hence,systemâsstatisticalsumis
Z=2exp3JT+6expâJT.sothattheprobabilitiesofhavingsome(notparticular)statesofthesetwokindsareequalto
W1=2exp3J/T2exp3J/T+6expâJ/T,W2=6expâJ/T2exp3J/T+6expâJ/T,withW1+W2=1.
SincetheproductskskâČ(withkâ kâČ)maytakeonlytwovalues:(+1)whenthespinskandkâČarealignedwitheachother,and(â1)otherwise,wemaywrite
KsâĄâšskskâČâ©=(+1)W++(â1)Wâ=W+âWâ,whereW±arethecorrespondingprobabilities,withW++Wâ=1.Ifthespinsareallaligned(State1 in the abovenomenclature), thenall skskâČ=1.However, if one of them ismisaligned (State2),thereisonly1/3chancethatanygivenpairofspinskandkâČisaligned(becausethisisthechancethatthespinnumberkâłâ k,kâČ,notinvolvedinthispair,ismisaligned).Hence,W+=W1+(1/3)W2,whileWâ=1âW+=(2/3)W2,andfinally:
Ks=W+âWâ=W1+13W2â23W2=1âexpâ4J/T1+3expâ4J/T,foranykâČâ k.
This result shows that in the low temperature limit, TâȘ J, themutual correlation coefficientapproaches1.This isnatural,becausetheprobabilityW1of the fullspinalignmentapproaches1,whileW2 isexponentiallysmall. Intheopposite limitofhightemperatures,exp{â4JT}â1â4JT,andthemutualcorrelationislow:
KsâJTâȘ1,forkâČâ k.Note, however, that âšskskâČâ© is positive for any ratio J/T. This is natural, because the ferromagneticcoupling,withJ>0,alwaysfavorsspinalignment.
Problem 5.5.* For a field-free 1D Ising system of N â« 1 âspinsâ, in thermal equilibrium attemperatureT,calculatethecorrelationcoefficientKsâĄâšslsl+nâ©,whereland(l+n)arethenumbersoftwospecificspinsinthechain.
Hint:Youmayliketostartwiththecalculationofthestatisticalsumforanopen-endedchainwitharbitraryN>1andarbitrarycouplingcoefficientsJk,andthenconsideritsmixedpartialderivativeoverapartoftheseparameters.
Solution:Foranopen-endedchainoutsideofexternal field(h=0),witharbitraryJk,Eq. (4.23)ofthelecturenotesmayberewrittenas
Em=ââk=1Nâ1Jksksk+1.Thestatisticalsumofsuchachainis
ZN=âmexpâEmT=âsk=±1,fork=1,2,âŠ,Nexpâk=1Nâ1JkTsksk+1âĄâsk=±1,fork=1,2,âŠ,Nâk=1N
(**)
(***)
(****)
â1expJkTsksk+1.Letusassumethatwealreadyknowthissum(of2N terms),anduse it tocalculate thestatisticalsumZN+1(of2N+1terms)forasimilarchainofZN+1spins.TheenergyofthenewsystemdiffersfromthatoftheoldoneonlybytheadditionoftheenergyÎ=âJNsNsN+1ofthenewcouplinglink.Foranyfixedsetofâoldâspins{s1,s2,âŠ,sN},theadditionalenergymaytakeonlytwovalues,±JN,thusreplacingeachtermofthesum(*)withtwoterms,withadditionalfactorsexp{âÎ/T}=exp{±JN/T}.Asaresult,weget
ZN+1=âsk=±1,fork=1,2,âŠ,N+1expâk=1Nâ1JkTsksk+1expâJNT+exp+JNTâĄZNĂ2coshJNT.
Nowwemayapplythissimplerecurrencerelationsequentially,startingfromtheeasyparticularcaseN=2,i.e.forthesystemwithjustonelink,whosestatisticalsumhasonlyfourterms,withtwoequalpairs:
Z2=âsk=±1,fork=1,2expJ1Ts1s2=2exp+J1T+2expâJ1TâĄ4coshJ1T.TheresultforanarbitraryNâ©Ÿ2is
ZN=4coshJ1TĂ2coshJ2TĂâŠĂ2coshJNâ1TâĄ2âk=1Nâ12coshJkT.
Notethatforauniformchain,withJ1=J2=âŠ=JNâ1âĄJ,thisexpressionbecomesZN=22coshJTNâ1,
whichisdifferentfromthatgivenbyEq.(4.88)ofthelecturenotes,foraringof(Nâ1)sites,forourcurrentcaseh=0,onlybytheinconsequentialfactorof2.
NowwearereadytocalculatethecoefficientsKsâĄâšslsl+nâ©,which(fornâ 0)describethemutualcorrelationofvaluesoftwospinsatthedistancenfromeachother.Foranopen-endchainoflengthN,againwitharbitrarycoefficientsJk,thegeneralEq.(2.7),withtheGibbs-distributionprobabilitiesWm=exp{âEm/T}/ZN,gives
Ks=1ZNâsk=±1,fork=1,2,âŠ,Nslsl+nexpâk=1Nâ1JkTsksk+1âĄ1ZNâsk=±1,fork=1,2,âŠ,Nslsl+nâk=1Nâ1expJkTsksk+1,
whereZN isgivenbyEq.(**).Letususethefactthatallsk2=1torewritethisexpression inthemathematically-equivalentform
Ks=1ZNâsk=±1,fork=1,2,âŠ,N(slsl+1)(sl+1sl+2)âŠ(sl+nâ1sl+n)âk=1Nâ1expJkTsksk+1,andcompareitwiththefollowingpartialderivative:
DnâĄânZNâJlâJl+1âŠâJl+nâ1.Ononehand,ifwedifferentiateZNinthelastformofEq.(*),andthenuseEq.(***)forKs,weget
DnâĄâsk=±1,fork=1,2,âŠ,NânâJlâJl+1âŠâJl+nâ1âk=1Nâ1expJkTsksk+1=1Tnâsk=±1,fork=1,2,âŠ,N(slsl+1)(sl+1sl+2)âŠ(sl+nâ1sl+n)âk=1Nâ1expJkTsksk+1âĄ1TnKsZN.
Ontheotherhand,ifweuseZNintheformEq.(**),thesamederivativegives
DnâĄânâJlâJl+1âŠâJl+nâ12âk=1Nâ12coshJkT=2âk=1lâ12coshJkTâk=ll+nâ11T2sinhJkTâk=l+nNâ12coshJkT.
Comparingthesetwoexpressions forDn,withZNagain taken fromEq. (**),wegetasurprisinglysimpleresult:
Ks=âk=ll+nâ1sinhJkT/âk=ll+nâ1coshJkTâĄâk=ll+nâ1tanhJkT,valid for arbitrary coupling coefficients Jk (both inside and outside of the interval [l, l + n]) andarbitrarypositionsofthesitesland(l+n)intheopenchain6.
Foraparticularcaseofauniformchain,theresultbecomesevensimpler,anddependsonlyonthedistancenbetweentheinvolvedsites:
Ks=Ksn=tanhnJTâĄexpânnc,wherethe(notnecessarilyinteger)constantnc,
ncâĄâ1lntanhJ/Tâ1/lnT/JâȘ1,atJâȘT,exp2J/T/2â«1,atJâ«T,playstheroleofthecorrelationradiusofthis1Dsystemâcf.Eq.(4.30)ofthelecturenotes(whosepre-exponentialfactor,aswellasEq.(4.31),arevalidonlyforsystemswithnonvanishingTc).
Problem5.6.WithintheframeworkofWeissâmolecular-fieldtheory,calculatethevarianceofspinfluctuationsinthed-dimensionalIsingmodel.Usetheresulttoderivetheconditionsofquantitativevalidityofthistheory.
Solution: Since in the Ising model, the variable sk2 = (±1)2 ⥠1 in any state of the system, itsstatistical average also equals 1 within any (reasonable:-) solution of the modelâincluding theWeissâapproximation.CombiningthisresultwithEq.(4.69)ofthelecturenotes,weget
sËk2=sk2ââšskâ©2=sk2âη2=1âtanh2(hef/T)âĄ1cosh2(hef/T).(Actually, we could get this result from Eq. (*) of the model solution of problem 5.2, with thereplacementhâhef,whichistheessenceoftheWeissâapproach.)
ThekeyassumptionofWeissâtheoryisâšsËk2â©âȘη2.ReviewingthedependenceofhefâĄh+2Jdηonparametersofthesystem,discussedinsection4.4(see,inparticular,figures4.8and4.9),wemayconcludethatforstablestationarystates,thisstronginequalityisfulfilled(andhencethetheoryisasymptoticallycorrect)at:
(i)lowtemperatures,TâȘ2Jd(meaning,withinthattheory,thatTâȘTc),atanyfieldh,and
(ii)highexternalfields,h2â«T2,atanyratio2Jd/T.
(*)
(**)
(***)
(*)
(**)
(***)
(****)
Problem5.7.CalculatethevarianceoffluctuationsoftheenergyofaquantumharmonicoscillatorwithfrequencyÏ,inthermalequilibriumattemperatureT,andexpressitviatheaveragevalueoftheenergy.
Solution:PluggingtheresultgivenbyEq.(2.72)ofthelecturenotesfortheaverageenergyEoftheoscillator,intheform
E=âÏeÎČâÏâ1,whereÎČâĄ1/T,intothegeneralEq.(5.19),weget
âšEË2â©=ââEâÎČ=âââÎČâÏeÎČâÏâ1=âÏeÎČâÏâ12eÎČâÏâĄâÏeâÏ/Tâ12eâÏ/T.
ItisstraightforwardtoverifythatEqs.(*)and(**)aresimplyrelated:âšEË2â©=âÏE+E2.
Thismeans,inparticular,thatthermsfluctuationofenergyisalwayslargerthanitsaveragevalue:ÎŽEâ©ŸE.
Note,however,thatthisrelation,aswellasEq.(***)assuch7,arevalidonlyifEisreferredtothe
ground-stateenergyâÏ/2oftheoscillator(asitisinEq.(2.72)ofthelecturenotes);itisinvalidforthetotalenergy,givenbyEq.(2.80),whichisreferredtothepotentialenergyâszero:
EtotalâĄE+âÏ2.Forthisreferencepoint,Eq.(***)takesadifferentform:
âšEË2â©=âšEtotalâ©2ââÏ22,sothattherelationbetweenthermsfluctuationofenergyanditsaverageisopposite:
ÎŽEâ©œâšEtotalâ©.
Problem5.8.ThespontaneouselectromagneticfieldinaclosedvolumeVisinthermalequilibriumwithtemperatureT.Assuming thatV issufficiently large,calculate thevarianceof fluctuationsofthe totalenergyof the field,andexpress the resultvia itsaverageenergyand temperature.HowlargeshouldthevolumeVbeforyourresultstobequalitativelyvalid?Evaluatethis limitationforroomtemperature.
Solution:Aswasdiscussed in section2.6(i)of the lecturenotes, thespontaneouselectromagneticfield in a sufficiently large closed volume V may be represented as a sum of independent fieldoscillatorswiththespectraldensity(2.83):
dN=VÏ2Ï2c3dÏ.InthethermalequilibriumattemperatureT, theaverageenergyofsuchanoscillator (referredtotheground-stateenergyâÏ/2)isgivenbyEq.(2.72),
E=âÏeâÏ/Tâ1,whilethecalculationofthevarianceof its fluctuationswasthetaskofthepreviousproblem,withthefollowingresult:
âšEË2â©=âÏeâÏ/Tâ12eâÏ/T.
Since the fluctuations of oscillators with different frequencies (and with different fieldpolarizations at the same frequency) are independent, and the fluctuations of their energies areuncorrelated,thevarianceofthetotalenergyEÎŁmaybeobtainedbythesummationofthepartialvariances:
EËÎŁ2=â«Ï=0Ï=ââšEË2â©dN.BysubstitutionofEqs.(*)and(***),thisintegralmaybereducedtoatableintegralâbutnotaverypleasingone.However,wemayre-usethesimplerintegralalreadyworkedoutinsection2.6forthecalculationoftheaveragetotalenergyoftheradiationâseeEq.(2.88):
âšEÎŁâ©=â«Ï=0Ï=âEdN=â«0ââÏeâÏ/Tâ1VÏ2Ï2c3dÏ=Ï215VT4â3c3âĄÏ215Vâ3c3ÎČâ4.Forthat,letususethegeneralEq.(5.19)towrite
EËÎŁ2=ââ«Ï=0Ï=ââEâÎČdNâĄââ«0ââEâÎČdNdÏdÏ.Since thedensityof statesdN/dÏ is temperature-independent,wemay take thepartialderivativeoperator â/âÎČ from under the integration sign, and notice that the remaining integral is just theaveragetotalenergy(****),sothat
EËÎŁ2=âââÎČâ«Ï=0Ï=âEdNâĄâââÎČâšEÎŁâ©=Ï215Vâ3c34ÎČâ5âĄ4TĂÏ215VT4â3c3.NowusingEq.(****)again,wemayreducetheresulttoaverysimpleform:
EËÎŁ2=4TâšEÎŁâ©,whichshowsthattherelativermsfluctuationofthetotalenergy,
ÎŽEÎŁâšEÎŁâ©âĄEËÎŁ21/2âšEÎŁâ©=2TâšEÎŁâ©1/2â1T3/2V1/2,decreaseswiththegrowthofbothtemperatureandvolume.(Thisresulthasimportantimplicationsfor accurate measurements of the fundamental anisotropy of the cosmic microwave backgroundradiation.)
Proceedingtothesecondquestionoftheassignment,notethatEq.(*)isstrictlyvalidonlyifdNâ« 1, i.e. if the volumeV ismuch larger that than the cube of thewavelength:Vâ«c3/Ï3 for allsubstantial frequencies. As was discussed in section 2.6(i), in thermal equilibrium, their scale isgivenbytherelation(2.87):âÏmaxâŒT,sothattherequiredconditionis
Vâ«cÏmax3âŒâcT3.Forroomtemperature(TâĄkBTKâ1.38Ă10â23JKâ1Ă300Kâ4Ă10â21J),theright-handsideof
(*)
(**)
(***)
(****)
thisrelationisoftheorderof10â15m3âĄ(10ÎŒm)3, i.e. issmallonthehumanscale,butnotquitemicroscopic.
Problem5.9.ExpressthermsuncertaintyoftheoccupancyNkofacertainenergylevelΔkby:
(i)aclassicalparticle,(ii)afermion,and(iii)aboson,
inthermodynamicequilibrium,viaitsaverageoccupancyâšNkâ©,andcomparetheresults.
Solutions: Aswas discussed in section 2.8 of the lecture notes, for a statistical ensemble of non-interactingparticleswemayusethegrandcanonicaldistributionforthesub-ensembleofparticleson the sameenergy level.HencewemayapplyEq. (5.24) of the lecturenotes, derived from thisdistribution,totheleveloccupancyNk:
NËk2=TââšNkâ©âÎŒ.(i)Foraclassicalparticle,thedependenceofâšNkâ©onthechemicalpotentialÎŒisgivenbythesameformula(5.25)asforthetotalnumberofparticles,
âšNkâ©âexpÎŒT,sothatwemayrepeatthe(verysimple)derivationofEq.(5.27)togetthesimilarresult:
NËk2=âšNkâ©,i.e.ÎŽNk=âšNkâ©1/2.(ii)â(iii)Forfermionsandbosons,âšNkâ©isgivenbyverysimilarexpressions(2.115)and(2.118),whichmaybemergedintoasingleformula(justaswasdoneinsection3.2):
âšNkâ©=1exp(ΔkâÎŒ)/T±1,withtheuppersignforfermionsandthelowersignforbosons.NowapplyingEq.(*)tothisformula,weget
NËk2=exp(ΔkâÎŒ)/Texp(ΔkâÎŒ)/T±12.Expressingtheexponent,participatinginbothpartsofthisfraction,fromEq.(***),
exp(ΔkâÎŒ)/T=1âšNkâ©â1,andpluggingthisexpressionintoEq.(****),wegetthefinalresult
NËk2=âšNkâ©1ââšNkâ©,i.e.ÎŽNk=âšNkâ©1ââšNkâ©1/2.
ComparingitwithEq.(**)weseethat,foragivenaveragevalueâšNkâ©oftheleveloccupancy,itsfluctuations in the case of fermions are smaller, and in the case of bosons, larger than those forclassicalparticles.(AlltheseexpressionstendtoeachotheratâšNkâ©â0,i.e. intheclassicallimitâseeEq.(3.1)ofthelecturenotes.)
Problem 5.10. Express the variance of the number of particles, âšNË2â©V,T,ÎŒ, via the isothermalcompressibility ÎșTâĄâ(1/V)(âV/âP)T,N of the same single-phase system, in thermal equilibrium attemperatureT.
Solution:AccordingtoEq.(5.24)ofthelecturenotes,âšNË2â©V,T,ÎŒ=TâNâÎŒV,T.
Butashasbeenprovedinthesolutionofproblem1.10,theisothermalcompressibilityofasingle-phasesystemmaybeexpressedviathesamepartialderivative:
ÎșT=VN2âNâÎŒV,T.Comparingtheaboveexpressions,weget
âšNË2â©V,T,ÎŒ=ÎșTTN2V.
Note,however,thatEq.(5.24)hasbeenderivedforthegrandcanonicalensembleoffixed-volumesystems,sothatourresultisalsovalidforsuchanensemble,i.e.ifthechemicalpotentialÎŒdoesnotfluctuate.(Incontrast,inacanonicalensemble,thenumberofparticlesdoesnotfluctuateatall,butÎŒmay.)
Problem 5.11.* Starting from theMaxwell distribution of velocities, calculate the low-frequencyspectraldensityoffluctuationsofthepressureP(t)ofanidealgasofNclassicalparticles,inthermalequilibrium at temperature T, and estimate their variance. Compare the former result with thesolutionofproblem3.2.
Hint:Youmayconsideracylindrically-shapedcontainerofvolumeV=LA (see the figure above),calculatefluctuationsoftheforce exertedbytheconfinedparticlesonitsplanelidofareaA,approximating it as a delta-correlated process (5.62), and then recalculate the fluctuations intothoseofthepressure .
Solution:Foronemolecule, theconstantc in theapproximate formula for the
force it exerts on the lid in the direction normal to its plane (see the figure above),may becalculatedastheintegral,
over a time interval larger than the collision duration Ïc, but smaller than the time intervalbetweentwosequentialcollisions,with beingthecomponentof theparticleâsvelocity
component in the direction normal to the lid. Due to particle independence, we can replace theaveraginginthisexpression,overastatisticalensembleofsimilarparticles(withthesame ),withthatoverthetimeintervalÎt:
âŠ=1Îtâ«âŠdt.Introducing,insteadofÏ,anewindependentvariabletâČâĄt+Ï,weget
Thetimeintegral inthelastformulais justthemomentumtransferredfromtheparticletothelidduringoneelasticcollision,equalto ,sothat
Forsuchashort-pulseprocessas (see,e.g.figure5.4ofthelecturenotes),itstimeaverage,
isnegligibleincomparisonwithitsrmsfluctuation:
(this almost evident fact will be proved a posteriori, in just a minute), so that Eq. (*) may berewrittenas
According to Eq. (5.62), this delta-correlated process has the following low-frequency spectraldensity:
(NowwemayproveEq.(**)bymakingtheestimateofthermsfluctuation,givenbyEq.(5.60):
whereÏmaxisthefrequencywherethespectraldensitystartstodropfromitslow-frequencyvalue.Besidessomenumericalfactoroftheorderof1(itsexactcalculationwouldrequireaspecificmodeloftheparticlecollisionswiththelid),thisfrequencyisjustthereciprocalcollisiontime,sothatwemaycontinueas
Foragastobehaveideally,theintervalsÎtbetweenlidhitsbymoleculeshavetobemuchlongerthanthehitdurationÏc,sothatthelastfactorhastobemuchlargerthan1,provingEq.(**).)
Thespectraldensitiesof independent force fluctuations fromdifferentmolecules (eachwith itsownvelocity )justaddup,sothatforthelow-frequencyspectraldensityofthenetforceweget
This statistical average may be readily calculated from the 1D Maxwell distribution, i.e. theGaussiandistributionwiththevariance :
Thisisatableintegral8,equalto1!/2=œ,sothatweget
and,finally,thespectraldensityofthepressure is
Anapproximatebutfairestimateofthepressurefluctuationvariancemaynowbeachieved,as
above, by replacing the integral participating in the last form of Eq. (5.60) with the productSP(0)ÏmaxâŒSP(0)/Ïc:9
(****)
(*)
This estimate differs from the âthermodynamicâ formula cited in a footnote in section 5.3 of thelecture notes. For an classical ideal gas, with the average pressure âšPâ© = NT/V, held at fixedtemperatureT,andhencewith(âââšPâ©/âV)T=NT/V2,thatformulayields
âšPË2â©=NT2V2.(WRONG!)Thedifferenceisduetothesharp-pulsenatureofthepressureforce(whichcannotbeaccountedforin the way the âthermodynamicâ formula is usually derived), extending the pressure fluctuationbandwidth to frequencies âŒ1/Ïc, much higher that the value implied by theâthermodynamicâ formula. (This fact, and the deficiency of the traditional derivation of Eq. (****),wererecognizedlongago10,butthisformulaisstillbeingcopiedfromoldtextbookstonewones.)
NowletuscompareEq.(***),rewrittenforthetotalforce exertedbyallmolecules,
withthedragcoefficientcalculatedinthesolutionofproblem3.2,η=NAV8mTÏ1/2T.
Weseethattheyaresimplyrelated:
In this equality, we may readily recognize the general relation (5.73) between the thermal
fluctuations and dissipation, i.e. the classical limit of the fluctuationâdissipation theorem (5.98)âthusprovidingagoodsanitycheckofEq.(***).
Problem5.12.Calculate the low-frequencyspectraldensityof fluctuationsof theelectriccurrentI(t)duetorandompassageofchargedparticlesbetweentwoconductingelectrodesâseethefigurebelow.Assume that theparticles areemitted, at random times,byoneof theelectrodes, andarefully absorbed by the counterpart electrode. Can your result be mapped on some aspect of theelectromagneticblackbodyradiation?
Hint:ForthecurrentI(t),usethesamedelta-correlated-processapproximationasfortheforceinthepreviousproblem.
Solution:Attheseconditions,thecurrentI(t)isasumofshort,independentpulses,withadurationÏc of the order of the time of particleâs passagebetween the electrodes.On the time scalemuchlargerthanÏcwecanwrite.
