Stochastic interpretation of the rate equation formulation of hopping transport theory

Abstract

Apart from a trivial factor, the (mn)th element of the Green matrix of the linearized rate equations is the Fourier transform of the probability that an electron placed on site n at time 0 will be found on site m at time t. Using the stochastic interpretation, it is shown that the system-averaged electrical conductivity has the Kubo-Greenwood form, the diffusivity is given by the Einstein relation and the thermopower may be obtained by extending the conventional transport formula for Bloch electrons to the hopping regime. A Dyson expansion is given for the (mn)th element of the Green matrix in which, apart from a trivial factor, the pth-order term is the probability that an electron placed on site n at time 0 will be found on site m at time t, having made p hops to get there. The dominant terms in the series at low site densities are summed and yield the pair approximation. An approximate summation of the self-avoiding walk terms in the series gives the same results for a random system as the theory of continuous-time random walks gives for a lattice.