Using bimodal probability distributions in the problems of Brownian diffusion

When analyzing nonlinear stochastic systems, we deal with the chains of differential equations for the moments or cumulants of dynamic variables. To disconnect such chains, the well-known cumulant approach, which is adequate to the quasi-Gaussian expansion of the higher-order moments is used. Howeve...

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Bibliographic Details
Published in:Radiophysics and quantum electronics 2006-08, Vol.49 (8), p.645-655
Main Author: Muzychuk, O V
Format: Article
Language:English
Subjects:
Brownian motion
Differential equations
Diffusion
Disengaging
Dynamic tests
Dynamical systems
Mathematical models
Normal distribution
Probability
Stochastic systems
Temporal logic
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Summary:When analyzing nonlinear stochastic systems, we deal with the chains of differential equations for the moments or cumulants of dynamic variables. To disconnect such chains, the well-known cumulant approach, which is adequate to the quasi-Gaussian expansion of the higher-order moments is used. However, this method is inefficient in the problems of Brownian diffusion in bimodal potential profiles, and the disconnection problem should be solved on the basis of bimodal probability distributions. To this end, we propose to construct bimodal model distributions, in particular, the bi-Gaussian distribution. Cumulants and the expansions of the higher-order moments for symmetric and nonsymmetric bi-Gaussian models. On this basis, we consider relaxation of probability characteristics of one-dimensional Brownian motion in the bimodal potential profile. The dependences of relaxation of the mean value and variance of particle coordinate on the potential barrier "power," the noise intensity, and the initial distribution of particles are analyzed numerically. In particular, it is shown that relaxation proceeds by stages with different temporal scales in the case of a powerful barrier.[PUBLICATION ABSTRACT]
ISSN:0033-8443
1573-9120
DOI:10.1007/s11141-006-0099-9