I(t)I(t+Ï)=CÎŽ(Ï).Integrating both parts of this equation over the time of one current pulse, and transforming itexactlyasinthemodelsolutionofthepreviousproblem,weget
C=â«I(t)I(t+Ï)dÏ=Îœâ«I(t)dt2>0,whereÎœ istheaverage(cyclic) frequencyofsinge-electronpassageevents.Butthe integral inthebrackets is just the electric chargeq of one particle. Taking into account that the product qÎœ isevidentlyequaltothetime-averaged(âdcâ)currentIÂŻ,weget
C=Îœq2=âŁqIÂŻâŁ,i.e.I(t)I(t+Ï)=âŁqIÂŻâŁÎŽ(Ï).(Since,unlikeinthepreviousproblem,allthepulseâareasââ«I(t)dtareequaltoq,andhencetoeachother, their statistical averaging isunnecessary.)NowEq. (5.62) of the lecturenotes immediatelyyieldsaconstantlow-frequencyspectraldensity:
SIÏ=SI(0)=âŁqIÂŻâŁ2Ï,forÏâȘ2ÏÎœ.UsingEq.(5.61),thisexpressionmayberecastintoitscommonform
âšIË2â©ÎÎœ=2âŁqIÂŻâŁÎÎœ.
ThisisthefamousSchottkyformulafortheshotnoise,whichwasbrieflymentionedinsection5.5ofthelecturenotesâseeEq.(5.82)anditsdiscussion.(Itwasfirstderivedin1918byWSchottkyusing a different approach, based on the Poisson probability distribution (5.31) of the number ofparticlespassingthroughthesystemduringatimeintervalÎtâ«Ïc.)
NowproceedingtotheelectromagneticradiationoffrequencyÏ,letsomesourceofit(say,asetofsimilaratomsbeingexternallyexcitedtoanenergylevelEe=Eg+âÏ)emititsquanta,eachwithenergyâÏ,independentlyofeachother,andthebackflowofradiationbenegligible.Thenreplacingthe above arguments for the electric current I(t) with those for the power flow of theelectromagnetic radiation (time-averaged over a time interval much larger than ), we mayexpect the spectral density of its fluctuations (frequently called the photon shot noise) to beexpressedbytheformulaanalogoustoEq.(*):
(*)
Boththedetailedquantumtheoreticalanalysisofradiation,andexperiments (mostly, inoptical
photoncounting)confirmthisresult.Unfortunately,itsmoreformalderivationwouldrequiremuchmoretimethanIhaveinthisseries,sothatIhavetoreferthereadertospecialliterature11.
Problem 5.13. 12 A very long, uniform string, of mass ÎŒ per unit length, is attached to a firmsupport, and stretched with a constant force (âtensionâ) âsee the figure below. Calculate thespectral density of the random force exerted by the string on the support point, within theplanenormaltoitslength,inthermalequilibriumattemperatureT.
Hint:Youmayassumethatthestringissolongthatatransversewave,propagatingalongitfromthesupportpoint,nevercomesback.
Solution: Temporarily, let us ignore the fluctuations, and assume that the support point is beingslightly moved, in the plane normal to the string, following some externally-fixed law q0(t),independentofthestringmotion.(Thismotionmaybetwo-dimensional,sothatgenerallyithastobedescribedwitha2Dvector.)Suchdisplacementimposestheboundarycondition,
q0,t=q0(t),onthetransversewavesq(z,t)thatareexcited,bythismotion,topropagatealongthestring.(Herezistheaxisdirectedalongthestring,withtheoriginattheattachmentpoint.)Classicalmechanicssays13thatifsuchwavesarenottoolarge14,theirdynamicsofobeysthelinearwaveequation
AttheconditiongivenintheHint15,thesolutionofthisequation,satisfyingtheboundarycondition(*),hastheformofawave,
traveling from the wall, with the velocity . Such a wave of string displacements isaccompaniedbythefollowingwaveofthetransverseforce16:
exerted by the ârightâ part of the string (as seen from the given point z) on its âleftâ part. (Theconstant iscalledthewaveimpedanceofthesystem.)
Applying this result to thepointz=0,wesee that thestringprovides, for thesupportpointâsmotion,thedamping(drag)forcedescribedbyEq.(5.64)ofthelecturenotes, ,withthedragcoefficient , i.e.hasthegeneralizedsusceptibility âseeEqs.(5.89)and(5.90)ofthelecturenotes17.Sincethisresultisvalidregardlessoftheactualmotionq0(t),wemayuseittospelloutEq.(5.98)forourparticularcase:
Inhuman-scalesystems,thedivergenceofthisresultatÏââiscutoff(atleast)bytheviolation
ofthealreadymentionedconditionâŁâq(z,t)/âzâŁâȘ1ofitsvalidity.Indeed,accordingtoEq.(**),thederivativeonitsleft-handsideofthisinequalityequals ,andforasinusoidaloscillationwithamplitudeAandfrequencyÏ,itsmagnitudegrowswithfrequencyas .
Problem5.14.18Eachoftwo3Dharmonicoscillators,withmassmandresonancefrequencyÏ0,hastheelectricdipolemomentd=qs,wheresistheoscillatorâsdisplacementfromitsequilibriumposition.UsetheLangevinformalismtocalculatetheaveragepotentialofelectrostaticinteractionof these twooscillators (a particular case of the so-calledLondondispersion force), separatedbydistance râ« (T/mÏ0
2)Âœ, in thermal equilibrium at temperature Tâ« âÏ0. Also, explain why theapproachusedinthesolutionoftheverysimilarproblem2.15isnotdirectlyapplicabletothiscase.
Hint:Youmayliketousethefollowingintegral:â«0â1âΟ2(1âΟ2)2+αΟ22dΟ=Ï4α.
Solution:LetustemporarilyprescribetoeachoscillatorsomenonvanishingdampingparameterÎŽâĄÎ·/2m.Iftheinteractionbetweentheoscillatorsisnegligible,intheclassicallimiteachofthemmaybedescribedbytheLangevinequation(5.65),withÎș=mÏ0
2,validforeachCartesiancomponentofthedisplacementvectors.Mergingtheseequationsintothevectorform,andmultiplyingalltermsbytheratioq/m,fortheelectricdipolemomentdoftheoscillatorweget
Since the Langevin forces exerted on each dipole by their dissipative environments are random,independent,andisotropic,soarethespontaneouslyinduceddipolemomentsd1,2.Asaresult,theenergyoftheirelectrostaticinteraction19,
U=14ÏΔ0r3(d1xd2x+d1yd2yâ2d1zd2z),forrâ«s1,2,vanishesatitsdirectstatisticalaveraging.
Thenon-zeroaverageLondondispersionforceappearsinthenextorderinthesmallparameter(q2/4ÏΔ0r3),andmaybeconvenientlydescribedasaresultofthefactthattheLorentzforceoftheelectricfield20
ofeachrandomdipoled=d1,2,atthelocationr2,1ofthecounterpartdipole,inducesinthelatteraproportional and correlated component of its dipole moment, so that the statisticalaverages and donotvanish,contributingtotheaverageinteractionenergy21
Letuscalculatethefirsttermofthissum,usingthefactthatthedifferentialequation(*)islinear,
anditishenceproductivetoFourier-expandbothitsright-handpartanditssolutionâforexample,asinEq.(5.52)ofthelecturenotes:
Then for theFourieramplitudesatanarbitrary frequencyÏ, theequationyields therelation (seealsoEq.(5.67)ofthelecturenotes):
FromthesimilarFourierimageofthe(alsolinear,butalgebraicratherthandifferential)Eq.(**),thecomplexamplitudeoftheelectricfieldatthelocationr2âĄrofthedipole2is
TheLorentzforce ofthisfieldshouldbeaddedtotheright-handsideofEq.(*),writtenfortheseconddipole.Sincethisforceiscompletelyindependentoftheenvironment-inducedforce ,andtheequationislinear,wemayuseit,intheform,
tocalculatethefield-inducedpartofthedipolemomentalone.ThesimilarFourierexpansionyields
Atthispointwehavetobecareful,becausetheinteractionenergyâšUâ©isaquadratic,ratherthan
linearform,sowehavetocalculateitusingallcomponentsoftheirFourierextensions.PerformingabsolutelythesamecalculationasatthederivationofEq.(5.60)ofthelecturenotes,weget
whereS12(Ï)isthemutualspectraldensityoftheoperands,definedsimilarlytoEq.(5.57):
(InplainEnglish,Eq.(****)saysthatthecontributionsofallfrequenciesintotheaverageinteractionareadditive.)InordertocalculateS12(Ï),wemayuseEq.(***)andthenEq.(**),getting
where in the last expression, the index 1 has been dropped for notation simplicity, because theparticipatingaveragesdonotdependonthedipolenumber.SinceallCartesiancomponentsofthespontaneouslyfluctuatingdipoled,describedbyEq.(*),havesimilarstatistics,theexpressioninthelastparenthesesisjustsixtimesoneofthemâsayâšdxÏdxÏâČâ©,with
sothat
(*)
According to its definition (5.55), the autocorrelation function on the right-hand side of this
relationshouldnotchangeatthesimultaneouschangeofsignsofthefrequenciesÏandÏâČ:
Hence our mutual correlation function has to change the sign of its imaginary part at suchfrequencychange.Asaresult,Eq.(****)mayberecastasfollows:
âšU12â©=â12â«0+âS12Ï+S12*ÏdÏâĄââ«0âReS12ÏdÏ,sothataddingtheindependentandequalcontributionâšU21â©,forthefullinteractionenergyweget
Sofar,ourresultisvalidforarbitrarytemperatures.AtTâ«âÏ0(andinthermalequilibrium),we
mayuseEq. (5.73a), i.e. theclassical limitof the fluctuationâdissipation theorem, for thespectraldensityofforce:
Withthissimplification,wemayusetheintegralprovidedintheHint22,withα=2ÎŽ/Ï0,tospellouttheresult:
U=â12q24ÏΔ0r321m3â«0â(Ï02âÏ2)(Ï02âÏ2)2+4ÎŽ2Ï222mÎŽÏTdÏâĄâ12q24ÏΔ0r321m32mÎŽÏT1Ï03â«0â(1âΟ2)(1âΟ2)2+(2ÎŽ/Ï0)2Ο22dΟ=
â12q24ÏΔ0r321m32mÎŽÏT1Ï03Ï4(2ÎŽ/Ï0)âĄâ3q24ÏΔ0r3mÏ022T.
ThisresultdoesnotdependontheoscillatorâsdampingÎŽ,confirmingthevalidityofourcurrentapproach. It coincides with the classical limit of the result obtained in the solution of Part QMproblems5.5and7.6,usingadifferent(andIhavetoconfess,morecompact)method.However,ourcurrent approach has advantages of being very physically transparent, and not requiring anartificial,temporaryintroductionofadifferenceofoscillatorparameters.
Finally,anattempt23tosolvethisproblembyadirectcalculationoftheaverageâšEâ©valueofthetotalenergyofthesystem,
E=p122m+p222m+Îșs122+Îșs222+U=p122m+p222m+Îșs122+Îșs222âq24ÏΔ0r3(s1xs2x+s1ys2yâ2s1zs2z),
using the sameclassical approachaswasused for the solutionof the (conceptually, very similar)problem2.15,yieldsasurprisingresult:
E=6T.This equality is in accordance with the equipartition theorem by describing the sum of classicalcontributionsT/2by the12half-degreesof freedomof the twonon-interacting3Doscillators, butgivesnoaverageinteractionenergy.
Theexplanationof this resultmaybe found in thesolutionofPartQM problem3.16: a simplecoordinatetransformshowsthatoursystemisexactlyequivalenttothatofsixnon-interacting1Dharmonicoscillatorswithslightlydifferentfrequencies,whichdependontheinteractionparameterÎŒâĄq2/4ÏΔ0r3mÏ0
2âȘ1.AtTâŒâÏ0,theaverageenergyofeachoscillator,andhencethenetaverageenergy âšEâ© of the system does depend on ÎŒ, thus describing the interaction energy. However, ifpursuingthelimitTâ«âÏ0,wetaketheenergiestoequalexactlyTfromtheverybeginning,asweimplicitly do in the classical approach, this dependence is lost, contradicting the (correct) resultobtainedabove.
Problem 5.15.*Within the van der Pol approximation24, calculatemajor statistical properties offluctuationsofclassicalself-oscillations,at:
(i)thefree(âautonomousâ)runoftheoscillator,and(ii)itsphaselockingbyanexternalsinusoidalforce,
assumingthatthefluctuationsarecausedbyaweaknoisewithasmoothspectraldensitySf(Ï). Inparticular,calculatetheself-oscillationlinewidth.
Solution: In the van der Pol approximation, the solution of the (weakly nonlinear) differentialequationdescribingquasi-sinusoidalself-oscillationsislookedforintheform25
qt=AtcosΚt=AtcosΩtâÏt.Intheabsenceoffluctuations,thedynamicsofÏ,i.e.ofthedifferencebetweenthefullphaseΚoftheoscillatorandthatofaweaksinusoidalphase-lockingforceoffrequencyΩ,maybedescribedby
(**)
(***)
(****)
thefollowingreduced(orâvanderPolâ,orâRWAâ)equationâsee,e.g.PartCMEq.(5.68):26
Ï=Ο+ÎcosÏ,whereΟ ⥠Ω â Ω0 is the detuning (the difference between Ω and the own frequency Ω0 of theoscillator), and the parameter ÎâȘ Ω is proportional to the phase-locking forceâs amplitude. (Asfollows from the elementary analysis of Eq. (**), this parameter, in particular, determines thefrequencyrangeofphase-lockingintheabsenceoffluctuations:ΩmaxâΩmin=2âŁÎâŁ.)
Inordertoaccountforthefluctuations,weneedtorecallthattheright-handsideofthereducedequation(**)resultsfromthefollowingtimeaveraging27,
1ΩAf0cosΚ¯,of the right-hand side f(t) of the initial differential equation of motion, taken in the â0thapproximationâthatignorestherelativelyslowevolutionoftheamplitudeAandthephaseÏ.Aswasdiscussed in section 5.5 of the lecture notes, within the Langevin formalism, a noise source isdescribedas an additional term, fË(t), in the right-hand of that initial equation ofmotion, so thattaking it intheformoftheFourierexpansion(5.52),weneedtoaddtotheright-handsideofthereducedequation(**)thefollowingterm:
1ΩAfËcosΩtâÏÂŻâĄ12ΩAâ«ââ+âfÏeâiÏ+ΩtâÏdÏ+â«ââ+âfÏeâiÏâΩt+ÏdÏÂŻ.
ThisexpressionshowsthattheLangevintermhastwocomponents,whichdifferfromtheoriginalnoise (besides the scaling front factor) only by shifting its frequency spectrum by ±Ω. The timeaveraginginthevanderPolmethodmaybecarriedoveranytimeperiodÎtmuchlargerthantheoscillationperiod2Ï/Ω,withtheonlyrequirementfor it tobestillmuchshorterthanthesmallesttime scale of the reduced equation(s) dynamics, in our case of the order of 1/max[âŁÎŸâŁ, Î]. Suchaveraging retains only low-frequency components of the averaged functionâin our case, thecomponents with the âmathematicalâ frequencies close to ±Ω, i.e. the âphysicalâ (positive)frequenciesclosetoΩ.Iftheinitialnoiseisindeedbroadband,itsspectraldensitySf(Ï),definedbyEq. (5.57), is virtually constantwithin suchanarrow interval.Hence, its additiongeneralizes thereducedequation(**)asfollows:
Ï=Ο+ÎcosÏ+ΟËt,whereΟË(t),atthe lowfrequenciesofour interest,maybetreatedasaprocesswithzeroaverageandaconstantspectraldensity
SΟÏâSΟ0=12ΩA2SfΩ+12ΩA2SfâΩ=12Ω2A2SfΩ.Accordingtothediscussionintheendofsection5.4ofthelecturenotes,thecorrelationfunctionofsuchaprocessmaybeapproximatedwithEq.(5.62):
KΟÏ=2ÏSΟ0ÎŽÏ=2ÎÎŽÏ,withÎâĄÏSΟ0=ÏSfΩ2Ω2A2.Nowwearereadytoconsiderthetwospecificcaseslistedintheproblemâsassignment.
(i)Thefree-running(âautonomousâ)modeoftheself-oscillatormaybedescribedbyEq.(***)withÎ=0,givingthelineardifferentialequation
Ï=Ο+ΟËt,whichmaybereadilyintegrated:
Ït=Οt+ÏËt+const,whereÏËtâĄâ«0tΟËtâČdtâČ,sothatwemayrewriteEq.(*)as
qt=AcosΩtâΟtâÏËt=AcosΩtâ(ΩâΩ0)tâÏËt=AcosΩ0tâÏËt.
In the absence of fluctuations, this expression describes coherent self-oscillations at the ownfrequencyoftheoscillator:
qtâŁÏË=0=Acos(Ω0t+const).Thenoise ΟË(t) induces fluctuations ÏË(t) of the phase around this deterministic evolution, whichobeyexactlythesameequationasthecoordinatefluctuationqË(t)ofafree1DâBrownianparticleâintheEinsteinâsproblemâseeEq.(5.74).Repeatingthecalculationsthatfollowedthisformula,weseethatthephaseÏoftheautonomousoscillatorperformsarandomwalk,withthediffusioncoefficientproportionaltothespectraldensityofthenoisesource:
ÏËt+ÏâÏËt2=2DÏ,withD=ÎâĄÏSfΩ2Ω2A2.
This phase diffusion has important implications for the observed self-oscillations process q(t).Indeed,letuscalculateitscorrelationfunctionKqÏâĄqtqt+Ï=A2cosΩ0tâÏËtcosΩ0t+ÏâÏËt+ÏâĄA22âšcos2Ω0t+Ω0ÏâÏËtâÏËt+Ï+cosÏËt+ÏâÏËtâΩ0Ïâ©.Sinceinastatisticalensembleofsimilarautonomousoscillators,thefullphaseΚoftheoscillationstakesallvalues(modulo2Ï)withequalprobability,thestatisticalaverageofthefirsttermvanishes,leavinguswith
KqÏ=A22cosÏËt+ÏâÏËtâΩ0ÏâĄA22Reexp{i[ÏËt+ÏâÏËtâΩ0Ï]}âĄA22Reexp{âiΩ0Ï}expiâ«tt+ÏΟËtâČdtâČâĄA22Reexp{âiΩ0Ï}ân=1Nexpiâ«t+(nâ1)Ï/Nt+nÏ/NΟËtâČdtâČ,
where the last representation is exactly valid for any integerN > 0. Since in our approximation(****),28 the function ΟË(t) is delta-correlated, the partial integrals in this product are statisticallyindependent,andourstatisticalaveragebreaksintoaproductofaverages:
KqÏ=A22Reexp{âiΩ0Ï}ân=1NΔn,withΔnâĄexpiâ«t+(nâ1)Ï/Nt+nÏ/NΟËtâČdtâČ.At sufficiently largeN, andhence sufficiently small intervalsdtâČ, the exponent in each ΔnmaybeexpandedintotheFourierseries,withonlythreeleadingtermskept,sothatΔnâlimNââ1+iâ«t+(nâ1)Ï/Nt+nÏ/NΟËtâČdtâČ+12iâ«t+(nâ1)Ï/Ntt+nÏ/NΟËtâČdtâČiâ«t+(nâ1)Ï/Nt+nÏ/NΟËtâłdtâłâĄlimNââ1+iâ«t+(nâ1)Ï/Nt+nÏ/NΟËtâČdtâČâ12â«t+(nâ1)Ï/Nt+nÏ/NdtâČâ«t+(nâ1)Ï/Nt+nÏ/NdtâłÎŸËtâČΟËtâł.
By the definition of the random function ΟË(t), its statistical average equals zero, so that the
secondterminthelastformofthisequalityvanishes,whiletheaverageinthelasttermisjustthecorrelationfunctionKΟ(tâČâtâł),sothatpluggingitfromEq.(****),wegetanexpressionindependentofthetimestepnumbern:
Δn=limNââ1âÎâ«t+(nâ1)Ï/Nt+nÏ/NdtâČâ«t+(nâ1)Ï/Nt+nÏ/NdtâČÎŽtâČâtâł=limNââ1âÎâ«t+(nâ1)Ï/Nt+nÏ/Ndtâł=limNââ1âÎÏN.
Hencethecorrelationfunctionoftheprocessq(t)maybecalculatedasKqÏ=limNââA22Reexp{âiΩ0Ï}1âÎÏNNâĄA22cosΩ0ÏĂlimNââ1âÎÏNN.
Butthelastlimitisjustexp{âÎÏ},29sothatwefinallygetaverysimpleresult,KqÏ=A22cosΩ0ÏexpâÎÏ,
withÎgivenbyEq.(****).Thisformuladescribesthegradualsuppressionofthecoherenceoftheself-oscillationsonatime
scaleâŒ1/Î.30NowletusapplytothisresulttheWienerâKhinchintheorem(5.58):SqÏ=1Ïâ«0âKqÏcosÏÏdÏ=A22Ïâ«0âcosÏÏcosΩ0ÏexpâÎÏdÏâĄA24ÏReâ«0âexpi(Ï+Ω0)ÏâÎÏ+expi(Ï
âΩ0)ÏâÎÏdÏ=A24ÏRe1i(Ï+Ω0)âÎ+1i(ÏâΩ0)âÎâĄA22ÏÎ(Ï2âΩ02)+Î2.
Thisexpressiondescribestheso-calledLorentzbroadeningoftheoscillationline,withthehalf-linewidth equal to the Î given by Eq. (****). Note again that according that relation, at smallfluctuations(givingÎâȘΩ0),forwhichthisdelta-correlatedapproximationofthenoiseisvalid,thelinewidthofoscillationsatfrequenciesÏâŒÎ©0isdeterminedbytheexternalforceâsintensitySf(Ω)atclosefrequencies,butitactsontheoscillatorviatheintensitySΟ(0)offrequencyfluctuationsatmuchlowerfrequenciesÏâŒÎâȘΩ0.Thisfactisimportant,becauseforsomeself-oscillators(suchasdc-voltage-biased Josephson junctions31), low-frequency external noisemaydirectlywobble theoscillationfrequency,andhenceprovideadditionallinebroadening.
