Abstract
A one-dimensional bistable flow driven by additive, exponentially correlated Gaussian noise is considered. The small relaxation time Fokker-Planck approximations, widely used in the recent literature, are derived and possible shortcomings of those approximation schemes are discussed. In particular, it is pointed out that higher order non-Fokker-Planck type contributions are generally of the same order as the Fokker-Planck terms. In principle, those contributions cannot be neglected if the global behavior of the probability solutions is to be described accurately. The result for the activation rate (Arrhenius factor), as evaluated from the approximative Fokker-Planck schemes, does not coincide in leading order in the correlation time τ of the noise with a computer simulation of the rate at low noise level. This result indicates that the wings of the stationary probability\(\bar p(x)\) are in leading order in τ not recovered correctly from the approximative Fokker-Planck schemes. Some implications of our study for adiabatic elimination procedures are also discussed.
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Hänggi, P., Marchesoni, F. & Grigolini, P. Bistable flow driven by coloured gaussian noise: A critical study. Z. Physik B - Condensed Matter 56, 333–339 (1984). https://doi.org/10.1007/BF01306642
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DOI: https://doi.org/10.1007/BF01306642