Approach to a Phase Transition in a One-Dimensional System

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Abstract
The manner in which a one‐dimensional system approaches a phase transition as the range of its attractive pairwise interactions becomes large is analyzed, as a continuation of studies on such systems by Kac, Uhlenbeck, Hemmer, and the author. Explicitly, consideration is given to an Ising ferromagnetic model and the corresponding lattice gas with interaction −αγ exp (−γ |i − j|) as γ → 0. One finds that the fluid isotherms are actually analytic, but that in a certain region they approach the zero slope characteristic of a phase transition in the essentially singular fashion exp (−const/γ).