Theory applied to Warm Dense Matter · 2017. 7. 11. · Redmer, Science 348, 1455 (2015) D...
Transcript of Theory applied to Warm Dense Matter · 2017. 7. 11. · Redmer, Science 348, 1455 (2015) D...
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TheoryappliedtoWarmDenseMatterLukeShulenburger
WhatisWarmDenseMatter?§ WarmDenseMatterisgenerallyassociatedwithstronglycoupledions
(Gii >1)andmoderatelydegenerateelectrons(q ~1)
§ Itistypicallyfoundatthejunctionofsolid,liquid,gas,andplasma.ThecomplicatedinterplayofthephysicalprocessesthatWDMshareswithitsneighborscreatesconsiderabledifficultiesfortheory.
§ q >>1isequivalenttowhereL isthethermaldeBrogliewavelength§ Fermi-Diracstatisticsforelectronicdegreesoffreedomstarttakingon
moreofaMaxwell-Boltzmanncharacter
WhereisWarmDenseMatterfound?
is closely connected with (P > 1 Mbar)
Somehistoryofthefield§ Conferencesdevotedtonon-idealplasmasandtheirstudy
havebeenaroundforalongtime§ ThePhysicsofNon-IdealPlasmasMeetingseriesstartedin1980informer
EastGermany§ TheStronglyCoupledCoulombSystemsmeetingshavealsohadalong-
standingcontingentinterestedinwhatisnowcalledWarmDenseMatter
§ Thefirst“WarmDenseMatter”meetingwasorganizedbyAndrewNg(whocoinedthename)andheldinVancouver,BCin2000.
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Whatmakesitdifficult?§ Thereisnosmallparameter
§ Plasmaexpansionsing,theplasmaparameter,failforWDM
§ Manydifferentaspectsofthephysicscontributeatacomparablelevelandmustbeincluded§ strongcorrelations§ ionization§ bondformationandbreaking§ complexpressureandtemperaturedependentchemistry
§ Computations/simulationsareoftenquitedemanding§ Massivelyparallelcomputationsarethenorm§ Shortcutsareverytempting
§ Experimentalconditionsareshortlivedandhardtodiagnose§ Whichexperimentalresultsshouldyoubelieve?
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Better EOSPhase diagrams
Time/Path dependence Approach
to phase equilibrium
Transport and mechanical properties
Mixtures
Material structure, Electronic, Ionic
Non-LTE
Electronic Excitation
Heating the Material
Ionic ExcitationCompressing the
Material
ScientificneedsinWarmDenseMatterResearch
Experimentalapproaches
Considersomethingas“simple”asthedeuteriumHugoniot
7Pierre Henri Hugoniot
One of these thingsis not like the others
Theprogresscontinuestothisday
§ Experimentaltechniquescontinuetoimproveinprecisionandaccuracy§ Understandingofmaterial
propertiesbootstrapsitself
§ Startingtodiscriminatethelimitationsofcurrenttheoreticalapproximations
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Knudson and Desjarlais, PRL 118, 035501 (2017)
Pres
sure
• Equations governing the properties of a material under any conditions are known
• Just need to solve the 3N dimensional partial differential equations• Approximations are necessary for real materials
QuantumCalculationsofferanappealingroute
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2ˆ !!!!
• Equations governing the properties of a material under any conditions are known
• Just need to solve the 3N dimensional partial differential equations• Approximations are necessary for real materials
• If we could do this accurately and efficiently, we could calculate any physical property
QuantumCalculationsofferanappealingroute
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HΨ(r1…rN ) = EΨ(r1…rN )
ååå-
+-
+Ñ
-=¹ Ii iI
I
ji jii
i
rReZ
rre
mH
,
222
21
2ˆ !!!!
Quantumcalculationsarenottrivial• RecastSchrodingerequationasanintegralproblemin3N
dimensions
● Massiveparallelismavailable,eachpointcanbecalculatedindependently
• Poorscaling
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RRR
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Quantumcalculationsarenottrivial• RecastSchrodingerequationasanintegralproblemin3N
dimensions
● Massiveparallelismavailable,eachpointcanbecalculatedindependently
• Poorscaling� 3dimensionsperelectron� 20pointsineachdirection� 209 ≈512billionpointsfor3electrons
� 3.8TBjusttostore!
