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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 <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
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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
  
<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>
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$$dx=vdt,dv=(F(t,x)-\beta v)dt+\sqrt{2\sigma}dW.$$
  
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
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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
  
<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>
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$$\partial_tf+v\cdot\nabla_xf+\operatorname{div}_v[(F(t,x)-\beta v)f-\sigma\nabla_vf]=0.$$
  
 
The Fokker–Planck term
 
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>
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$$Lf=\operatorname{div}_v[\beta vf+\sigma\nabla_vf]$$
  
 
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]]].
 
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]]
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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]]
  
<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>
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$$F=-\nabla_x\phi,\quad-\Delta_x\phi=\lambda\int fdv.$$
  
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.
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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 <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.
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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 <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
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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
  
<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>
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$$\mathcal E(t)=\iint\frac{|v|^2}{2}fdxdv+\frac1\lambda\int\frac{|F|^2}{2}dx.$$
  
Existence of strong solutions (with a bounded force <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
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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
  
<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>
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$$A(t)=\mathcal E(t)+\frac\sigma\beta\iint f\ln fdxdv,$$
  
 
which satisfies
 
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>
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$$\frac{dA}{dt}=-\beta\iint\left|v\sqrt f+2\frac\sigma\beta\nabla_v\sqrt f\right|^2dxdv.$$
  
 
The same analysis has been generalized to the case of a bounded domain in [[#References|[a1]]].
 
The same analysis has been generalized to the case of a bounded domain in [[#References|[a1]]].

Latest revision as of 08:31, 22 August 2014

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
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=33073
This article was adapted from an original article by F. Bouchut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article