Approximate probability distributions for stochastic systems: maximum entropy method

Abstract

The effective analytical methods for stochastic nonlinear dynamical systems are applicable only in some simple cases. If one deals with more complex systems and with the so-called real life applications the approximate methods and numerical integration are necessary. In this paper we present the possible approaches to approximate characterization of the probability distributions of stochastic nonlinear systems. Starting from the description of the basic properties of such systems, the most notable recent efforts to evaluation of their probability distributions are presented with emphasis on the maximum entropy method. This method, originated in its simple classical form in statistical physics, when suitably generalized, allows complicated stochastic systems to be treated successfully using information contained in the equations for statistical moments of the solution (or response). In this paper, the general scheme of the method is presented both for stationary and nonstationary distributions and then its numerical implementation is expounded. Nonlinear stochastic oscillatory systems are treated in detail and the obtained probability distributions are shown graphically in comparison with the exact solutions and with the simulation results.