Avalanche dynamics in evolution, growth, and depinning models

Abstract

The dynamics of complex systems in nature often occurs in terms of punctuations, or avalanches, rather than following a smooth, gradual path. A comprehensive theory of avalanche dynamics in models of growth, interface depinning, and evolution is presented. Specifically, we include the Bak-Sneppen evolution model, the Sneppen interface depinning model, the Zaitsev flux creep model, invasion percolation, and several other depinning models into a unified treatment encompassing a large class of far from equilibrium processes. The formation of fractal structures, the appearance of 1/f noise, diffusion with anomalous Hurst exponents, L\'evy flights, and punctuated equilibria can all be related to the same underlying avalanche dynamics. This dynamics can be represented as a fractal in d spatial plus one temporal dimension. The complex state can be reached either by tuning a parameter, or it can be self-organized. We present two exact equations for the avalanche behavior in the latter case. (1) The slow approach to the critical attractor, i.e., the process of self-organization, is governed by a ``gap'' equation for the divergence of avalanche sizes. (2) The hierarchical structure of avalanches is described by an equation for the average number of sites covered by an avalanche. The exponent \ensuremath{\gamma} governing the approach to the critical state appears as a constant rather than as a critical exponent. In addition, the conservation of activity in the stationary state manifests itself through the superuniversal result \ensuremath{\eta}=0. The exponent \ensuremath{\pi} for the L\'evy flight jumps between subsequent active sites can be related to other critical exponents through a study of ``backward avalanches.'' We develop a scaling theory that relates many of the critical exponents in this broad category of extremal models, representing different universality classes, to two basic exponents characterizing the fractal attractor. The exact equations and the derived set of scaling relations are consistent with numerical simulations of the above mentioned models. \textcopyright{} 1996 The American Physical Society.