# Vlasov-Poisson-Fokker-Planck system

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

The linear Fokker–Planck operator is a model for a certain type of collision in a gas or plasma of particles, which are assumed to be described by a kinetic distribution function . This means that the number of particles having their positions and velocities at time is given by . The linear Fokker–Planck collision process has been introduced by S. Chandrasekhar [a6]. The main assumption is that the collisional effects take the form of a stochastic perturbation in Newton's laws of classical mechanics, which are written as Here, denotes the standard Wiener process, is a diffusion coefficient, is a friction parameter and is an external force (per mass unit). This random perturbation can be interpreted as the result of interactions with a thermal bath at temperature , with the mass of particles and the Boltzmann constant. Writing the local conservation of the number of particles , one obtains the Vlasov–Fokker–Planck equation The Fokker–Planck term can also be viewed as a simplified version of the Fokker–Planck–Landau collision operator, which is quadratic as the Boltzmann collision operator, see [a9] or [a7].

There are two situations where the Vlasov–Fokker–Planck equation is relevant: for charged particles and for gravitational systems. When dealing with a self-consistent force field , one obtains in both cases the Poisson equation The constant is positive in the Coulombic case and negative in the Newtonian case. See [a8], [a10] for discussions on this model in the latter gravitational case.

Concerning the mathematical study of the Vlasov–Poisson–Fokker–Planck system, the situation is comparable with the Vlasov–Poisson system, which is obtained when . The main differences are that there are no characteristics, and that the Laplacian term gives rise to smoothing effects. Actually, the Vlasov–Fokker–Planck operator is hypo-elliptic as soon as ; when , the Green function can be computed explicitly.

Weak solutions in the whole space , can be obtained exactly as for the Vlasov–Poisson case, using an estimate of the energy Existence of strong solutions (with a bounded force ) is obtained in [a2], and smoothing effects are provided in [a3]. For a bounded domain , with boundary conditions, existence of weak solutions is obtained in [a5]. Concerning the asymptotics, it is proved in [a4] that tends to a stationary solution when (at least in the Coulombic case), by using the decrease of the free energy which satisfies The same analysis has been generalized to the case of a bounded domain in [a1].

How to Cite This Entry:
Vlasov-Poisson-Fokker-Planck system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vlasov-Poisson-Fokker-Planck_system&oldid=16992
This article was adapted from an original article by F. Bouchut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article