Algebraic Methods in Statistical Mechanics

Abstract

There are rather few models in statistical mechanics which have been solved exactly and many of these can be formulated in vertex form (see Chap. 5 and Baxter 1982a). In the absence of such solutions, two alternative methods of attack have been to determine ‘rigorous’ conditions for the existence of phase transitions, or bounds on the regions of phase space within which phase transitions can exist (Ruelle 1969, Griffiths 1972, Sinai 1982). A common feature of many of these approaches is that information about the properties of an infinite system is obtained by investigating an equivalent finite system (or system of reduced dimensionality). This can be taken to be the unifying theme of the methods described in this chapter. In Sects. 4.3–4.8 it is convenient, although not essential, to described the ideas in terms of a lattice fluid model. The equivalence between the spin-½ Ising model with nearest-neighbour interactions and a simple lattice fluid with nearest-neighbour pair interactions between particles was first shown by Lee and Yang (1952). This relationship holds between any one-component lattice fluid of particles of chemical potential μ and a spin-½ system on the same lattice in a magnetic field 𝓗. It follows that the methods and results described here can be ‘translated’ into a spin-½ formulation.