(ii)Forthephase-lockedoscillator(Îâ 0,âŁÎŸâŁ<âŁÎâŁ),intheabsenceoffluctuations,Eq.(**)describesatransientprocessinwhichthephaseÏapproachesaconstant:
ÏtâÏ0,withcosÏ0=âΟÎ,i.e.ÎsinÏ0=(Î2âΟ2)1/2,sothattheoscillationsq(t)=Acos[ΩtâÏ(t)]settletothefrequencyΩoftheexternalforce,ratherthantoΩ0âthis isexactlywhat iscalled âphase lockingâ (or âsynchronizationâ). In thiscase, smallnoise(notstrongenoughtodisruptthephaselocking32)causesnotthephasediffusion,butrathersmall,limitedfluctuationsofthephasearoundtheconstantvalueÏ0.Inordertofindtheirspectraldensity,wemaylinearizeEq.(***)bytakingÏ(t)=Ï0+ÏË(t),expandingthenonlinearfunctioncosÏintotheTaylorseriesinsmallÏË(t),andkeepingonlythetwoleadingterms.Thisyieldsthelineardifferentialequation
ÏË+(ÎsinÏ0)ÏË=ΟËt,whichmaybesolvedaswasdiscussedinsection5.5,bytheFourierexpansionofthefunctionsÏË(t)andΟË(t).FortheirFourierimages,thelinearizedequationyieldstherelation
âiÏÏÏ+(ÎsinÏ0)ÏÏ=ΟÏ,i.e.ÏÏ=1âiÏ+ÎsinÏ0ΟÏ,whichallowsus toexpress thespectraldensityofphase fluctuationsvia thatof thenoisesource,givenbyEq.(***):
SÏÏ=âŁ1âiÏ+ÎsinÏ0âŁ2SΟÏâ1Î2sin2Ï0+Ï2SΟ0=SfΩ2Ω2A21(Î2âΟ2)+Ï2.Plugging this expression into the general Eq. (5.60), we may readily calculate the fluctuationsâvariance33:
âšÏË2â©=2â«0âSÏÏdÏ=SfΩΩ2A2â«0âdÏ(Î2âΟ2)+Ï2=ÏSfΩ2Ω2A21(Î2âΟ2)1/2.
Notethatthefluctuationsaresmallest inthecenterofthephase-lockingregion(Ο=0),wheretheholdoftheexternalforceontheoscillatorsâphaseismostfirm,andgrowinfinitelytowardeitheredgeof theregion(Οâ±Î)where thephase lockingeffect ismost fragile.However, inanycase,thesephasefluctuationsaremuchsmallerthanthoseintheautonomousoscillator,analyzedinTask(i),withtheirinfinitevariance.Thisisnatural,becausethephaselockingessentiallydoesnotallowthe oscillationâs instant frequency Κ=ΩâÏ to fluctuate slowly, i.e. deviate significantly from thefrequencyΩofthelockingforce:
SΚÏ=SÏÏ=Ï2SÏÏ=SfΩ2Ω2A2Ï2(Î2âΟ2)+Ï2â0,atÏâ0.
Problem5.16.Calculatethecorrelationfunctionofthecoordinateofa1DharmonicoscillatorwithsmallOhmicdampingatthermalequilibrium.Comparetheresultwiththatfortheautonomousself-oscillator(thesubjectofthepreviousproblem).
Solution:ThespectraldensityoffluctuationsoftheoscillatorâscoordinateisgivenbyEq.(5.68)ofthelecturenotes34.Withthesimplification(5.69),validatlowdampingandÏâÏ0,itbecomes
Fromhere,thecorrelationfunctionmaybefoundusingtheFouriertransform(5.59):35
Sinceatlowdamping(ÎŽâȘÏ0)thisintegralconvergesrapidlynearthepointÏ=Ï0,i.e.nearΟ=0,wemayformallyextendthelimitsofintegrationoverΟfromââto+â,andget
Thisisatableintegral36equalto(Ï/2ÎŽ)exp{âÎŽÏ},sothatusingthefluctuation-dissipationtheorem(5.98)for ,withintheOhmicmodelofdissipation(Ï(Ï)=iηÏ),
wefinallyget
(Asasanitycheck,atÏ=0thisexpressiongivesthesameresultforKq(0)âĄâšq2â©asEq.(2.78)withÏ=Ï0=(Îș/m)1/2.)
Notethatthefunctionalform,Kq(Ï)=Kq(0)cosÏ0ÏeâÎŽÏ,
ofourcurrentresult (quantitativelyvalidonly for lowdamping,ÎŽâȘÏ0), issimilar to that for theautonomousself-oscillatoranalyzed in thepreviousproblem,despite thequitedifferentphysicsofthesetwoprocessesâtheexternalnoiseâfiltrationâbyapassiveoscillatorinourcurrentcaseversusthe noise broadening of the line of an active self-oscillator. (The difference between these twoprocesses may be revealed by other statistical measuresâfor example, their time-averagedprobabilitydistributionsw(q).)
Notealsothatthisproblemmaybealsosolvedusinga1DversionofEq.(5.177),calculatingtheprobability distribution w(q, p, Ï) from the 1D version of the FokkerâPlanck equation (5.149).However,asshouldbeclearfromthesolutionofthesimilarproblemforhighdamping,carriedoutintheendofsection5.8,thiswayissubstantiallylonger,sothereisnogoodmotivationforapplyingittothislinearsystem,forwhichtheLangevinformalism,usedabove,givesasimplerapproach.
Problem5.17.Consideraverylong,uniform,two-wiretransmissionline(seethefigurebelow)witha wave impedance , which allows propagation of TEM electromagnetic waves with negligibleattenuation,inthermalequilibriumattemperatureT.Calculatethevariance ofthevoltagebetweenthewireswithinasmallintervalÎÎœofcyclicfrequencies.
Hint:AsanE&Mreminder37, intheabsenceofdispersivematerials,TEMwavespropagatewithafrequency-independentvelocity(equaltothespeedcoflight,ifthewiresareinvacuum),withthevoltage andthecurrent I (seethe figureabove)relatedas ,where is thelineâswaveimpedance.
Solution:Since ina travelingTEMwave, theelectricandmagnetic fields,andhence thevoltage and the current I(t) vary in time simultaneously (âin phaseâ), the instantaneous power
transferredbyasinusoidalwavethroughacross-sectionofthetransmissionlineis
where andIÏarethevoltageandcurrentamplitudes,relatedbythewaveimpedance ,andthetwosignsdescribetwopossibledirectionsofthewavepropagation.Thusthetime-averagepowerofthewaveis
Sincethewavetravelswithvelocityc,itsaverageenergyperlengthLofthelineis
On the other hand, according to Eq. (2.80) of the lecture notes, in thermal equilibrium the
statistical-ensembleaverage(whichinthiscaseincludesthetimeaveraging)ofthisenergyshouldbeequalto
E=âÏ2cothâÏ2T,sothatforeachwavemode,
SincethevelocitycoftheTEMwavesdoesnotdependonfrequency,theirwavenumberisrelated
withfrequencyask=±Ï/c,sothatthenumberofTEMmodescorrespondingtoasmallintervalÎÏ
âĄ2ÏÎÎœofphysical(positive)frequenciesisÎN=2LÎk2Ï=2LÎÏ2ÏcâĄ2LÎÎœc,
where the front factor 2 describes two possible intervals of k (positive and negative), i.e. twopossibledirectionsofwavepropagationâseeEq. (*).Since the intensities ofwaveswithdifferentfrequenciesanddirectionssumupindependently,therequiredvarianceofthevoltageis
Problem5.18.Nowconsiderasimilar longtransmissionlinebutterminated,atoneend,withanimpedance-matchingOhmicresistor .Calculatethevariance ofthevoltageacrosstheresistor, and discuss the relation between the result and the Nyquist formula (5.81), includingnumericalfactors.
Hint: Take into account that a load with resistance absorbs incident TEM waves withoutreflection.
Solution:Suchanimpedance-matchedresistorcannotchancethewavestatistics,andhenceEq.(**)ofthemodelsolutionofthepreviousproblem,with replacedwithR,givestherequestedvoltagevariance.Inparticular,intheclassicallimitâÏâȘT,
It may look like this result contradicts the Nyquist formula (5.81b), which gives a twice largernumericalfactor.
In order to resolve this paradox, we should notice that in the Langevin-approach analysis ofsection5.5,wehaverepresentedtheenvironmentalforceasthesum
and then argued that for theOhmic dissipation, the forceâs average over a thermally-equilibriumensembleofenvironments,withthesamemotionq(t),maybeexpressedbyEq.(5.64),
Forthetransitiontotheelectriccircuitcase(seethediscussionfollowingfigure5.9ofthelecture
notes),wemayrepresenttheaboveresultas
where the variance of the second term (within the cyclic frequency interval ÎÎœ) is given by Eq.(5.81b),
From the point of the electric circuit theory38, Eq. (**) is the algebraic representation of anequivalentcircuitincludingadeterministicresistorRandanideal(internal-resistance-free)voltagesourcewiththeemfâsvariancegivenbyEq.(***),connectedinseriesâseethesolid-linepartofthefigurebelow.Ifthevoltage ismeasuredwithanidealvoltmeter(withinfiniteinternalresistance),thenthecurrentIinthecircuitvanishes,andthevarianceofthemeasuredvoltageisindeedgivenbyEq.(***).39
Inourcurrentproblemweare,however,discussingvoltagemeasurementsinadifferentcircuit,
inwhich the ânoisyâ resistor is connected to a semi-infinite transmission line. ForTEMwaves, itslumpedequivalentcircuitforoutcomingwaves(thosegeneratedbyfluctuationsintheloadresistorand disappearing at infinity) may be obtained40 by complementing the equivalent circuit with anoise-freeresistorofmagnitude ,representingthetransmissionlineâseethedashed-linepartofthe figure above. This equivalent circuit clearly shows that for , the randomemf is equallydividedbetweentheinternalandexternalresistances,sothatthevarianceoftheoutcomingwavevoltageis
In equilibrium, incoming waves have equal voltage variance, so that adding these two
(incoherent)contributions,werecoverourinitialresult(*)obtainedfrommodecounting41.
Problem5.19.Anoverdampedclassical1Dparticle escapes fromapotentialwellwitha smoothbottom,butasharptopofthebarrierâseethefigurebelow.PerformthenecessarymodificationoftheKramersformula(5.139).
(*)
(*)
Solution: In thiscase, thequadraticapproximation (5.135) is inapplicable,andhas tobereplacedwiththelinearone:
U(qâ©œq2)âU(q2)âF(q2âq),whereFâĄdU/dqatq=q2â0istheinternalslopeofthepotentialatitssharpedge.Now,takingintoaccountthestronginequality(5.127),theintegralontheright-handsideofEq.(5.131)maybecalculatedas
â«qâČqâłexpU(q)âU(q1)Tdqââ«ââq2expU(q2)âF(q2âq)âU(q1)Tdq=expU(q2)âU(q1)Tâ«ââq2expFT(qâq2)dq=TFexpU0T.
Comparing this result with Eq. (5.136) for a smooth-edge well, we see that the necessarymodificationofEq.(5.139)affectsonlythepre-exponentialcoefficient,
2ÏTÎș21/2âTF,i.e.2Ïη(Îș1Îș2)1/2â2ÏTÎș11/2ηF,ratherthantheArrheniusexponent.
Problem5.20.Perhapsthesimplestmathematicalmodelofthediffusionisthe1Ddiscreterandomwalk:eachtimeintervalÏ,aparticleleaps,withequalprobability,toanyoftwoadjacentsitesofa1Dlatticewithaspatialperioda.Provethattheparticleâsdisplacementduringatimeintervaltâ«ÏobeysEq.(5.77)ofthelecturenotes,andcalculatethecorrespondingdiffusioncoefficientD.
Solution:Aparticleâsdisplacementattimet=NÏ,i.e.afterNrandomleaps,isevidentlyÎq=ân=1Nasn,
wheresnisarandomnumberthatmaytakejusttwovalues,±1,withequalprobabilityW±=Âœ,andhencewiththevanishingstatisticalaverage42:âšsnâ©=0.Fromthisexpression,wemaycalculatethedisplacementsquared,anditsaverage:
âšÎq2â©=ân,nâČ=1NasnasnâČ=a2ân,nâČ=1NâšsnsnâČâ©.Sincedifferentvaluessnarestatisticallyindependent,anymutualcorrelationcoefficientâšsnsnâČâ©withn â nâČ equals the product of averages, and hence is equal to zero43. As a result, nonvanishingcontributionstotheright-handsideofEq.(*)aregivenonlybyNtermswithn=nâČ,i.e.âšsn2â©.Butsn2
equals1foranysignofsn,sothatEq.(*)isreducedtoâšÎq2â©=a2NâĄa2Ït.
Due to the condition tâ« Ï, this result is approximately valid not only for discrete valuesNÏ
(where it isexact),butalso forany times t.Comparing itwithEq. (5.77),wesee that thismodelindeeddescribesthe1Ddiffusion,withthefollowingdiffusioncoefficient:
D=a22Ï.Inaccordancewithcommonsense,itgrowswiththejumpsizea,andwiththefrequency1/Ïofthejumps.
Problem 5.21. A classical particle may occupy any of N similar sites. Its interaction with theenvironment induces random, incoherent jumps from theoccupiedsite toanyother site,with thesame time-independent rate Î. Calculate the correlation function and the spectral density offluctuationsoftheinstantoccupancyn(t)(equaltoeither1or0)ofasite.
Solution: Performing an evident generalization of the master equations for two-level systems,discussed in sections 4.5 and 5.8, to the multi-site case, we may write the following balance ofprobabilitiesWjfortheparticletooccupyan(arbitrarynumbered)jthsite:
Wj=âjâČ=1jâČâ jNÎWjâČâ(Nâ1)ÎWj.SincethesumofallWjâČoverjâČ(includingjâČ=j)shouldbeequalto1,thesumontheright-handsideof the above equation is equal to Î(1 â Wj), so that this master equation is actually a lineardifferentialequationforonevariable:
Wj=Î(1âWj)â(Nâ1)ÎWjâĄÎ(1âNWj),andmaybereadilysolvedforarbitraryinitialconditions,givingtheresultfunctionallysimilartoEq.(5.171)ofthelecturenotes44:
W(t)=W(0)eâNÎt+W(â)(1âeâNÎt),withW(â)=1N,Thelimiting,stationaryvalueW(â)oftheprobabilityimmediatelyyieldsthe(ratherobvious)resultfortheaveragesiteoccupancy:
n=ân=0,1nW(â)=1N.
Now we can use the general Eq. (5.167) to calculate the correlation function of the instantoccupancy(forthestationarycase,impliedbytheassignment45):
n(t)n(t+Ï)=ân=1,0nW(â)ânâČ=1,0nâČW(Ï)âŁW(0)=1.Sinceoneof thetwopossibleoccupancynumbers iszero,onlyoneterm(withn=nâČ=1)of fourgivesanonvanishingcontributiontothissum:
n(t)n(t+Ï)=W(â)W(Ï)âŁW(0)=1.
Usingthegeneralsolution(*)withthereplacementoftwithÏ,andW(0)=1,wegetn(t)n(t+Ï)=1NeâNÎÏ+1N(1âeâNÎÏ)âĄ1N1N+1â1NeâNÎÏ.
Asasanitycheck,aparticularcaseofthisresult,n(t)n(t)âĄâšn2â©=1N,
maybereadilyverifiedbyasimplercalculation:âšn2â©=ân=1,0n2Wâ=1N.
Nowwearereadytocalculatethecorrelationfunctionoftheoccupancyfluctuations:
Kn(Ï)âĄnË(t)nË(t+Ï)âĄn(t)ânn(t+Ï)ân=n(t)n(t+Ï)ân2=1N1N+1â1NeâNÎÏâ1N2âĄNâ1N2eâNÎÏ,andusetheWienerâKhinchintheorem(5.58)tofindtheirspectraldensity:
Sn(Ï)=1ÏReâ«0+âKn(Ï)eiÏÏdÏ=Nâ1ÏN2Reâ«0+âe(âNÎ+iÏ)ÏdÏ=Nâ1ÏN2Re1âNÎ+iÏâĄNâ1ÏNÎNÎ2+Ï2.
Theresultdescribesazero-frequency-centeredLorentzian line (typical forsuchproblemsâsee,
e.g.thetwo-stateproblemsolvedinsection5.8ofthelecturenotes),withthecut-offfrequency(i.e.thefluctuationbandwidth)NÎ.Asanadditionalsanitycheck,itshowsthatforasystemconsistingofjust one site (N = 1),Sn(Ï) = 0, i.e. the site occupancy does not fluctuateâof course. As a lessobviouscorollary,thelow-frequencyfluctuationintensity
Sn(0)=1ÏÎNâ1N3,asafunctionofN,reachesitsmaximumalreadyfortwositesandthendecreases.
NotealsothatfortheparticularcaseN=2,theresult isapplicabletotheEhrenfestâsdog-fleasystem(seeproblem2.1)with justone flea.However, in thiscasetheentropydoesnotchange intime,becauseithasthelargestpossiblevalueS=lnN=ln2fromtheverybeginning.
References[1]BurgessR1973Phys.Lett.A4437[2]LaxM1968FluctuationandCoherentPhenomenainClassicalandQuantumPhysics(GordonandBreach)[3]LouisellW1990QuantumStatisticalPropertiesofRadiation(Wiley)[4]VystavkinAetal1974Rev.Phys.Appl.979
1Forthesumcalculation,thewell-knownEq.(A.8b)isused.2HereEq.(A.9a)isusedforthesumcalculation.3Note that these twocasesmaybeconsideredas thenon-interacting limitsof, respectively, theIsingmodel (4.23)and theclassical limitof theHeisenbergmodel(4.21),whoseanalysiswithintheWeissapproximationwasthesubjectofproblem4.18.4See,e.g.sections1.1and4.5ofthelecturenotes,inparticularEq.(1.3),andthediscussionleadingtoEq.(4.90).5Notethatforasysteminanexternalfield,whichmayhaveηâĄâšskâ©â 0,amoreappropriatedefinitionofthecorrelationcoefficient(whichensuresitsdecayatâŁkâkâČâŁââ)isKsâĄâšsËksËkâČâ©âĄâšskskâČâ©ââšskâ©âšskâČâ©.6ThisresultisalsovalidforaclosedIsingring,butonlyifthesitedistancenismuchsmallerthanringâslengthN.(ThisiswhyforaringwithN=3,consideredofthepreviousproblem,Eq.(****)givesthecorrectresultonlyinthelimitJâȘT,whenstrongfluctuationssuppressthedifferencebetweenopenstringsandclosedrings.)ForauniformringwithNâ«1sites,thegeneralexpressionforKsmaybecalculated(evenforhâ 0)usingthetransfermatrixapproachdiscussedinsection4.5ofthelecturenotes;fordetailssee,e.g.section5.3inthebookbyYeomans,citedinthelecturenotes.7Thatformulawasfirstobtainedasearlyasin1909byAEinsteinfromthePlanckâsradiationlaw(whichdoesnottakethegroundstateenergyintoaccount),andisreproducedinsometextbookswithoutproperqualification.Noteagainthatthegroundstateenergyisnotonlymeasurable,butalsoresponsibleforseveralimportantphenomenaâseethediscussioninsection2.6ofthelecturenotes.8See,e.g.Eq.(A.36e)withn=1.9NotethatEq.(**)fortheforceimposedbyasinglemolecule,whichwasprovedabove,doesnotmeanthatÎŽPâȘâšPâ©.Indeed,duetotheindependenceofmolecularhits, ,while ,sothatfortheusualâastronomicalâvaluesNâŒ1023,theratioÎŽP/âšPâ©â1/N1/2ismuchsmallerthan1.10See,e.g.[1]andreferencestherein.Inbrief,theâthermodynamicâderivationimpliesacontinuous,uniformspreadofthemomentum , transferred fromeachparticletothepistonduringonehit,overallthetimeperiodÎt=2L/âŁvâŁbetweentheadjacenthits.Suchaspreadcouldbeachieved,forexample,byreplacingtheusualhardpistonwithaconducting,voltage-biasedlid,inducinganelectricfieldthatwouldpresschargedparticlesofthegastotheoppositelidofthecylinder.Iamnotawareofanypracticalimplementationofsuchasystem.11See,e.g.either[2]or[3].12Thisproblem,conceptually important for thequantummechanicsofopensystems,wasoffered inchapter7PartQMof thisseries,and is repeatedhere for thebenefitofreaderswho,foranyreason,skippedthatcourse.13See,e.g.PartCM section6.4, inparticularEq. (6.40),with ,andanarbitraryconstantd.Due to the linearityof thisequation,valid foreachCartesiancomponentofthe2Dvectorq(z,t),itisvalidforthevectorasthewholeaswell.14ThequantitativeconditionofthissmallnessisâŁâq(z,t)/âzâŁâȘ1.Forthermalfluctuations,inreal-lifeconditions,thisrequirementisalwayssatisfied.15Thisconditionisquiterealisticifthewavespropagatewithsomeattenuationâsee,e.g.PartCMsection6.6.(Ifthisattenuationisnonvanishingbutnottoohigh,itdoesnotaffecttheforthcomingfluctuationanalysis.)16See,e.g.PartCMEqs.(6.45)and(6.47).17This result should not be too surprising, because the support pointâsmotion induces travelingwaves of the string,which carry away from it (âto infinityâ) themechanicalpower âsee.e.g.PartCMEq.(6.49).18Thisproblem,forthecaseofarbitrarytemperature,wasthesubjectofPartQMproblem7.6,withproblem5.15ofthatpartservingasthebackground.However,themethodusedinthemodelsolutionsofthoseproblemsrequiresonetoprescribe,totheoscillators,differentfrequenciesÏ1andÏ2atfirst,andonlyafter thismoregeneralproblemhasbeensolved,pursuethelimitÏ1âÏ2.ThegoalofthisproblemistodemonstratethattheLangevinformalismenablesasolutiontakingÏ1=Ï2âĄÏ0fromtheverybeginning.19See,e.g.PartEMEq.(3.16),inwhichthedipolemomentsaredenotedasp1,2.20See,e.g.PartEMEq.(3.13).Inthesecondformofthisexpression,thez-axisisassumedtobedirectedalongthevectorr.21See,e.g.PartEMEq.(3.15b).NotethefactorsÂœ,whichareduetotheinducednatureofthemomentsdË2,1.22Actually,itmaybereadilyworkedoutbydifferentiation,overaparameter,ofthefollowing(generallyuseful)tableintegral:
â«0âdΟ(aΟ)2+2bΟ+1=Ï23/2(a+b)1/2,fora+b>0,butIdidnotwanttodistractthereaderâsattentionfromphysics.23Thisisaveryusefuladditionalexercise,highlyrecommendedtothereader.24See,e.g.PartCM sections 5.2â5.5.Note that in quantummechanics, a similar approach is called the rotating-waveapproximation (RWA)âsee, e.g.Part QMsections6.5,7.6,9.2,and9.4.25Here,incontrasttoPartCMsection5.4,capitallettersΩandΩ0areisusedtodenotethefrequenciesofthephaselockingforceandtheoscillator,todistinguishthemfromfrequenciesÏoftheFouriercomponentsoffluctuations.26See,e.g.PartCMEq.(5.41).Inthisapproximation,theoscillationamplitudeA(t)maybeconsideredconstantâseePartCMEq.(5.71)anditsdiscussion.27See,e.g.thesecondofPartCMEqs.(4.57a).28Forourcurrentautonomouscase,withÎ=0,thisapproximationisvalidifthespectraldensitySf(Ï)ofthenoiseisvirtuallyconstantinafrequencyinterval(around
theself-oscillationfrequencyΩ0)thatislargerthantheoscillationlinewidthwearecurrentlycalculatingâseebelow.29See,e.g.Eq.(A.2a)withn=âN/ÎÏ.30NotethatthiscalculationessentiallyrepeatsthederivationofEq.(7.89)inPartQMsection7.3.Thisisnatural,becausethequantumstatedephasing,describedinthatsection,isessentiallythedecoherenceofthefundamentaloscillationsofthequantum-mechanicalwavefunctionintime(withfrequencyE/â),undertheeffectoflow-frequencyexternalfluctuationsimposedbytheenvironment.31See,e.g.briefdiscussionsinPartEMsection6.5andPartQMsections1.6and2.8.32Justforthereaderâsreference:Suchagradualdestructionofphaselocking,describedwithintheframeworkofEq.(***),isoneofmostfamousanalyticallysolvablenonlinearproblemsofthefluctuationtheory.Thissolution,firstfoundin1958byRStratonovichandthenrepeatedlyre-discoveredbyothers,maybeexpressedeitherviasomeexoticBesselfunctions(ofanimaginary,continuousorder),orasaseriesincludingtheâusualâmodifiedBesselfunctionsIk(ofanintegerorderk),orjustasthesolutionofaverysimplesystemoflinearalgebraicequationsâsee,e.g.section2in[4].33Ifyouneedto,seeEq.(A.32a).34Pleasenoteagainthat,aswasnotedatthederivationofthatresult,thedirectstatisticalaverageofthenoise-inducedoscillationsq(t)yieldsexactzero,sothatthecorrelationfunctionisthesimplestquantitativetime-domaincharacteristicofthenoise-inducedrandomoscillations.35Notethatthisdiscussionisvalidforthegeneral(quantum)case,âÏâŒT,onlyifbothKq(Ï)andSq(Ï)areunderstood in thesenseof thesymmetrizedfunctionsdefinedbyEqs.(5.95)â(5.96).36See,e.g.Eq.(A.38).37See,e.g.PartEMsection7.6.38See,e.g.PartEMsections4.1and6.6.39Bytheway,thisequivalentcircuitgivesanalternativewaytoderiveEq.(5.81c)ofthelecturenotes.Indeed,iftheresistorisconnectedtoanidealammeter(withzerointernalresistance),weseethatthevoltage vanishes,whilethefluctuationcurrentbecomesequalto ,withthevariance
TogetherwithEq.(5.81b),thisrelationimmediatelygivesEq.(5.81c).40See,e.g.PartEMsection7.6.41Bytheway,thisisexactlythewayHNyquistfirstderivedhistheoremâcorrectlyfortheclassicallimitâÏâȘTandwithaâsmallâerrorforthegeneral(quantum)case.ThiserrorwascorrectedlaterbyHCallenandTWelton,whousedadifferentapproach(andconsideredamoregeneralsituation).42Thisprocessisevidentlymemory-free,i.e.ergodic,sothatthisaveragingmaybeunderstoodaseitherthatovertheensembleofmanydifferentrandomwalks,orovertheensembleofleapsinasinglewalkwithNâ«1steps.43Notethatthesimilarargumentwasusedinsection5.1ofthelecturenotestoderiveEq.(5.12).44Itisevidentthatinthisuniformsystemalltheresultsareindependentofthesitenumber,sothatfromthispointontheindexjisdropped.45Note again the somewhat counter-intuitive nature of Eq. (5.167): it expresses an average for a stationary process via the probability evolution laws for non-stationarycase(withspecialinitialconditions).
