Evolution of the system with multiplicative noise

Abstract

The governed equations for the order parameter, one- and two-time correlators are obtained for systems with white multiplicative noise. We consider the noise whose amplitude depends on stochastic variable as xa where 0<a<1. It turns out that the equation for autocorrelator includes an anomalous average of the power-law function with the fractional exponent 2a. Determination of this average for the stochastic system with a self-similar phase space is performed. It is shown that at a>1/2, when the system is disordered, the correlator behaves in the course of time non-monotonically, whereas the autocorrelator increases monotonically. At a<1/2 the phase portrait of the system divides into two domains: at small initial values of the order parameter, the system evolves to a disordered state, as above; within the ordered domain it is attracted to the point with finite values of the autocorrelator and order parameter. The long-time asymptotes are defined to show that, within the disordered domain, the autocorrelator decays hyperbolically and the order parameter behaves as a power-law function with fractional exponent −2(1 − a). Correspondingly, within the ordered domain, the behaviour of both dependencies is exponential with an index proportional to −tlnt. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3036