States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures

Abstract

This paper is devoted to the solution of the problem of characterizing correlation measures arising naturally in classical statistical mechanics of point particles. A correlation measure ρ must be related to a (not necessarily unique) probability measure μ over an infinite particle configuration space X by the formula μ(H)=∝N _H (ξ)dμ where {N _H } is a certain family of integer valued random variables. We prove that there are three conditions, namely (S) symmetry, (P) positivity, and (N) normalization, which together are sufficient as well as necessary for ρ to be a correlation measure. The main operative condition is (P), which says that ξφ(x)dρ≧0 must hold for every function φ for which Sφ(ξ)≧0 identically for ξ ε X, where φ → Sφ is a certain linear operator whose properties we study. Condition (P) gives rise to a large class of inequalities satisfied by the ρ-measures of certain sets. The theory is also generalized to the case when there is a group of translations acting in the one-particle space, the concern then being with measures ρ as well as μ that are invariant with respect to the group.