Iso-g(2) Processes in Equilibrium Statistical Mechanics†

The pair correlation function g(2)(r) in a classical many-body system depends in a nontrivial way both on the number density F and on the pair interactions ν(r), and a long-standing goal of statistical mechanics has been to predict these effects quantitatively. The present investigation focuses on a restricted circumstance whereby simultaneous isothermal changes in F and ν(r) have exactly canceling effects on g(2). By appealing to the isothermal compressibility relation, we establish that an upper limit for density increase exists for this “iso-g(2)” process, and at this limit in three dimensions the correspondingly modified pair interaction develops a long-ranged Coulombic character. Using both the standard hypernetted chain and Percus-Yevick approximations, we have examined the iso-g(2) process for rigid rods in one dimension that starts at zero density, and maintains the simple step-function pair correlation during density increase, a process that necessarily terminates at a covering fraction of one-half. These results have been checked with detailed Monte Carlo simulations. We have also estimated the effective pair potentials that are required for the corresponding rigidsphere model in three dimensions, for which the simple step-function pair correlation can be maintained up to a covering fraction of one-eighth.