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Dynamical systems: A unified colored-noise approximation

Peter Jung and Peter Hänggi
Phys. Rev. A 35, 4464(R) – Published 1 May 1987
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Abstract

By use of an adiabatic elimination procedure and a time scaling t^=τ1/2t, 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)=τ1/2-τ1/2[f1(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.

  • Received 19 January 1987

DOI:https://doi.org/10.1103/PhysRevA.35.4464

©1987 American Physical Society

Authors & Affiliations

Peter Jung and Peter Hänggi

  • Lehrstuhl für Theoretische Physik, Universitat Augsburg, Memminger Strasse 6, D-8900 Augsburg, Federal Republic of Germany

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Issue

Vol. 35, Iss. 10 — May 1987

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