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DFT– TheMostCommonApproximation§ Threeinsightsunderpinthedevelopmentofthemost
commonlyusedtheory§ PhysicalInsight
§ Wavefunctionisnotanobservablebutthedensityis§ Replacethe3Ndimensionalwavefunctionwiththe3dimensionaldensity
§ Canapproximatekineticenergyanddevelopasensibledensitybysolvingfornoninteracting electronsinaneffectivepotential
§ Areasonableapproximationistomaketheeffectivepotentialasimplefunctionofthedensity
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N(r) = ∫Ψ(r,r2..rN)dr2���drN
V(r) ∝ 1/r
Ion
DFT– TheMostCommonApproximation§ Threeinsightsunderpinthedevelopmentofthemost
commonlyusedtheory§ PhysicalInsight
§ Wavefunctionisnotanobservablebutthedensityis§ Replacethe3Ndimensionalwavefunctionwiththe3dimensionaldensity
§ Canapproximatekineticenergyanddevelopasensibledensitybysolvingfornoninteracting electronsinaneffectivepotential
§ Areasonableapproximationistomaketheeffectivepotentialasimplefunctionofthedensity§ Thisisthedensityfunctional
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N(r) = ∫Ψ(r,r2..rN)dr2���drN
V(r) ∝ 1/r
Ion
(Kohn-Sham)DFTinpractice§ Startbychoosingapproximation(functional)§ MakeBorn-Oppenheimerapproximation
§ Chooseachunkofmaterialtostudy(generallyperiodicboundaryconditions)
§ Solveforelectronsinpresenceofstaticions§ Note:Mermin approximationmeansthermalgroundstate
§ Hamiltonianisnowset.Solveforsingleparticlesolutionsinsomebasis(physiciststendtopreferplanewaves,chemistsgaussians)§ Note:allofthesesolutionsneedtobeorthogonalforFermionslike
electrons,soneedO(N3)diagonalizationofamatrix
§ Combinesingleparticlewavefunctionstogetdensityandthusinteractionpotential§ Usethistogetnewapproximationofpotentialanditeratetoself-
consistency 15
WhatdoesDFTallowustocalculate?
§ Formally,wegetE(R)andrelatedquantitieslikeF(R)§ Ifweareintegratingovertheelectrons,thisisgenerally
enough§ Structure,Diffusion…
§ Forelectronicproperties,thingsaremorecomplicated§ Formallyspeaking,needanewtypeoffunctional§ Practically,weoftenuse(abuse)thesingleparticlewavefunctions
§ Conductivity,Opacity…
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DFTisaverysuccessfultechniqueforstudyingWDM
§ CarefulDFT/QMDcalculationscancomplementexperimentbyprovidingadditionalinformation
§ Thisisespeciallypowerfulwhenexperimentscanvalidateapproximations
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Root, LNS, Lemke, Dolan, Mattsson and Desjarlais, PRL 115, 198501 (2015)
Shock melting of diamond Phase diagram of MgO
Knudson, Desjarlais and Dolan, Science 322, 1823 (2008)
DFTisnotperfect
§ Scalingwithtemperature§ Memory~T3
§ CPUtime~T4.5
§ Approximationsarenotoriouslydifficulttoimprove
§ Theseshortcomingsarenotjustacademic!
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Knudson, Desjarlais, Becker, Lemke, Cochrane, Savage, Bliss, Mattsson and
Redmer, Science 348, 1455 (2015)
D2 liquid-liquid phase transition
FRONTIERSINCALCULATINGPROPERTIESOFWDM:EXTENDINGRANGEOFVALIDITY
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AverageAtomApproximation
§ Athightemperatures,thepresenceofnearbyatomsislessrelevant§ Solveforanisolated“averageatom”in
abackgroundofelectrons§ Selfconsistentlychangeionizationstate
withtemperature§ Fewerelectronson”atom”moreinbackground
§ InadditiontoEOS,ageneralizationtohandlevariousionicstatesinaplasmacanbeusedtocalculatetransportproperties
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Bailey et al. Nature 517, 56 (2015)
Fe opacity at stellar conditions
AverageAtomApproximation
§ Athightemperatures,thepresenceofnearbyatomsislessrelevant§ Solveforanisolated“averageatom”in
abackgroundofelectrons§ Selfconsistentlychangeionizationstate
withtemperature§ Fewerelectronson”atom”moreinbackground
§ InadditiontoEOS,ageneralizationtohandlevariousionicstatesinaplasmacanbeusedtocalculatetransportproperties
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Bailey et al. Nature 517, 56 (2015)
Fe opacity at stellar conditions
What about higher densities / lower temperatures?See for example talk by C. Starrett
OrbitalFreeDFT§ Kohn-Shamapproximationisnotonly
approachtoDFT§ Cansolveeverythingwithoutcalculating
singleparticlestates§ Approximationbecomesmuchmore
difficult§ Needtogetkineticenergyandelectronic
entropyfromdensityalone!§ Notoriouslydifficulttogetmoleculesto
bind§ However,thisisanoldtechnique
§ Thomas-Fermiapproximationisperhapsearliestexample
§ Typicallyworkswellathightemperatures
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SiO2 from OFDFT and KSDFT
Sjostrom and Crockett, PRB 92, 115104 (2015)
OrbitalFreeDFT§ Kohn-Shamapproximationisnotonly
approachtoDFT§ Cansolveeverythingwithoutcalculating
singleparticlestates§ Approximationbecomesmuchmore