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IOPPublishing
StatisticalMechanicsProblemswithsolutionsKonstantinKLikharev
Chapter6
Elementsofkinetics
Problem6.1.UsetheBoltzmannequationintherelaxation-timeapproximationtoderivetheDrudeformulaforthecomplexacconductivityÏ(Ï),andgiveaphysicalinterpretationoftheresultâstrendathighfrequencies.
Solution: For a uniform system, in a uniformly distributed ac field , in the Boltzmann equation(6.18)wemayneglectthetermproportionaltoârw,buthavetoretainthetermâw/ât.Then,inthesamelow-fieldapproximationaswasusedtoderiveEq.(6.25),wegetthefollowinggeneralizationtotime-dependentcases:
Lookingforthevariables andwËassinusoidalfunctionsoftime(proportionaltoexp{âiÏt}),wegetfromEq.(*)thefollowingrelationbetweentheircomplexamplitudes:
ComparingthisresultwithEqs.(6.25),weseethattheonlychangeduetothenon-zerofrequencyÏisthefactor(1âiÏÏ)inthedenominatorofexpressionsfortheprobabilityperturbation,andhenceforthecomplexamplitudejÏoftheelectriccurrent j,givenbyEq. (6.26).Hence, thesamefactorappears in the complex conductivityÏ(Ï), defined by a relation similar to Eq. (6.28), but for thecomplexamplitudesjÏand :
whereÏ(0)isgivenbyEq.(6.29)âandhencebytheDrudeformula(6.32).Forthepurposeofinterpretation,letusrewritethisresultas
Ï(Ï)=ÏâČ(Ï)+iÏâł(Ï),withÏâČ(Ï)=Ï(0)1+(ÏÏ)2,Ïâł(Ï)=Ï(0)ÏÏ1+(ÏÏ)2.It shows that the real part ÏâČ of the conductivity, responsible in particular for the Joule heatgeneration,dropsfastassoonasthefieldfrequencyexceedsthereciprocalrelaxationtime1/Ï(inpractical conductors, from âŒ1011 to âŒ1013 sâ1), while its imaginary part Ïâł first grows withfrequency, and then starts dropping as well, but slower, as 1/ÏÏ. The latter behavior, with Ï-independentimaginaryconductivity
Ïâł(Ï)âÏ(0)ÏÏâĄq2nmÏ,atÏÏâ«1,correspondstocollision-freeoscillationsofparticledisplacements,inducedbytheexternalacfield.
NownotethatthefrequencydependenceofthecurrentdensityatÏÏâ«1issimilartothatofthecurrent in a lumped inductance L (with voltage , and hence ). Due to thissimilarity, the Ïâł given by Eq. (**) is called the kinetic inductance of a conductor, because, incontrast to the usual âmagneticâ inductance, it is due to the finite massm (inertia) of the chargecarriers, rather than the magnetic field it induces. This effect is especially noticeable insuperconductors,whoselinearelectrodynamicsmaybeapproximatelydescribedasthatoftheusualconductors,butwithnegligiblescattering, i.e.withÏ=â,sothatEq.(**),withnandmreplacedwithcertaineffectiveparameters,isvalidinaverybroadrangeoffrequenciesstartingfromzero1.
Problem 6.2. Apply the variable separation method2 to the driftâdiffusion equation (6.50) tocalculate the time evolution of the particle density distribution in an unlimited uniform medium,providedthatatt=0,theparticlesarereleasedfromtheiruniformdistributioninaplanelayerofthickness2a:
n=n0,forâaâ©œxâ©œ+a,0,otherwise.Solution: In the absence of a drift-inducing field, âU = 0, the driftâdiffusion equation (6.50) isreducedtothesimplediffusionequationsimilartoEq.(5.116),
ânât=Dâ2n,withD=ÏTm.Thisequationisisotropic,sothattheinitial1Ddistributionn=n(x,0)remainsone-dimensionalatalllatertimes,anditsevolutionmaybedescribedbythe1Dversionofthediffusionequation:
ânât=Dâ2nâx2.
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(****)
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Letuslookforthesolutionofthisequationinthevariable-separatedformnx,t=âkTktXkx.
Plugging a particular term of this series into Eq. (*), and dividing both parts of the equation byDTkXk,weget
1DTkdTkdt=1Xkd2Xkdx2=constâĄâk2.SolvingthetworesultingsimpleordinarydifferentialequationsforTkandXk,weget
Tk=akexp{âDk2t},Xk=bkcoskx,with (so far) arbitrary ak and bk.3 Since the length of the diffusion segment (ââ â©œ x â©œ + â) isinfinite,thespectrumofeigenvalueskiscontinuous,sothatpluggingthesesolutionsintoEq.(**),weneedtoreplacethesummationoverkwithintegration:
nx,t=â«ââ+âckexp{âDk2t}coskxdk,whereckâĄakbk.
Whatremainsistofindthefunctionckfromtheinitialcondition(att=0):â«ââ+âckcoskxdkâĄ12â«ââ+âck(eikx+eâikx)dk=nx,0âĄn0,forâaâ©œxâ©œ+a,0,otherwise.
AsusualforthereciprocalFouriertransform,letusmultiplybothpartsofthisequationbyeâikâČx,andintegratetheresultoverthewholex-axis.Changingtheorderof integrationontheleft-handside,weget
12â«ââ+âdkckâ«ââ+âdx[ei(kâkâČ)x+ei(k+kâČ)x]=â«ââ+ân(x,0)eâikâČxdxâĄn0â«âa+acoskâČxdx.The inner integral on the left-hand side equals 2Ï[ÎŽ(k â kâČ) + ÎŽ(k + kâČ)],4 so that the outerintegrationiseasy,giving
12(ckâČ+câkâČ)=n02Ïâ«âa+acoskâČxdxâĄn0ÏsinkâČakâČ.
AccordingtoEq.(***),ckâČhastobeanevenfunctionofkâČ,sothat(droppingtheprimesignatk),thisresultbecomes
ck=n0Ïsinkak,andpluggingitintoEq.(***),wemayfoldtheintegralinthatformulatothepositivesemi-axis.Asaresult,weget
nx,t=2n0Ïâ«0+âexp{âDk2t}sinkakcoskxdk.
Plots of this distribution for several values of the normalized time , where , are
showninthefigureabove.Theplotsshowthattheinitiallyrectangulardistributionoftheparticledensityfirstsmearsattheedgesveryfast,butthelater(at )spreadoftheparticlesbecomesslowerandslowerwithtime.Thisisverynaturalinthelightofthebasiclaw(5.77)ofdiffusionofasingle particle (equivalent to the delta-functional initial distribution ofn), in our current notationreading
ÎŽx=2Dt1/2.Reversing thesamestatement into the timedomain,wemaysay that thecharacteristic timeofasubstantialchangeoftheparticledistributionisnot ,butratherÎtâŒ(Îx)2/D,whereÎxisthespatialwidthofthesharpfeature(s)ofthedistribution.(At ,thisisthefullwidthÎxâ«aofthewholedistribution,sothat .)
Problem6.3.SolvethepreviousproblemusinganappropriateGreenâsfunctionforthe1Dversionofthediffusionequation,anddiscusstherelativeconvenienceoftheresults.
Solution: The spatialâtemporal Greenâs function of any linear, homogeneous, partial differentialequationin1+1dimensions(1coordinate+time)maybedefinedbythefollowinggeneralformulaforthesolutionoftheequationatt>t0:
nx,t=â«ââ+ân(x0,t0)G(x,t;x0,t0)dx0.Appliedtothedelta-functionalinitialconditions,thisdefinitionyields
n(x,t)=G(x,t;x0,t0),ifn(x,t0)=ÎŽ(xâx0).
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Butsuchasolution(witht0=0)ofthediffusionequation,givenbyEq.(*)ofthepreviousproblem,andequivalenttoEq.(5.114)ofthelecturenotes,
ânât=Dâ2nâx2,forââ<x<+â,wasalreadydiscussedinsection5.6âseeEqs.(5.112)and(5.113).Fromtheserelations,
G(x,t;x0,t0)=12Ï1/2ÎŽxtexpâ(xâx0)22ÎŽxt2,withÎŽxt=2D(tât0)1/2,sothatEq.(*),witht0=0,becomes
nx,t=14ÏDt1/2â«ââ+ân(x0,0)expâ(xâx0)24Dtdx0.
Fortheinitialconditionsspecifiedinthepreviousproblem,nx,0=n0,forâaâ©œxâ©œ+a,0,otherwise,
wegetnx,t=n04ÏDt1/2â«âa+aexpâ(xâx0)24Dtdx0.
This integral may be readily expressed via a difference of two values of the so-called error
functionerfζâĄ2Ï1/2â«0ζexp{âΟ2}dΟ;
however, for most practical purposes the explicit integral form (**) is preferable. In particular, ityields exactly the same plots of function n(x, t) as shown in the model solution of the previousproblemâdespitethesubstantialdifferenceoftheexpressionforms.Indeed,theresult(****)oftheprevious problem is just the Fourier-integral expansion of Eq. (**). However, for practicalcalculations, there isabigdifferencebetweenthesetwo integral forms: thereal-space integral inEq.(**)convergesfasteratrelativelysmalltimes, whilethereciprocal-spaceintegral,obtainedfromthevariableseparation,convergesfasterat ,whenonlyrelativelysmallvaluesoftheeffectivewavenumberk,withâŁkâŁâŒ1/(Dt)1/2âȘ1/a,givenoticeablecontributionsintoit.
Problem6.4.*Calculatethedcelectricconductanceofanarrow,uniformconductinglinkbetweentwo bulk conductors, in the low-voltage and low-temperature limit, neglecting the electroninteractionandscatteringinsidethelink.
Solution: As was discussed in section 3.3 of the lecture notes, atT â 0 any fermions (includingelectrons) in equilibrium fill all eigenstates up to the Fermi energy ΔF. As we know from thediscussioninsection6.3(see,inparticular,figure6.5c),ifconductorsareweaklyconnected,withnovoltageappliedbetweenthem,theirFermilevelsbecomealigned.Thedcvoltage appliedbetweentwoconductorsshiftsalltheirenergyspectraby ,sothatthesingle-particleenergydiagramofthewholesystemlooksassketchedinthefigurebelow.
Let us assume that the link is uniform5; then in the absence of scattering, the longitudinal
componentΔxoftheelectronâsenergyisconserved.Inthiscase,asthefigureaboveshows,itcanmovefromconductor1toconductor2onlyifthisenergyiswithintherange6
whereΔ℠istheeigenenergycorrespondingtothetransversecomponentÏâ„(y,z)ofthestationaryorbitalwavefunction
Ïr=aÏâ„y,zexp{ikxx},sothatthefullenergyoftheelectronisΔx+Δâ„,where
Δx=â2kx22m.
If theappliedvoltage is low, , thenumberNofdifferentelectronstateswith thesametransversewavefunctionÏâ„(y,z),maybecalculatedas
where the front factor of 2 is due to two possible electron spin states with the same âorbitalâwavefunctionÏ(r),whiledN/dΔxisthe1Ddensityoftheorbitalstates,whichmaybecalculatedas7
dNdΔx=dNdkx/dΔxdkxâĄl2Ï/dâ2kx2/2mdkxâĄl2Ï/â2kxmâĄl2Ï/21/2âΔx1/2m1/2.Withthisresult,Eq.(*)becomes
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Eachofthesetraveling-wavestatescarriestheprobabilitycurrent8
Iw=âkxmâŁaâŁ2â«âŁÏâ„y,zâŁ2dydz.ThewaveâsamplitudeâŁaâŁhastobecalculatedfromthenormalizationcondition
â«âŁÏâŁ2d3râĄâ«0lâŁaeikxxâŁ2dxâ«âŁÏâ„y,zâŁ2dydzâĄlâŁaâŁ2â«âŁÏâ„y,zâŁ2dydz=1.Fromhere,weget
âŁaâŁ2â«âŁÏâ„y,zâŁ2dydz=1l,i.e.Iw=âkxlm=1l2Δxm1/2.
NowwemayuseEqs.(**)and(***)tocalculatethefullelectriccurrentcarriedbyonepopulatedtransversemodeÏâ„(y,z)(frequentlycalledtheballisticchannel):
Thismeans that theelectricconductancedue toonepopulatedchannel isgivenbyawonderfullysimpleexpression:
ThisresulthadbeenderivedbyRLandauerin1957,butattractedcommonattentiononlyinthe
late1980s,whentheeffectoflongitudinalconductancequantizationwasobservedexperimentallyinnarrow links formed by negatively biased gate electrodes in 2D electron gas in semiconductorheterojunctionsâsee,e.g.thefigurebelow.Asthenegativegatevoltageisreduced,thelinkâswidthwisincreased,sothatthetransversequantizationenergyΔâ„âÏ2â2/2mw2+constisreduced,andat certain gate voltage values, new and new ballistic transverse channels become populated,increasingtheconductancebydiscretestepsequaltoGq.
The geometry of a typical conductance quantization experiment using a semiconductorheterojunction,anditsresult.VG1&G2isthevoltageappliedtotheâgateâelectrodesG1andG2(markedintheinset)usedtosqueezethe2Delectrongasfromunderthemandthusform(andcontrol thewidthof)aquasi-1Dconducting linkbetweentwobroaderconductingelectrodes.Adaptedfrom[1].©IOP,reproducedwithpermission.
The most important feature of Eq. (****) is its independence of the electron massm, channeldimensions,andanyotherparametersoftheusedsample.Asimilar(but,duetothesuppressionofback scattering by magnetic field, much more robust and hence more precise) conductancequantization takes place at the quantum Hall effect9. Note also the similar effect of thermalconductancequantization10.
SinceEq. (****)wasderivedneglectingelectronscattering, it isalso interestingto thinkaboutthephysicsofthequantumconductanceGq,inparticularinthecontextofthecorrespondingJouleheat power . The extra energy , picked up by each electron during its passagethroughtheballisticchannel,isturnedtoheatnotinsidethechannel(wherethereisnoscatteringand hence no energy dissipation), but somewhere inside of one of the bulk electrodes, due to agraduallossofthegainedenergyviainelastic(e.g.electronâphonon)interactions.ThisisonemoretwistoftherelationbetweentheelasticandinelasticscatteringatOhmicconductance,whichwasdiscussedintheendofsection6.2ofthelecturenotes.
Problem 6.5. Calculate the effective capacitance (per unit area) of a broad plane sheet of adegenerate2Delectrongas,separatedbydistancedfromametallicgroundplane.
Solution:Accordingtothesolutionofproblem3.20(seealsoproblem3.8),inthedegeneratelimit(TâȘÎŒ)theFermienergyofagasofNparticlesconfinedtoa2DsheetofareaAis
ΔFâĄÎŒâŁTâ0=Ïâ2mNA.At that calculation, the Coulomb interaction effects have been neglected. The most important oftheseeffects11isthatthedistributedelectricchargeofthegas,withthearealdensity
ÏâĄQA=qNA,createsauniformelectricfieldwithmagnitude inthegapbetweenthegasandthegroundplane, where Îș is the dielectric constant of the material filling the gap12. As a result, theelectrostaticpotentialofthelayer(relativetothatofthegroundplane)becomes
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sothattheelectrochemicalpotential(6.40)ofthelayerbecomesequaltoÎŒâČâĄqÏ+ÎŒ=q2dÎșΔ0NA+Ïâ2mNAâĄÏqdÎșΔ0+Ïâ2mq2.
Aswasdiscussed in section6.3, thenetelectrochemicalpotentialÎŒâČ (dividedbyq) iswhatwe
usuallymeasureasthevoltage , inthiscasebetweentheelectrongasandthegroundplane.Aswe know from the elementary electrostatics, capacitance C is just the ratio , so that thecapacitanceperunitarea(C/A)istheratio ,andEq.(*)mayberepresentedas
CAâ1=dÎșΔ0+Ïâ2mq2.In thecircuit theory language, thismeans that theeffectivecapacitanceC of the2Delectrongassheetisaconnection,inseries:
1C=1Ce+1Cq,oftheusual,âelectrostaticâcapacitance
Ce=ÎșΔ0dA,andtheso-calledquantumcapacitance13
Cq=mq2Ïâ2A.
Thephysicsofthiseffectisprettystraightforward:byitsdefinition,theelectrochemicalpotentialis the average energynecessary to addoneparticle from the environment (in our case, from theground plane) to the system. In our case such addition requires, first, to overcome the Coulombrepulsionoftheelectronsalreadyinthesheetand,second,togivetoonemoreparticletheFermienergyofthegasΔFtofillthelowestquantumstatenotyetoccupiedâjustaboveΔF.Inthesimplemodelanalyzedabove, theseenergyadditionsare independent, leading to theadditionof the twocontributionstotheeffectivepotentialÎŒâČ,andhencetotheadditionofthereciprocalcapacitancesdescribingeachoftheenergycomponents.
Forfreeelectrons(with âŁqâŁ=eâ1.6Ă10â19Candm=meâ0.91Ă10â30kg), thequantumcapacitance is quite macroscopic, Cq/A â 0.67 F mâ2, and becomes even smaller (so that theimportantfraction1/Cqbecomeslarger)insemiconductorswithlowereffectiveelectronmassâe.g.mâ0.2me in the conduction band of Si. Because of that, its effects may be noticeable in somemodern electron devicesâmost importantly, in the ubiquitous silicon field-effect transistors withtheirverythingateinsulatinglayers.(ForcomparisonwiththeaboveestimateofCq/A,theratioCe/Aformodernhigh-Îșgate-oxidelayersmaybeoftheorderof0.1Fmâ2.)
Problem 6.6. Give a quantitative description of the dopant atom ionization, which would beconsistentwiththeconductionandvalencebandoccupationstatistics,usingthesamesimplemodelofann-dopedsemiconductorasinsection6.4ofthelecturenotes(seefigure6.7a),andtakingintoaccountthatthegroundstateofthedopantatomistypicallydoublydegenerate,duetotwopossiblespin orientations of the bound electron. Use the results to verify Eq. (6.65), within the displayedlimitsofitsvalidity.
