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 $w_{i,}$ $i=1,\dots ,N,$ is studied using computer simulations. The time evolution of the $w_i\text{'}\mathrm{s}$ combines a random multiplicative dynamics $w_i\mathrm{}(t+1\mathrm{})=\lambda w_i\mathrm{}\left(t\right)\mathrm{}$ at the individual level with a global coupling through a constraint which does not allow the $w_i\text{'}\mathrm{s}$ to fall below a lower cutoff given by $c\overline{w},$ where $\overline{w}$ is their momentary average and $0<c<\mathrm{}1\mathrm{}$ is a constant. The dynamic variables $w_i$ are found to exhibit a power-law distribution of the form $p\left(w\right)\mathrm{}\sim w^{-1-\alpha }.$ The exponent $\alpha \mathrm{}(c,N)\mathrm{}$ is quite insensitive to the distribution $\Pi \left(\lambda \right)$ of the random factor $\lambda ,$ but it is nonuniversal, and increases monotonically as a function of c. The “thermodynamic” limit $\overrightarrow{N}\infty$ and the limit of decoupled free multiplicative random walks $\overrightarrow{c}0$ do not commute: $\alpha \mathrm{}(0,N)=0\mathrm{}$ for any finite N while $\alpha (c,\infty )>~1$ (which is the common range in empirical systems) for any positive c. The time evolution of $\overline{w}\mathrm{}\left(t\right)\mathrm{}$ exhibits intermittent fluctuations parametrized by a (truncated) Lévy-stable distribution $L_{\alpha }\mathrm{}\left(r\right)\mathrm{}$ with the same index $\alpha .$ This nontrivial relation between the distribution of the $w_i\text{'}\mathrm{s}$ at a given time and the temporal fluctuations of their average is examined, and its relevance to empirical systems is discussed.