Lecture 30 PHYS 416 Thursday December 9 Fall 2021 1.7.16 ...

Lecture 30 PHYS 416 Thursday December 9 Fall 2021

1. 7.16 White dwarfs, neutron stars, and black holes2. 12.2 Scale invariance: RG equations3. 12.33 Pandemic

Copyright Oxford University Press 2006 v2.0 --

Exercises 205

photon dominated, because there are currentlymany more photons than atoms.We can also roughly estimate the relative contri-butions of photons and atoms to other propertiesof the Universe.(c) Calculate formulæ for the entropy S, the in-ternal energy E, and the pressure P for the pho-ton gas in a volume V and temperature T . Forsimplicity, write them in terms of the Stefan–Boltzmann constant55 σ = π2k4

B/60!3c2. Ig-nore the zero-point energy in the photon modes56

(which otherwise would make the energy andpressure infinite, even at zero temperature).(Hint: You will want to use the grand free en-ergy Φ for the photons. For your information,∫∞0

x3/(ex − 1) dx = π4/15 = −3∫∞0

x2 log(1 −e−x) dx, where the last integral can be integratedby parts to get the first integral.)(d) Calculate formulæ for the entropy, mass-energy57 density, and pressure for an ideal gasof hydrogen atoms at density nmatter and thesame volume and temperature. Can we ignorequantum mechanics for the atomic gas? As-semble your results from parts (c) and (d) intoa table comparing photons to atoms, with fourcolumns giving the two analytical formulæ andthen numerical values for V = 1 cm3, the cur-rent microwave background temperature, and thecurrent atom density. Which are dominatedby photons? By atoms? (Hint: You willwant to use the Helmholtz free energy A forthe atoms. More useful constants: σ = 5.67 ×10−5 erg cm−2 K−4 s−1, and mH ≈ mp = 1.673×10−24 g.)Before the decoupling time, the coupled light-and-matter soup satisfied a wave eqn [79]:

ρ∂2Θ∂t2

= B∇2θ. (7.90)

Here Θ represents the local temperature fluctu-ation ∆T/T . The constant ρ is the sum of threecontributions: the matter density, the photonenergy density E/V divided by c2, and a contri-bution P/c2 due to the photon pressure P (thiscomes in as a component in the stress-energy ten-sor in general relativity).

(e) Show that the sum of the two photon con-tributions to the mass density is proportional toE/(c2V ). What is the constant of proportional-ity?The constant B in our wave eqn 7.90 is the bulkmodulus: B = −V (∂P/∂V )|S.58 At decoupling,the dominant contribution to the pressure (andto B) comes from the photon gas.(f) Write P as a function of S and V (eliminat-ing T and E), and calculate B for the photongas. Show that it is proportional to the photonenergy density E/V . What is the constant ofproportionality?Let R be the ratio of ρmatter to the sum of thephoton contributions to ρ from part (e).(g) What is the speed of sound in the Universebefore decoupling, as a function of R and c?(Hint: Compare with eqn 10.78 in Exercise 10.1as a check for your answer to parts (e)–(g).)Exercise 10.1 and the ripples-in-fluids animationat [180] show how this wave equation explainsmuch of the observed fluctuations in the mi-crowave background radiation.

(7.16) White dwarfs, neutron stars, and blackholes. (Astrophysics, Quantum) ⃝3As the energy sources in large stars are con-sumed, and the temperature approaches zero,the final state is determined by the competitionbetween gravity and the chemical or nuclear en-ergy needed to compress the material.A simplified model of ordinary stellar matter isa Fermi sea of non-interacting electrons, withenough nuclei to balance the charge. Let usmodel a white dwarf (or black dwarf, since we as-sume zero temperature) as a uniform density ofHe4 nuclei and a compensating uniform densityof electrons. Assume Newtonian gravity. As-sume the chemical energy is given solely by theenergy of a gas of non-interacting electrons (fill-ing the levels to the Fermi energy).(a) Assuming non-relativistic electrons, calculatethe energy of a sphere with N zero-temperature

law says that a black body radiates power σT 4 per unit area, where σ is the Stefan–Boltzmann.

(eqn 7.42) with µ = 0 rather than as harmonic oscillators (eqn 7.23).57That is, be sure to include the mc2 for the hydrogen atoms into their contribution to the energy density.58The fact that one must compress adiabatically (constant S) and not isothermally (constant T ) is subtle but important (IsaacNewton got it wrong). Sound waves happen too fast for the temperature to equilibrate. Indeed, we can assume at reasonablylong wavelengths that there is no heat transport (hence we may use the adiabatic modulus). All this is true both for air andfor early-Universe photon gasses.


