Variational Formulations of Equilibrium Statistical Mechanics

Abstract

Thermodynamical functions for classical and quantum systems are expressed in terms of the one‐particle density n₁ and the two‐particle correlation matrix C₁₂ (or quantities in direct relation to them). Use is made of topological relations valid for the diagram representations of the grand partition function expansions. The result considered as a functional of n₁ and C₁₂ is stationary under independent variations δn₁ and δC₁₂. In particular, the entropy functional of a classical system no longer contains any reference to the equilibrium parameters (or to the interactions) and the second functional derivative is a negative definite matrix. The entropy functional of a quantum system conserves traces of the equilibrium parameters in the Lee‐Yang formulation; the Green's function formulation does not, but in this case the second functional derivative is no longer a negative definite matrix.