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Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume–Emery–Griffiths model

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  • Ellis, Richard S.
  • Touchette, Hugo
  • Turkington, Bruce

Abstract

We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume–Emery–Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the BEG model, we show that for values of the mean energy where the microcanonical entropy is nonconcave, the microcanonical distributions of states are nowhere realized in the canonical ensemble. Moreover, we show that for values of the mean energy where the microcanonical entropy is strictly concave, the equilibrium microcanonical distributions of states can be put in one-to-one correspondence with equivalent canonical equilibrium distributions of states. Our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by Ellis et al. (J. Stat. Phys. 101 (2000) 999).

Suggested Citation

  • Ellis, Richard S. & Touchette, Hugo & Turkington, Bruce, 2004. "Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume–Emery–Griffiths model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(3), pages 518-538.
    • Handle: RePEc:eee:phsmap:v:335:y:2004:i:3:p:518-538
    DOI: 10.1016/j.physa.2003.11.028
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    References listed on IDEAS

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    1. Lynden-Bell, D., 1999. "Negative specific heat in astronomy, physics and chemistry," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 263(1), pages 293-304.
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    1. Touchette, Hugo & Ellis, Richard S. & Turkington, Bruce, 2004. "An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 138-146.
    2. Touchette, H. & Costeniuc, M. & Ellis, R.S. & Turkington, B., 2006. "Metastability within the generalized canonical ensemble," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(1), pages 132-137.
    3. Casetti, Lapo & Kastner, Michael, 2007. "Partial equivalence of statistical ensembles and kinetic energy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 384(2), pages 318-334.

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