Simple theory of relaxation from instabilities

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Abstract
This paper gives a method for the study of relaxation at instabilities described by Langevin equations. The method uses backward-diffusion equations which are conditioned on the initial value of the dynamical variables, instead of the Fokker-Planck (forward) equation. By using the backward-diffusion equation, we obtain simple characterizations of relaxation properties such as the mean time to leave an observation region about the instability, the probability of reaching a preferred steady state, the mean time to reach it, and the rate of reaching it. Ensemble effects are included by averaging over the initial distribution. This kind of approach complements previous studies which were based on the Fokker-Planck equation. The interaction of noise, strength of the instability, and size of the observation region is studied. A number of examples are given (lasers, ferromagnets, classical molecular scattering, spontaneous optical resolution) and the methods developed here are applied to calculate properties of spontaneous optical resolution.