Title:

Randomwalk theory and statistical mechanics of lattice systems

It has been found elsewhere that when approximate relations for the twoparticle correlation functions of classical statistical mechanics, such as the PercusYevick and the meanspherical approximations, are applied to the lattice gas models with nearestneighbour interactions simple expressions are obtained for the total correlation function in terms of the lattice Green's function. Since many of the properties of random walks on a lattice can be described by the lattice Green's function, it follows that these systems, at least when treated under these approximations, may be analysed in terms of the language of random walks. Here the theory of random walks on lattices is appropriately extended to show that the relationship between the correlation functions and the lattice Green's function is not dependent upon the employment of these approximations, nor to a particular range or form of the potential function. Instead, this relationship is shown to be an alternative form of the OrnsteinZernike relation between the direct and total correlation functions. The direct correlation function is directly related to the probability of a single step, whereas the total correlation function is given by the firstpassage time probabilities of the random walks. Thermodynamic properties, such as the isothermal compressibility, are also interpreted in terms of randomwalk concepts. The randomwalk formulation is then extended to include the study of ordered phases in latticegas models and hence orderdisorder transitions in these systems. Also, an asymptotic form for the lattice Green's function is derived to illustrate how the form of decay of the total correlation function at large distances depends on the properties of the potential function. To show that the randomwalk interpretation of the OrnsteinZernike relation is not restricted to lattice systems, we define analogous randomwalk functions for continuous space. These lead to a straightforward generalization of most expressions for discrete space; the relationship between the continuousspace total correlation and Green's functions has the same form as that for the lattice systems. We also explore the possibility of obtaining randomwalk properties of a (lattice or continuousspace) system, not from the existing approximate expressions for the direct correlation function, but instead from a generalised OrnsteinZernike relation based on a maximum principle of classical statistical mechanics. Finally, we choose a few specific latticegas models to show how the method describes their different properties, such as the behaviour of the total correlation function or that of an order disorder phase transition.
