Power - Law Distributions and L evy - Stable IntermittentFluctuations in Stochastic Systems of Many

A generic model of stochastic autocatalytic dynamics with many degrees of freedom wi, i = 1; : : : ; N is studied using computer simulations. The time evolution of the wi's combines a random multiplicative dynamics wi(t + 1) = wi(t) at the individual level with a global coupling through a constraint which does not allow the wi's to fall below a lower cuto given by c w, where w is their momentary average and 0 < c < 1 is a constant. The dynamic variables wi are found to exhibit a power-law distribution of the form p(w) w 1 . The exponent (c;N) is quite insensitive to the distribution ( ) of the random factor , but it is non-universal, and increases monotonically as a function of c. The "thermodynamic" limit N ! 1 and the limit of decoupled free multiplicative random walks c ! 0 do not commute: (0; N) = 0 for any nite N while (c;1) 1 (which is the common range in empirical systems) for any positive c. The time evolution of w(t) exhibits intermittent uctuations parametrized by a (truncated) L evy-stable distribution L (r) with the same index . This non-trivial relation between the distribution of the wi's at a given time and the temporal uctuations of their average is examined and its relevance to empirical systems is discussed. Typeset using REVTEX 2