Journal of Statistical Physics

, Volume 147, Issue 5, pp 1020–1040 | Cite as

Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems

  • Takashi Mori


Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called “exactness of the mean-field theory”. It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the α-Potts model with annealed vacancies and the α-Potts model with invisible states.


Long-range interaction Mean-field theory Ensemble inequivalence 



The author is grateful to Prof. Seiji Miyashita for useful comments and Dr. Shu Tanaka for careful reading of the manuscript. He acknowledges JSPS for financial support (Grant No. 227835).


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Physics, Graduate School of ScienceThe University of TokyoTokyoJapan

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