Bistable kinetic model driven by correlated noises: Steady-state analysis.

Abstract

A simple rule to obtain the Fokker-Planck equation for a general one-dimensional system driven by correlated Gaussian white noises is proved by the functional method. The Fokker-Planck equation obtained in this paper is applied to the bistable kinetic model. We find the following for the steady state. (1) In the \ensuremath{\alpha}-D parameter plane (\ensuremath{\alpha} is the strength of the additive noise and D is the multiplicative noise strength), the critical curve separating the unimodal and bimodal regions of the stationary probability distribution (SPD) of the model is shown to be affected by \ensuremath{\lambda}, the degree of correlation of the noises. As \ensuremath{\lambda} is increased, the area of the bimodal region in the \ensuremath{\alpha}-D plane is contracted. (2) When we take a point fixed in the \ensuremath{\alpha}-D plane and increase \ensuremath{\lambda}, the form of the SPD changes from a bimodal to a unimodal structure. (3) The positions of the extreme value of the SPD of the model sensitively depend on the strength of the multiplicative noise, and weakly depend on the additive noise strength. (4) For \ensuremath{\lambda}=1, the case of perfectly correlated noises, when the parameters \ensuremath{\alpha} and D take values in the neighborhood of the line \ensuremath{\alpha}=D in the \ensuremath{\alpha}-D plane, the SPD's corresponding to the points \ensuremath{\alpha}/Dg1 and \ensuremath{\alpha}/D1 exhibit a very different shape of divergence. Therefore, the ratio \ensuremath{\alpha}/D=1 plays the role of a ``critical ratio.''