Statistical Mechanics of the Parallel Hard Squares in Canonical Ensemble

Abstract

The Monte Carlo method was used to generate canonical‐ensemble isotherms for N=25, 64, 100, and 400 parallel hard squares in a two‐dimensional periodic box. The Monte Carlo realizations with their chain lengths ranging up to ${\text{∼ 8× 10}}^5$ trials per particle were employed to ascertain statistical accuracy of the calculated pressures. The resulting isotherms are monotone increasing functions of the density with a possibility of a higher‐order transition but without showing any sign of a first‐order phase transition, with a density increment greater than 0.005_ρ0 (_ρ0 is the close‐packed density). This departs remarkably from the behavior of the hard‐disk isotherm, which displayed a van der Waals‐like loop for $\mathrm{N}\text{≳ 72}$ . Instead, it resembles isotherms of the lattice‐gas systems (e.g., a square‐lattice system with the first and second nearest‐neighbor exclusion, or a system of dimers) which, like the present case, contain a residual degree of freedom at close packing. By numerically integrating the Monte Carlo isotherms, the constant which corrects the entropy expression of the self‐consistent free‐volume theory at close packing was evaluated. The resulting value $\text{(2\hspace{0.17em}}\ln \text{2+0.268± 0.004)}$ for the N=400 case agrees satisfactorily with the corresponding value (2 ln2+0.26042) predicted from the cell‐cluster theory for the checkerboard structure. The N dependence of the Monte Carlo isotherms was examined in both the low‐ and the high‐density regions. In the low‐density region, the Monte Carlo data are in satisfactory agreement with theoretically predicted values, whereas, in the high‐density region, the MC data revealed existence of the N ‐dependent contribution to the pressure attributable to the correlated slidings of rows or columns of particles. This can occur even at 95% of the close‐packed density for $\mathrm{N}\text{≤ 100}$ .