# 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} }

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…

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