1999/07/26 by Ofer Malcai, Ofer Biham, Sorin Solomon · 5 citations
Computer Science · Economics, Econometrics and Finance · Physics and Astronomy · #Complex Systems and Time Series Analysis #Nonlinear Dynamics and Pattern Formation #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.60.1299
7 pages, 4 figures
arxiv created 1999/07/26 · openalex publication_date 1999/08/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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)=\ensuremathλwi(t) at the individual level with a global coupling through a constraint which does not allow the wi'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 wi are found to exhibit a power-law distribution of the form p(w)\ensuremath∼w^\ensuremath-1\ensuremath-\ensuremathα. The exponent \ensuremathα(c,N) is quite insensitive to the distribution \ensuremathΠ(\ensuremathλ) of the random factor \ensuremathλ, but it is nonuniversal, and increases monotonically as a function of c. The ``thermodynamic'' limit \stackrel\ensuremath→N\ensuremath∞ and the limit of decoupled free multiplicative random walks \stackrel\ensuremath→c0 do not commute: \ensuremathα(0,N)=0 for any finite N while \ensuremathα(c,\ensuremath∞)>~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) L'evy-stable distribution L_\ensuremathα(r) with the same index \ensuremathα. 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.