206 Quantum statistical mechanics

non-interacting electrons and radius R.59 Cal-culate the Newtonian gravitational energy of asphere of He4 nuclei of equal and opposite chargedensity. At what radius is the total energy min-imized?A more detailed version of this model was stud-ied by Chandrasekhar and others as a modelfor white dwarf stars. Useful numbers: mp =1.6726 × 10−24 g, mn = 1.6749 × 10−24 g, me =9.1095 × 10−28 g, ! = 1.05459 × 10−27 erg s,G = 6.672 × 10−8 cm3/(g s2), 1 eV = 1.60219 ×10−12 erg, kB = 1.3807 × 10−16 erg/K, and c =3× 1010 cm/s.(b) Using the non-relativistic model in part (a),calculate the Fermi energy of the electrons in awhite dwarf star of the mass of the Sun, 2 ×1033 g, assuming that it is composed of helium.(i) Compare it to a typical chemical binding en-ergy of an atom. Are we justified in ignoringthe electron–electron and electron–nuclear inter-actions (i.e., chemistry)? (ii) Compare it to thetemperature inside the star, say 107 K. Are wejustified in assuming that the electron gas is de-generate (roughly zero temperature)? (iii) Com-pare it to the mass of the electron. Are weroughly justified in using a non-relativistic the-ory? (iv) Compare it to the mass difference be-tween a proton and a neutron.The electrons in large white dwarf stars are rel-ativistic. This leads to an energy which growsmore slowly with radius, and eventually to anupper bound on their mass.(c) Assuming extremely relativistic electrons withε = pc, calculate the energy of a sphere of non-interacting electrons. Notice that this energycannot balance against the gravitational energyof the nuclei except for a special value of themass, M0. Calculate M0. How does your M0

compare with the mass of the Sun, above?A star with mass larger than M0 continues toshrink as it cools. The electrons (see (iv) inpart (b) above) combine with the protons, stay-ing at a constant density as the star shrinks intoa ball of almost pure neutrons (a neutron star,often forming a pulsar because of trapped mag-netic flux). Recent speculations [141] suggestthat the ‘neutronium’ will further transform intoa kind of quark soup with many strange quarks,forming a transparent insulating material.

For an even higher mass, the Fermi repulsionbetween quarks cannot survive the gravitationalpressure (the quarks become relativistic), andthe star collapses into a black hole. At thesemasses, general relativity is important, going be-yond the purview of this text. But the basiccompetition, between degeneracy pressure andgravity, is the same.

(7.17) Eigenstate thermalization. (Quantum) ⃝pFootnote 10 on page 183 discusses many-bodyeigenstates as ‘weird delicate superpositions ofstates with photons being absorbed by the atomand the atom emitting photons’. Many-bodyeigenstates with a finite energy density can befar more comprehensible – they often correspondto equilibrium quantum systems.Look up ‘eigenstate thermalization’. Find a dis-cussion that discusses a single pure state of alarge system A+B, tracing out the ‘bath’ B andleaving a density matrix for a small subsystemA.

(7.18) Is sound a quasiparticle? (Condensed mat-ter) ⃝pSound waves in the harmonic approximationare non-interacting – a general solution is givenby a linear combination of the individual fre-quency modes. Landau’s Fermi liquid theory(footnote 23, page 188) describes how the non-interacting electron approximation can be effec-tive even though electrons are strongly coupledto one another. The quasiparticles are electronswith a screening cloud; they develop long life-times near the Fermi energy; they are describedas poles of Greens functions.(a) Do phonons have lifetimes? Do their life-times get long as the frequency goes to zero?(Look up ‘ultrasonic attenuation’ and Gold-stone’s theorem.)(b) Are they described as poles of a Green’s func-tion? (See Section 9.3 and Exercise 10.9.)Are there analogies for phonons to the screeningcloud around a quasiparticle? A phonon screen-ing cloud would be some kind of collective, non-linear movement of atoms that behaved at longdistances and low frequencies like an effective,harmonic interaction. In particular, the effectivescattering between these quasiphonons should be

59You may assume that the single-particle eigenstates have the same energies and k-space density in a spherethey do for a cube of volume V ; just like fixed versus periodic boundary conditions, the boundary does notproperties.

















What are avalanches?


Exercises 231

∑α Pβαρ

(n)α is less than or equal to the free en-

ergy for ρ(n). You may use the properties of theMarkov transition matrix P , (0 ≤ Pαβ ≤ 1 and∑

α Pαβ = 1), and detailed balance (Pαβρ∗β =

Pβαρ∗α, where ρ

∗α = exp(−Eα/kBT )/Z).

(Hint: You will want to use µα = Pαβ ineqn 8.25, but the entropy will involve Pβα, whichis not the same. Use detailed balance to convertfrom one to the other.)

(8.13) Hysteresis and avalanches.48 (Complexity,Computation) ⃝4A piece of magnetic material exposed to an in-creasing external field H(t) (Fig. 8.14) will mag-netize (Fig. 8.15) in a series of sharp jumps, oravalanches (Fig. 8.16). These avalanches ariseas magnetic domain walls in the material arepushed by the external field through a ruggedpotential energy landscape due to irregularitiesand impurities in the magnet. The magnetic sig-nal resulting from these random avalanches iscalled Barkhausen noise.We model this system with a non-equilibriumlattice model, the random field Ising model. TheHamiltonian or energy function for our system is

H = −∑

⟨i,j⟩

Jsisj −∑

i

(H(t) + hi

)si, (8.27)

where the spins si = ±1 lie on a square or cubiclattice with periodic boundary conditions. Thecoupling and the external field H are as in thetraditional Ising model (Section 8.1). The dis-order in the magnet is incorporated using therandom field hi, which is independently chosenat each lattice site from a Gaussian probabilitydistribution of standard deviation R:

P (h) =1√2πR

e−h2/2R2

. (8.28)

We are not interested in thermal equilibrium;there would be no hysteresis! We take the op-posite extreme; we set the temperature to zero.We start with all spins pointing down, and adi-abatically (infinitely slowly) increase H(t) from−∞ to ∞.

