Power-law distributions and Lévy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements.

Abstract

A generic model of stochastic autocatalytic dynamics with many degrees of freedom w(i,) i=1, em leader,N, is studied using computer simulations. The time evolution of the w(i)'s combines a random multiplicative dynamics w(i)(t+1)=lambdaw(i)(t) at the individual level with a global coupling through a constraint which does not allow the w(i)'s to fall below a lower cutoff given by cw, where w is their momentary average and 0<c<1 is a constant. The dynamic variables w(i) are found to exhibit a power-law distribution of the form p(w) approximately w(-1-alpha). The exponent alpha(c,N) is quite insensitive to the distribution Pi(lambda) of the random factor lambda, but it is nonuniversal, and increases monotonically as a function of c. The "thermodynamic" limit N-->infinity and the limit of decoupled free multiplicative random walks c-->0 do not commute: alpha(0,N)=0 for any finite N while alpha(c,infinity)>or=1 (which is the common range in empirical systems) for any positive c. The time evolution of w(t) exhibits intermittent fluctuations parametrized by a (truncated) Lévy-stable distribution L(alpha)(r) with the same index alpha. This nontrivial relation between the distribution of the wi's at a given time and the temporal fluctuations of their average is examined, and its relevance to empirical systems is discussed.