Abstract
We study the relation between the microcanonical, canonical, and grand canonical ensembles in the thermodynamic limit when the system becomes infinite. They are equivalent if there is only one phase in the system. In general it is shown that there is a unique limit of the microcanonical state being a mixture of pure phases if the microcanonical restrictions determine the volume fractions of the phases uniquely, and then the Gibbs phase rule is valid. In this context we show how to define the set of order parameters associated with the state of the system in a natural way.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. L. Dobrushin and B. Tirozzi, The central limit theorem and the problem of equivalence of ensembles,Comm. Math. Phys. 54:173 (1977).
R. Griffiths and D. Ruelle, Strict convexity (“continuity”) of the pressure in lattice systems,Comm. Math. Phys. 23:169 (1971).
O. Lanford, Entropy and equilibrium states in classical statistical mechanics, inStatistical Mechanics and Mathematical Problems, A. Lenard, ed. (Springer Lecture Notes in Physics 20).
J. Lebowitz and A. Martin-Lof, On the uniqueness of the equilibrium state for Ising spin systems,Comm. Math. Phys. 25:276 (1972).
A. Martin-Löf, Lectures on statistical mechanics and the foundations of thermodynamics, Res. Rep. no. 105, Dept. of Math. Statistics, University of Stockholm (1977), to appear inSpringer Lecture Notes in Physics.
A. Messger and S. Miracle-Solé, Equilibrium states of the two-dimensional Ising model in the two-phase region,Comm. Math. Phys. 40:187–196 (1975).
R. A. Minlos and Ja. G. Sinai, The phenomenon of “phase separation” at low temperatures in some lattice models of a gas. I,Math. USSR-Sbornik 2:335 (1967);II,Trans. Moscow Math. Soc. 19:121 (1968).
R. T. Rockafellar,Convex Analysis (Princeton University Press, 1970).
D. Ruelle,Statistical Mechanics (Benjamin, 1969).
R. L. Thompson, Equilibrium states on thin energy shells, Mem. AMS no. 150 (1974).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Martin-Löf, A. The equivalence of ensembles and the Gibbs phase rule for classical lattice systems. J Stat Phys 20, 557–569 (1979). https://doi.org/10.1007/BF01012899
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01012899