SN

Fig. 8.14 Barkhausen noise experiment. By in-creasing an external magnetic field H(t) (bar magnetapproaching), the magnetic domains in a slab of ironflip over to align with the external field. The re-sulting magnetic field jumps can be turned into anelectrical signal with an inductive coil, and then lis-tened to with an ordinary loudspeaker. Barkhausennoise from our computer experiments can be heardon the Internet [91].

Our rules for evolving the spin configuration aresimple: each spin flips over when doing so woulddecrease the energy. This occurs at site i whenthe local field at that site

J∑

j nbr to i

sj + hi +H(t) (8.29)

changes from negative to positive. A spin canbe pushed over in two ways. It can be triggeredwhen one of its neighbors flips (by participatingin a propagating avalanche) or it can be trig-gered by the slow increase of the external field(starting a new avalanche).

-1.0 -0.5 0.0 0.5 1.0Magnetization M

-3

-2

-1

0

1

2

3

App

lied

mag

netic

fiel

d H/J

48This exercise is largely drawn from [92]. It and the associated software were developed in collaboration with ChristopherMyers. Computational hints can be found at the book Web site [167].


Exercises 231

∑α Pβαρ

(n)α is less than or equal to the free en-

ergy for ρ(n). You may use the properties of theMarkov transition matrix P , (0 ≤ Pαβ ≤ 1 and∑

α Pαβ = 1), and detailed balance (Pαβρ∗β =

Pβαρ∗α, where ρ

∗α = exp(−Eα/kBT )/Z).

(Hint: You will want to use µα = Pαβ ineqn 8.25, but the entropy will involve Pβα, whichis not the same. Use detailed balance to convertfrom one to the other.)

(8.13) Hysteresis and avalanches.48 (Complexity,Computation) ⃝4A piece of magnetic material exposed to an in-creasing external field H(t) (Fig. 8.14) will mag-netize (Fig. 8.15) in a series of sharp jumps, oravalanches (Fig. 8.16). These avalanches ariseas magnetic domain walls in the material arepushed by the external field through a ruggedpotential energy landscape due to irregularitiesand impurities in the magnet. The magnetic sig-nal resulting from these random avalanches iscalled Barkhausen noise.We model this system with a non-equilibriumlattice model, the random field Ising model. TheHamiltonian or energy function for our system is

H = −∑

⟨i,j⟩

Jsisj −∑

i

(H(t) + hi

)si, (8.27)

where the spins si = ±1 lie on a square or cubiclattice with periodic boundary conditions. Thecoupling and the external field H are as in thetraditional Ising model (Section 8.1). The dis-order in the magnet is incorporated using therandom field hi, which is independently chosenat each lattice site from a Gaussian probabilitydistribution of standard deviation R:

P (h) =1√2πR

e−h2/2R2

. (8.28)

We are not interested in thermal equilibrium;there would be no hysteresis! We take the op-posite extreme; we set the temperature to zero.We start with all spins pointing down, and adi-abatically (infinitely slowly) increase H(t) from−∞ to ∞.

SN

Fig. 8.14 Barkhausen noise experiment. By in-creasing an external magnetic field H(t) (bar magnetapproaching), the magnetic domains in a slab of ironflip over to align with the external field. The re-sulting magnetic field jumps can be turned into anelectrical signal with an inductive coil, and then lis-tened to with an ordinary loudspeaker. Barkhausennoise from our computer experiments can be heardon the Internet [91].

Our rules for evolving the spin configuration aresimple: each spin flips over when doing so woulddecrease the energy. This occurs at site i whenthe local field at that site

J∑

j nbr to i

sj + hi +H(t) (8.29)

changes from negative to positive. A spin canbe pushed over in two ways. It can be triggeredwhen one of its neighbors flips (by participatingin a propagating avalanche) or it can be trig-gered by the slow increase of the external field(starting a new avalanche).

-1.0 -0.5 0.0 0.5 1.0Magnetization M

-3

-2

-1

0

1

2

3

App

lied

mag

netic

fiel

d H/J

48This exercise is largely drawn from [92]. It and the associated software were developed in collaboration with ChristopherMyers. Computational hints can be found at the book Web site [167].


232 Calculation and computation

Fig. 8.15 Hysteresis loop with subloops forour model. As the external field is raised and low-ered (vertical), the magnetization lags behind—thisis called hysteresis. The magnetization curves herelook macroscopically smooth.

Fig. 8.16 Tiny jumps: Barkhausen noise. Blow-ing up a small portion of Fig. 8.15, we see that themagnetization is growing in a series of sharp jumps,or avalanches.

