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

A generic model of stochastic autocatalytic dynamics with many degrees of freedom [Formula Presented] [Formula Presented] is studied using computer simulations. The time evolution of the [Formula Presented] combines a random multiplicative dynamics [Formula Presented] at the individual level with a global coupling through a constraint which does not allow the [Formula Presented] to fall below a lower cutoff given by [Formula Presented] where [Formula Presented] is their momentary average and [Formula Presented] is a constant. The dynamic variables [Formula Presented] are found to exhibit a power-law distribution of the form [Formula Presented] The exponent [Formula Presented] is quite insensitive to the distribution [Formula Presented] of the random factor [Formula Presented] but it is nonuniversal, and increases monotonically as a function of c. The “thermodynamic” limit [Formula Presented] and the limit of decoupled free multiplicative random walks [Formula Presented] do not commute: [Formula Presented] for any finite N while [Formula Presented] (which is the common range in empirical systems) for any positive c. The time evolution of [Formula Presented] exhibits intermittent fluctuations parametrized by a (truncated) Lévy-stable distribution [Formula Presented] with the same index [Formula Presented] This nontrivial relation between the distribution of the [Formula Presented] at a given time and the temporal fluctuations of their average is examined, and its relevance to empirical systems is discussed.