Solution:Forspellingouttheelectroneutralitycondition(6.62),n=p+n+,
we need to express the concentrations n,p, and n+ via the parameters ÎŒ andT (in equilibrium,common for all components of the system), at the given energies ΔV,ΔC, andΔDâsee figure 6.7a,reproducedbelow.Aswasarguedinsection6.4ofthelecturenotes,atTâȘÎâĄÎ”CâΔV,Eqs.(6.58),
n=nCexpÎŒâΔCT,p=nVexpΔVâÎŒT,
are independent of thedopant excitation statistics, soweneedonly to express thenumbern+ ofactivated (ionized) dopants via the above parameters and the full concentrationnD of thedopantatoms.
This may be done similarly to the calculation ofN0 in the solution of problem 3.9.14 With theassumptiongivenintheassignment,thedonoratommaybeineitherofthreedifferentstates:oneionizedstate,withoutanelectron,withacertainenergyΔa,andanyoftwogroundstates,withoneelectronofanyspinorientation,andwiththeenergyΔa+ΔD.ApplyingthegeneralEqs.(2.106)and(2.107)toagrandcanonicalensembleofsuchsystems,forthecorrespondingprobabilitieswemaywrite
W0=1ZGexpâΔaT,W1=1ZGexpÎŒâ(ΔD+Δa)T,with
ZG=expâΔaT+2expÎŒâ(ΔD+Δa)TâĄexpâΔaT1+2expÎŒâΔDT,sothattheprobabilitiesW0,1areindependentofΔa(astheyshouldbe):
W0=11+2exp(ÎŒâΔD)/T,W1=exp(ÎŒâΔD)/T1+2exp(ÎŒâΔD)/T.Fromhere,theaveragenumberofionizedatoms(perunitvolume)is
(****)
n+=nDW0=nD1+2exp(ÎŒâΔD)/T.Asasanitycheck,Eq.(***)showsthatatfixedÎŒandT,thefractionn+/nDof theactivateddonorsincreasesasΔDisincreasedâasitshouldaccordingtotheenergydiagramshownabove.
Withthisexpression,theelectroneutralitycondition(*)takestheformnCexpÎŒâΔCT=nVexpΔVâÎŒT+nD1+2exp(ÎŒâΔD)/T.
Generally, this transcendental equation for ÎŒ cannot be solved analytically. However, it may bereadilyusedtoplottheelectrondensityn,givenbythefirstofEqs.(**),asafunctionofnD,withthechemicalpotentialÎŒusedastheparameterâseethesolidredlineinthefigurebelow,calculatedforparameterstypicalforsemiconductors.(Notethelogâlogscaleoftheplot,coveringmanyordersofmagnitudeofbothdensities.)
TheplotshowsthatatTâȘÎ,therearethreedistinctbranchesofnasafunctionofthedopant
densitynD.
(i) If the density is very low, nD âȘ ni,15 the last term in Eq. (****) is negligible, so that thesemiconductor remainspractically intrinsic,withnâpâni, and the chemical potential near themidgap:ÎŒâ(ΔC+ΔV)/2.
(ii)Inthe(mosttypical)casewhennDisincreasedwellbeyondni,thenumbern+ofactivatedatomsandtheelectrondensitynarevirtuallyequaltonD(andhencetemperature-independent),withthehole density p decreasing accordingly, and the chemical potential gradually rising toward theconductionbandedgeâseeEqs.(6.65)ofthelecturenotes:
nânD,pâni2nDâȘn,ÎŒâΔCâTlnnCnD.
Note that this result means that the donor atoms may be fully activated even if the apparentionizationenergy(ΔCâΔD)ismuchlargerthanTâasit isintheexampleshownintheplotabove.Themathematicalexplanationofthiscounter-intuitivefact isgivenbyEq.(***):whatis importantfor the full activation is theFermi levelÎŒ tobewell (bya fewT) belowΔD, andaccording toEq.(****),thevalueofÎŒparticipating inthisexpressiondependsontheconductionandvalencebandstatisticsaswell.ThehandwavingphysicalinterpretationIcanofferisthatatnDâ«ni,therelativelyabundantelectronswiththeenergyΔDmayreadilygodowntotheFermilevel(playingtheroleofaneffectiveexternalparticlesourceâcf. figure2.13of the lecturenotes)witha lowerenergyÎŒ,andfromtherebere-distributedintotheconductionandvalencebandsâmostlytotheformerone.Suchinterpretation of the Fermi level as a virtual reservoir of particles is generally useful for a semi-quantitativeanalysisofothersystemsaswell.
(iii)Finally,ifnDbecomessohighthattheÎŒexpressedbyEq.(6.65)ofthelecturenotesenterstheT-widevicinityofthedopinglevelΔD,thelastterminEq.(****)becomessomewhatlowerthannD,causing a proportional reduction ofnâsee the deviation of the solid red line from the (straight)dashed one in the top right corner of the figure above. However, as the plot shows, for typicalparametervaluesthiseffectisrelativelyminor,andEq.(6.65)isvirtuallyprecisewithinmanyordersofmagnitude.
Problem6.7.Generalizethesolutionof thepreviousproblemtothecasewhenthen-dopingofasemiconductor is complemented with its simultaneous p-doping by nA acceptor atoms per unitvolume, whose energy ΔA â ΔV of activation, i.e. of accepting an additional electron and hencebecominganegativeion,ismuchlessthanthebandgapÎâseethefigurebelow.
(*)
(**)
(***)
(****)
Solution:Inthiscase,theelectroneutralityconditionshouldalsotakeintoaccountthedensitynâofnegativelyionizedacceptorions,becoming
n+nâ=p+n+.Thedensitynâmaybecalculatedjustliken+wascalculatedinthesolutionofthepreviousproblem,onlytakingintoaccountthedifferencebetweentheelectronsandholes.Theresultis16
nâ=nA1+2exp(ΔAâÎŒ)/T.Withthisexpression,andtheformulasforn,p,andn+giveninthesolutionofthepreviousproblem,Eq.(*)becomes
nCexpÎŒâΔCT+nA1+2exp(ΔAâÎŒ)/T=nVexpΔVâÎŒT+nD1+2exp(ÎŒâΔD)/T.
ThistranscendentalequationforÎŒcannotbesolvedanalyticallyinthegeneralcase.However,itmay be used to plot, for example, the reciprocal dependence of the donor doping level nD, as afunctionoftheÎŒityields,forseveralvaluesofnAâseethefigurebelow.
Theplotsclearlyshowthatat theusualconditionsTâȘÎandnD,nAâȘnC,nV, there are three
distinctrangesofdoping,whereEq.(**)yieldssimpleresults:
(i) At nD ⌠nA, the value of Ό is somewhere around the midgap, the exponents in the termsproportionaltonAandnDaremuchsmallerthan1(showingthatthedopantatomsofbothtypesarefullyactivated),andEq.(**)isreducedto
n+nA=p+nD.ThisequationissimilartoEq.(6.63)ofthelecturenotes,justwithnDreplacedwiththedifference(nDânA).Withthisreplacement,itssolutionisgivenbyEq.(6.64):
n=nDânA2+(nDânA)24+ni21/2,p=nAânD2+(nDânA)24+ni21/2âĄnni2,whereniistheintrinsiccarrierdensitynigivenbyEq.(6.60):
ni=(nCnV)1/2expâÎ2T.Themostimportantfeatureofthisresultisthat
nâp=nDânA,sothatthesignof(nâp),i.e.oftheeffectivechargeofcarriers,maybecontrolledbydoping.Suchcompensated semiconductors are convenient for some special applications, but in mostsemiconductordevices,oneofthefollowingtwolimitsisused.
(ii)Ifni,nAâȘnD(butnDisstillmuchlowerthantheeffectivedensitynCofstatesintheconductionband),thesituationisreducedtothestrongn-dopinganalyzedinsection6.4ofthelecturenotes,whereEqs.(6.65)arevalid:
ÎŒâΔCâTlnnCnD,nânD,pâni2nDâȘn.The dashed straight line in the figure above shows the first of these approximate equalities. TherelativelyminordeviationsfromitinthetoprightcorneroftheplotsareduetothecloseapproachofÎŒtoΔDâseethepreviousproblem.
(iii)Intheoppositelimitofdominatingp-doping(ni,nDâȘnAâȘnV),Eq.(**)isreducedtotheequallysimpleEqs.(6.67)ofthelecturenotes:
ÎŒâΔV+TlnnVnA,nânA,nâni2nAâȘp,sothatthecarrierdensitiesandtheFermilevelinthiscasearevirtuallyindependentofnDâseetheleft,nearly-verticaltailsoftheplotsinthefigureabove17.
Problem 6.8. A nearly-classical gas of N particles with mass m, is in thermal equilibrium attemperatureT,inaclosedcontainerofvolumeV.Atsomemoment,anorificeofaverysmallareaAisopen inthecontainerâswall,allowingtheparticlestoescape intothesurroundingvacuum18. InthelimitofverylowdensitynâĄN/V,usesimplekineticargumentstocalculatethermsvelocityofthe escaping particles during the time when the number of escaped particles is still negligible.FormulatethelimitsofvalidityofyourresultsintermsofV,A,andthemeanfreepathl.
Solution:Assumingthe linearsizesof theorificearemuchsmaller thanthoseof thecontainer,sothatAâȘV2/3,letuscalculatetheaveragevelocityoftheparticlesthathittheorificeareaduringatimeintervalÎtâ«A1/2/ ,where âĄ(3T/m)1/2istheirrmsvelocityinequilibriumâseeEq.(3.9)ofthelecturenotes.Forthat,letusalsoassumethat,simultaneously,Îtismuchlessthan ,sothattheparticle collisions at their path to the orificemaybeneglected; thenonly theparticles flyingdirectly toward the hole area may hit itâsee the figure below. Moreover, the velocities of suchparticlesshouldsatisfythecondition â©Ÿr/Ît,whereristheinitialdistanceoftheparticlefromthehole.(Ifthelinearsizescaleoftheorifice,A1/2,ismuchlessthanthisr,itisnotimportantexactlywhichpartoftheholewearespeakingabouthere.)
Thenumberofsuchparticles,withvelocitieswithinasmallrange , isproportionalto
:
wheretheprobabilitydensityw( )obeystheMaxwelldistribution,
(Sincetheparticlesfromeachpointmayreachtheholeareaonlyiftheyflyinacertaindirection,thedistribution shouldbe for oneCartesian component of the velocity only.)As a result,wemaywrite
wherec is some âconstantââwhichmaydependon theareaof theholeand thevelocitydirection.Now the average 2 of the molecules hitting the hole area, from a certain direction, may becalculatedas19
Thisresultdoesnotincludethevelocitydirection,andhenceisvalidforthewholeparticleflux.It
also is independent of the time interval Ît, but since particle collisions were neglected at itsderivation,itmaylooklikeitisonlyvalidforintervalswithintheinitiallyassumedrange
However,ifthelossofparticlesissufficientlysmall,asassumedintheassignment,theeffusiondoes
not change the statistical distribution of the particles, because their mutual scattering tends torestore it at each point. (The process is frequently called thermalization.) Hence Eq. (**) isapplicable to each sequential time interval after the orifice opening, even at tâ« A1/2/ . Note,howeverthattheinterval(***)disappearsiflisreducedtoapproachthescaleofthelinearsizesoftheorifice,A1/2.HencetheimportantconditionofvalidityofouranalysisisA1/2âȘl.(Wehavealsoneglected possible reflections of the particles from the orifice areaâthe assumption correct inparticularifthewallthicknessismuchsmallerthanA1/2.)
NotethataccordingtoEq.(**),theescapingparticlesare,onaverage,hotterthanthegasasawhole:
Hencetheeffusiontendstocoolthegas,andthemaintenanceofitsthermalequilibriumwithfixedtemperatureTwouldrequireasupplyofheatfromanexternalsource.
Problem6.9.Forthesystemanalyzedinthepreviousproblem,calculatetherateofparticleflowthroughtheorifice(theso-calledeffusionrate).Discussthelimitsofvalidityofyourresult.
Solution:Letuscalculatetheeffusionrate,whichmaybedefinedasÎâĄâdNdt>0,
usingthesameballisticapproachasinthemodelsolutionofthepreviousproblem.Considerasmallgroupofparticleshavingacertainvelocityv,withangleΞtothedirectionnormaltothewallwiththe orifice in itâsee the figure below. The area dAâČ of transverse displacements (normal to thevelocityvectorv),leadingtotheparticleâspassagethroughtheorifice,isAcosΞ,whiletherangedrofradialdistances,leadingtosuchapassageduringasmalltimeintervaldt,isdr= ,sothatthenumberofsuchescapingparticlesis
wherenâĄN/Visthespatialdensityoftheparticles,andw(v)isthe3DMaxwelldistribution,givenbyEqs.(3.5)and(3.6)ofthelecturenotes,rewrittenintermsofvelocitiesv=p/m:
Since inEq.(*)isthemagnitudeoftheparticlevelocity,theproduct cosΞparticipatinginthat
expression is just its Cartesian component, , normal to the wall, so that the summation of thecontributions(*)overallvelocitiesmayberepresentedintheCartesianform
where vâŁâŁ is the velocity within the plane of the wall, and the integration is limited only to thevelocitiesdirectedtowardthewall,i.e. â©Ÿ0.Sincew(v)mayberepresentedasaproductofthreesimilarCartesiandistributions(seeEq.(3.5)again),eachofthemnormalizedto1,theinnerintegralis
sothatweneedtoactuallyintegrateonlyinonedirection:
sothattheeffusionrateis
where =(3T/m)1/2isthermsvelocityoftheparticlesâseeEq.(3.9)ofthelecturenotes20.ThisresultisonlyvalidifthecharacteristiceffusiontimeÏ=N/Îissufficientlylong:
i.e.theconditionassumedfromtheverybeginning.AnotherconditionofapplicabilityofEq.(**)isthat the density n of particles and their temperature T are kept constant (which may require acontrolmechanismwiththeresponsetimemuchshorterthanÏ).Inaddition,justasinthepreviousproblem,theresultrequiresthemeanfreepathtobemuchlongerthanthelinearsizeoftheorifice:lâ«A1/2(andthewallthicknesssmallerthanA1/2).Notethatsincethemeanfreepath inatypicalgasatambientconditionsisverysmall(e.g.âŒ70nminair),thisconditionmaybefulfilledonlyfor
(**)
extremelysmallorifices.However,itistypicallysatisfiedintheso-calledmolecularovens,usedforemittingultra-pureatomicandmolecularbeamsinhighvacuum(inparticular,forepitaxialthin-filmdeposition21andisotopeseparation),whereEq.(**)servesasthebaselineformulafortheeffusionrate.
Problem6.10.Usesimplekineticargumentstoestimate:
(i)thediffusioncoefficientD,(ii)thethermalconductivityÎș,and(iii)theshearviscosityη,
ofanearly-idealclassicalgaswithmeanfreepathl.ComparetheresultforDwiththatcalculatedinsection6.3ofthelecturenotesfromtheBoltzmann-RTAequation.
Hint: In fluid dynamics, the shear viscosity (frequently called simply âviscosityâ) is defined as thecoefficientηintherelation22
where is the jâČth Cartesian component of the tangential force between two parts of a fluid,separated by an imaginary interface normal to some directionnj (with jâ jâČ, and hencenjâ„ njâČ),exerted over an elementary area dAj of this surface, and v(r) is the velocity of the fluid at theinterface.
Solution:Thecommonapproachtothecalculationofallthesekineticcoefficientsistoconsider,justasmentioned intheHint,an imaginaryplane interface,normal tosomeaxisnj.Letusconsiderasubsetofparticles,withthenumberdnperunitvolume,whosevelocityinthedirectionnormaltotheinterfaceisclosetosomevalue .Ifthegasisinequilibrium,thenduringasmalltimeintervaldt,asmallareadAjoftheinterfacewillbecrossedonlybysuchparticleswithinthedistancefromitâhalfoftheminonedirection(theplussignbelow),andhalfintheoppositeone(theminussign):
Thisexpressionisthebasisfortherequiredparticularestimates.
(i)According toEq. (6.52)of the lecturenotes, thediffusioncoefficientDmaybedefinedvia thelinearrelationbetweenthedensityjnoftheaverageflowofparticles,andthesmalldensitygradientthatcausesthisflow(i.e.thediffusion):
jn=âDân.FortheelementaryareadAjofourimaginaryinterface,thismeansthattherateofthenetparticleflowthroughitis
dInâĄ(jn)jdAj=âDânârjdAj.
Forourmodel,theleft-handsideofthisequalityisjustthesumoffractions(dN+âdNâ)/dtforparticleswithallvelocities .WiththedirectsubstitutionofEq.(*)wewouldgetzeroresult,butatanon-zerogradientofn,andhencedn,wehavetomodifythatrelationas
wherer±aretwopointsinwhichaparticlecrossingtheinterfacehaditslastscatteringevents(andhence, on average, equilibrated with other particles at this location)23. Considering the gradientâ(dn)/ârjsufficientlysmall(onthescaleofdn/l),wemayTaylor-expanddninsmall(r±)j(withtherjoftheinterfacetakenforthereference),andlimittheexpansiontotwoleadingterms:
Subtractingthem,weget
where rj is the jth Cartesian component of the vector râĄr+ â râ. Though nominally r± are thepoints of the last scattering events of two different particles (before each of them crosses theinterface),rshouldhaveapproximatelythesamestatisticsastwicethevectorofasingleparticleâsdisplacementbetweenitstwosequentialscatteringevents.Sincerjiscorrelatedwith : ,butthestatisticsofthescatteringtimesÏshouldbeindependentofthatofthevelocitydirections,wemayusethevelocityâsisotropytowrite
whereÏ,justasintheapproximateBoltzmannequation(6.17),meanstheaveragescatteringtime.Asaresult,summing(dN+âdNâ)overallmolecules,andassumingÏtobeindependentofthe
particleenergy(asitisintheBoltzmann-RTAequation)24,weget
andcomparisonwithEq.(**)yields
(***)
whereatthelaststep,themeanfreepathâsdefinition(6.51c)wasused.ThisestimateagreeswithEq.(6.51b)ofthelecturenotes.However,inviewoftheapproximate
treatmentofthecollisionstatisticsinthisanalysis,andthephenomenologicalnatureofEq.(6.17),such an exact agreement of the numerical coefficient cannot be considered more than a happycoincidence.
(ii)NowletususethesameapproachforcalculationofthethermalconductivitycoefficientÎș,whichmaybedefinedviaEq.(6.105)ofthelecturenotes.LetusassumethatthechemicalpotentialÎŒofthegasanditselectrochemicalpotentialÎŒâČareconstant25;thenthisrelationissimply
jh=âÎșâT,i.e.(jh)j=âÎșâTârj,where jh is the energy flow density. In our simple model, we may calculate the jth Cartesiancomponentofthedensityasthesum(overallvelocities )ofthecontributions26
dN+Δ(râ)âdNâΔ(râ)dAjdt,wheredN±nowmaybetakendirectlyfromEq.(*),butthedifferenceoftheparticleenergiesΔontheoppositesidesof the interface,dueto thegradientâT/ârj,has tobe taken intoaccount. If thegradient is sufficiently small (much smaller than T/l), we may treat the energy as the particleconcentrationintheprevioustask,getting
Δ(râ)âΔ(r+)âââΔârjrj.Now the summation of contributions from all particles, again with the assumption of constant Ï,yields
Though one of three Cartesian components of the translational kinetic energy of theparticleiscorrelatedwith ,forasimpleestimatewemayignorethiscorrelation27,taking
wherecVâĄââšÎ”â©/âTisthespecificheatperparticle28.NowthecomparisonwithEq.(***)yields
A comparison of this result with that following from the Boltzmann-RTA equation will be the
subjectofproblem6.12.
(iii)Instatisticalphysics,thevelocityvinthedefinitionofη,givenintheHint,shouldbeunderstoodasthestatisticalaverageâšvâ©oftheparticlevelocities:
where is the average force exertedby the âupperâ part of thegas (with rj>0) on its âlowerâcounterpart.(Theforceexertedontheâupperâpartisevidently .)Inourmodel,eachparticleofthesubsetsdN+anddNâ carries, through the interface, a tangentialmechanicalmomentumwiththejâČthcomponentequaltomvjâČ.Asaresult,thecontributionofthednmoleculeswiththevelocitiesclosetosomevintothenetmomentumtransferredacrosstheareadAjâČduringthetimeintervaldtinthepositivedirection,i.e.totheaverage ,is
Nowactingjustasintheprevioustasksofthisproblem,inthepresenceofasmallgradientwemaywrite
sothatthesummationoverallparticlesgivesthefollowingestimate29
ComparingthisexpressionwithEq.(****),wegettheviscositycoefficient
Problem 6.11. Use simple kinetic arguments to relate the mean free path l in a nearly-idealclassicalgas,with the fullcross-sectionÏofmutualscatteringof itsparticles30.Use the result toevaluate the thermal conductivity and the viscosity coefficient estimates, made in the previousproblem,forthemolecularnitrogen,withthemolecularmassmâ4.7Ă10â26kgandtheeffective(âvan der Waalsâ) diameter def â 4.5 Ă 10â10 m, at ambient conditions, and compare them withexperimentalresults.
Solution: Let us consider scattering of a uniform, parallel flux of particles by a single scatteringcenter.Bydefinition31,itsfullcross-sectionofis
(*)
(**)
ÏâĄaveragenumberofscatteredparticlesaveragenumberofincidentparticlesperunitarea.Inamediumwitharelatively lownumbernâȘÏâ3/2ofsimilarscatteringcentersperunitvolume,the scattering events may be considered as independent. Let us consider a slab of areaA andasmall thickness dx, made of such a medium. The number of scatterers in it is nAdx. The totalscatteringarea,asseenbytheincidentparticlespropagatingalongtheaxisx,isÏnAdx.Duetotheassumedscatteringeventindependence,thefractionoftheparticlesscatteredbyallthesecenters,is(ÏnAdx)/AâĄÏndx.Thismeansthatthefluxjnofstill-unscatteredparticlesisreduced,atthissmalldistance,byâjnÏndx,givingthefollowinglawofitsdecayalongthepropagationaxis:
âjnâx=âÏnjn.
