Diffusion in a bistable potential: A systematic WKB treatment |
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Authors: | B. Caroli C. Caroli B. Roulet |
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Affiliation: | (1) Groupe de Physique des Solides de l'Ecole Normale Supérieure (associé au Centre National de la Recherche Scientifique), Université Paris VII, Paris, France;(2) Département de Physique, UER de Sciences Exactes et Naturelles, Université de Picardie, Amiens, France |
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Abstract: | We study the distributionP of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, in the limit of constant and small diffusion coefficient. A systematic WKB analysis of the lowest eigenmodes of the equivalent Schrödinger-like equation yields the following results: the final approach to equilibrium is governed by the Kramers high-viscosity rate, which is shown to be exact in this limit; for intermediate times, we show that Suzuki's scaling statement does give the correct behavior for the transition between the one-peak and the two-peak structure forP. However, the intermediate time domain also contains a second half, whereP enters the diffusive equilibrium regions, characterized by a time scale of the same order as Suzuki's time. |
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Keywords: | Nonlinear Fokker-Planck equation instability diffusion |
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