Avalanche dynamics in evolution, growth, and depinning models

Journal Article · Mon Jan 01 04:00:00 EST 1996 · Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

DOI:https://doi.org/10.1103/PhysRevE.53.414· OSTI ID:283777

Paczuski, M; Maslov, S; Bak, P ^[1]

---

1. Department of Physics, Brookhaven National Laboratory, Upton, New York 11973 (United States)

Dynamics of complex systems in nature often occurs in terms of punctuations, or avalanches, rather than following a smooth, gradual path. Theory of avalanche dynamics in models of growth, interface depinning, and evolution is presented. Specifically, we include the Bak-Sneppen evolution model, Sneppen interface depinning model, 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/{ital f} noise, diffusion with anomalous Hurst exponents, L{acute e}vy flights, and punctuated equilibria can all be related to the same underlying avalanche dynamics. This dynamics can be represented as a fractal in {ital 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 {ital 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 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 {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 L{acute e}vy flight jumps between subsequent active sites can be related to other critical exponents through a study of {open_quote}{open_quote}backward avalanches.{close_quote}{close_quote} 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. (Abstract Truncated)

• Research Organization: Brookhaven National Laboratory
• DOE Contract Number: AC02-76CH00016
• OSTI ID: 283777
• Journal Information: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics Journal Issue: 1 Vol. 53; ISSN PLEEE8; ISSN 1063-651X
• Country of Publication: United States
• Language: English

Related Subjects

55 BIOLOGY AND MEDICINE
BASIC STUDIES
99 GENERAL AND MISCELLANEOUS
ATTRACTORS
BIOLOGICAL EVOLUTION
DYNAMICAL SYSTEMS
DYNAMICS
FRACTALS
GROWTH
MATHEMATICAL MODELS
NONEQUILIBRIUM