Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels

We present general and rigorous results showing that the microcanonical and canonical ensembles are equivalent at all three levels of description considered in statistical mechanics—namely, thermodynamics, equilibrium macrostates, and microstate measures—whenever the microcanonical entropy is concave as a function of the energy density in the thermodynamic limit. This is proved for any classical many-particle systems for which thermodynamic functions and equilibrium macrostates exist and are defined via large deviation principles, generalizing many previous results obtained for specific classes of systems and observables. Similar results hold for other dual ensembles, such as the canonical and grand-canonical ensembles, in addition to trajectory or path ensembles describing nonequilibrium systems driven in steady states.

[1]  M. Kastner.  Nonequivalence of ensembles for long-range quantum spin systems in optical lattices. , 2010, Physical review letters.

[2]  W. Sullivan,et al.  The equivalence of ensembles for lattice systems: Some examples and a counterexample , 1994 .

[3]  P. Chleboun,et al.  Condensation in Stochastic Particle Systems with Stationary Product Measures , 2013, 1306.3587.

[4]  Hugo Touchette.  Ensemble equivalence for general many-body systems , 2011 .

[5]  D. Haar,et al.  Statistical Physics , 1971, Nature.

[6]  Richard S. Ellis,et al.  The theory of large deviations: from Boltzmann's 1877 calculation to equilibrium macrostates in 2D turbulence , 1999 .

[7]  Imre Csisźar,et al.  The Method of Types , 1998, IEEE Trans. Inf. Theory.

[8]  G. Wannier,et al.  OBSERVATIONS ON THE SPHERICAL MODEL OF A FERROMAGNET , 1965 .

[9]  Fluctuations of composite observables and stability of statistical systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  John T. Lewis,et al.  Entropy, concentration of probability and conditional limit theorems , 1995 .

[11]  R. Rockafellar.  Convex Analysis: (pms-28) , 1970 .

[12]  Steven Orey,et al.  Large Deviations for the Empirical Field of a Gibbs Measure , 1988 .

[13]  O. Lanford.  ENTROPY AND EQUILIBRIUM STATES IN CLASSICAL STATISTICAL MECHANICS , 1973 .

[14]  H. Touchette.  The large deviation approach to statistical mechanics , 2008, 0804.0327.

[15]  Ofer Zeitouni,et al.  Microcanonical Distributions, Gibbs States, and the Equivalence of Ensembles , 1991 .

[16]  I. Csiszár.  Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem , 1984 .

[17]  W. Thirring,et al.  Systems with negative specific heat , 1970 .

[18]  Nonequivalent ensembles and metastability , 2005, cond-mat/0501390.

[19]  M. Lax.  Relation Between Canonical and Microcanonical Ensembles , 1955 .

[20]  Editors , 1986, Brain Research Bulletin.

[21]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[22]  H. Touchette.  Equivalence and Nonequivalence of the Microcanonical and Canonical Ensembles: A Large Deviations Study , 2003 .

[23]  D. Ruelle.  Statistical Mechanics: Rigorous Results , 1999 .

[24]  G. Schütz,et al.  Discontinuous Condensation Transition and Nonequivalence of Ensembles in a Zero-Range Process , 2008, 0801.1310.

[25]  Hugo Touchette,et al.  An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles , 2004 .

[26]  Srinivasa R. S. Varadhan,et al.  Asymptotic probabilities and differential equations , 1966 .

[27]  Hans-Otto Georgii,et al.  Large Deviations and Maximum Entropy Principle for Interacting Random Fields on $\mathbb{Z}^d$ , 1993 .

[28]  CNRS,et al.  Statistical mechanics and dynamics of solvable models with long-range interactions , 2009, 0907.0323.

[29]  Paul C. Shields,et al.  Two divergence-rate counterexamples , 1993 .

[30]  G. Eyink,et al.  Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence , 1993 .

[31]  Negative magnetic susceptibility and nonequivalent ensembles for the mean-field $φ^4$ spin model , 2007, cond-mat/0702004.

[32]  Hugo Touchette,et al.  Nonequilibrium microcanonical and canonical ensembles and their equivalence. , 2013, Physical review letters.

