Lattice-gas physics of Widom's problem

Abstract

Widom has shown that potential distribution theory can be used to reformulate the equilibrium statistical mechanics of simple lattice models, in terms of a set of nearest neighbour occupation numbers for sites surrounding an empty cell. For lattices with coordination number three, the lowest value allowing for non-trivial phase transitions of the Ising class, the four occupation numbers involved can be expressed entirely in terms of the chemical potential and the energy, as functions of temperature and density. The discovery of one more independent equation of this class would generate full solutions to non-trivial Ising models. In a continuum model such a result would follow from the virial equation for the pressure, which mysteriously is problematic for lattice-gas models. Thus, Widom's important problem can be stated in terms of a search for an equation for the pressure of a lattice-gas in terms of nearest neighbour occupation numbers. This paper considers one possible route to the desired solution, which so far can be made to work exactly only in one-dimensional models. In addition, it considers whether or not inverse potential distribution theory could be used to obtain an independent result, finding instead that this simply rewrites the problem in terms of nearest neighbour occupations for sites surrounding a filled cell. Some general aspects of the difficulties encountered are discussed.