Dynamical systems: A unified colored-noise approximation

Abstract

By use of an adiabatic elimination procedure and a time scaling t^=${\tau }^{\mathrm{-}1/2}$t, where τ denotes the correlation time of colored noise ɛ(t), one arrives at a novel colored-noise approximation which is exact both for τ=0 and τ=∞. The theory is implemented for one-dimensional flows of the type ẋ=f(x)+g(x)ɛ(t). The approximation has the form of a Smoluchowski dynamics which is valid in regions of state space for which the damping γ(x,τ)=${\tau }^{-1/2}$-${\tau }^{1/2}$[f’1(g’/g)f] is positive and large; and times t≫${\tau }^{1/2}$/γ(x,τ). This novel Smoluchowski dynamics combines the advantageous features of a recent decoupling theory that does not restrict the value of τ, together with those occurring in the small-correlation-time theory due to Fox. The approximative theory is applied to a nonlinear model for a dye laser driven by multiplicative noise. Excellent agreement for the stationary probability is obtained between numerical exact solution and the novel approximative theory.