Monte Carlo Calculations for Hard Disks in the Isothermal-Isobaric Ensemble

Abstract

The $\mathrm{N\; p\; T}$ ensemble for hard disks is formulated as an equivalent $\mathrm{N\; V\; T}$ ensemble for a pseudopotential interaction in a reduced configuration space, with exact relations for small periodic systems emphasized. The Monte Carlo method originally devised by Metropolis et al. is then adapted to the calculation of the equation of state and radial distribution function. The present paper then describes some results obtained for a small system (12 hard disks), as an example of the method, and with emphasis upon the statistical reliability of the results and on the effect of different pseudorandom‐number‐generating procedures. At least for such small systems it is possible to obtain, as a byproduct, the $\mathrm{N\; V\; T}$ ‐ensemble equation of state over a range of densities from an $\mathrm{N\; p\; T}$ ‐ensemble calculation at a single pressure. Subsequent papers will describe results already in hand for larger systems, including reasonably reliable extrapolations to the “thermodynamic” limit.