difficult§ Needtogetkineticenergyandelectronic
entropyfromdensityalone!§ Notoriouslydifficulttogetmoleculesto
bind§ However,thisisanoldtechnique
§ Thomas-Fermiapproximationisperhapsearliestexample
§ Typicallyworkswellathightemperatures
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SiO2 from OFDFT and KSDFT
Sjostrom and Crockett, PRB 92, 115104 (2015)
Work continues to improve functionals and implementationSee for example work of E. Carter, T. Sjostrom, S. Trickey and K. Burke
PathIntegralMonteCarlo
§ ReturntodirectsolutionofmanybodySchrodingerequation
§ UseFeynmanformulationtocastasseriesofpathintegrals
§ SampleoverpathintegralsusingMonteCarlotechniques
§ Veryexpensive,potentiallyveryaccurate§ Difficultiesduetosymmetryof
particlesandergodicity(pathsgettangledatlowtemperatureorhighdensity)
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Hugoniot of Deuterium
Militzer and Ceperley, PRL, 85, 1890 (2000)
PathIntegralMonteCarlo
§ ReturntodirectsolutionofmanybodySchrodingerequation
§ UseFeynmanformulationtocastasseriesofpathintegrals
§ SampleoverpathintegralsusingMonteCarlotechniques
§ Veryexpensive,potentiallyveryaccurate§ Difficultiesduetosymmetryof
particlesandergodicity(pathsgettangledatlowtemperatureorhighdensity)
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Hugoniot of Deuterium
Militzer and Ceperley, PRL, 85, 1890 (2000)
Work continues to improve approximations and expand validity for WDMSee for example work of Bonitz, Ceperely and talk by B. Militzer
FRONTIERSINCALCULATINGPROPERTIESOFWDM:IMPROVINGAPPROXIMATIONS
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GroundstateMonteCarloApproaches§ Improvetreatmentof
electronicinteractionbystochasticsamplingofmanybodySchrodingerequation
§ Canbehighlyaccurate,butveryexpensive
§ Inaccuraciesareoftenduetoapproximationsnecessarytoimprovecomputationalcost§ Methodisverywellsuited
tosupercomputersthough,sothisshouldimprove
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Mazzola and Sorella, PRL 114, 105701 (2015)
BetterDFTfunctionals
§ DFTgets10’softhousandsofcitationsyearly§ Lotsofworkonimproving
functionalsforgroundstate/ambientproperties
§ Functionalsforfinitetemperature§ OlderworkbyDharma-
Wardana andPerrot(1984)§ RecentPIMCcalculationsfor
referencesystemhavespurredwork
§ Trickey groupandBurkegroupbothhavenewapproximations
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Browne et al. PRL 110, 146405 (2013)
FRONTIERSINCALCULATINGPROPERTIESOFWDM:BEYONDEQUATIONOFSTATE
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TransportProperties:Kubo-Greenwood
§ CancalculateelectronicandthermalconductivityusingDFT
§ CalculateimaginarypartofdielectricfunctionusingtheKubo-Greenwoodrelation§ Linearresponse§ Matrixelementsofsingle
particlewavefunctions
§ WorkswellinWDMregime§ Issuesifbandgapisnot
closed(Helium?)
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Desjarlais, Kress, and Collins, Phys. Rev. E 66, 025401 (2002)
Aluminum electrical conductivity
Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding
theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat
constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior
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Born-Oppenheimer TDDFT
Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding
theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat
constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior
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Born-Oppenheimer TDDFT
Transportproperties:Energytransfer§ Stoppingafastmovingion§ Understandinghowionsareslowedisessentialtounderstanding
theenergybalanceininertialconfinementfusion§ Asatestproblem,wedragahydrogenionthroughaluminumat
constantvelocityandmeasuretheforceontheion§ Generationofplasmons necessarytocapturetheproperbehavior
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Born-Oppenheimer TDDFT
FRONTIERSINCALCULATINGPROPERTIESOFWDM:WHERETOFROMHERE
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Whatproblemsseemripe?
§ WhiteDwarfs§ Cosmochronology – ageofwhitedwarfsthroughcoolingrate
§ EquationsofstateforC,O,andHeatextremeconditions§ Thermalconductivities,viscosities,diffusioncoefficients
§ Asteroseismology – modelpulsatingwhitedwarf,toinfer§ Totalmassandmasscomposition§ Interiorrotationprofile§ Surfacetemperature§ Structuraldetails
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SummaryComments
§ WarmDenseMatterresearchisgrowingrapidlyandaddressingimportantproblemsininertialfusion,planetaryscience,andmanyotherfields
§ WarmDenseMatterhasbeenveryslowtoyieldtopurelytheoreticaldescription
§ Advancesinelectronicstructureandmoleculardynamicsmethodshaveplayedanenormousroleinadvancingourunderstandingofwarmdensematter
§ Thisisgoingtobeaveryinterestingweek!36