[33]  H. Georgii.  Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction , 1994 .

[34]  Hugo Touchette,et al.  Nonequilibrium Markov Processes Conditioned on Large Deviations , 2014, 1405.5157.

[35]  D. Stroock.  Microcanonical Distributions for one Dimensional Lattice Gases , 1991 .

[36]  Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles , 2000, math/0012081.

[37]  J. Gibbs.  Elementary Principles in Statistical Mechanics , 1902 .

[38]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[39]  R. S. Ellis,et al.  The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble , 2004 .

[40]  T. C. Dorlas.  Statistical Mechanics: Fundamentals and Model Solutions, , 1999 .

[41]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[42]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[43]  J. Barré,et al.  Long-range one-dimensional gravitational-like interaction in a neutral atomic cold gas , 2012, 1202.1258.

[44]  Peter Sollich,et al.  Large deviations and ensembles of trajectories in stochastic models , 2009, 0911.0211.

[45]  L. Boltzmann,et al.  Über die Eigenschaften monozyklischer und anderer damit verwandter Systeme , 2012 .

[46]  E. Olivieri,et al.  Large deviations and metastability: Large deviations and statistical mechanics , 2005 .

[47]  M. Kastner.  Nonequivalence of ensembles in the Curie–Weiss anisotropic quantum Heisenberg model , 2010, 1005.5050.

[48]  M. Kastner,et al.  Microcanonical Analysis of the Curie–Weiss Anisotropic Quantum Heisenberg Model in a Magnetic Field , 2013, 1311.1306.

[49]  I. Csiszár.  $I$-Divergence Geometry of Probability Distributions and Minimization Problems , 1975 .

[50]  R. Evans,et al.  Comment on `Detailed balance has a counterpart in non-equilibrium steady states' , 2004, 0901.4879.

[51]  Hans Föllmer,et al.  Random fields and diffusion processes , 1988 .

[52]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[53]  H. Georgii.  The equivalence of ensembles for classical systems of particles , 1995 .

[54]  Pierre-Henri Chavanis,et al.  PHASE TRANSITIONS IN SELF-GRAVITATING SYSTEMS , 2006 .

[55]  Stefan Adams.  Complete Equivalence of the Gibbs Ensembles for One-Dimensional Markov Systems , 2001 .

[56]  D. Lynden-Bell,et al.  On the negative specific heat paradox , 1977 .

[57]  Astronomer Royal,et al.  The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems , 1968 .

[58]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[59]  M. A. Cayless.  Statistical Mechanics (2nd edn) , 1977 .

[60]  D. Lynden-Bell,et al.  NEGATIVE SPECIFIC HEAT IN ASTRONOMY, PHYSICS AND CHEMISTRY , 1998, cond-mat/9812172.

[61]  W. Sullivan,et al.  Large deviations and the thermodynamic formalism: A new proof of the equivalence of ensembles , 1994 .

[62]  Asymptotic equivalence of classical ensembles by the method of the maximum , 1970 .

[63]  J. P. Garrahan,et al.  Dynamics on the way to forming glass: bubbles in space-time. , 2009, Annual review of physical chemistry.

[64]  Daniel W. Stroock,et al.  Microcanonical distributions for lattice gases , 1991 .

[65]  Diego Garlaschelli,et al.  Breaking of Ensemble Equivalence in Networks. , 2015, Physical review letters.

[66]  R. Ellis.  An overview of the theory of large deviations and applications to statistical mechanics , 1995 .

[67]  V. Gurarie.  The equivalence between the canonical and microcanonical ensembles when applied to large systems , 2007 .

[68]  R. Dobrushin,et al.  The central limit theorem and the problem of equivalence of ensembles , 1977 .

[69]  Asymptotic Equivalence of Equilibrium Ensembles of Classical Statistical Mechanics , 1971 .

[70]  Mark Kac,et al.  On the van der Waals Theory of the Vapor‐Liquid Equilibrium. I. Discussion of a One‐Dimensional Model , 1963 .

[71]  Condensation in the Zero Range Process: Stationary and Dynamical Properties , 2003, cond-mat/0302079.