Metastable States of a Finite Lattice Gas

Abstract

Some recent papers on condensation were based in part on the conjecture that the probability function $W\left(N_1\right)$ for the number of particles $N_1$ in a finite volume at given temperature and chemical potential $\mu$ per particle has two maxima for a finite range of these variables. We have investigated the validity of this conjecture for a finite version (square of $N$ sites with toroidal connection) of the two-dimensional lattice gas of Lee and Yang. Considering at first only the values $N_1<~3$ and $N_1>~N-3\mathrm{}$, we show that $W\left(N_1\right)$ has at least two maxima for values of $\mu$ in the neighborhood of the transition value at temperatures smaller than or approximately equal to $\frac{3.5T_c}{\ln N}$ where $T_c$ is the critical temperature of the infinite model, while the upper points of the first three saltus of the most probable density of the finite model approximate the density of the infinite model in the gas region with a relative error of order $N^{-\frac{1}{2}}$. (An analogous result holds by symmetry for the liquid region.) To extend these results to a larger range of numbers $N_1$ we consider a histogram obtained from $W\left(N_1\right)$ by summation over relatively narrow groups of numbers in the range $N_1<~n$, where $n$ is an arbitrary integer $<N^{\frac{1}{2}}$. We show that at sufficiently low temperatures this histogram has a maximum centered on $N{\rho }_n$ where ${\rho }_n$ is the $n\mathrm{th}$ partial sum of the Mayer series for the density of the infinite model in powers of the fugacity. The finite model thus provides a physical interpretation for the extrapolation of the density (by means of a partial sum of the Mayer series) beyond the transition value of the fugacity.