Dynamical systems: A unified colored-noise approximation

By use of an adiabatic elimination procedure and a time scaling t^=τ^-1/2t, where {τ} denotes the correlation time of colored noise {ε}(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 x ̇ \.{}{}=f(x)+g(x){ε}(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})=τ^-1/2-τ1/2[f'1(g'/g)f] is positive and large; and times t{}τ1/2/{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.