Isostructural phase transitions due to core collapse. I. A one-dimensional model

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Abstract
This is the first of a two part study of a statistical mechanical system that exhibits an isostructural phase transition at high density and pressure. Here, in Paper I, we consider a one‐dimensional fluid; in Paper II we go on to treat the same system in three dimensions, using the approximation methods developed and tested here for one of the systems studied exactly by Stell and Hemmer [J. Chem. Phys. 56, 4274 (1972)]. Our pair potential φ (r) consists of a hard core of diameter d plus a shoulder of constant positive magnitude V0 for d<r<d (1+λ), to which a weak long‐range attraction term is added. We study the results of first‐order perturbation theory in V0, as well as in f0=exp(−βV0)−1, β= (kT)−1, and find that used jointly they yield remarkably accurate results that promise to be similarly accurate in three dimensions, where an exact theory is lacking. In particular, we note that first‐order perturbation theory in V0 becomes rigorously exact in one‐dimension for fixed β as the density ρ approaches close packing; we argue (nonrigorously) that this result can be expected in any dimension. The first‐order theory in exp(−βV0)−1 is correspondingly good in the opposite limit of ρ→0.