View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100401.png" />. This means that the number of particles having their positions and velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100402.png" /> at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100403.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100404.png" />. 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100405.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100406.png" /> denotes the standard [[Wiener process|Wiener process]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100407.png" /> is a diffusion coefficient, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100408.png" /> is a friction parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v1100409.png" /> is an external force (per mass unit). This random perturbation can be interpreted as the result of interactions with a thermal bath at temperature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004010.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004011.png" /> the mass of particles and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004012.png" /> the Boltzmann constant. Writing the local conservation of the number of particles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004013.png" />, one obtains the Vlasov–Fokker–Planck equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004014.png" /></td> </tr></table>

The Fokker–Planck term

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004015.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004016.png" />, one obtains in both cases the [[Poisson equation|Poisson equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004017.png" /></td> </tr></table>

The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004018.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004019.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004020.png" />; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004021.png" />, the [[Green function|Green function]] can be computed explicitly.

Weak solutions in the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004023.png" /> can be obtained exactly as for the Vlasov–Poisson case, using an estimate of the energy

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004024.png" /></td> </tr></table>

Existence of strong solutions (with a bounded force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004025.png" />) is obtained in [[#References|[a2]]], and smoothing effects are provided in [[#References|[a3]]]. For a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004027.png" /> with boundary conditions, existence of weak solutions is obtained in [[#References|[a5]]]. Concerning the asymptotics, it is proved in [[#References|[a4]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004028.png" /> tends to a stationary solution when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004029.png" /> (at least in the Coulombic case), by using the decrease of the free energy

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004030.png" /></td> </tr></table>

which satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v110/v110040/v11004031.png" /></td> </tr></table>

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>