Ontheotherhand,thesamedecayoftheincidentfluxmaybedescribedinthelanguageoftherelaxation-timeapproximationâseeEq. (6.17)of the lecturenotes. Inthepicturewherethe initialflux of the particles had been initially uniform over the volume, and then the scattering wassuddenly turnedoneverywhere, itdescribesaspace-uniformdecayof thenon-equilibriumpartoftheprobabilitydensityw,andallitsfunctionalsincludingjn,withthetimeconstantÏ:
âjnât=âjnÏ.
The comparison of these two expressions yields the following relation between the passeddistancedxinthefirstonecaseandthepassedtimedtinthesecondcase
dxdt=1ÏnÏ.If the incident particles move in the same direction with the same velocity amidst immobilescatteringcenters,thenwemayalsowritedx/dt= .Iftheparticlevelocitiesdifferindirectionandmagnitude,butthescatteringcentersarestillimmobile,itisfairertoreplace inthisrelationwithits rmsvalue (3.9): .However, in thegaswhere thescatterersaresimilarparticles,andmovewithsimilarvelocities,abetterestimateofdx/dt isgivenby thermsvalueofrelative velocity vrel ⥠v1 â v2 of the mutually scattering particles. This value may be readilycalculatedassumingtheindependenceofthevectorsv1,2:
where withoutanindexreferstothevelocityofasingleparticleinanimmobile(âlabâ)referenceframe.Inthiscasewemaywritetheestimate
andcomparingitwithEq.(*),gettheapproximateequality
Nowusingthedefinitionofthemeanfreepass,givenbyEq.(6.51b)ofthelecturenotes, ,wefinallyget
lâ12nÏ,i.e.nl=12Ï.
Forapproximateestimates,manymoleculesmaybetreatedashardsphereswithcertaineffective(âvanderWaalsâ)diameterdef.(Thisisessentiallythehardballmodelthatwasdiscussedinsection3.5ofthelecturenotes,withr0=def/2.)Inthisapproximation,themoleculespassingbyeachotherdonotscatteriftheirimpactparameter32islargerthandef,sothat
Ï=Ïdef2.For many molecules, def is virtually constant within a broad range of kinetic energies Δ of thecollidingparticles;inthiscaseEq.(**)showsthatlistemperature-independent,whiletherelaxationtimeÏisnot:
Thismeans,inparticular,thattheconstant-Ïapproximation,usedinmostofchapter6ofthelecturenotes,cannotworkwellinthiscase.Note,however,thatthisscalingisvalidonlyforaclassicalgas,andonlyformutualscattering,andonlywithinthehardballmodel33.Forexample,itisnotvalidforthe important cases of impurity or phonon scattering of electrons in metals. (Unfortunately, adetaileddiscussionofthesecasesiswellbeyondtheframeworkofthiscourse,andIhavetorefertheinterestedreadertooneofthetextbooksonsolidstatephysics34.)
Nowreturningtotheestimatesmadeinthesolutionofthepreviousproblem,
wemayuseEq.(**)torewritethemas
Sincefornearly-idealgases,withtheirrelativelysmallgasdensityn=N/VâȘÏâ3/2,Ïisindependentofthedensity,soaretheheatconductivityandtheviscositycoefficient.
Thediatomicmoleculeofnitrogen(N2), thebasiccomponentof theairwebreathe,atambientconditions(TK=300K,i.e.TâĄkBTKâ4.1Ă10â21J)hasfivethermallyexcited,essentiallyclassicalhalf-degreesoffreedom(threecorrespondingtothetranslationalmotion,andtwototherotationalmotion), so its specific heatcV = 5/2âsee the second line of Eq. (3.31). With this value, and theparametersgivenintheassignment(whichgive,inparticular,thecross-sectionÏâÏ(def)2â6.4Ă10â19m2),Eqs.(***)yield
(*)
(**)
Îșâ4.8Ă1020Wmâ Jâ0.66Ă10â2Wmâ K,ηâ0.90Ă10â5kgmâ s.
Thesenumbersmaybecomparedwithexperimentalvaluesforairthatarelisted,respectively,intable6.1ofthelecturenotes:Îșâ2.6Ă10â2W(m·K)â1,andinPartCMtable8.1:35ηâ1.8Ă10â5
kg (m·s)â1. We see that our approximate estimates give the correct orders of magnitude of bothparameters,butitwouldbenaĂŻvetoexpectfromthemmoreexactvalues.Indeed,onthetopoftheapproximationsmadeinthesetwoproblems,thegivenvalueofdefisalsoapproximate,andonemayfindinliteratureothervalues,whichdifferfromtheonegivenabovebyasmuchas50%.Thereasonforthisuncertaintyisthatdefismeasuredratherindirectly,forexamplebyfittingtheexperimentalequationofstateP(V,T)byvariousmodels incorporating theeffectivevolumeb=(4Ï/3)def
3âseeEqs.(3.98)and(4.2).
Perhaps a more important role of Eqs. (***) is to give the correct (though also approximate)functionaldependenceofthekineticparametersontemperature:Îș,ηâT1/2,andtheirindependenceofthegasdensityn.
Problem6.12.UsetheBoltzmann-RTAequationtocalculatethethermalconductivityofanearly-idealclassicalgas,measuredinconditionswhentheappliedthermalgradientdoesnotcreateanetparticle flow.Compare the resultwith that following from the simple kinetic arguments (problem6.10),anddiscusstheirrelation.
Solution:Foranon-degenerategas,theconditionspecifiedintheassignmentshouldbetakenveryseriously.Indeed,letusfirstforgetaboutit,andcalculateÎșdirectlyfromEq.(6.107)ofthelecturenotes:
Îș=gÏ2Ïâ34Ï3â«0â(8mΔ3)1/2(ΔâÎŒ)T2ââNΔâΔdΔ.Foraclassicalgas,wemayusethehigh-temperaturelimitofEqs.(2.115)and(2.118):
NΔ=expÎŒâΔT,givinginparticular
ââNΔâΔ=1TexpÎŒâΔT.
ThisexponentiallydecayingfunctionofΔprovidesafastconvergenceoftheintegralparticipatinginEq.(*)atenergiesâŒT.SinceaccordingtoEq.(3.34),âŁÎŒâŁintheclassicalgasismuchlargerthanT,thefactor(ΔâÎŒ)2inthatformulamaybeapproximatedwithÎŒ2.Asaresult,weget
ÎșâgÏ2Ïâ34Ï3ÎŒ2Tâ«0â(8mΔ3)1/2eÎŒâΔ/TdΔ.Thisexpressiondiffersonlyby theextra factorÎŒ2/Tq2 fromEq. (6.31) forÏ (in thecorrespondinglimitforthefunctionâšN(Δ)â©),sothatweimmediatelymayuseEq.(6.32)towrite
ÎșâÎŒ2Tq2Ï=ÎŒT2nÏTm.
However,intheconditionsspecifiedintheassignment,thisapparentresultiswrong.Indeed,theÎșgivenbyEq.(*)isthecoefficientdefinedbyEq.(6.105);intheabsenceoftheelectricfield(orjustforcharge-freeparticles),whenÎŒâČ=ÎŒ,itreads
where the last form isobtainedusingEq. (6.108).This relation is reduced to theFourier lawEq.(6.114),
jh=âÎșâT,only ifâÎŒ=0.However, for a gas of unchargedparticles, this condition is hard to implement inpractice.Itismucheasiertoensurethattheappliedtemperaturegradientdoesnotresultinthenetparticleflow.(Forexample,wemayheatoneendofalongsealedtube,thermallyinsulatedfromthelateralsidesâseethefigurebelow36.)
In order to analyze this situation, let us rewrite Eq. (6.97), with ÎŒâČ = ÎŒ, for the particle flow
densityjn=j/q:
Itshowsthatinconditionswhenthereisnoparticleflow, jn=0,theapplicationofatemperaturegradientunavoidablycreatesagradientofthechemicalpotential37:
PluggingthisexpressionintoEq.(***),weseethattheeffectivethermalconductancediffersfromtheÎșgivenbyEq.(*):
According to Eqs. (6.101) and (6.110), valid atTâȘ ΔF, for a degenerate Fermi gas this thermalconductivityreductionisoftheorderofÎș(T/ΔF)2,i.e.negligible,sothatinthiscase,Eq.(6.109)andhencetheWiedemannâFranzlaw(6.110)isvalidregardlessofthemeasurementconditions.
However,foraclassicalgasthesituationisdifferent.Indeed,pluggingthesimpleaboveformula
(****)
(**)
for[âââšN(Δ)/âΔ]intoEq.(6.98),andworkingouttwosimpleintegrals38,wereadilyget39
ComparingthisexpressionwithEq.(**),weseethattheleadingtermofthecorrection,proportionalto(ÎŒ/T)2â«1,exactlycancelsthecrudeapproximationforÎș.Hence,thisâseedâconductivityshouldbe re-calculated more carefully from Eq. (*), keeping all terms in the parentheses (Δ â ÎŒ)2. Astraightforwardcalculation40yieldsthefollowingfiniteresult:
Îșef=52nÏTm,whichissignificantlylowerthantheÎșgivenbyEq.(**).
Inordertocomparethisresultwiththeestimateobtainedinthesolutionofproblem6.10,letusrewritethelatterusingEq.(3.9), :41
ÎșâcVnÏTm.NowweshouldtakeintoaccountthatatthederivationofEqs.(6.106)and(6.107)fromEq.(6.104)insection6.5,theparticleâsenergyΔwasassociatedwithitskineticenergyonly,thusneglectingthepossiblethermalexcitationofitsinternaldegreesoffreedom.Thusforafaircomparison,weshouldtakecVequaltothecorrespondingvalue3/2,sothatthisestimatebecomes
Îșâ32nÏTm,WeseethatthisresultdiffersfromEq.(****)onlybyanumericalfactorof(5/2)/(3/2)â1.7;aswasdiscussed in the model solution of the previous problem, this is not too bad for such a crudeestimate,evenwithoutacorrectionforthelargerheatcapacityofthediatomicnitrogenmolecule:cV=5/2insteadofcV=3/2impliedbyourtheory.(Letmeleavetheanalysisofhowsuchcorrectionshouldbemade,forthereaderâsadditionalexercise.)
Neverthelesswe should remember that, aswasnoted in the lecturenotes, theBoltzmann-RTAequation may give unreliable numerical factors in its results for classical gases, because itsassumption of energy-independent scattering time Ï is frequently too crude for the broaddistributionofparticleenergiesinsuchsystems.
Problem 6.13. Use the heat conduction equation (6.119) to calculate the time evolution oftemperature in the center of a uniform solid sphere of radius R, initially heated to a uniformlydistributed temperature Tini, and at t = 0 placed into a heat bath that keeps its surface attemperatureT0.
Solution:Duetothesphericalsymmetryofthesystem,Eq.(6.119),âTât=DTâ2T,withDTâĄÎșcV,
isreducedto42
âTât=DT1r2âârr2âTâr.IntroducingthetemperatureâsdeviationTË(r,t)âĄTâT0fromtheboundaryvalueT(R,t)=T0,wemayformulateourboundaryproblemasfollows:
âTËât=DT1r2âârr2âTËâr,withTËR,t=0,andTËr,0=TiniâT0.
Separatingthevariables,i.e.lookingforthegeneralsolutionintheform
forthenthpartialsolutionweget
whereλnistheseparationconstant.Theresultingordinarydifferentialequationfor iselementary,giving
whilethatfor maybereducedtothewell-known1DHelmholtzequation,d2fndr2+kn2fn=0,withkn2âĄÎ»nDT,
by the substitution .43 This equation, with the boundary conditions and, i.e. fn(0) = fn(R) = 0, immediately yields the eigenfunctions and eigenvalues of the
problem:fnrâsinknr,withknR=Ïn,i.e.kn=ÏnR,λnâĄDTkn2=DTÏnR2,
wheren=1,2,âŠ,sothatthegeneralsolution(*)ofourboundaryproblemmaybespelledoutasTËr,t=ân=1âCnrsinÏnrRexpân2tÏ.
HeretheconstantÏ,definedasÏâĄR2Ï2DTâĄR2cVÏ2Îș,
physicallyisthetimescaleofthethermalrelaxationofthesphere,whiletheexpansioncoefficientsCnhavetobechosentosatisfytheinitialconditionTË(r,0)=TiniâT0,givingthesystemofequations
ân=1âCnrsinÏnrR=TiniâT0,for0â©œrâ©œR.
Thissystemmaybesolved,asusualatthereciprocalFouriertransform,bymultiplyingbothpartsofthisequationbyasimilareigenfunctionwithanarbitraryindexnâČ,inourcasebyrsin(ÏnâČr/R),andtheirintegrationovertheinterval[0,R].Atthisintegration,alltermswithnâČâ nunderthesumonthe left-hand side vanish due to eigenfunctionsâ orthogonality, while the term with nâČ = n yields
(*)
(**)
(***)
Cn(R/2).Asaresult,weget
Cn=2R(TiniâT0)â«0RrsinÏnrRdr=2R(TiniâT0)â«01sinÏnΟΟdΟ.Thisintegralmaybereadilyworkedoutbyparts,giving
Cn=2R(TiâT0)â1nâ1Ïn.PluggingthisresultintoEq.(**),forthecenterofthesphere(râ0)wegetthefinalresult:
T(0,t)âĄT0+2(TiniâT0)ân=1ââ1nâ1expân2tÏ.
Thistimedependenceisplottedwithasolidlineinthefigurebelow.Itshowsthattheinfluenceofhighereigenfunctions(withn>1)issignificantonlyovertheinitialperiodoftherelaxation,wheretheirsuperpositiondescribestheeffectivedelayoftheprocessbyâŒ0.7Ï.
Problem 6.14. Suggest a reasonabledefinition of the entropyproduction rate (perunit volume),andcalculatethisrateforastationarythermalconduction,assumingthatitobeystheFourierlaw,inamaterialwithnegligiblethermalexpansion.Giveaphysicalinterpretationoftheresult.Doesthestationary temperature distribution in a sample correspond to the minimum of the total entropyproductioninit?
Solution:Incontrasttotheconservedphysicalvariables,theentropydensitysâĄdS/dVsatisfiesonlyageneralizedcontinuityequation:
âsât+ââ js=rs,(wherejs is theentropycurrentdensity),and itsright-handside isexactlywhatmayrationallybecalledtheentropyproductionrate.Inthestationary(time-independent)situation,thisrelationyields
rs=ââ js.AccordingtothefundamentalEq.(1.19),withtemperature-independentvolumeV,wemaywritedS=dQ/T,sothatjsmaybecalculatedjustasjh/T,wherejhistheheatflowdensity,andEq.(*)yields
rs=ââ jhTâĄ1Tââ jhâ1T2âTâ jh.
Incontrasttotheentropy,theinternalenergy,andhence(intheabsenceofmechanicalworkandthe Joule heat generation) the heat energy, is a conserved variable, so it satisfies the continuityequationwithzeroright-handside:
âuât+ââ jh=0,sothat inastationarysituation,thefirsttermontheright-handsideofEq.(**)vanishes,andtherelationisreducedto
rs=â1T2âTâ jh.NowusingtheFourierlaw(6.114),jh=âÎșâT,wefinallyget
rs=Îș1T2âTâ âTâĄÎșâTT2.
Thephysicsofthisresultmaybeseenmoreclearlyonasimple,1Dexampleofauniformheat
flow,withthetotalpower ,throughalayerofareaAandsmallthicknessdx,undertheeffectofsmalltemperaturedifferencedT=(âT/âx)dxâseethefigureabove.TheheatflowsintothelayerthroughtheleftboundarykeptatsometemperatureT,butleavesitthroughtherightboundaryata
lowertemperature,TâdT,sothatthecorrespondingentropyflows, ,and ,arenotequal,evenif theheatpowerflows are.Therateof the totalentropyproduction in the layer isdeterminedbythedifferenceoftheseflows:
AfterpluggingtheFourierconductionlawforthis1Dsituation, ,fortheentropyproductionspecificrate,rsâĄdRs/dV=dRs/(Adx),weobtain
i.e.thesameresultaswewouldgetforthissituationfromthemoregeneralEq.(***).NownotethataccordingtoEq.(6.119)ofthelecturenotes,
âTât=DTâ2T,thestationarytemperaturedistributioninauniformsampleofvolumeVobeystheLaplaceequationâ2T = 0. However, it is well known44, that this equation is equivalent to the requirement of theminimumofthefollowingfunctional:
â«VâT2d3r.Comparingthisexpressionwiththefullrateofentropyproductioninthesample,followingfromEq.(***),
RsâĄâ«Vrsd3r=Îșâ«VâTT2d3r,we see that if the temperature gradient is so low that T â const, the stationary distribution oftemperaturecorrespondstotheminimumofRs.However,thiscomparisonalsoshowsthatthisso-calledminimumentropyproductionprinciple isnotexact.(Itmaybealsoviolatedbythesampleâsnon-uniformity.)
References[1]RösslerCetal2011NewJ.Phys.13113006[2]SchwabKetal2000Nature404974[3]ShokleyW1950ElectronsandHolesinSemiconductors(VanNostrand)[4]SmithD1995Thin-FilmDeposition(McGraw-Hill)[5]ZimanJ1979PrinciplesoftheTheoryofSolids2nded(CambridgeUniversityPress)[6]AshcroftNandMerminN1976SolidStatePhysics(W.B.Saunders)
1Formorediscussionofthisissuesee,e.g.PartEMsection6.4.2Anintroductiontothismethodmaybefound,forexample,inPartEMsection2.5.3AnotherpossiblecontributiontothefunctionXk(x),proportionaltosinkx,hasbeendroppedbecausethatfunctionevidentlyshouldretainitsinitialsymmetry:Xk(âx)=Xk(x).4See,e.g.Eq.(A.88).5TheWKBapproximation (discussed, e.g. inPartQM section2.4)maybeused to show that the result of this analysis is alsovalid for the so-called âadiabaticâchannelswhosecross-section is slowlychangingalong the length.Moreover, strictly speaking the result isonly valid for suchadiabatic channels,with smoothedinterfacesbetweenthebulkconductorsandthechannel,becauseonlyforsuchgeometrytheelectronscatteringatthelinkentrance/exit(theeffectartfullysweptunderthecarpetintheprovidedsolution:-)isnegligible.6Theleftinequalityensuresthatthisstateinconductor2isempty,andhenceavailableforoccupationbythetravelingelectron.7See,e.g.PartQMEq.(1.100).8See,e.g.PartQMEq.(2.5)withâÏ/âx=kx.9See,e.g.PartQMsection3.2.10See,e.g.[2]andreferencestherein.11AnothereffectistheCoulombinteractionoftheelectronswithinthegas,leading,inparticular,totheirscattering.Insolids,thiseffectistypicallylessimportant,duetothecompensatingpositivechargeoftheatomiclattice.Incontrast,thechargeQdiscussedinthisproblemistheuncompensatedchargeofadditionalelectrons.12Ifthisformulaisnotevident,thereadermayconsultPartEMsections2.2and3.4.13NotethatbothCeandCqarealwayspositive,regardlessofthechargeoftheparticles(e.g.electrons).14This fact emphasizes a broad similarity between the thermal activation of atomic/molecular condensation centers in usual gases, and of dopant atoms insemiconductors.15NoteagainthatniisaverystrongfunctionoftemperatureâseethesecondofEqs.(6.60).16Note that in somesemiconductors thedegeneracyofelectronson the levelΔAmaybedifferent from2. (InSi, it is equal to4.) In this case, the factor2 in theexpressionfornâ,andhenceinEq.(**),shouldbereplacedwiththeproperdegeneracyfactor.However,thischangeofthepre-exponentialfactorhasvirtuallynoeffectontheresultspresentedbelow,inparticularontheplotsofÎŒversusnD.17Foramoredetaileddiscussionofsemiconductordopingstatistics(aswellassomeotherissuesdiscussedinsection6.4),Icanrecommendtheclassicalmonograph[3].18Inchemistryandrelatedfields,thisprocessisfrequentlycalledeffusion.19BothinvolveddimensionlessintegralsareofthetypeEq.(A.36e),withn=2andn=1,respectively.20Notethatthesameresult(**)maybealsoobtainedinaslightlydifferentway(actually,usedinmosttextbooks),byconsideringwhatfractionfofparticles,inanelementaryvolumed3r=r2drdΩ,withr= andhencedr= ,hasvelocitiesdirectedtowardtheorifice(theanswerisf=AcosΞ/4Ïr2),and then integrating theresultingparticlenumberdN=nfdr3overallvelocitiesinsphericalcoordinates,withd3 = .Letmeleavethecompletionofthisapproachforthereaderasasimple,butusefuladditionalexercise.21See,e.g.chapters6and7in[4].22See,e.g.PartCMEq.(8.56).Notethedifferencebetweentheshearviscositycoefficientηconsideredinthisproblem,andthedragcoefficientηwhosecalculationwasthetaskofproblem3.2.Despitethesimilartraditionalnotation,andbelongingtothesamerealm(kinematicfriction),thesecoefficientshavedifferentdefinitions,andevendifferentdimensionalities.23Thisassumptionisperhapsthelargestsourceofimprecisionofnumericalcoefficientsinourestimates.24Asthesolutionofthenextproblemwillshow,inmanycasesthisisnotaverygoodassumption,andmaycostusonemorenumericalfactoroftheorderof1.25Accordingtothedefinition(6.40)ofÎŒâČ,foragasofcharge-freeparticles,ÎŒâČ=ÎŒ,sothesetwoconditionsareequivalent.26ComparingthisexpressionwithEq.(6.104)ofthelecturenotes,pleaserememberthatourcurrentcalculationisforÎŒ=const,sothattheterms,proportionaltothechemicalpotentialonbothsidesoftheinterface,cancel.27Itsaccountwouldrequireaspecificationofthethermally-activatedinternaldegreesoffreedomoftheparticle.28Asareminder,accordingtotheequipartitiontheorem,forfreeclassicalparticleswithnegligiblethermalexcitationof their internaldegreesoffreedom,i.e.withthreehalf-degreesoffreedom,cV=3/2âseeEq.(3.31).29Notethatherethefactoringoftheaveragesismoreâcleanâ(lessapproximate)thanintheprevioustask,becausethetwoCartesiancomponentsofthevelocity, and
,areindependent.30Iamsorrytouseforthecross-sectionthesameletterasfortheelectricOhmicconductivity.(Bothnotationsareverytraditional.)Letmehopethiswouldnotleadtoconfusion,becausetheconductivityisnotdiscussedinthisproblem.31Thisdefinitioniscommonforparticlescatteringdescriptionisclassicalandquantummechanics,andmapsonasimilardefinitionatthewavescatteringâsee,e.g.
