Correlation Functions and the Coexistence of Phases

Abstract

We discuss the existence, continuity and other properties of the canonical and grand canonical many‐particle correlation functions n_s(r₁, … r_s) in the thermodynamic limit of classical and quantum mechanical systems. If the pressure of the system for fixed T is constant in the range of specific volume v_a to v_b, one expects physically to observe the coexistence of two separated phases. In terms of the correlation functions this is expressed by $$n_s{\mathrm{(v)=x}}_a{\mathrm{(v}}_a{\mathrm{/v)n}}_s{\mathrm{(v}}_a{\mathrm{)+x}}_b{\mathrm{(v}}_b{\mathrm{/v)n}}_s{\mathrm{(v}}_b)$$ , where x and x_b are the mole fractions of the phases so that v = x_av_a + x_bv_b. With the aid of various lemmas on convex functions we prove that such a ``separation of the phases'' follows rigorously from statistical mechanics provided the correlation functions are ``well defined'' in an appropriate sense.