Anomalous Diffusion and Fractons in Disordered Structures

Abstract

In this lecture I review the developments of recent years in our understanding of transport properties in disordered structures. The breakthrough in this area came about largely by the discovery of anomalous diffusion1–4 and by the concept of fractons introduced by Alexander and Orbach.1 These are based on the fundamental concept of fractals introduced and developed by Mandelbrot5 to characterize disordered structures in nature. Since then (1982), the interest of the scientific community in this area has increased dramatically, and as a consequence hundreds of papers have been published on this subject. The existence of anomalous diffusion for the case of singular transition rates on regular lattices was known earlier (see e.g., the review articles of Alexander et al6 and Weiss and Rubin7). The fact that disorder in the lattice structure can lead to anomalies in the transport properties was the surprising result presented in Refs.1–4. Although today there is a much better understanding of the dynamical properties of disordered structures, there still remain many open questions, and the static properties as well as the dynamical properties are not fully understood. In this lecture I will focus on several basic dynamical properties such as conductivity, diffusion and fractons. I will also discuss the concept of chemical distance and the probability density of random walks on fractals which contain much information about the dynamical properties. Relevant experiments will be mentioned only briefly since they will be discussed in detail in the talks of Jorgen Kjems and Eric Courtens. Effects of additional physical disorder which are different from the structural disorder will be discussed in the talk of Armin Bunde.