Statistical Mechanics of Hard-Particle Systems

Abstract

The concept of the equivalence of a hard particle (depending on the number of dimensions: hard sphere, disk, or rod, with no attractive interaction) of diameter a in a system of hard particles, and a spherical cavity of radius at least a, is extended to define distribution functions, γn(r1, ···rn), for sets of n cavities in a hard-particle system. A new theorem, relating the excess chemical potential of a hard-particle system to the distribution function of two cavities at zero separation, γ2(r12)|τ12–0, as well as corresponding theorems for higher-order distributions, are derived. The theorems for γ2(r12) and γ3(r1, r2, r3), and some simple physical arguments are used to obtain an approximation to the exact Born–Green–Yvon equation for γ2(r12), the “pair stress approximation” (PSA). The resulting simple equation of state yields very good approximate values for the fourth and fifth virial coefficients for hard disks, and for the fourth virial coefficient for hard spheres. The corresponding zero-separation theorems for higher-order distribution functions yield successively higher- and higher-order approximations to the equation of state and pair distribution function of hard-particle systems. The next-order approximation after PSA results in an integral equation whose solution would yield equations of state exact through the fourth virial coefficient. A rough approximation to the solution of that equation results in an equation of state which agrees very well with molecular dynamics computer calculations, within the entire fluid range, both for hard spheres and hard disks. The equation of state yields minimum specific volumes for both systems, between the fluid and crystal specific volumes at phase transitions observed by molecular dynamics computer calculations. All these approximations become exact for hard rods in one dimension, and the method yields the known expressions for the equation of state and pair distribution function for this system in a particularly simple way.