(a) Set up lattices s[m][n] and h[m][n] on thecomputer. (If you do three dimensions, addan extra index to the arrays.) Fill the formerwith down-spins (−1) and the latter with ran-dom fields (real numbers chosen from the distri-bution 8.28). Write a routine FlipSpin for thelattice, which given i and j flips the spin froms = −1 to s = +1 (complaining if it is alreadyflipped). Write a routine NeighborsUp whichcalculates the number of up-neighbors for thespin (implementing the periodic boundary con-ditions).On the computer, changing the external field in-finitely slowly is easy. To start a new avalanche(or the first avalanche), one searches for the un-flipped spin that is next to flip, jumps the fieldH to just enough to flip it, and propagates theavalanche, as follows.

Lattice

12 12 15 15 19 19 76 1020 6

3 57 8 9

13

1 2 4

152019

14181716

11 12

2421 22 23 25

11 17 10 20 18

End of shell

Queue

Fig. 8.17 Avalanche propagation in the hys-teresis model. Left: a propagating avalanche. Spin13 triggered the avalanche. It triggered the first shellof spins 14, 8, and 12, which then triggered the sec-ond shell 15, 19, 7, 11, and 17, and finally the thirdshell 10, 20, 18, and 6. Right: the first-in–first-out

queue, part way through flipping the second shell.(The numbers underneath are the triggering spinsfor the spins on the queue, for your convenience.)The spin at the left of this queue is next to flip. No-tice that spin 20 has been placed on the queue twice(two neighbors in the previous shell). By placing amarker at the end of each shell in the queue, we canmeasure the number of spins flipping per unit ‘time’during an avalanche (Fig. 8.18).

Propagating an avalanche

(1) Find the triggering spin i for the nextavalanche, which is the unflipped site withthe largest internal field J

∑j nbr to i sj +hi

from its random field and neighbors.

(2) Increment the external field H to minus thisinternal field, and push the spin onto a first-in–first-out queue (Fig. 8.17, right).

(3) Pop the top spin off the queue.

(4) If the spin has not been flipped,49 flip it andpush all unflipped neighbors with positivelocal fields onto the queue.

(5) While there are spins on the queue, repeatfrom step (3).

(6) Repeat from step (1) until all the spins areflipped.

(b) Write a routine BruteForceNextAvalanche

for step (1), which checks the local fields of all ofthe unflipped spins, and returns the location ofthe next to flip.(c) Write a routine PropagateAvalanche thatpropagates an avalanche given the triggeringspin, steps (3)–(5), coloring the spins in the dis-play that are flipped. Run a 300 × 300 systemat R = 1.4, 0.9, and 0.7 (or a 503 system atR = 4, R = 2.16, and R = 2) and display theavalanches. If you have a fast machine, youcan run a larger size system, but do not overdoit; the sorted list algorithm below will dramati-cally speed up the simulation.

49You need to check if the spin is flipped again after popping it off the queue; spins can be put onto the queue more than onceduring an avalanche (Fig. 8.17).






Lattice

12 12 15 15 19 19 76 1020 6

3 57 8 9

13

1 2 4

152019

14181716

11 12

2421 22 23 25

11 17 10 20 18

End of shell

Queue




















Lattice

12 12 15 15 19 19 76 1020 6

3 57 8 9

13

1 2 4

152019

14181716

11 12

2421 22 23 25

11 17 10 20 18

End of shell

Queue
















Exercises 233

0 500 1000 1500 2000 2500 3000Time t (in avalanche shells)

0

100

200

300

dM/d

t ~ v

olta

ge V

(t)

Fig. 8.18 Avalanche time series. Number of do-mains flipped per time step for the avalanche shownin Fig. 12.5. Notice how the avalanche almost stopsseveral times; if the forcing were slightly smaller com-pared to the disorder, the avalanche would have sep-arated into smaller ones. The fact that the disorderis just small enough to keep the avalanche growing isthe criterion for the phase transition, and the causeof the self-similarity. At the critical point, a partialavalanche of size S will on average trigger anotherone of size S.

There are lots of properties that one mightwish to measure about this system: avalanchesizes, avalanche correlation functions, hystere-sis loop shapes, average pulse shapes duringavalanches, . . . It can get ugly if you put allof these measurements inside the inner loop ofyour code. Instead, we suggest that you try thesubject–observer design pattern: each time a spinis flipped, and each time an avalanche is finished,the subject (our simulation) notifies the list ofobservers.(d) Build a MagnetizationObserver, whichstores an internal magnetization starting at −N ,adding two to it whenever it is notified. Buildan AvalancheSizeObserver, which keeps trackof the growing size of the current avalanche af-ter each spin flip, and adds the final size to ahistogram of all previous avalanche sizes whenthe avalanche ends. Set up NotifySpinFlip andNotifyAvalancheEnd routines for your simula-tion, and add the two observers appropriately.Plot the magnetization curve M(H) and theavalanche size distribution histogram D(S) forthe three systems you ran for part (c).

(8.14) Hysteresis algorithms.50 (Complexity, Com-putation) ⃝4As computers increase in speed and memory, thebenefits of writing efficient code become greaterand greater. Consider a problem on a system ofsize N ; a complex algorithm will typically runmore slowly than a simple one for small N , butif its time used scales proportional to N and thesimple algorithm scales as N2, the added com-plexity wins as we can tackle larger, more ambi-tious questions.

