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Uspekhi Fizicheskikh Nauk, 1994, Volume 164, Number 8, Pages 811–844
DOI: https://doi.org/10.3367/UFNr.0164.199408b.0811
(Mi ufn993)
 

This article is cited in 149 scientific papers (total in 149 papers)

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Nonlinear Brownian motion

Yu. L. Klimontovich

Lomonosov Moscow State University, Faculty of Physics
Abstract: The theory of Brownian motion as described by nonlinear Langevin equations and the corresponding Fokker—Planck equations is discussed. The general problems of the theory of nonlinear Brownian motion considered are: Brownian motion in a medium with nonlinear friction; the critical analysis of three forms of the relevant Langevin and Fokker—Planck equations (Ito's form, Stratonovich's form, and the kinetic form); the Smoluchowski equations and master equations for different cases; two methods of transition from master equation to Fokker—Planck equation; master equations for one-step processes; traditional and nontraditional definition of transition probabilities; evolution of free energy and entropy in Brownian motion; Lyapunov functionals. The following particular examples are considered: Brownian motion in self-oscillatory systems; H-theorem for the van der Pol oscillator; S-theorem; oscillator with inertial nonlinearity; bifurcation of energy of the limiting cycle; oscillator with multistable stationary states; oscillators in discrete time; bifurcations of energy of the limiting cycle and the period of oscillations; criterion of instability upon transition to discrete time, based on the H-theorem; Brownian motion of quantum atoms oscillators in the equilibrium electromagnetic field; Brownian motion in chemically reacting systems; partially ionised plasmas; the Malthus—Verhulst process.
Received: July 1, 1994
English version:
Physics–Uspekhi, 1994, Volume 37, Issue 8, Pages 737–766
DOI: https://doi.org/10.1070/PU1994v037n08ABEH000038
Bibliographic databases:
Document Type: Article
PACS: 05.40.+j
Language: Russian


Citation: Yu. L. Klimontovich, “Nonlinear Brownian motion”, UFN, 164:8 (1994), 811–844; Phys. Usp., 37:8 (1994), 737–766
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