From shape to randomness: A classification of Langevin stochasticity |
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Authors: | Iddo Eliazar Morrel H. Cohen |
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Affiliation: | 1. Holon Institute of Technology, P.O. Box 305, Holon 58102, Israel;2. Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019, USA;3. Department of Chemistry, Princeton University, Princeton, NJ 08544, USA |
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Abstract: | The Langevin equation–perhaps the most elemental stochastic differential equation in the physical sciences–describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise. The Langevin equation is, in effect, a mechanism that maps the stochastic white-noise input to a stochastic output: a stationary steady state distribution in the case of potential wells, and a transient extremum distribution in the case of potential gradients. In this paper we explore the degree of randomness of the Langevin equation’s stochastic output, and classify it à la Mandelbrot into five states of randomness ranging from “infra-mild” to “ultra-wild”. We establish closed-form and highly implementable analytic results that determine the randomness of the Langevin equation’s stochastic output–based on the shape of the Langevin equation’s potential field. |
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Keywords: | Langevin dynamics Geometric Langevin dynamics Potential wells Potential gradients Stochastic equilibria Stochastic extrema Mild randomness Wild randomness |
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