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

@article{Touchette2015EquivalenceAN,
  title={Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels},
  author={Hugo Touchette},
  journal={Journal of Statistical Physics},
  year={2015},
  volume={159},
  pages={987-1016}
}
  • H. Touchette
  • Published 26 March 2014
  • Physics
  • Journal of Statistical Physics
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… 
Ensemble Dependence of Fluctuations: Canonical Microcanonical Equivalence of Ensembles
We study the equivalence of microcanonical and canonical ensembles in continuous systems, in the sense of the convergence of the corresponding Gibbs measures and the first order corrections. We are
Strong ensemble nonequivalence in systems with local constraints
The asymptotic equivalence of canonical and microcanonical ensembles is a central concept in statistical physics, with important consequences for both theoretical research and practical applications.
Equivalence of ensembles in Curie-Weiss models using coupling techniques
We consider equivalence of ensembles for two mean field models: the discrete, standard Curie-Weiss model and its continuum version, also called the mean-field spherical model. These systems have two
Equivalence of nonequilibrium ensembles in turbulence models.
TLDR
The equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence is tested and it is verified that the equivalence is obeyed for the mean values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic.
Emergence and Breaking of Duality Symmetry in Thermodynamic Behavior: Repeated Measurements and Macroscopic Limit
Thermodynamic laws are limiting behavior of the statistics of repeated measurements of an arbitrary system with a priori probability distribution. A duality symmetry arises, between
Breaking of Ensemble Equivalence in Networks.
TLDR
It is shown that ensemble nonequivalence can manifest itself also in random graphs with topological constraints, and it is found that, while graphs with a given number of links are ensemble equivalent, graphsWith a given degree sequence are not.
Hamiltonian of Mean Force for Strongly-Coupled Systems
A central assumption in macroscopic thermodynamics is the weak coupling approximation, which posits that the equilibrium properties of a system are not influenced by the interactions with its
First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy
Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics. This tutorial focuses on the derivation of the first and second law for isolated and open
Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations.
Thermodynamics as limiting behaviors of statistics is generalized to arbitrary systems with probability a priori where the thermodynamic infinite-size limit is replaced by a multiple-measurement
...
...

References

SHOWING 1-10 OF 76 REFERENCES
Nonequilibrium microcanonical and canonical ensembles and their equivalence.
TLDR
Using this theory, conditions for the equivalence of nonequilibrium ensembles are provided, generalizing those found for equilibrium systems, and construct driven physical processes that generate theseEnsembles, and rederive in a simple way known and new product rules for their transition rates.
Equivalence and Nonequivalence of the Microcanonical and Canonical Ensembles: A Large Deviations Study
3 4 SUMMARY This thesis presents an in-depth study of statistical mechanical systems having microcano-nical equilibrium properties, i.e., energy-dependent equilibrium properties, which cannot be put
Ensemble equivalence for general many-body systems
It has been proved for a class of mean-field and long-range systems that the concavity of the thermodynamic entropy determines whether the microcanonical and canonical ensembles are equivalent at the
Nonequivalence of ensembles in the Curie?Weiss anisotropic quantum Heisenberg model
The microcanonical entropy s(e, m) as a function of the energy e and the magnetization m is computed analytically for the anisotropic quantum Heisenberg model with Curie–Weiss-type interactions. The
Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles
We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive
Large deviations and the thermodynamic formalism: A new proof of the equivalence of ensembles
In statistical mechanics the problem of the equivalence of ensembles goes back to Boltzmann and Gibbs. Here it is the problem of proving that, in the thermodynamic limit, the microcanonical measures
The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble
This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a
Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction
SummaryFor Gibbsian systems of particles inRd, we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable,
The equivalence between the canonical and microcanonical ensembles when applied to large systems
A straightforward technique is suggested that demonstrates that a microcanonical ensemble and canonical ensemble behave in exactly the same way in the thermodynamic limit. The canonical distribution
...
...