Approximate probability distributions for stochastic systems: maximum entropy method

The effective analytical methods for stochastic nonlinear dynamical systems are applicable only in some simple cases. If one deals with more complex s ystems and with the so-called rear life applications the approximate method s and numerical integration are necessary. In this paper we present the pos sible approaches to approximate characterization of the probability distrib utions of stochastic nonlinear systems. Starting from the description of th e basic properties of such systems, the most notable recent efforts to eval uation of their probability distributions are presented with emphasis on th e maximum entropy method. This method, originated in its simple classical f orm in statistical physics, when suitably generalized, allows complicated s tochastic systems to be treated successfully using information contained in the equations for statistical moments of the solution (or response). In th is paper, the general scheme of the method is presented both for stationary and nonstationary distributions and then its numerical implementation is e xpounded. Nonlinear stochastic oscillatory systems are treated in detail an d the obtained probability distributions are shown graphically in compariso n with the exact solutions and with the simulation results. (C) 1999 Elsevi er Science S.A. All rights reserved.