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 ρ and on the pair interactions v(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 ρ and v(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 rigid-sphere model in three dimensions, for which the simple step-function pair correlation can be maintained up to a covering fraction of one-eighth.