Monte Carlo Procedure for Statistical Mechanical Calculations in a Grand Canonical Ensemble of Lattice Systems

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Abstract
The Monte Carlo method of estimating statistical mechanical averages in the petite canonical ensemble, described by Rosenbluth et al., Wood and Parker (for fluid systems), and Salsburg et al. (for lattice models), has been extended to a general multicomponent lattice model in a restricted grand canonical ensemble. The procedure is applied to the two-dimensional traingular lattice gas with periodic boundary conditions at a supercritical temperature (βε= —ln2), and numerical results are presented for the energy, specific heat, density, isothermal compressibility, thermal-expansion coefficient, and grand partition function (pressure) at Δ=0.1, 0.0, —0.1, —0.2, —0.3, —0.4, —0.6, —0.8 (with B=100); at B=16, 25, 36, 49, 64, 100, 196 (with Δ=0.0); where Δ=βμ—3βε; where ε is the nearest-neighbor interaction; β=(kT)—1, where μ is the chemical potential; and B is the number of sites. These are compared (where possible) with previous (Salsburg et al.) and additional petite-ensemble Monte Carlo results. This comparison emphasizes the different B dependence of intensive properties in these two ensembles and is supplemented by an asymptotic analysis of this difference. Properties in both ensembles display the type of irregular B dependence predicted by Lebowitz and Percus for very small systems. For the larger lattices, properties in the petite ensemble show a stronger B dependence than in the grand ensemble, which is in quantitative agreement with the leading term in the asymptotic analysis. A comparison with the exact analytical results (B= ∞, Δ=0) indicates that the accuracy of the Monte Carlo procedure for the grand ensemble can be reliably estimated by a statistical analysis of partial averages over the Markov chain.