equivalence of ensembles for general many-body...

Equivalence of ensembles for generalmany-body systems

Hugo Touchette

School of Mathematical SciencesQueen Mary, University of London

Open Statistical Physics MeetingOpen University, Milton Keynes, UK

March 2011

Supported by RCUK

Hugo Touchette (QMUL) Ensemble equivalence March 2011 1 / 17

Outline

1 Equilibrium ensembles

2 Result to be generalised

3 New result

4 Applications

5 Conclusion


Microcanonical vs canonicalN-particle systemHamiltonian: H(ω)Macrostate: M(ω)

Microcanonical u = H/N ME

Pu(ω) = const · δΛ|u

Density of states:

Ω(u) =

∫δ(H(ω)− uN) dω

Entropy:

s(u) = limN→∞

1

Nln Ω(u)

Equilibrium states:

Eu = mu

Canonical β CE

Pβ(ω) = e−βH(ω)/Z (β)

Partition function:

Z (β) =

∫e−βH(ω) dω

Free energy:

ϕ(β) = limN→∞

− 1

Nln Z (β)

Equilibrium states:

Eβ = mβHugo Touchette (QMUL) Ensemble equivalence March 2011 3 / 17

Equivalence of ensembles

ME?= CE

Thermodynamic level

u?←→ β

s(u)?←→ ϕ(β)

Macrostate level

Eu ?←→ Eβ

Thermo equivalence?←→ Macrostate equivalence

Main result

Short-range systems have equivalent ensembles

Long-range systems may have nonequivalent ensembles

Related to concavity of s(u)


Thermodynamic equivalence

Microcanonical Canonical

s(u)

u

slope= β

slope= uϕ(β)

β

s(u) = infββu − ϕ(β) ϕ(β) = inf

uβu − s(u)

s ←→ ϕu ←→ β

s = ϕ∗ ϕ = s∗


Thermodynamic nonequivalence

s

u

ϕ

β u

s**

Nonconcave Always concaves

ϕ = s∗

s∗∗ = ϕ∗

s 6= s∗∗

s∗∗(u) = concave envelope of s(u)

Related to first-order phase transitions


Macrostate equivalence of ensembles[Eyink & Spohn JSP 1993; Ellis, Haven & Turkington JSP 2000]

Theorem1 Equivalence:

s(u) strictly concave ⇒ Eu = Eβ for some β ∈ R2 Nonequivalence:

s(u) nonconcave ⇒ Eu 6= Eβ for all β ∈ R3 Partial equivalence:

s(u) concave (not strictly) ⇒ Eu ⊆ Eβ

Assumptions

1 H(ω) can be expressed as a function of M(ω)I Energy representation function h(m)

2 Entropy s(m) for M(ω):I s(u) = sup

m:h(m)=us(m)


Applications[Campa, Dauxois & Ruffo Phys Rep 2009]

Gravitational systemsI Lynden-Bell, Wood, Thirring (1960-)I Cohen & Ispolatov (2002)I Chavanis (2002, 2006)

Spin systems

I Blume-Emery-Griffiths modelI Mean-field Potts model (q ≥ 3)I Mean-field φ4 modelI Quantum spins (Kastner 2010)

Free electron laser (Florence-Lyon group)

2D turbulence modelI Point-vortex models, OnsagersI Kiessling & Lebowitz (1997)I Ellis, Haven & Turkington (2002)

Networks (Barre & Goncalves 2007)

s(u)

u1

2− 1

4− 1

6−

1

0

0.5

1

1.5

-1-0.5

00.5

ε=0.16

ε=0.08

ε=-0.04

ε=0.04

ε=0

m

s(ε,m)

traces over the single-spin Hilbert spaces C2, which can beeasily performed. (iv) The resulting high-dimensionalcomplex integral can be solved in the thermodynamic limitN ! 1, for example, by the method of steepest descent.

The final result for the microcanonical entropy of theCurie-Weiss anisotropic quantum Heisenberg model in thethermodynamic limit is

s!e;m" # ln2$ 12%1$ f!e;m"& ln%1$ f!e;m"&

$ 12%1' f!e;m"& ln%1' f!e;m"& (5)

with

f!e;m" #!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!m2

"1$ !3

!?

#$ 2e

!?

s; (6)

and !? # maxf!1;!2g [16], where s!e;m" is defined on thesubset of R2 for which

0<m2!!? $ !3" $ 2e < !? and 2e <$m2!3: (7)

The result is remarkably simple, in the sense that anexplicit expression for s!e; m" can be given. This is incontrast to the canonical ensemble, where g!"; h" is givenimplicitly as the solution of a maximization [13]. Plots ofthe domains and graphs of s!e;m" are shown in Fig. 1 for anumber of coupling strengths !?, !3.

Nonequivalence of ensembles.—On a thermodynamiclevel, equivalence or nonequivalence of the microcanoni-cal and the canonical ensembles is related to the concavityor nonconcavity of the microcanonical entropy [5]. Byinspection of rows three to seven in Fig. 1 [or by simpleanalysis of the results in (5)–(7)], the entropy s for !? >!3 is seen to be a concave function on a domain which is aconvex set. For !? < !3, the domain is not a convex setand therefore the entropy is neither convex nor concave. Inthe latter case, microcanonical and canonical ensemblesare not equivalent, in the sense that it is impossible toobtain the microcanonical entropy s!e;m" from the canoni-cal Gibbs free energy g!"; h", although the converse isalways possible by means of a Legendre-Fencheltransform.

