Vlasov-Poisson-Fokker-Planck system

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 [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, $\sigma>0$ 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 [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 $F$, one obtains in both cases the 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 [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 $\sigma=\beta=0$. 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 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 [a2], and smoothing effects are provided in [a3]. For a bounded domain $x\in\Omega$, $v\in\mathbf R^3$ with boundary conditions, existence of weak solutions is obtained in [a5]. Concerning the asymptotics, it is proved in [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 [a1].

References

[a1]	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)
[a2]	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
[a3]	F. Bouchut, "Smoothing effect for the non-linear Vlasov–Poisson–Fokker–Planck system" J. Diff. Eq. , 122 (1995) pp. 225–238
[a4]	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
[a5]	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)
[a6]	S. Chandrasekhar, "Stochastic problems in physics and astronomy" Rev. Mod. Phys. , 15 (1943) pp. 1–89
[a7]	S. Chapman, T.G. Cowling, "The mathematical theory of non-uniform gases" , Cambridge Univ. Press (1939)
[a8]	M.K.H. Kiesling, "On the equilibrium statistical mechanics of isothermal classical gravitating matter" J. Stat. Phys. , 55 (1989) pp. 203–257
[a9]	E.M. Lifshitz, L.P. Pitaevskii, "Physical kinetics" , Pergamon (1981)
[a10]	T. Padmanabhan, "Statistical mechanics of gravitating systems" Phys. Rep. , 188 (1990) pp. 285–362