Order-disorder statistics. I

Abstract

In 1941 Kramers & Wannier discussed the statistical mechanics of a two-dimensional Ising model of a ferromagnetic. By making use of a ‘screw transformation’ they showed that the partition function was the largest eigenvalue of an infinite matrix of simple characteristic structure. In the present paper an alternative method is used for deriving the partition function, and this enables the ‘screw transformation’ to be generalized to apply to a number of problems of classical statistical mechanics, including the three-dimensional Ising model. Distant neighbour interactions can also be taken into account. The relation between the ferromagnetic and order-disorder problems is discussed, and it is shown that the partition function in both cases can be derived from a single function of two variables. Since distant neighbour interactions can be taken into account the theory can be formally applied to the statistical mechanics of a system of identical particles.