Dynamical systems: A unified colored-noise approximation.

Abstract

By use of an adiabatic elimination procedure and a time scaling t^=${\ensuremath{\tau}}^{\mathrm{\ensuremath{-}}1/2}$t, where \ensuremath{\tau} denotes the correlation time of colored noise \ensuremath{\varepsilon}(t), one arrives at a novel colored-noise approximation which is exact both for \ensuremath{\tau}=0 and \ensuremath{\tau}=\ensuremath{\infty}. The theory is implemented for one-dimensional flows of the type x\ifmmode \dot{}\else \.{}\fi{}=f(x)+g(x)\ensuremath{\varepsilon}(t). The approximation has the form of a Smoluchowski dynamics which is valid in regions of state space for which the damping \ensuremath{\gamma}(x,\ensuremath{\tau})=${\ensuremath{\tau}}^{\ensuremath{-}1/2}$-${\ensuremath{\tau}}^{1/2}$[f'1(g'/g)f] is positive and large; and times t\ensuremath{\gg}${\ensuremath{\tau}}^{1/2}$/\ensuremath{\gamma}(x,\ensuremath{\tau}). This novel Smoluchowski dynamics combines the advantageous features of a recent decoupling theory that does not restrict the value of \ensuremath{\tau}, 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.