The number of configurations of an assembly with long-distance order

Abstract

The standard method of studying the equilibrium properties of an assembly of physical systems is to construct certain characteristic functions of the assembly, such as the partition function or the grand partition function. From these functions, the equilibrium properties can be immediately deduced. Conversely, if the equilibrium properties are given or obtained by some indirect method, we can argue backward and obtain information about the characteristic functions. In an earlier paper (Chang 1939 a ), we have chosen a problem of adsorption in which the expression for the grand partition function of the adsorbed phase contains the number of ways g ( N, n, X ) of arranging n particles upon a lattice of N sites so that the number of pairs of nearest neighbours formed by the particles is X . By comparing the equilibrium properties given by Peierls (1936) by using Bethe’s method (Bethe 1935) and the standard equations giving the equilibrium properties in terms of the grand partition function, we have been able to obtain a formula for g ( N, n, X ). It goes without saying that the formula can only be approximately correct.