PartCM(3.70),PartEMEq.(8.39),andPartQMEq.(3.59).32Thisparameterisdefinedasthesmallestdistancebetweentheparticlecentersintheabsenceofscatteringâsee,e.g.PartCMsection3.5.33Asperhapsthemostimportantexample,attheso-calledRutherfordscatteringbyaCoulombpotentialUâ1/r,thefullcross-sectionÏisinfinite,atleastinthelimitnâ0âsee,e.g.PartCMEq.(3.73).34See,e.g.either[5]or[6].35ThattableusestraditionalunitsmPa·sâĄ(10â3Nmâ2)·sâĄ10â3kg(m·s)â1.36Foragasofchargedparticles,forexamplesofelectronsinametal,thiscondition,jnâĄj/q=0,maybeimposedsimplybydisconnectingalongsample,madeofthismetal,fromanexternalconductingelectriccircuitâsee,e.g.figure6.12ofthelecturenotes.37NotethatthisisessentiallyEq.(6.102),onlywithÎŒ=ÎŒâČ.38TheyarebothofthetypeEq.(A.34a),onewiths=7/2,andtheotheronewiths=5/2âseealsoEq.(A.34e).39ItisinstructivetocomparethisexpressionwithEq.(6.101)ofthelecturenotes,validintheopposite,degeneratelimitTâȘÎŒ.40Thecalculationmaybesimplifiedtakingintoaccountthatinthecasejn=0wearepursuing,wemayuseEq.(6.104)withoutthesecondtermintheparenthesesâseetheremarkimmediatelyfollowingthatformula.41Notethatatitsderivationwehaveignoredtheparticleflow,sothatthecomparisonoftheresultingÎșwithÎșefismoreorlessfairâatleastwithintheframeworkofverycrudeassumptionsmadeatthederivation.42See,e.g.Eq.(A.67)withâ/âΞ=â/âÏ=0.43Thisfactiswellknownfromelectrodynamicsandquantummechanicsâsee,e.g.eitherPartEMsection8.1,inparticular,Eqs.(8.7)â(8.8),orPartQMsection3.1,inparticularEqs.(3.4)â(3.7).44Foraproof,see,e.g.thesolutionofPartEMproblem1.16,withthereplacementÏâT.
(A.1)
(A.2a)
(A.2b)
(A.3)
(A.4a)
(A.4b)
(A.5)
(A.6)
(A.7)
(A.8a)
(A.8b)
(A.9a)(A.9b)
(A.10a)
(A.10b)
(A.11a)
(A.11b)
IOPPublishing
StatisticalMechanicsProblemswithsolutionsKonstantinKLikharev
AppendixA
Selectedmathematicalformulas
Thisappendix listsselectedmathematical formulasthatareusedinthis lecturecourseseries,butnotalwaysrememberedbystudents(andsomeinstructors:-).
A.1ConstantsEuclideancircleâslength-to-diameterratio:
Ï=3.141592653âŠ;Ï1/2â1.77.Naturallogarithmbase:
eâĄlimnââ1+1nn=2.718281828âŠ;fromthatvalue,thelogarithmbaseconversionfactorsareasfollows(Ο>0):
lnΟlog10Ο=ln10â2.303,log10ΟlnΟ=1ln10â0.434.TheEuler(orâEulerâMascheroniâ)constant:
ÎłâĄlimnââ1+12+13+âŠ1nâlnn=0.5771566490âŠ;eÎłâ1.781.
A.2Combinatorics,sums,andseries(i)Combinatorics
Thenumberofdifferentpermutations,i.e.orderedsequencesofkelementsselectedfromasetofndistinctelements(nâ©Ÿk),is
PknâĄnâ (nâ1)âŻ(nâk+1)=n!(nâk)!;inparticular,thenumberofdifferentpermutationsofallelementsoftheset(n=k)is
Pkk=kâ (kâ1)âŻ2â 1=k!.Thenumberofdifferentcombinations,i.e.unorderedsequencesofkelementsfromasetofnâ©Ÿkdistinctelements,isequaltothebinomialcoefficient
CknâĄnkâĄPknPkk=n!k!(nâk)!.Inanalternative,verypopular âball/boxlanguageâ,nCk isthenumberofdifferentwaystoputinabox,inanarbitraryorder,kballsselectedfromndistinctballs.Ageneralizationofthebinomialcoefficientnotionisthemultinomialcoefficient,
Cnk1,k2,âŠklâĄn!k1!k2!âŠkl!,withn=âj=1lkj,which,inthestandardmathematicallanguage,isanumberofdifferentpermutationsinamultisetof l distinctelement types fromann-elementsetwhichcontainskj(j=1, 2,âŠl)elements of each type. In the âball/box languageâ, the coefficient (A.6) is the number ofdifferentwaystodistributendistinctballsbetweenldistinctboxes,eachtimekeepingthenumber (kj) of balls in the jth box fixed, but ignoring their order inside the box. ThebinomialcoefficientnCk(A.5)isaparticularcaseofthemultinomialcoefficient(A.6)forl=2-countingtheexplicitboxforthefirstone,andtheremainingspaceforthesecondbox,sothatifk1âĄk,thenk2=nâk.One more important combinatorial quantity is the number Mn
(k) of ways to place nindistinguishableballsintokdistinctboxes.ItmaybereadilycalculatedfromEq.(A.5)asthenumberofdifferentwaystoselect(kâ1)partitionsbetweentheboxesinanimaginedlinearrowof(kâ1+n)âobjectsâ(ballsintheboxesandpartitionsbetweenthem):
Mn(k)=Ckâ1nâ1+kâĄkâ1+n!kâ1!n!.(ii)Sumsandseries
Arithmeticprogression:r+2r+âŻ+nrâĄâk=1nkr=n(r+nr)2;
inparticular,atr=1itisreducedtothesumofnfirstnaturalnumbers:1+2+âŻ+nâĄâk=1nk=n(n+1)2.
Sumsofsquaresandcubesofnfirstnaturalnumbers:12+22+âŻ+n2âĄâk=1nk2=n(n+1)(2n+1)6;
13+23+âŻ+n3âĄâk=1nk3=n2(n+1)24.TheRiemannzetafunction:
ζ(s)âĄ1+12s+13s+âŻâĄâk=1â1ks;theparticularvaluesfrequentlymetinapplicationsare
ζ32â2.612,ζ2=Ï26,ζ52â1.341,ζ3â1.202,ζ(4)=Ï490,ζ5â1.037.Finitegeometricprogression(forrealλâ 1):
1+λ+λ2+âŻ+λnâ1âĄâk=0nâ1λk=1âλn1âλ;inparticular, ifλ2<1, theprogressionhasa finite limitatnââ (called thegeometricseries):
(A.12)
(A.13)
(A.14a)
(A.14b)
(A.15a)
(A.15b)
(A.16a)(A.16b)
(A.17a)(A.17b)(A.17c)
(A.18a)(A.18b)(A.18c)
(A.18d)
(A.19)
(A.20)
(A.21)
(A.22)
(A.23)
(A.24)
(A.25)
(A.26)
(A.27)
(A.28)(A.29a)(A.29b)
(A.30a)(A.30b)(A.30c)(A.30d)
(A.31a)(A.31b)
limnâââk=0nâ1λk=âk=0âλk=11âλ.Binomialsum(ortheâbinomialtheoremâ):
1+an=âk=0nCknak,wherenCkarethebinomialcoefficientsdefinedbyEq.(A.5).TheStirlingformula:
limnââlnn!=n(lnnâ1)+12ln(2Ïn)+112nâ1360n3+âŠ;formostapplicationsinphysics,thefirstterm1issufficient.TheTaylor(orâTaylorâMaclaurinâ)series:foranyinfinitelydifferentiablefunctionf(Ο):
limΟËâ0f(Ο+ΟË)=f(Ο)+dfdΟ(Ο)ΟË+12!d2fdΟ2(Ο)ΟË2+âŻ=âk=0â1k!dkfdΟk(Ο)ΟËk;notethat formanyfunctionsthisseriesconvergesonlywithina limited,sometimessmallrangeofdeviationsΟË.Forafunctionofseveralarguments,f(Ο1,Ο2,âŠ,ΟN),thefirsttermsoftheTaylorseriesare
limΟËkâ0f(Ο1+ΟË1,Ο2+ΟË2,âŻ)=fΟ1,Ο2,âŻ+âk=1NâfâΟkΟ1,Ο2,âŻÎŸËk+12!âk,kâČ=1Nâ2fâkΟâΟkâČΟËkΟËkâČ+âŻTheEulerâMaclaurinformula,validforanyinfinitelydifferentiablefunctionf(Ο):
âk=1nf(k)=â«0nf(Ο)dΟ+12f(n)âf(0)+16â 12!dfdΟ(n)âdfdΟ(0)â130â 14!d3fdΟ3(n)âd3fdΟ3(0)+142â 16!d5fdΟ5(n)âd5fdΟ5(0)+âŻ;thecoefficientsparticipatinginthisformulaaretheso-calledBernoullinumbers2:
B1=12,B2=16,B3=0,B4=130,B5=0,B6=142,B7=0,B8=130,âŻ
A.3BasictrigonometricfunctionsTrigonometricfunctionsofthesumandthedifferenceoftwoarguments3:
cosa±b=cosacosbâsinasinb,sina±b=sinacosb±cosasinb.Sumsoftwofunctionsofarbitraryarguments:
cosa+cosb=2cosa+b2cosbâa2,cosaâcosb=2sina+b2sinbâa2,sina±sinb=2sina±b2cos±bâa2.Trigonometricfunctionproducts:
2cosacosb=cos(a+b)+cos(aâb),2sinacosb=sin(a+b)+sin(aâb),2sinasinb=cos(aâb)âcos(a+b);fortheparticularcaseofequalarguments,b=a,thesethreeformulasyieldthefollowingexpressionsforthesquaresoftrigonometricfunctions,andtheirproduct:
cos2a=121+cos2a,sinacosa=12sin2a,sin2a=121âcos2a.Cubesoftrigonometricfunctions:
cos3a=34cosa+14cos3a,sin3a=34sinaâ14sin3a.Trigonometricfunctionsofacomplexargument:
sin(a+ib)=sinacoshb+icosasinhb,cos(a+ib)=cosacoshbâisinasinhb.Sumsoftrigonometricfunctionsofnequidistantarguments:
âk=1nsincoskΟ=sincosn+12Οsinn2Ο/sinΟ2.
A.4GeneraldifferentiationFulldifferentialofaproductoftwofunctions:
d(fg)=(df)g+f(dg).Fulldifferentialofafunctionofseveralindependentarguments,f(Ο1,Ο2,âŠ,Οn):
df=âk=1nâfâΟkdΟk.CurvatureoftheCartesianplotofa1Dfunctionf(Ο):
ÎșâĄ1R=d2f/dΟ21+df/dΟ23/2.
A.5GeneralintegrationIntegrationbyparts-immediatelyfollowsfromEq.(A.22):
â«g(A)g(B)fdg=fgBAââ«f(A)f(B)gdf.Numerical(approximate)integrationof1Dfunctions:thesimplesttrapezoidalrule,
â«abf(Ο)dΟâhfa+h2+fa+3h2+âŻ+fbâh2=hân=1Nfaâh2+nh,hâĄbâaN.has relatively low accuracy (error of the order of (h3/12)d2f/dΟ2 per step), so that thefollowingSimpsonformula,
â«abf(Ο)dΟâh3f(a)+4f(a+h)+2f(a+2h)+âŻ+4f(bâh)+f(b),hâĄbâa2N,whoseerrorperstepscalesas(h5/180)d4f/dΟ4,isusedmuchmorefrequently4.
A.6Afew1Dintegrals5(i)Indefiniteintegrals:
Integralswith(1+Ο2)1/2:â«1+Ο21/2dΟ=Ο21+Ο21/2+12lnâŁÎŸ+(1+Ο2)1/2âŁ,
â«dΟ1+Ο21/2=lnâŁÎŸ+1+Ο21/2âŁ,â«dΟ1+Ο23/2=Ο1+Ο21/2.Miscellaneousindefiniteintegrals:
â«dΟΟ(Ο2+2aΟâ1)1/2=arccosaΟâ1âŁÎŸâŁa2+11/2,â«sinΟâΟcosΟ2Ο5dΟ=2Οsin2Ο+cos2Οâ2Ο2â18Ο4,â«dΟa+bcosΟ=2a2âb21/2tanâ1aâba2âb21/2tanΟ2,fora2>b2.â«dΟ1+Ο2=tanâ1Ο.
(ii)Semi-definiteintegrals:Integralswith1/(eΟ±1):
â«aâdΟeΟ+1=ln1+eâa,
(A.32a)(A.32b)
(A.32c)
(A.33a)(A.33b)
(A.34a)
(A.34b)
(A.34c)
(A.34d)
(A.34e)
(A.35a)(A.35b)
(A.35c)(A.35d)
(A.35e)
(A.36a)
(A.36b)(A.36c)
(A.36d)
(A.36e)
(A.37)(A.38)(A.39)
(A.40)(A.41)
(A.42a)(A.42b)
(A.43)
(A.44)
(A.45)
(A.46)
(A.47)
(A.48)
(A.49a)
(A.49b)
â«a>0âdΟeΟâ1=ln11âeâa.(iii)Definiteintegrals:
Integralswith1/(1+Ο2):6â«0âdΟ1+Ο2=Ï2,
â«0âdΟ1+Ο23/2=1;moregenerally,
â«0âdΟ1+Ο2n=Ï22nâ3!!2nâ2!!âĄÏ21â 3â 5âŠ2nâ32â 4â 6âŠ2nâ2,forn=2,3,âŠIntegralswith(1âΟ2n)1/2:
â«01dΟ1âΟ2n1/2=Ï1/22nÎ12n/În+12n,â«011âΟ2n1/2dΟ=Ï1/24nÎ12n/Î3n+12n,where Î(s) is the gamma-function, which is most often defined (for Re s > 0) by thefollowingintegral:
â«0âΟsâ1eâΟdΟ=Î(s).Thekeypropertyofthisfunctionistherecurrencerelation,validforanysâ 0,â1,â2,âŠ:
Î(s+1)=sÎ(s).Since,accordingtoEq. (A.34a),Î(1)=1,Eq.(A.34b) fornon-negative integers takes theform
Î(n+1)=n!,forn=0,1,2,âŻ(where0!âĄ1).Becauseofthis,forintegers=n+1â©Ÿ1,Eq.(A.34a)isreducedto
â«0âΟneâΟdΟ=n!.Other frequently met values of the gamma-function are those for positive semi-integerarguments:
Î12=Ï1/2,Î32=12Ï1/2,Î52=12â 32Ï1/2,Î72=12â 32â 52Ï1/2,âŠ.Integralswith1/(eΟ±1):
â«0âΟsâ1dΟeΟ+1=(1â21âs)Î(s)ζ(s),fors>0,â«0âΟsâ1dΟeΟâ1=Î(s)ζ(s),fors>1,whereζ(s)istheRiemannzeta-functionâseeEq.(A.10).Particularcases:fors=2n,
â«0âΟ2nâ1dΟeΟ+1=22nâ1â12nÏ2nB2n,â«0âΟ2nâ1dΟeΟâ1=(2Ï)2n4nB2n.whereBnaretheBernoullinumbersâseeEq. (A.15).Fortheparticularcases=1 (whenEq.(A.35a)yieldsuncertainty),
â«0âdΟeΟ+1=ln2.Integralswithexp{âΟ2}:
â«0âΟseâΟ2dΟ=12Îs+12,fors>â1;forapplicationsthemostimportantparticularvaluesofsare0and2:
â«0âeâΟ2dΟ=12Î12=Ï1/22,â«0âΟ2eâΟ2dΟ=12Î32=Ï1/24,althoughwewillalsorunintothecasess=4ands=6:
â«0âΟ4eâΟ2dΟ=12Î52=3Ï1/28,â«0âΟ6eâΟ2dΟ=12Î72=15Ï1/216;foroddintegervaluess=2n+1(withn=0,1,2,âŠ),Eq.(A.36a)takesasimplerform:
â«0âΟ2n+1eâΟ2dΟ=12În+1=n!2.Integralswithcosineandsinefunctions:
â«0âcosΟ2dΟ=â«0âsinΟ2dΟ=Ï81/2.â«0âcosΟa2+Ο2dΟ=Ï2aeâa.â«0âsinΟΟ2dΟ=Ï2.Integralswithlogarithms:
â«01lna+1âΟ21/2aâ1âΟ21/2dΟ=Ïaâa2â11/2,foraâ©Ÿ1.â«01ln1+1âΟ1/2Ο1/2dΟ=1.IntegralrepresentationsoftheBesselfunctionsofintegerorder:
Jn(α)=12Ïâ«âÏ+Ïei(αsinΟânΟ)dΟ,sothateiαsinΟ=âk=âââJk(α)eikΟ;In(α)=1Ïâ«0ÏeαcosΟcosnΟdΟ.
A.73Dvectorproducts(i)Definitions:
Scalar(âdot-â)product:aâ b=âj=13ajbj,
whereajandbjarevectorcomponentsinanyorthogonalcoordinatesystem.Inparticular,thevectorsquared(thesameasthenormsquared):
a2âĄaâ a=âj=13aj2âĄâ„aâ„2.Vector(âcross-â)product:
aĂbâĄn1(a2b3âa3b2)+n2(a3b1âa1b3)+n3(a1b2âa2b1)=n1n2n3a1a2a3b1b2b3,where {nj} is the set of mutually perpendicular unit vectors7 along the correspondingcoordinatesystemaxes8.Inparticular,Eq.(A.45)yields
aĂa=0.(ii)Corollaries(readilyverifiedbyCartesiancomponents):
Doublevectorproduct(theso-calledbacminuscabrule):aĂ(bĂc)=b(aâ c)âc(aâ b).
Mixedscalarâvectorproduct(theoperandrotationrule):aâ bĂc=bâ cĂa=câ aĂb.
Scalarproductofvectorproducts:aĂbâ cĂd=aâ cbâ dâaâ dbâ c;
intheparticularcaseof twosimilaroperands(say,a=candb=d), the last formula isreducedto
aĂb2=(ab)2â(aâ b)2.
(A.50)
(A.51)
(A.52)
(A.53)
(A.54)
(A.55)
(A.56)
(A.57)
(A.58)
(A.60)
(A.61)
(A.62)
(A.63)
(A.64)
(A.66)
(A.67)
(A.68)
(A.69)
A.8Differentiationin3DCartesiancoordinatesDefinitionofthedel(orânablaâ)vector-operatorâ:9
ââĄâj=13njâârj,where rj is a set of linear and orthogonal (Cartesian) coordinates along directionsnj. Inaccordancewiththisdefinition,theoperatorâactingonascalar functionofcoordinates,f(r),10givesitsgradient,i.e.anewvector:
âfâĄâj=13njâfârjâĄgradf.Thescalarproductofdelbyavectorfunctionofcoordinates(avectorfield),
f(r)âĄâj=13njfj(r),compiled formally followingEq. (A.43), is a scalar functionâthedivergence of the initialfunction:
ââ fâĄâj=13âfjârjâĄdivf,whilethevectorproductofâandf,formedinaformalaccordancewithEq.(A.45),isanewvector-thecurl(inEuropeantradition,calledrotoranddenotedrot)off:
âĂfâĄn1n2n3ââr1ââr2ââr3f1f2f3=n1âf3âr2ââf2âr3+n2âf1âr3ââf3âr1+n3âf2âr1ââf1âr2âĄcurlf.One more frequently met âproductâ is (f·â)g, where f and g are two arbitrary vectorfunctionsofr.ThisproductshouldbealsounderstoodinthesenseimpliedbyEq.(A.43),i.e.asavectorwhosejthCartesiancomponentis
fâ âgj=âjâČ=13fjâČâgjârjâČ.
A.9TheLaplaceoperatorâ2âĄâ·âExpressioninCartesiancoordinatesâintheformalaccordancewithEq.(A.44):
â2=âj=13â2ârj2.Accordingtoitsdefinition,theLaplaceoperatoractingonascalarfunctionofcoordinatesgivesanewscalarfunction:
â2fâĄââ âf=divgradf=âj=13â2fârj2.On the other hand, acting on a vector function (A.52), the operator â2 returns anothervector:
â2f=âj=13njâ2fj.Note that Eqs. (A.56)â(A.58) are only valid in Cartesian (i.e. orthogonal and linear)coordinates,butgenerallynotinother(evenorthogonal)coordinatesâsee,e.g.Eqs.(A.61),(A.64),(A.67)and(A.70)below.
A.10Operatorsâandâ2inthemostimportantsystemsoforthogonalcoordinates11
(i)Cylindrical12coordinates{Ï,Ï,z}(seefigurebelow)maybedefinedbytheirrelationswiththeCartesiancoordinates:
Gradientofascalarfunction:âf=nÏâfâÏ+nÏ1ÏâfâÏ+nzâfâz.
TheLaplaceoperatorofascalarfunction:â2f=1ÏââÏÏâfâÏ+1Ï2â2fâÏ2+â2fâz2,
Divergenceofavectorfunctionofcoordinates(f=nÏfÏ+nÏfÏ+nzfz):ââ f=1ÏâÏfÏâÏ+1ÏâfÏâÏ+âfzâz.
Curlofavectorfunction:âĂf=nÏ1ÏâfzâÏââfÏâz+nÏâfÏâzââfzâÏ+nz1Ïâ(ÏfÏ)âÏââfÏâÏ.
TheLaplaceoperatorofavectorfunction:â2f=nÏâ2fÏâ1Ï2fÏâ2Ï2âfÏâÏ+nÏâ2fÏâ1Ï2fÏ+2Ï2âfÏâÏ+nzâ2fz.
(ii)Sphericalcoordinates{r,Ξ,Ï}(seefigurebelow)maybedefinedas:
Gradientofascalarfunction:âf=nrâfâr+nΞ1râfâΞ+nÏ1rsinΞâfâÏ.
TheLaplaceoperatorofascalarfunction:â2f=1r2âârr2âfâr+1r2sinΞââΞsinΞâfâΞ+1(rsinΞ)2â2fâÏ2.
Divergenceofavectorfunctionf=nrfr+nΞfΞ+nÏfÏ:ââ f=1r2âr2frâr+1rsinΞâ(fΞsinΞ)âΞ+1rsinΞâfÏâÏ.
