An upper bound on the critical temperature for a continuous system with short-range interaction

Abstract

A classical gas with short-range interaction in the grand canonical ensemble is studied. Ifp(β, z) denotes the thermodynamic pressure at inverse temperatureβ and activityz, then it follows from the Mayer expansion thatp(β, z) is infinitely differentiable providedβ andβz are sufficiently small. Here it is shown that there existsβ 0>0 such thatp(β, z) is infinitely differentiable ifβ<β 0 andz>0. One can interpret this result as saying that (β 0)−1 is an upper bound on the critical temperature for the system.