+4.0+0.9−1.1

−19.9

4

638

7

295

+14.9 1

0

Sorted list

12

3,48

2 3

5 6

97

4

hi Spin #Lattice

= 1.1H

1

−1.4−2.5−6.6

+5.5

Fig. 8.19 Using a sorted list to find the nextspin in an avalanche. The shaded cells have alreadyflipped. In the sorted list, the arrows on the rightindicate the nextPossible[nUp] pointers—the firstspin that would not flip with nUp neighbors at thecurrent external field. Some pointers point to spinsthat have already flipped, meaning that these spinsalready have more neighbors up than the correspond-ing nUp. (In a larger system the unflipped spins willnot all be contiguous in the list.)

In the hysteresis model (Exercise 8.13), thebrute-force algorithm for finding the nextavalanche for a system with N spins takes atime of order N per avalanche. Since there areroughly N avalanches (a large fraction of allavalanches are of size one, especially in three di-mensions) the time for the brute-force algorithmscales as N2. Can we find a method which doesnot look through the whole lattice every time anavalanche needs to start?We can do so using the sorted list algorithm:we make51 a list of the spins in order of theirrandom fields (Fig. 8.19). Given a field range

50This exercise is also largely drawn from [92], and was developed with the associated software in collaboration with ChristopherMyers.51Make sure you use a packaged routine to sort the list; it is the slowest part of the code. It is straightforward to write yourown routine to sort lists of numbers, but not to do it efficiently for large lists.


338 Continuous phase transitions

many different microscopic models lead to the same low-energy, long-wavelength theory.

Fig. 12.5 A medium-sizedavalanche (flipping 282 785 do-mains) in a model of avalanches andhysteresis in magnets [166] (see Exer-cises 8.13, 12.13 and Fig. 12.11). Theshading depicts the time evolution: theavalanche started in the dark regionin the back, and the last spins to flipare in the upper, front region. Thesharp changes in shading are real, andrepresent sub-avalanches separatedby times where the avalanche almoststops (see Fig. 8.18).

The behavior near continuous transitions is unusually independent ofthe microscopic details of the system—so much so that we give a newname to it, universality. Figure 12.6(a) shows that the liquid and gasdensities ρℓ(T ) and ρg(T ) for a variety of atoms and small moleculesappear quite similar when rescaled to the same critical density and tem-perature. This similarity is partly for mundane reasons: the interactionsbetween the molecules is roughly the same in the different systems up tooverall scales of energy and distance. Hence argon and carbon monoxidesatisfy

ρCO(T ) = AρAr(BT ) (12.1)

for some overall changes of scale A, B. However, Fig. 12.6(b) showsa completely different physical system—interacting electronic spins inmanganese fluoride, going through a ferromagnetic transition. The mag-netic and liquid–gas theory curves through the data are the same if weallow ourselves to not only rescale T and the order parameter (ρ and M ,respectively), but also allow ourselves to use a more general coordinatechange

ρAr(T ) = A(M(BT ), T ) (12.2)

which untilts the axis.2 Nature does not anticipate our choice of ρ and

2Here B = TMc /T ℓg

c is as usualthe rescaling of temperature andA(M,T ) = a1M + a2 + a3T =(ρcρ0/M0)M + ρc(1 + s)− (ρcs/T

ℓgc )T

is a simple shear coordinate transfor-mation from (ρ, T ℓg) to (M,TM ). Asit happens, there is another correctionproportional to (Tc − T )1−α, whereα ∼ 0.1 is the specific heat expo-nent. It can also be seen as a kindof tilt, from a pressure-dependent ef-fective Ising-model coupling strength.It is small for the simple molecules inFig. 12.6(a), but significant for liquidmetals [64]. Both the tilt and this 1−αcorrection are subdominant, meaningthat they vanish faster as we approachTc than the order parameter (Tc−T )β .

T for variables. At the liquid–gas critical point the natural measureof density is temperature dependent, and A(M,T ) is the coordinatechange to the natural coordinates. Apart from this choice of variables,this magnet and these liquid–gas transitions all behave the same at theircritical points.This would perhaps not be a surprise if these two phase diagrams

had parabolic tops; the local maximum of an analytic curve generically3

3The term generic is a mathematicalterm which roughly translates as ‘ex-cept for accidents of zero probability’,like finding a function with zero secondderivative at the maximum.

looks parabolic. But the jumps in magnetization and density near Tc

both vary as (Tc−T )β with the same exponent β ≈ 0.3264 . . . , distinctlydifferent from the square-root singularity β = 1/2 of a generic analyticfunction.Also, there are many other properties (susceptibility, specific heat,

correlation lengths) which have power-law singularities at the criticalpoint, and all of the exponents of these power laws for the liquid–gassystems agree with the corresponding exponents for the magnets. Thisis universality. When two different systems have the same singular prop-erties at their critical points, we say they are in the same universalityclass. Importantly, the theoretical Ising model (despite its drastic sim-plification of the interactions and morphology) is also in the same univer-sality class as these experimental uniaxial ferromagnets and liquid–gassystems—allowing theoretical physics to be directly predictive in realexperiments.To get a more clear feeling about how universality arises, consider

site and bond percolation in Fig. 12.7. Here we see two microscopicallydifferent systems (left) from which basically the same behavior emerges(right) on long length scales. Just as the systems approach the threshold


338 Continuous phase transitions

many different microscopic models lead to the same low-energy, long-wavelength theory.