Curlofasimilarvectorfunction:
(A.70)
(A.71)
(A.72)
(A.73)
(A.74a)
(A.74b)
(A.75)(A.76)(A.77)
(A.78)
(A.79)
(A.80)
(A.81)
(A.82)
(A.83)
(A.84a)
(A.84b)
(A.85)
(A.86)
(A.87a)
(A.87b)
(A.88)
(A.89)
(A.90)
âĂf=nr1rsinΞâ(fÏsinΞ)âΞââfΞâÏ+nΞ1r1sinΞâfrâÏââ(rfÏ)âr+nÏ1râ(rfΞ)ârââfrâΞ.TheLaplaceoperatorofavectorfunction:
â2f=nrâ2frâ2r2frâ2r2sinΞââΞ(fΞsinΞ)â2r2sinΞâfÏâÏ+nΞâ2fΞâ1r2sin2ΞfΞ+2r2âfrâΞâ2cosΞr2sin2ΞâfÏâÏ+nÏâ2fÏâ1r2sin2ΞfÏ+2r2sinΞâfrâÏ+2cosΞr2sin2ΞâfΞâÏ.
A.11Productsinvolvingâ(i)Usefulzeros:
Foranyscalarfunctionf(r),âĂâfâĄcurlgradf=0.
Foranyvectorfunctionf(r),ââ âĂfâĄdivcurlf=0.
(ii)TheLaplaceoperatorexpressedviathecurlofacurl:â2f=âââ fââĂâĂf.
(iii)Spatialdifferentiationofaproductofascalarfunctionbyavectorfunction:Thescalar3DgeneralizationofEq.(A.22)is
ââ fg=âfâ g+fââ g.Itsvectorgeneralizationissimilar:
âĂfg=âfĂg+fâĂg.
(iv)Spatialdifferentiationofproductsoftwovectorfunctions:âĂfĂg=fââ gâfâ âgâââ fg+gâ âf,
âfâ g=fâ âg+gâ âf+fĂâĂg+gĂâĂf,ââ fĂg=gâ âĂfâfâ âĂg.
A.12Integro-differentialrelations(i)ForanarbitrarysurfaceSlimitedbyclosedcontourC:
TheStokestheorem,validforanydifferentiablevectorfieldf(r):â«SâĂfâ d2râĄâ«SâĂfnd2r=âźCfâ drâĄâźCfÏdr,
whered2r âĄnd2r is the elementary area vector (normal to the surface), and dr is theelementarycontourlengthvector(tangentialtothecontourline).
(ii)ForanarbitraryvolumeVlimitedbyclosedsurfaceS:Divergence(orâGaussâ)theorem,validforanydifferentiablevectorfieldf(r):
â«Vââ fd3r=âźSfâ d2râĄâźSfnd2r.Greenâstheorem,validfortwodifferentiablescalarfunctionsf(r)andg(r):
â«Vfâ2gâgâ2fd3r=âźSfâgâgâfnd2r.Anidentityvalidforanytwoscalarfunctionsfandg,andavectorfieldjwithâ·j=0(alldifferentiable):
â«Vf(jâ âg)+g(jâ âf)d3r=âźSfgjnd2r.
A.13TheKroneckerdeltaandLevi-Civitapermutationsymbols
TheKroneckerdeltasymbol(definedforintegerindices):ÎŽjjâČâĄ1,ifjâČ=j,0,otherwise.
TheLevi-Civitapermutationsymbol(mostfrequentlyusedfor3integerindices,eachtakingoneofvalues1,2,or3):
ΔjjâČjâłâĄ+1,iftheindicesfollowintheâcorrectâ(âevenâ)order:1â2â3â1â2âŠ,â1,iftheindicesfollowintheâincorrectâ(âoddâ)order:1â3â2â1â3âŠ,0,ifanytwoindicescoincide.RelationbetweentheLevi-CivitaandtheKroneckerdeltaproducts:
ΔjjâČjâłÎ”kkâČkâł=âl,lâČ,lâł=13ÎŽjlÎŽjlâČÎŽjlâłÎŽjâČlÎŽjâČlâČÎŽjâČlâłÎŽjâłlÎŽjâłlâČÎŽjâłlâł;summationofthisrelation,writtenfor3differentvaluesofj=k,overthesevaluesyieldstheso-calledcontractedepsilonidentity:
âj=13ΔjjâČjâłÎ”jkâČkâł=ÎŽjâČkâČÎŽjâłkâłâÎŽjâČkâłÎŽjâłkâČ.
A.14Diracâsdelta-function,signfunction,andtheta-functionDefinitionof1Ddelta-function(forreala<b):
â«abf(Ο)ÎŽ(Ο)dΟ=f(0),ifa<0<b,0,otherwise,wheref(Ο)isanyfunctioncontinuousnearΟ=0.Inparticular(iff(Ο)=1nearΟ=0),thedefinitionyields
â«abÎŽ(Ο)dΟ=1,ifa<0<b,0,otherwise.Relationtothetheta-functionΞ(Ο)andsignfunctionsgn(Ο)
Ύ(Ο)=ddΟΞ(ζ)=12ddΟsgn(Ο),where
Ξ(Ο)âĄsgn(Ο)+12=0,ifΟ<0,1,ifΟ>1,sgn(Ο)âĄÎŸâŁÎŸâŁ=â1,ifΟ<0,+1,ifΟ>1.Animportantintegral13:
â«ââ+âeisΟds=2ÏÎŽ(Ο).3D generalization of the delta-function of the radius-vector (the 2D generalization issimilar):
â«VfrÎŽrd3r=f(0),if0âV,0,otherwise;itmayberepresentedasaproductof1Ddelta-functionsofCartesiancoordinates:
ÎŽ(r)=ÎŽ(r1)ÎŽ(r2)ÎŽ(r3).
A.15TheCauchytheoremandintegralLet a complex function be analyticwithin a part of the complex plane , that is limitedby aclosedcontourCandincludespoint .Then
ThefirstoftheserelationsisusuallycalledtheCauchyintegraltheorem(ortheâCauchyâGoursat
theoremâ),andthesecondoneâtheCauchyintegral(ortheâCauchyintegralformulaâ).
A.16Literature(i) Properties of some special functions are briefly discussed at the relevant points of thelecturenotes;inthealphabeticalorder:Airyfunctions:PartQMsection2.4;Besselfunctions:PartEMsection2.7;Fresnelintegrals:PartEMsection8.6;Hermitepolynomials:PartQMsection2.9;Laguerrepolynomials(bothsimpleandassociated):PartQMsection3.7;Legendrepolynomials,associatedLegendre functions:PartEM section2.8, andPartQMsection3.6;Sphericalharmonics:PartQMsection3.6;SphericalBesselfunctions:PartQMsections3.6and3.8.
(ii)Formoreformulas,andtheirdiscussion,Icanrecommendthefollowinghandbooks14:
HandbookofMathematicalFormulas[2];TablesofIntegrals,Series,andProducts[3];MathematicalHandbookforScientistsandEngineers[4];IntegralsandSeriesvolumes1and2[5];ApopulartextbookMathematicalMethodsforPhysicists[6]maybealsousedasaformulamanual.
Many formulas are also available from the symbolic calculation modules of thecommerciallyavailablesoftwarepackageslistedinsection(iv)below.
(iii)ProbablythemostpopularcollectionofnumericalcalculationcodesarethetwinmanualsbyWPressetal[1]:NumericalRecipesinFortran77;NumericalRecipes[inC++âKKL].
My lecture notes include very brief introductions to numerical methods of differentialequationsolution:ordinarydifferentialequations:PartCM,section5.7;partialdifferentialequations:PartCMsection8.5andPartEMsection2.11,whichincludereferencestoliteratureforfurtherreading.
(iv) The following are the most popular software packages for numerical and symboliccalculations,allwithplottingcapabilities(inthealphabeticalorder):Maple(www.maplesoft.com/products/maple/);MathCAD(www.ptc.com/engineering-math-software/mathcad/);Mathematica(www.wolfram.com/mathematica/);MATLAB(www.mathworks.com/products/matlab.html).
References[1]PressWetal1992NumericalRecipesinFortran772ndedn(Cambridge:CambridgeUniversityPress)
PressWetal2007NumericalRecipes3rdedn(Cambridge:CambridgeUniversityPress)[2]AbramowitzMandStegunI(eds)1965HandbookofMathematicalFormulas(NewYork:Dover),andnumerouslaterprintings.
Anupdatedversionofthiscollectionisnowavailableonlineathttp://dlmf.nist.gov/.[3]GradshteynIandRyzhikI1980TablesofIntegrals,Series,andProducts5thedn(NewYork:Academic)[4]KornGandKornT2000MathematicalHandbookforScientistsandEngineers2ndedn(NewYork:Academic)[5]PrudnikovAetal1986IntegralsandSeriesvol1(BocaRaton,FL:CRCPress)
PrudnikovAetal1986IntegralsandSeriesvol2(BocaRaton,FL:CRCPress)[6]ArfkenGetal2012MathematicalMethodsforPhysicists7thedn(NewYork:Academic)[7]DwightH1961TablesofIntegralsandOtherMathematicalFormulas4thedn(London:Macmillan)
1Actually,thisleadingtermwasderivedbyAdeMoivrein1733,beforeJStirlingâswork.2NotethatdefinitionsofBk(orrathertheirsignsandindices)varyevenamongthemostpopularhandbooks.3Iamconfidentthatthereaderisquitecapableofderivingtherelations(A.16)byrepresentingtheexponentintheelementaryrelationei(a±b)=eiae±ibasasumofitsreal and imaginary parts, Eqs. (A.18) directly from Eqs. (A.16), and Eqs. (A.17) from Eqs. (A.18) by variable replacement; however, I am still providing theseformulastosavehisorhertime.(Quiteafewformulasbelowareincludedbecauseofthesamereason.)4Higher-orderformulas(e.g.theBoderule),andotherguidanceincludingready-for-usecodesforcomputercalculationsmaybefound,forexample,inthepopularreferencetextsbyWHPressetal[1].Inaddition,someadvancedcodesareusedassubroutinesinthesoftwarepackageslistedinthesamesection.Insomecases,theEulerâMaclaurinformula(A.15)mayalsobeusefulfornumericalintegration.5Apowerful(andfree)interactiveonlinetoolforworkingoutindefinite1Dintegralsisavailableathttp://integrals.wolfram.com/index.jsp.6Eq.(A.32a)followsimmediatelyfromEq.(A.30d),andEq.(A.32b)fromEq.(A.29b)âacouplemoreexamplesofthe(intentional)redundancyinthislist.7Otherpopularnotationsforthisvectorsetare{ej}and{rËj}.8It is easy to use Eq. (A.45) to check that the direction of the product vector corresponds to the well-known âright-hand ruleâ and to the even more convenientcorkscrewrule:ifwerotateacorkscrewâshandlefromthefirstoperandtowardthesecondone,itsaxismovesinthedirectionoftheproduct.9Onecanrunintothefollowingnotation:ââĄâ/âr,whichisconvenientissomecases,butmaybemisleadinginquiteafewothers,soitwillbenotusedinthesenotes.10Inthis,andfournextsections,allscalarandvectorfunctionsareassumedtobedifferentiable.11SomeotherorthogonalcurvilinearcoordinatesystemsarediscussedinPartEM,section2.3.12Inthe2Dgeometrywithfixedcoordinatez,thesecoordinatesarecalledpolar.13Thecoefficientinthisrelationmaybereadilyrecalledbyconsideringitsleft-handpartastheFourier-integralrepresentationoffunctionf(s)âĄ1,andapplyingEq.(A.85)tothereciprocalFouriertransform
f(s)âĄ1=12Ïâ«ââ+âeâisΟ[2ÏÎŽ(Ο)]dΟ.
14Onapersonalnote,perhaps90%ofallformulaneedsthroughoutmyresearchcareerweresatisfiedbyatiny,wonderfullycompiledoldbook[7],usedcopiesofwhich,ratheramazingly,arestillavailableontheWeb.
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AppendixB
Selectedphysicalconstants
Thelistednumericalvaluesoftheconstantsarefromthemostrecent(2014)InternationalCODATArecommendation(see,e.g.http://physics.nist.gov/cuu/Constants/index.html),besidesanewerresultforkBâsee[1].Pleasenotetherecentlyannounced(but,bythisvolumeâspresstime,notyetofficial)adjustment of the SI values - see, e.g. https://www.nist.gov/si-redefinition/meet-constants. Inparticular, thePlanckconstantwill alsogetadefinite value (within the interval specified in tableB.1),enablinganew,fundamentalstandardofthekilogram.
TableB.1.
Symbol Quantity SIvalueandunit Gaussianvalueandunit Relativermsuncertaintyc speedoflightin
freespace2.99792458Ă108msâ1
2.99792458Ă1010cmsâ1
0(definedvalue)
G gravitationconstant
6.6741Ă10â11m3
kgâ1sâ26.6741Ă10â8cm3gâ1sâ2
âŒ5Ă10â5
â Planckconstant 1.05457180Ă10â34Js
1.05457180Ă10â27ergs
âŒ2Ă10â8
e elementaryelectriccharge
1.6021762Ă10â19C
4.803203Ă10â10statcoulomb
âŒ6Ă10â9
me electronâsrestmass
0.91093835Ă10â30kg
0.91093835Ă10â27g
âŒ1Ă10â8
mp protonâsrestmass 1.67262190Ă10â27kg
1.67262190Ă10â24g
âŒ1Ă10â8
ÎŒ0 magneticconstant 4ÏĂ10â7NAâ2 â 0(definedvalue)
Δ0 electricconstant 8.854187817Ă10â12Fmâ1
â 0(definedvalue)
kB Boltzmannconstant
1.380649Ă10â23JKâ1
1.3806490Ă10â16ergKâ1
âŒ2Ă10â6
Comments:1. Thefixedvalueofcwasdefinedbyaninternationalconventionin1983,inordertoextendthe
official definition of the second (as âthe duration of 9 192 631 770 periods of the radiationcorresponding to the transitionbetween the twohyperfine levelsof thegroundstateof thecesium-133 atomâ) to that of the meter. The values are back-compatible with the legacydefinitionsof themeter (initially,as1/40000000thof theEarthâsmeridian length)and thesecond (for a long time, as 1/(24Ă 60Ă 60)= 1/86 400th of the Earthâs rotation period),withintheexperimentalerrorsofthosemeasures.
2. Δ0andΌ0arenotreallythefundamentalconstants;intheSIsystemofunitsoneofthem(say,Ό0)isselectedarbitrarily1,whiletheotheroneisdefinedviatherelationΔ0Ό0=1/c2.
3. TheBoltzmannconstantkB isalsonotquite fundamental,because itsonly role is tocomplywiththeindependentdefinitionofthekelvin(K),asthetemperatureunitinwhichthetriplepointofwaterisexactly273.16K.IftemperatureisexpressedinenergyunitskBT(asisdone,forexample,inPartSMofthisseries),thisconstantdisappearsaltogether.
4. Thedimensionless finestructure (âSommerfeldâsâ)constantα isnumerically the same inanysystemofunits:
αâĄe2/4ÏΔ0âcinSIunitse2/âcinGaussianunitsâ7.297352566Ă10â3â1137.03599914,andisknownwithamuchsmallerrelativermsuncertainty(currently,âŒ3Ă10â10)thanthoseofthecomponentconstants.
References[1]GaiserCetal2017Metrologia54280[2]NewellD2014Phys.Today6735â41
1NotethattheselectedvalueofÎŒ0maybechanged(abit)inafewyearsâsee,e.g.,[2].
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Bibliography
This section presents a partial list of textbooks and monographs used in the work on the EAPseries1,2.
PartCM:ClassicalMechanicsFetterALandWaleckaJD2003TheoreticalMechanicsofParticlesandContinua(NewYork:Dover)GoldsteinH,PooleCandSafkoJ2002ClassicalMechanics3rdedn(Reading,MA:AddisonWesley)GrangerRA1995FluidMechanics(NewYork:Dover)JoséJVandSaletanEJ1998ClassicalDynamics(Cambridge:CambridgeUniversityPress)LandauLDandLifshitzEM1976Mechanics3rdedn(Oxford:Butterworth-Heinemann)LandauLDandLifshitzEM1986TheoryofElasticity(Oxford:Butterworth-Heinemann)LandauLDandLifshitzEM1987FluidMechanics2ndedn(Oxford:Butterworth-Heinemann)SchusterHG1995DeterministicChaos3rdedn(NewYork:Wiley)SommerfeldA1964Mechanics(NewYork:Academic)SommerfeldA1964MechanicsofDeformableBodies(NewYork:Academic)
PartEM:ClassicalElectrodynamicsBatyginVVandToptyginIN1978ProblemsinElectrodynamics2ndedn(NewYork:Academic)GriffithsDJ2007IntroductiontoElectrodynamics3rdedn(EnglewoodCliffs,NJ:Prentice-Hall)JacksonJD1999ClassicalElectrodynamics3rdedn(NewYork:Wiley)LandauLDandLifshitzEM1984ElectrodynamicsofContinuousMedia2ndedn(Auckland:Reed)LandauLDandLifshitzEM1975TheClassicalTheoryofFields4thedn(Oxford:Pergamon)PanofskyWKHandPhillipsM1990ClassicalElectricityandMagnetism2ndedn(NewYork:Dover)StrattonJA2007ElectromagneticTheory(NewYork:Wiley)TammIE1979FundamentalsoftheTheoryofElectricity(Paris:Mir)ZangwillA2013ModernElectrodynamics(Cambridge:CambridgeUniversityPress)
PartQM:QuantumMechanicsAbersES2004QuantumMechanics(London:Pearson)AulettaG,FortunatoMandParisiG2009QuantumMechanics(Cambridge:CambridgeUniversityPress)CapriAZ2002NonrelativisticQuantumMechanics3rdedn(Singapore:WorldScientific)Cohen-TannoudjiC,DiuBandLaloëF2005QuantumMechanics(NewYork:Wiley)ConstantinescuF,MagyariEandSpiersJA1971ProblemsinQuantumMechanics(Amsterdam:Elsevier)GalitskiVetal2013ExploringQuantumMechanics(Oxford:OxfordUniversityPress)GottfriedKandYanT-M2004QuantumMechanics:Fundamentals2ndedn(Berlin:Springer)GriffithD2005QuantumMechanics2ndedn(EnglewoodCliffs,NJ:PrenticeHall)LandauLDandLifshitzEM1977QuantumMechanics(NonrelativisticTheory)3rdedn(Oxford:Pergamon)MessiahA1999QuantumMechanics(NewYork:Dover)MerzbacherE1998QuantumMechanics3rdedn(NewYork:Wiley)MillerDAB2008QuantumMechanicsforScientistsandEngineers(Cambridge:CambridgeUniversityPress)SakuraiJJ1994ModernQuantumMechanics(Reading,MA:Addison-Wesley)SchiffLI1968QuantumMechanics3rdedn(NewYork:McGraw-Hill)ShankarR1980PrinciplesofQuantumMechanics2ndedn(Berlin:Springer)SchwablF2002QuantumMechanics3rdedn(Berlin:Springer)
PartSM:StatisticalMechanicsFeynmanRP1998StatisticalMechanics2ndedn(Boulder,CO:Westview)HuangK1987StatisticalMechanics2ndedn(NewYork:Wiley)KuboR1965StatisticalMechanics(Amsterdam:Elsevier)LandauLDandLifshitzEM1980StatisticalPhysics,Part13rdedn(Oxford:Pergamon)LifshitzEMandPitaevskiiLP1981PhysicalKinetics(Oxford:Pergamon)PathriaRKandBealePD2011StatisticalMechanics3rdedn(Amsterdam:Elsevier)PierceJR1980AnIntroductiontoInformationTheory2ndedn(NewYork:Dover)PlishkeMandBergersenB2006EquilibriumStatisticalPhysics3rdedn(Singapore:WorldScientific)SchwablF2000StatisticalMechanics(Berlin:Springer)YeomansJM1992StatisticalMechanicsofPhaseTransitions(Oxford:OxfordUniversityPress)
Multidisciplinary/specialtyAshcroftWNandMerminND1976SolidStatePhysics(Philadelphia,PA:Saunders)BlumK1981DensityMatrixandApplications(NewYork:Plenum)BreuerH-PandPetruccioneE2002TheTheoryofOpenQuantumSystems(Oxford:OxfordUniversityPress)CahnSBandNadgornyBE1994AGuidetoPhysicsProblems,Part1(NewYork:Plenum)CahnSB,MahanGDandNadgornyBE1997AGuidetoPhysicsProblems,Part2(NewYork:Plenum)Cronin J A,GreenbergD F and Telegdi V L 1967University ofChicagoGraduate Problems in Physics (Reading,MA: AddisonWesley)
HookJRandHallHE1991SolidStatePhysics2ndedn(NewYork:Wiley)JoosG1986TheoreticalPhysics(NewYork:Dover)KayeGWCandLabyTH1986TablesofPhysicalandChemicalConstants15thedn(London:LongmansGreen)KompaneyetsAS2012TheoreticalPhysics2ndedn(NewYork:Dover)LaxM1968FluctuationsandCoherentPhenomena(London:GordonandBreach)LifshitzEMandPitaevskiiLP1980StatisticalPhysics,Part2(Oxford:Pergamon)
NewburyNetal1991PrincetonProblemsinPhysicswithSolutions(Princeton,NJ:PrincetonUniversityPress)PaulingL1988GeneralChemistry3rdedn(NewYork:Dover)TinkhamM1996IntroductiontoSuperconductivity2ndedn(NewYork:McGraw-Hill)WaleckaJD2008IntroductiontoModernPhysics(Singapore:WorldScientific)ZimanJM1979PrinciplesoftheTheoryofSolids2ndedn(Cambridge:CambridgeUniversityPress)
1The list does not include the sources (mostly, recent original publications) cited in the lecture notes and problem solutions, and themathematics textbooks andhandbookslistedinsectionA.16.2Recently several high-quality teaching materials on advanced physics became available online, including R. Fitzpatrickâs text on Classical Electromagnetism(farside.ph.utexas.edu/teaching/jk1/Electromagnetism.pdf), B Simonsâ âlecture shrunksâ on Advanced Quantum Mechanics (www.tcm.phy.cam.ac.uk/âŒbds10/aqp.html),andDTongâslecturenotesonseveraladvancedtopics(www.damtp.cam.ac.uk/user/tong/teaching.html).