Introduction

In the last two chapters we have studied kinetic theories of relaxation and diffusion as specific representations of the general master equation (6.2.1). Such analyses allowed us to obtain insight into a number of aspects of these processes, especially in dilute systems, e.g. the L-coefficients, the way correlations in atomic movements enter into diffusion coefficients, the relation of diffusion to relaxation rates and so on. In this chapter and the next we turn to another well-established body of theory, namely the theory of random walks. This too can be presented in the context of the general analysis of Chapter 6 and additional insights obtained.

The basic model or system which is analysed in the mathematical theory of discrete random walks is that of a particle (or ‘walker’) which moves in a series of random jumps or ‘steps’ from one lattice site to another. It can be used to represent physical systems such as interstitial atoms (e.g. C in α-Fe) or point defects moving through crystal lattices under the influence of thermal activation, as long as the concentrations of these species are low enough that their movements do not interfere with one another. The mathematical theory of such random walks has received considerable attention and is well recorded in many books and articles (recent examples include Barber and Ninham, 1970 and Haus and Kehr, 1987). For this reason it would be superfluous (and impractical) to go over all the same ground again here. Nevertheless there are various results which are useful in the theory of atomic transport either directly (e.g. in the evaluation of transport coefficients accordings to eqns.