Ultrametricity and Memory in a Solvable Model of Self-Organized Criticality

Abstract

Slowly driven dissipative systems may evolve to a critical state where long periods of apparent equilibrium are punctuated by intermittent avalanches of activity. We present a self-organized critical model of punctuated equilibrium behavior in the context of biological evolution, and solve it in the limit that the number of independent traits for each species diverges. We derive an exact equation of motion for the avalanche dynamics from the microscopic rules. In the continuum limit, avalanches propagate via a diffusion equation with a nonlocal, history-dependent potential representing memory. This nonlocal potential gives rise to a non-Gaussian (fat) tail for the subdiffusive spreading of activity. The probability for the activity to spread beyond a distance $r$ in time $s$ decays as $\sqrt{\frac{24}{\pi }}{s}^{-3/2}{x}^{1/3}\exp [-\frac{3}{4}{x}^{1/3}]$ for $x=\frac{r^4}{s}\gg 1$. The potential represents a hierarchy of time scales that is dynamically generated by the ultrametric structure of avalanches, which can be quantified in terms of ``backward'' avalanches. In addition, a number of other correlation functions characterizing the punctuated equilibrium dynamics are determined exactly.

44 pages, Revtex, (12 ps-figures included)

References in corpus (4)

• Avalanche Dynamics in Evolution, Growth, and Depinning Models
• Exact Results for Spatio-Temporal Correlations in a Self-Organized Critical Model of Punctuated Equilibrium
• Criticality in Simple Models of Evolution
• Mass-extinction: Evolution and the effects of external influences on unfit species

Cited by in corpus (13)

• Optimization with Extremal Dynamics
• Nature's Way of Optimizing
• Coherent noise, scale invariance and intermittency in large systems
• Self-Organized Criticality and Punctuated Equilibria
• Aging in a Model of Self-Organized Criticality
• The Bak-Sneppen Model on Scale-Free Networks
• Anomalous Diffusion on the Hanoi Networks
• d_c=4 is the upper critical dimension for the Bak-Sneppen model
• Cluster formation in complex multi-scale systems
• Aging Exponents in Self-Organized Criticality
• Entropic Sampling and Natural Selection in Biological Evolution
• Boundary Spatiotemporal Correlations in a Self-Organized Critical Model of Punctuated Equilibrium
• Hysteretic response to different modes of ramping an external field in sparse and dense Ising spin glasses