The free energy of a macroscopic system

Abstract

The statistical mechanics of classical and quantum mechanical systems interacting with many-body forces are investigated in the canonical and grand canonical ensembles. Under various general conditions on the attractive and repulsive parts of the potential energy and on the shapes of the domains Ω_k confining the system, it is shown that the canonical free energy per particle and the grand canonical pressure have unique limits for infinite systems which are convex monotonie functions of the specific volume and chemical potential respectively, and satisfy the expected thermodynamic relations.

For pure pair forces with potential ϕ(r) sufficient conditions are: ϕ(r)⪴D ₁/r^3+ɛ as r→0, |ϕ(r)|⪴D ₂/r^3+ɛ as r → ∞ (ε>0), and ϕ(r)≧-w₀ all r; the domains Ω_k may be constructed from a finite set of bounded domains of arbitrary shape by any sequence of isotropic expansions such that the volume V(Ω_k) approaches infinity with k.