Fig. 12.5 A medium-sizedavalanche (flipping 282 785 do-mains) in a model of avalanches andhysteresis in magnets [166] (see Exer-cises 8.13, 12.13 and Fig. 12.11). Theshading depicts the time evolution: theavalanche started in the dark regionin the back, and the last spins to flipare in the upper, front region. Thesharp changes in shading are real, andrepresent sub-avalanches separatedby times where the avalanche almoststops (see Fig. 8.18).

The behavior near continuous transitions is unusually independent ofthe microscopic details of the system—so much so that we give a newname to it, universality. Figure 12.6(a) shows that the liquid and gasdensities ρℓ(T ) and ρg(T ) for a variety of atoms and small moleculesappear quite similar when rescaled to the same critical density and tem-perature. This similarity is partly for mundane reasons: the interactionsbetween the molecules is roughly the same in the different systems up tooverall scales of energy and distance. Hence argon and carbon monoxidesatisfy

ρCO(T ) = AρAr(BT ) (12.1)

for some overall changes of scale A, B. However, Fig. 12.6(b) showsa completely different physical system—interacting electronic spins inmanganese fluoride, going through a ferromagnetic transition. The mag-netic and liquid–gas theory curves through the data are the same if weallow ourselves to not only rescale T and the order parameter (ρ and M ,respectively), but also allow ourselves to use a more general coordinatechange

ρAr(T ) = A(M(BT ), T ) (12.2)

which untilts the axis.2 Nature does not anticipate our choice of ρ and

2Here B = TMc /T ℓg

c is as usualthe rescaling of temperature andA(M,T ) = a1M + a2 + a3T =(ρcρ0/M0)M + ρc(1 + s)− (ρcs/T

ℓgc )T

is a simple shear coordinate transfor-mation from (ρ, T ℓg) to (M,TM ). Asit happens, there is another correctionproportional to (Tc − T )1−α, whereα ∼ 0.1 is the specific heat expo-nent. It can also be seen as a kindof tilt, from a pressure-dependent ef-fective Ising-model coupling strength.It is small for the simple molecules inFig. 12.6(a), but significant for liquidmetals [64]. Both the tilt and this 1−αcorrection are subdominant, meaningthat they vanish faster as we approachTc than the order parameter (Tc−T )β .

T for variables. At the liquid–gas critical point the natural measureof density is temperature dependent, and A(M,T ) is the coordinatechange to the natural coordinates. Apart from this choice of variables,this magnet and these liquid–gas transitions all behave the same at theircritical points.This would perhaps not be a surprise if these two phase diagrams

had parabolic tops; the local maximum of an analytic curve generically3

3The term generic is a mathematicalterm which roughly translates as ‘ex-cept for accidents of zero probability’,like finding a function with zero secondderivative at the maximum.

looks parabolic. But the jumps in magnetization and density near Tc

both vary as (Tc−T )β with the same exponent β ≈ 0.3264 . . . , distinctlydifferent from the square-root singularity β = 1/2 of a generic analyticfunction.Also, there are many other properties (susceptibility, specific heat,

correlation lengths) which have power-law singularities at the criticalpoint, and all of the exponents of these power laws for the liquid–gassystems agree with the corresponding exponents for the magnets. Thisis universality. When two different systems have the same singular prop-erties at their critical points, we say they are in the same universalityclass. Importantly, the theoretical Ising model (despite its drastic sim-plification of the interactions and morphology) is also in the same univer-sality class as these experimental uniaxial ferromagnets and liquid–gassystems—allowing theoretical physics to be directly predictive in realexperiments.To get a more clear feeling about how universality arises, consider

site and bond percolation in Fig. 12.7. Here we see two microscopicallydifferent systems (left) from which basically the same behavior emerges(right) on long length scales. Just as the systems approach the threshold

Sethna 12.2 Scale invariance

Will demonstrate some of the concepts behind the Renormalization Group by deriving power law relationships for the distribution of avalanche sizes, D(S).

Near the critical point, the distribution will be:

! " = $"% ⁄' (.

When disorder with size scale R is included, the distribution is:

! ", + = "%,-( + − +( "0,

where - isaparticularscalingfunction.

FortherandomfieldIsingmodel(Problem12.28)theavalanchesizedistributionhastheform

P M = NM%O/QR%MSQ,

whichwewillusein12.33Pandemicnexttime.

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Mean time between contacts !" = ⁄1 &

Mean time until removal !' = ⁄1 (

⁄!' !" = ⁄& ( = )*= # of contacts by an infected person before removal

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Important questions: • How “transmissible” is the disease?• How many contacts does an infected person have?• How variable is this number of contacts?

within-location parameters are largely based on intui-tion, and future versions of this model will includeestimates based on data. We found that the structure ofthese simulated networks is robust to small changes inthese values. Each school or hospital is sub-divided intoclassrooms or wards. Pairs of students or patients withinthese sub-units were connected with higher probabilitythan pairs associated with different sub-units. Teachersare assigned to classrooms and connected stochasticallyto appropriate students. Caregivers are assigned wardsand then connected to appropriate patients. There arealso low probability neighborhood contacts betweenindividuals from different households3.

