A STATISTICAL-MODEL FOR COLLECTIVE INSTABILITIES

We introduce a simple stochastic model for the collective evolution of a population of elements which can assume only a finite number n of s tates ('damage') d = k/n for 0 less than or equal to k<n. The evolutio n is given by a probabilistic rule which depends only on a function of the damage, and diverges for d=1. We consider mostly cases where a ho mogeneous evolution (same damage for all elements) is unstable. Our ai m is to characterize the final state of the system (i.e. the statistic al distribution of the damage) in the thermodynamic limit. A non-trivi al scaling with the number of damage states is observed. The scaling v ariable (1-d)n beta accounts for the n-dependence of most properties o f the model. The exponent beta is only a function of the singularity o f the probability, and its expression is obtained from a mapping to a convection-diffusion problem. The dependence on the number of elements leads to logarithmic corrections which are discussed.