Statistical Mechanics of Hard-Particle Systems
- 1 April 1968
- journal article
- research article
- Published by AIP Publishing in The Journal of Chemical Physics
- Vol. 48 (7), 3139-3155
- https://doi.org/10.1063/1.1669587
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 in a system of hard particles, and a spherical cavity of radius at least is extended to define distribution functions, for sets of 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, as well as corresponding theorems for higher‐order distributions, are derived. The theorems for and and some simple physical arguments are used to obtain an approximation to the exact Born–Green–Yvon equation for 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.
Keywords
This publication has 23 references indexed in Scilit:
- Generalizations of the Virial and Wall Theorems in Classical Statistical MechanicsJournal of Mathematical Physics, 1966
- Equations of state for fluid and crystalline hard discsPhysica, 1965
- Triplet Correlations in Hard SpheresPhysical Review Letters, 1964
- Phase Transition in Elastic DisksPhysical Review B, 1962
- Analysis of Classical Statistical Mechanics by Means of Collective CoordinatesPhysical Review B, 1958
- Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard SpheresThe Journal of Chemical Physics, 1957
- Molecular Distribution Functions in a One-Dimensional FluidThe Journal of Chemical Physics, 1953
- Radial Distribution Function of a Gas of Hard Spheres and the Superposition ApproximationPhysical Review B, 1952
- Radial Distribution Functions and the Equation of State of a Fluid Composed of Rigid Spherical MoleculesThe Journal of Chemical Physics, 1950
- The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic SpheresPhysical Review B, 1936