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

Abstract

A generic model of stochastic autocatalytic dynamics with many degrees of freedom $w_i$ $i=1,...,N$ is studied using computer simulations. The time evolution of the $w_i$'s combines a random multiplicative dynamics $w_i(t+1) = λw_i(t)$ 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 $c \cdot \bar w$, where $\bar w$ 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 $p(w) \sim w^{-1-α}$. The exponent $α(c,N)$ is quite insensitive to the distribution $Π(λ)$ of the random factor $λ$, but it is non-universal, and increases monotonically as a function of $c$. The "thermodynamic" limit, N goes to infty and the limit of decoupled free multiplicative random walks c goes to 0, do not commute: $α(0,N) = 0$ for any finite $N$ while $ α(c,\infty) \ge 1$ (which is the common range in empirical systems) for any positive $c$. The time evolution of ${\bar w (t)} $ exhibits intermittent fluctuations parametrized by a (truncated) Lévy-stable distribution $L_α(r)$ with the same index $α$. This non-trivial relation between the distribution of the $w_i$'s at a given time and the temporal fluctuations of their average is examined and its relevance to empirical systems is discussed.