This network offers a high degree of realism but isquite complex. We therefore use two simpler networks toprovide additional insight. One is a random network witha Poisson degree distribution in which individualsconnect to others independently and uniformly atrandom (Fig. 1B). Neither the simulated nor the Poissonnetwork, however, contains a significant fraction ofsuperspreaders. Therefore we also study a network witha (truncated) power-law degree distribution (Fig. 1C), aform much discussed in recent work on networkepidemiology (Barabasi and Albert, 1999; Pastor-Sator-ras and Vespignani, 2001). This network has a ‘‘heavytail’’ of superspreaders (Fig. 2) and, as we will see, theseindividuals can have a profound effect on outbreakpatterns despite being few in number. Network theoristsoften refer to such networks as scale-free because of theabsence of a typical degree in the network.

We define the transmissibility of a disease, T, to be theaverage probability that an infectious individual willtransmit the disease to a susceptible individual withwhom they have contact. T summarizes core aspects ofdisease transmission including the rate at which contactstake place between individuals, the likelihood that acontact will lead to transmission, the duration of theinfectious period, and the susceptibility of individuals toSARS. The epidemic threshold, which, in an uncorre-lated network, is given by

Tc ¼hki

hk2i" hki;

where hki and hk2i are the mean degree and mean squaredegree of the network, respectively, is the minimumtransmissibility (T) required for an outbreak to become alarge-scale epidemic.

We choose the parameters of the Poisson and powerlaw networks so that all three networks share the sameepidemic threshold Tc. Let pk denote the probability thata randomly selected individual in a network has degreek. Then, the Poisson network is given by pk ¼

ðmk=k!Þ expð"mÞ with mean contact number m=19.6;and the power law network is given by

pk ¼0 for k ¼ 0;

Ck"a exp "kk

! "

for k40;

(

with distribution parameters k=94.2 and a=2. Here wefixed a and solved for k. The results described below arequalitatively similar for a large range of values of a.Truncation of the power law distribution raises theepidemic threshold of the network to values comparableto those found for urban networks.

To generate the two idealized networks, we begin witha specified number of vertices and choose degrees forthese vertices at random from the desired degreedistribution. Then we connect random pairs of vertices,until the chosen degrees are exhausted. This often yieldsimperfect graphs with loops connecting vertices tothemselves or redundant edges that connect the sametwo vertices more than once. We remove these imperfec-tions using an algorithm suggested by Maslov et al.(2001) in which we select at random two edges connect-ing, for example, vertex pairs AB and CD, and swapthem so that they now connect AC and BD, unless thiswould create a new loop or double edge, in which case wedo nothing. This process occasionally eliminates loopsand repeated edges and by repeating it a sufficiently largenumber of times (depending on the network size) we canproduce a network with none at all.

2.1. Epidemiological analysis

Given the degree distribution of a network, we canuse tools based on percolation theory (Pastor-Satorras

ARTICLE IN PRESS

Fig. 2. Cumulative degree distributions for simulated urban, Poisson,and power law networks. As described in the text, these share the sameepidemic threshold (Tc). Each line gives the probability that arandomly chosen individual (vertex) will have at least the number ofcontacts (degree) indicated on the x-axis. The degree distribution forthe urban network is nearly exponential for degrees greater than ten.

3The authors will provide copies of the simulation software and adetailed list of parameter values to any interested readers upon request.

L.A. Meyers et al. / Journal of Theoretical Biology 232 (2005) 71–8174




Tc ¼hki

hk2i" hki;




pk ¼0 for k ¼ 0;

Ck"a exp "kk

! "

for k40;

(





ARTICLE IN PRESS







Tc ¼hki

hk2i" hki;




pk ¼0 for k ¼ 0;

Ck"a exp "kk

! "

for k40;

(





ARTICLE IN PRESS







Tc ¼hki

hk2i" hki;




pk ¼0 for k ¼ 0;

Ck"a exp "kk

! "

for k40;

(





ARTICLE IN PRESS




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In [ ]: def pandemicInstance(R0=1., i0 = 1): I = i0 t = 0; Itraj = [I] size = i0 while I!=0: I = random.poisson(R0*I) size += ... Itraj.append(...) return size, Itraj size = 0;while size < 1000000: size, traj = pandemicInstance(...) plot(traj)title("size is "+ str(size))xlabel("Time in shells")ylabel("Infected in this shell")show()

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In [ ]: def pandemicEnsemble(N, R0=1., i0 = 1): sizes = [] durations = [] for n in range(N): I = ... size = i0 while I!=0: I = random.poisson(R0*I) size += ... sizes += [...] return sizes sizes = pandemicEnsemble(10**4, R0=...)hist(sizes, bins=100)title("Epidemic size histogram")xlabel("Size")ylabel("Counts")show()

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Ro=0.9999


Exercises 233

0 500 1000 1500 2000 2500 3000Time t (in avalanche shells)

0

100

200

300dM

/dt ~

vol

tage

V(t)

Fig. 8.18 Avalanche time series. Number of do-mains flipped per time step for the avalanche shownin Fig. 12.5. Notice how the avalanche almost stopsseveral times; if the forcing were slightly smaller com-pared to the disorder, the avalanche would have sep-arated into smaller ones. The fact that the disorderis just small enough to keep the avalanche growing isthe criterion for the phase transition, and the causeof the self-similarity. At the critical point, a partialavalanche of size S will on average trigger anotherone of size S.

