Nonequilibrium transitions driven by external dichotomous noise

Abstract

The stationary probability densityP _s for a class of nonlinear one-dimensional models driven by a dichotomous Markovian process (DMP)I _t , can be calculated explicitly. For the specific case of the Stratonovich model, x=ax −-x ³ +I _t x, the qualitative shape ofP _s and its support is discussed in the whole parameter region. The location of the maxima ofP _s shows a behavior similar to order parameters in continuous phase transitions. The possibility of a noiseinduced change from continuous to a discontinuous transition in an extended model, in which the DMP couples also to the cubic term, is discussed. The time-dependent moments 〈x_t ⁿ〉 can be represented as an infinite series of terms, which are determined by a recursion formula. For negative even moments the series terminates and the long-time behavior can be obtained analytically. As a function of the physical parameters, qualitative changes of this behavior may occur which can be partially related to the behavior ofP _s . All results reproduce those for Gaussian white noise in the corresponding limit. The influence of the finite correlation time and the discreteness of the space of states of the DMP are discussed. An extensive list of references is contained in U. Behn, K. Schiele, and A. Teubel,Wiss. Z. Karl-Marx-Univ. Leipzig, Mathem.-Naturwiss. R. 34:602 (1985).