NONLINEAR BROWNIAN-MOTION

A survey is presented of problems of Brownian motion which are described by nonlinear Langevin and corresponding Fokker-Planck equations. The theory of Brownian motion today is one of the main divisions of the statistical theory of open systems. Fluctuations of any internal thermodynamic parameters, density, velocity and temperature in hydrodynamics, and distribution functions in the kinetic theory are actually regarded as ''Brownian particles'' enganged in unceasing irregular motion. The following general problems of nonlinear Brownian motion are considered: Brownian motion in a medium with nonlinear friction: the critical analysis of three forms of the corresponding Langevin and Fokker-Planck equations (Ito form, Stratonovich form, and Kinetic form); Smoluchowski equations and Master equations for different cases, two types of transition from Master equation to Fokker-Planck equation. The Master equations for one-step processes; the traditional and the nontraditional definition of transition probabilities, evolution of free energy and entropy in Brownian motion; Lyapunov functionals. In order to illustrate the efficiency of the general theory the following concrete examples are considered: Brownian motion in self-oscillatory systems; H-theorem for the Van der Pol oscillator; self-organization in the Van der Pol oscillator, S-theorem; oscillator with inertial nonlinearity; bifurcation of energy of limiting cycle; oscillator with multistable stationary states; oscillators in discrete time; bifurcations of energy of limiting cycle and period of oscillations; criterion of instability upon transition to discrete time. based on H-theorem; Brownian motion of quantum atom-oscillators in the equilibrium electromagnetic field: Brownian motion in chemically reacting systems; partially ionised plasma; Malthus-Verhulst process.