AVALANCHE DYNAMICS IN EVOLUTION, GROWTH, AND DEPINNING MODELS

The dynamics of complex systems in nature often occurs in terms of pun ctuations, or avalanches, rather than following a smooth, gradual path . A comprehensive theory of avalanche dynamics in models of growth, in terface depinning, and evolution is presented. Specifically, we includ e the Bak-Sneppen evolution model, the Sneppen interface depinning mod el, the Zaitsev flux creep model, invasion percolation, and several ot her depinning models into a unified treatment encompassing a large cla ss of far from equilibrium processes. The formation of fractal structu res, the appearance of 1/f noise, diffusion with anomalous Hurst expon ents, Levy flights, and punctuated equilibria can all be related to th e same underlying avalanche dynamics. This dynamics can be represented as a fractal in d spatial plus one temporal dimension. The complex st ate can be reached either by tuning a parameter, or it can be self-org anized. We present two exact equations for the avalanche behavior in t he latter case. (1) The slow approach to the critical attractor, i.e., the process of self-organization, is governed by a ''gap'' equation f or the divergence of avalanche sizes. (2) The hierarchical structure o f avalanches is described by an equation for the average number of sit es covered by an avalanche. The exponent 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 eta = 0. The exponent pi for the Levy flight jumps between subsequent active sites can be related to other critical exponents through a study of ''backwa rd avalanches.'' We develop a scaling theory that relates many of the critical exponents in this broad category of extremal models, represen ting different universality classes, to two basic exponents characteri zing the fractal attractor. The exact equations and the derived set of scaling relations are consistent with numerical simulations of the ab ove mentioned models.