On the relationship between bifurcation points of the Kirkwood-Salsburg equation and phase transitions

Abstract

In this paper we explore the Vlasov conjecture on the relationship between bifurcation points and phase transitions. Because of the availability of the exact results of Ruelle, we focus our attention on the Kirkwood‐Salsburg hierarchy, and recognizing that the first equation in this hierarchy is of the form of a Lichtenstein‐Lyapunov nonlinear operator equation, we use a fundamental theorem of Krasnosel'skii to determine, under a suitable closure, bifurcation points for the same system considered by Ruelle. A special example is treated—that of a one‐dimensional system of hard rods— and our main conclusion follows from the results of this study: namely, that ``in this one‐dimensional system'' the bifurcation point does not seem to be related to the onset of a phase transition.