The physical interpretation of ensemble equivalence isthat every thermodynamic equilibrium state of the systemthat can be probed by fixing certain values for e and m canalso be probed by fixing the corresponding values of theinverse temperature"!e;m" and the magnetic field h!e;m".In the situation !? < !3 where nonequivalence holds, thisis not the case: only equilibrium states corresponding tovalues of (e, m) for which s coincides with its concaveenvelope can be probed by fixing (", h); macrostatescorresponding to other values of (e, m), however, are notaccessible as thermodynamic equilibrium states when con-trolling temperature and field in the canonical ensemble. Inthis sense, microcanonical thermodynamics can be consid-ered not only as different from its canonical counterpart,but also as richer, allowing us to probe equilibrium states ofmatter which are otherwise inaccessible. The realization ofa long-range quantum spin system by means of a cold

dipolar gas in an optical lattice offers the unique andexciting possibility to study such states in a fully controlledlaboratory setting [17].Thermodynamic equivalence of models.—Let us leave

aside for a moment the question of experimental realiza-

FIG. 1 (color online). Domains (left) and graphs (right) of themicrocanonical entropy s!e;m" of the anisotropic quantumHeisenberg model for some combinations of the couplings !?,!3. From top to bottom: !!?;!3" # !1=4; 1", (9=10, 1), (1, 1), (1,9=10), (1, 1=2), (1, 1=5), (1, 0). For the domains, the abscissa isthe energy e and the ordinate is the magnetization m, and theentropy is defined on the shaded area.

PRL 104, 240403 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending18 JUNE 2010

240403-3


Step 1: Equilibrium large deviations

[Lanford 1973; Ellis 1985; HT Phys Rep 2009]

Microcanonical

Large deviation principle:

Pu(m) e−NI u(m)

Rate function:

I u(m) = limN→∞

− 1

Nln Pu(m)

Equilibrium states:

Eu = m : I u(m) = 0

Canonical

Large deviation principle:

Pβ(m) e−NIβ(m)

Rate function:

Iβ(m) = limN→∞

− 1

Nln PN,β(m)

Equilibrium states:

Eβ = m : Iβ(m) = 0


Step 2: Energy decomposition of CE

CE = mixture of MEs

Pβ(ω) =e−βH(ω)

Z (β), Pu(ω) = const · δΛ|u

Energy conditioning:Pβ(ω|u) = Pu(ω)

Energy decomposition:

Pβ(ω) =

∫Pβ(ω|u)Pβ(u) du =

∫Pu(ω)Pβ(u) du

Energy LDP:

Pβ(u) e−NJβ(u), Jβ(u) = βu − s(u)− ϕ(β)

Equilibrium energy:Uβ = u : Jβ(u) = 0


Step 3: Representation formula

Theorem

Iβ(m) = infuJβ(u) + I u(m)

Proof:

Pβ(m) =

∫Pβ(m|u)Pβ(u) du

=

∫Pu(m)Pβ(u) du

∫

e−N[I u(m)+Jβ(u)] du

e−N infuJβ(u)+I u(m)


Main result

Assumption

LDPs for Pβ(m) and Pu(m) ⇒

I u(m), Iβ(m) existEu, Eβ exists(u), ϕ(β) exist

Theorem1 Equivalence:

s(u) strictly concave ⇒ Eu = Eβ for some β ∈ R2 Nonequivalence:

s(u) nonconcave ⇒ Eu 6= Eβ for all β ∈ R3 Partial equivalence:

s(u) concave (not strictly) ⇒ Eu ⊆ Eβ


Proof

Iβ(m) = infuJβ(u) + I u(m)

Eu = m : I u(m) = 0Eβ = m : Iβ(m) = 0Uβ = u : Jβ(u) = 0

Result 1

Iβ(Euβ ) = 0

Result 2

I uβ (mβ) = 0

s

u

Strictly concaveUβ = u⇒ Eβ = Eu

s

u

Nonconcaveu /∈ Uβ ∀β ∈ R⇒ Eu ∩ Eβ = ∅

s

u

Concave not strictlyUβ = u, u′, . . .⇒ Eu ⊆ Eβ


Applications

Mean-field and non-mean-field systems

Long-range systems

Gravitating systems

Short-range systems

Mixed short/long-range systems

Macrostates with no energy representation functions

More than one macrostate for given model


Example 1: 1D α-Ising model[Barre, Bouchet, Dauxois & Ruffo JSP 2005; Dyson CMP 1969]

1D spin model:

H =J

N1−α

N∑i>j=1

1− SiSj

|i − j |α, J > 0, Si = ±1

Mean-field limit for 0 ≤ α < 1

“Mean-field” macrostate: Magnetization profile m(x)

Standard magnetization:

M =1

N

N∑i=1

Si

No energy representation function for M

Entropy s(u) is known

Equivalence (either strict or partial)


Example 2: Short/long-range models[Campa, Giansanti, Mukamel & Ruffo 2006]

Coupled rotators (generalized HMF):

H =N∑

i=1

p2i

2+

J

2N

N∑i ,j=1

[1− cos(θi − θj)]︸︷︷︸Long-range

−KN∑

i=1

cos(θi+1 − θi )︸︷︷︸Short-range

Macrostate:

M =1

N

√√√√( N∑i=1

cos θi

)2

+

(N∑

i=1

sin θi

)2

No representation function for M

Entropy s(u) is known

Nonequivalence for −0.25 < K < K1 ≈ −0.168


Conclusions

Thermodynamic equivalence ⇔ Macrostate equivalence

Applicable to any many-body system

Applicable to any (valid) macrostate

Same result forI Canonical / grand-canonicalI Any other dual ensembles

Works for higher-dimensional Hamiltonian / macrostatesI e.g., turbulence models: H = circulation, entrosphy

Future work:

Other notion of equivalence, e.g., relative entropy

Quantum?


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