Finite size effects for the simulation of phase coexistence in the Gibbs ensemble near the critical point

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Abstract
In the Gibbs ensemble gas-liquid phase coexistence can be studied by obtaining the density distribution function in a finite system from the study of two subsystems exchanging particles. The temperature dependence of the peak of this distribution function is studied near the critical point, and Monte Carlo simulations for the simple special case of the two-dimensional lattice gas model are presented. This case is a "restricted Gibbs ensemble" where the particle numbers of the two systems fluctuate but their volume fluctuations are suppressed. From formal analysis and physical arguments, we predict that the density difference of the peak positions vanishes according to a classical power law [1 - T/T(c)(L)]1/2, where T(c)(L) is a shifted critical temperature of the finite system of linear dimension L, for temperatures within a regime where fluctuations are significant (L does not exceed the correlation length xi there). This behavior is verified by Monte Carlo simulations for L x L lattices with periodic boundary conditions and L in the range from 4 less-than-or-equal-to L less-than-or-equal-to 20. It is also shown that T(c)(L) approaches the critical point of the infinite system from above, according to a law T(c)(L) - T(c)(infinity) approximately L - 1/nu, nu being the correlation length exponent, as expected. Our discussion also applies to the standard grand canonical ensemble of the lattice gas or the equivalent canonical ensemble of an Ising magnet, respectively. Some Monte Carlo results for this ensemble are also presented. They are consistent with the theoretical predictions.