View source for Vlasov-Poisson-Fokker-Planck system

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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 $f(t,x,v)\geq0$. This means that the number of particles having their positions and velocities $(x,v)\in D$ at time $t$ is given by $\int_Df(t,x,v)dxdv$. The linear Fokker–Planck collision process has been introduced by S. Chandrasekhar [[#References|[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

$$dx=vdt,dv=(F(t,x)-\beta v)dt+\sqrt{2\sigma}dW.$$

Here, $W$ denotes the standard [[Wiener process|Wiener process]], $\sigma>0$ is a diffusion coefficient, $\beta\geq0$ is a friction parameter and $F$ is an external force (per mass unit). This random perturbation can be interpreted as the result of interactions with a thermal bath at temperature $kT=m\sigma/\beta$, with $m$ the mass of particles and $k$ the Boltzmann constant. Writing the local conservation of the number of particles $\int_Df(t,x,v)dxdv$, one obtains the Vlasov–Fokker–Planck equation

$$\partial_tf+v\cdot\nabla_xf+\operatorname{div}_v[(F(t,x)-\beta v)f-\sigma\nabla_vf]=0.$$

The Fokker–Planck term

$$Lf=\operatorname{div}_v[\beta vf+\sigma\nabla_vf]$$

can also be viewed as a simplified version of the Fokker–Planck–Landau collision operator, which is quadratic as the Boltzmann collision operator, see [[#References|[a9]]] or [[#References|[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 $F$, one obtains in both cases the [[Poisson equation|Poisson equation]]

$$F=-\nabla_x\phi,\quad-\Delta_x\phi=\lambda\int fdv.$$

The constant $\lambda$ is positive in the Coulombic case and negative in the Newtonian case. See [[#References|[a8]]], [[#References|[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 $\sigma=\beta=0$. 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 $F\in C^\infty$; when $F=0$, the [[Green function|Green function]] can be computed explicitly.

Weak solutions in the whole space $x\in\mathbf R^3$, $v\in\mathbf R^3$ can be obtained exactly as for the Vlasov–Poisson case, using an estimate of the energy

$$\mathcal E(t)=\iint\frac{|v|^2}{2}fdxdv+\frac1\lambda\int\frac{|F|^2}{2}dx.$$

Existence of strong solutions (with a bounded force $F$) is obtained in [[#References|[a2]]], and smoothing effects are provided in [[#References|[a3]]]. For a bounded domain $x\in\Omega$, $v\in\mathbf R^3$ with boundary conditions, existence of weak solutions is obtained in [[#References|[a5]]]. Concerning the asymptotics, it is proved in [[#References|[a4]]] that $f$ tends to a stationary solution when $t\to\infty$ (at least in the Coulombic case), by using the decrease of the free energy

$$A(t)=\mathcal E(t)+\frac\sigma\beta\iint f\ln fdxdv,$$

which satisfies

$$\frac{dA}{dt}=-\beta\iint\left|v\sqrt f+2\frac\sigma\beta\nabla_v\sqrt f\right|^2dxdv.$$

The same analysis has been generalized to the case of a bounded domain in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.L. Bonilla,   J.A. Carrillo,   J. Soler,   "Asymptotic behavior of an initial-boundary value problem for the Vlasov–Poisson–Fokker–Planck system"  ''SIAM J. Appl. Math.'' , '''to appear'''  (1997)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Bouchut,   "Existence and uniqueness of a global smooth solution for the Vlasov–Poisson–Fokker–Planck system in three dimensions"  ''J. Funct. Anal.'' , '''111'''  (1993)  pp. 239–258</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Bouchut,   "Smoothing effect for the non-linear Vlasov–Poisson–Fokker–Planck system"  ''J. Diff. Eq.'' , '''122'''  (1995)  pp. 225–238</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. Bouchut,   J. Dolbeault,   "On long time asymptotics of the Vlasov–Fokker–Planck equation and of the Vlasov–Poisson–Fokker–Planck system with coulombic and newtonian potentials"  ''Diff. Int. Eq.'' , '''8'''  (1995)  pp. 487–514</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.A. Carrillo,   "Global weak solutions of the absorption and reflection-type initial-boundary value problems for the Vlasov–Poisson–Fokker–Planck system"  ''submitted''  (1996)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Chandrasekhar,   "Stochastic problems in physics and astronomy"  ''Rev. Mod. Phys.'' , '''15'''  (1943)  pp. 1–89</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Chapman,   T.G. Cowling,   "The mathematical theory of non-uniform gases" , Cambridge Univ. Press  (1939)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M.K.H. Kiesling,   "On the equilibrium statistical mechanics of isothermal classical gravitating matter"  ''J. Stat. Phys.'' , '''55'''  (1989)  pp. 203–257</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  E.M. Lifshitz,   L.P. Pitaevskii,   "Physical kinetics" , Pergamon  (1981)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  T. Padmanabhan,   "Statistical mechanics of gravitating systems"  ''Phys. Rep.'' , '''188'''  (1990)  pp. 285–362</TD></TR></table>