There are lots of properties that one mightwish to measure about this system: avalanchesizes, avalanche correlation functions, hystere-sis loop shapes, average pulse shapes duringavalanches, . . . It can get ugly if you put allof these measurements inside the inner loop ofyour code. Instead, we suggest that you try thesubject–observer design pattern: each time a spinis flipped, and each time an avalanche is finished,the subject (our simulation) notifies the list ofobservers.(d) Build a MagnetizationObserver, whichstores an internal magnetization starting at −N ,adding two to it whenever it is notified. Buildan AvalancheSizeObserver, which keeps trackof the growing size of the current avalanche af-ter each spin flip, and adds the final size to ahistogram of all previous avalanche sizes whenthe avalanche ends. Set up NotifySpinFlip andNotifyAvalancheEnd routines for your simula-tion, and add the two observers appropriately.Plot the magnetization curve M(H) and theavalanche size distribution histogram D(S) forthe three systems you ran for part (c).

(8.14) Hysteresis algorithms.50 (Complexity, Com-putation) ⃝4As computers increase in speed and memory, thebenefits of writing efficient code become greaterand greater. Consider a problem on a system ofsize N ; a complex algorithm will typically runmore slowly than a simple one for small N , butif its time used scales proportional to N and thesimple algorithm scales as N2, the added com-plexity wins as we can tackle larger, more ambi-tious questions.

+4.0+0.9−1.1

−19.9

4

638

7

295

+14.9 1

0

Sorted list

12

3,48

2 3

5 6

97

4

hi Spin #Lattice

= 1.1H

1

−1.4−2.5−6.6

+5.5

Fig. 8.19 Using a sorted list to find the nextspin in an avalanche. The shaded cells have alreadyflipped. In the sorted list, the arrows on the rightindicate the nextPossible[nUp] pointers—the firstspin that would not flip with nUp neighbors at thecurrent external field. Some pointers point to spinsthat have already flipped, meaning that these spinsalready have more neighbors up than the correspond-ing nUp. (In a larger system the unflipped spins willnot all be contiguous in the list.)

In the hysteresis model (Exercise 8.13), thebrute-force algorithm for finding the nextavalanche for a system with N spins takes atime of order N per avalanche. Since there areroughly N avalanches (a large fraction of allavalanches are of size one, especially in three di-mensions) the time for the brute-force algorithmscales as N2. Can we find a method which doesnot look through the whole lattice every time anavalanche needs to start?We can do so using the sorted list algorithm:we make51 a list of the spins in order of theirrandom fields (Fig. 8.19). Given a field range

50This exercise is also largely drawn from [92], and was developed with the associated software in collaboration with ChristopherMyers.51Make sure you use a packaged routine to sort the list; it is the slowest part of the code. It is straightforward to write yourown routine to sort lists of numbers, but not to do it efficiently for large lists.

From recent discussion of avalanches: SIR model near transition point:

Suggests self-similarity, same universality class....








Not than informative...

In [ ]: def intlogspace(start, stop, num=50, endpoint=True, base=10.0): realBins = logspace(start, stop, num, endpoint, base) bins = unique(realBins.astype(int)) return bins

In [ ]: sizes = pandemicEnsemble(10000, R0=0.99) def logbinnedHist(vals, nBins=30, exponent=3/2): maxval = max(vals) bins = intlogspace(0,int(log10(maxval))+1,nBins) widths = (bins[1:]-bins[:-1]) counts, edges = histogram(vals, bins=bins) hist_norm = counts/(widths*len(vals)) plot(bins, (exponent-1) * bins**(-exponent), color='green', linewidth=3, label="Power law -"+str(exponent)) bar(bins[:-1], hist_norm, widths, bottom=10**(-9)) # bottom avoids problems with log(0) xscale('log') yscale('log') # or 'log=True' in bar) legend() logbinnedHist(sizes)title("Epidemic size probability distribution")xlabel("Size S")ylabel("P(S)")show()

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In [ ]: def logbinnedHistScaling(vals,R0=0.9,C=0.45): num = len(vals) maxval = max(vals) bins = intlogspace(0,int(log10(maxval))+1,30) widths = (bins[1:]-bins[:-1]) centers = (bins[1:]+bins[:-1])/2 counts, edges = histogram(vals, bins=bins) counts_rescaled = ...*counts/(num*widths) plot(centers, C*exp(-centers*(1-R0)**2/2), color='green', linewidth=3, label="Universal prediction") plot(centers, ..., label = "Rescaled data") minY = max(min(counts_rescaled),10**(-4)) ylim(minY,10**0) title("Epidemic size distribution scaling plot") xlabel("Size S") ylabel(r"S**(3/2) P(S)") xscale('log') yscale('log') sizes = pandemicEnsemble(10000, R0=0.9)logbinnedHistScaling(sizes,R0=0.9)








Pretty darned good fit!

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Lecture 30 PHYS 416 Thursday December 9 Fall 2021 1.7.16 ...

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