Nonlinear Brownian motion

Abstract

Contents 1. Introduction 2. Two ways of describing Brownian motion 2.1 The Langevin equation; 2.2 The Fokker-Planck equation 3. Brownian motion in a medium with nonlinear friction. Three forms of the Fokker-Planck equation 4. The Fokker-Planck equation for a Boltzmann gas 5. The Smoluchowski equation. The master equation 6. Two ways of transition from the master equation to the Fokker-Planck equation 6.1 The kinetic form of the Fokker-Planck equation; 6.2 Stationary solution of the Fokker-Planck equation 7. The master equation for a system of atoms in an electromagnetic field 8. Brownian motion of quantum atom oscillators 8.1 The master equation; 8.2 The Fokker-Planck equation 9. Master equations for one-step processes 9.1 Traditional definition of transition probability; 9.2 Nontraditional definition of transition probability 10. Spatial diffusion. The Einstein-Smoluchowski equation 10.1 Spatial diffusion. The Langevin method; 10.2 Diffusion of a Brownian particle in an external field; 10.3 Comparison of stationary distributions in 'linear' and 'nonlinear' thermostats 11. Hydrodynamic description of Brownian motion 12. Evolution of free energy and entropy in Brownian motion.

Relativistic Brownian motion: From a microscopic binary collision model to the Langevin equation

“…However, in general, it cannot be expected that such a 'truncated' LE yields the correct relaxation behavior and/or the correct stationary solution [59,62]. Furthermore, also under such simplifying approximations, the results will depend on the choice of the discretization rule [50,53,54,55,56,57,58,59,60] because of the multiplicative coupling between 2 D(P ) and ζ(t). Loosely speaking, this discretization dilemma is the price that one has to pay for mapping the large number of collisions between t and t + τ into a single instant of time.…”

“…II B) suggests that the "transport-form", i.e. the post-point discretization rule [50,58,59,60] should be preferable in the relativistic case as well.…”

Influence of external potentials on heterogeneous diffusion processes

“…Commonly used values of the parameter γ are γ = 0 corresponding to pre-point Itô convention, γ = 1/2 corresponding to mid-point Stratonovich convention and γ = 1 corresponding to post-point Hänggi-Klimontovich [52,53], kinetic or isothermal convention [54][55][56]. The integration convention should be determined from the experimental data or derived from another model [57].…”

On the Attractors of Two-Dimensional Rayleigh Oscillators Including Noise

“…The dynamics of the systems studied here is based on Langevin equations, known from the theory of conventional Brownian motion (Hänggi et al, 1990;Langevin, 1908). For the two-dimensional space we get four first order coupled differential equations in the phase space {x 1 , x 2 , v 1 , v 2 }:…”

“…Simple models composed of active Brownian particles were studied in many ear- * Electronic address: udo.erdmann@physik.hu-berlin.de lier works (e.g. Helbing and Molnár, 1995;Klimontovich, 1994;Schienbein and Gruler, 1993).…”