Journal of Statistical Physics

, Volume 159, Issue 5, pp 987–1016 | Cite as

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

  • Hugo Touchette


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.


Microcanonical and canonical ensembles  Equivalent and nonequivalent ensembles Large deviation theory  Entropy Long-range systems 



I would like to thank many colleagues who have provided useful ideas, comments, and support during the last 12 years that I worked on long-range systems and nonequivalent ensembles: Julien Barré, Freddy Bouchet, Raphael Chetrite, Thierry Dauxois, Rosemary J. Harris, Michael Kastner, Cesare Nardini, and Stefano Ruffo. I especially want to thank Richard S. Ellis for introducing me to many gems and subtleties of large deviations. The present paper owes much to his work.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.National Institute for Theoretical Physics (NITheP)StellenboschSouth Africa
  2. 2.Department of PhysicsStellenbosch UniversityStellenboschSouth Africa
  3. 3.Institute of Theoretical PhysicsStellenbosch UniversityStellenboschSouth Africa

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