Variational Formulations of Equilibrium Statistical Mechanics

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.