Diffusion Coefficients of Single and Many Particles in Lattices with Different Forms of Disorder

Topics: random transition rates, random barriers, random traps, Miller-Abrahams model, diffusion coefficient, coefficient of collective diffusion, effective medium theory, site-exclusion model Abstract A survey is given on asymptotic diffusion coefficients of particles in lattices with random transition rates. Exact and approximate results for single particles are reviewed. A recent exact expression in d = 1 which includes occupation factors is discussed. The utilization of the result is demonstrated for the Miller-Abrahams model and a model of random barriers combined with random traps. Exact and approximate results for the site-exclusion model in disordered lattices are also given. 1. Introduction Transport processes of particles in disordered materials exhibit a variety of interesting phenomena , such as a strong reduction of the asymptotic diffusion coefficients, anomalous frequency dependence of the conductivity, dispersive transport, etc. The explanation of the transport processes of single and many particles in disordered materials has been a challenge to theory; it is of great practical importance as well. This paper focuses on the asymptotic diffusion coefficients of single and many particles for different models of disordered lattices. Quite different models of disorder were introduced to describe particle transport in crystals with point defects, as well as in amorphous materials and glasses. Exact results will be presented as far as possible ; these results are mainly but not always restricted to the dimension d = 1. Recently a general exact expression for the diffusion coefficient of single particles in d = 1 has been derived [1, 2, 3]. The insight obtained from this derivation can serve as a basis for the effective-medium approximation in higher dimensions. The derivation of diffusion coefficients of many particles where multiple occupancy of sites is excluded will be restricted to the coefficient of collective diffusion which appears in Fick's law. Exact results for this diffusion coefficient in disordered lattices are scarce. It will be shown that a properly formulated effective-medium approximation yields reasonable results in higher-dimensional disordered lattices.