Some Problems In Rigorous Equilibrium Statistical Mechanics

Rechtman, Raul Mauricio (1978). Some Problems In Rigorous Equilibrium Statistical Mechanics. MPhil thesis The Open University.



In this dissertation we deal with some problems of classical statistical mechanics. In chapter 1 we review the problem of the thermodynamic equivalence of ensembles for equilibrium ensembles of classical statistical mechanics. We show how the problem of the thermodynamic equivalence of ensembles is solved with the help of generalized Legendre transforms (defined in Appendix A).

In chapter 2 we present some new results concerning the continuity of the thermodynamic limit temperature and the derivation of the Gibbs canonical distribution. For a classical system of interacting particles we prove, in the microcanonical ensemble formalism of statistical mechanics, that the thermodynamic limit inverse temperature, is a continuous function of the energy density. We also prove that the inverse temperature of a system approaches the thermodynamic limit inverse temperature as the volume of the system increases indefinitely. We also show that the probability distribution for a system of fixed size in thermal contact with a large system approaches the Gibbs canonical distribution as the size of the large system increases indefinitely, if the composite system is distributed microcanonically.

In chapter 3 we present a review of the duplicate variable method as a simple and useful tool for proving correlation inequalities for the Ising ferromagnet and other lattice systems. Roughly speaking, the method consists in expressing a product of correlation functions as an expectation of a suitable function over a larger space. New variables are introduced through a transformation that is usually orthogonal, chosen in such a way that the correlation inequality we set to prove appears explicitly.

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