Vlasov equation

Vlasov equation is a system of non-linear integro-differential equations describing dynamics of plasma consisting of charged particles with long-range (for example, Coulomb) interaction. The equations were first suggested for description of plasma by Anatoly Vlasov in 1938 [cite journal|author=A. A. Vlasov|title=On Vibration Properties of Electron Gas|journal=J. Exp. Theor. Phys.|volume=8 (3)|pages=291|year=1938|url= http://ufn.ru/ru/articles/1967/11/f/] (see also [cite journal|author=A. A. Vlasov|title=The Vibrational Properties of an Electron Gas|journal=Sov. Phys. Usp.|volume=10|pages=721|year=1968|url= http://www.iop.org/EJ/abstract/0038-5670/10/6/R01] ) and later discussed by him in details in the monograph [cite book|last=A. A. Vlasov|title=Theory of Vibrational Properties of an Electron Gas and Its Applications|year=1945] .

Difficulties of the standard kinetic approach

First, Vlasov argues that the standard kinetic approach based on Boltzmann equation has difficulties when applied for description of plasma with long-range Coulomb interaction. He mentions the following problems arising when applying the kinetic theory based on pair collisions to plasma dynamics:

# Theory of pair collisions disagree with the discovery by John Rayleigh, Irving Langmuir and Lewi Tonks of natural vibrations in electron plasma.
# Theory of pair collisions formally not applicable to Coulomb interaction due to the divergence of the kinetic terms.
# Theory of pair collisions can not explain experiments by Harrison Merrill and Harold Webb on anomalous electron scattering in gaseous plasma [cite journal|author=H. J. Merrill and H. W. Webb|title=Electron Scattering and Plasma Oscillations|journal=Phys. Rev.|volume=55|pages=1191|year=1939|url=http://prola.aps.org/abstract/PR/v55/i12/p1191_1]

Vlasov suggests that these difficulties originate from the long-range character of Coulomb interaction.

The Vlasov-Maxwell system of equations

Then, instead of collision-based kinetic description for interaction of charged particles in plasma, Vlasov suggests to use self-consistent collective field created by charged plasma particles. Such description uses distribution functions $f\_e(vec\{r\},vec\{p\},t)$ and $f\_i(vec\{r\},vec\{p\},t)$ for electrons and (positive) plasma ions. The distribution function $f\_\{alpha\}(vec\{r\},vec\{p\},t)$ for species $alpha$ describes the number of particles of the species $alpha$ having approximately the momentum $vec\{p\}$ near the position $vec\{r\}$ at time $t$. Instead of the Boltzmann equation, the following system of equations was proposed for description of charged components of plasma (electrons and positive ions):

:$frac\{partial\; f\_e\}\{partial\; t\}\; +\; vec\{v\}\; frac\{partial\; f\_e\}\{partialvec\{x\; -\; eBigl(vec\{E\}+frac\{1\}\{c\}\; [vec\{v\},vec\{B\}]\; Bigr)\; frac\{partial\; f\_e\}\{partialvec\{p\; =\; 0$:$frac\{partial\; f\_i\}\{partial\; t\}\; +\; vec\{v\}\; frac\{partial\; f\_i\}\{partial\; vec\{x\; +\; eBigl(vec\{E\}+frac\{1\}\{c\}\; [vec\{v\},vec\{B\}]\; Bigr)\; frac\{partial\; f\_i\}\{partial\; vec\{p\; =\; 0$:$\{\; m\; rot\}vec\{B\}=frac\{4pivec\{j\{c\}+frac\{1\}\{c\}frac\{partialvec\{E\{partial\; t\},quad\; \{\; m\; rot\}vec\{E\}=-frac\{1\}\{c\}frac\{partialvec\{B\{partial\; t\}$:$\{\; m\; div\}vec\{E\}=4pi\; ho,quad\; \{\; m\; div\}vec\{B\}=0$:$ho=eint(f\_i-f\_e)dvec\{p\},quad\; vec\{j\}=eint(f\_i-f\_e)vec\{v\}dvec\{p\}$

Here $e$ is the electron charge, $c$ is the speed of light, $vec\{E\}(vec\{r\},t)$ and $vec\{B\}(vec\{r\},t)$ represent collective self-consistent electromagnetic field created in the point $vec\{r\}$ at time moment $t$ by all plasma particles. The essential difference of this system of equations from equations for particles in external electromagnetic field is that self-consistent electromagnetic field depends in a complex way on the distribution functions of electrons and ions $f\_e(vec\{r\},vec\{p\},t)$ and $f\_i(vec\{r\},vec\{p\},t)$.

The Vlasov-Poisson equation

The Vlasov equations is a system of non-linear integro-differential equations. If the fluctuations of the distribution functions around the equilibrium are small, this system of equations can be linearized. Such linearization produces Vlasov-Poisson equations which describe the dynamics of plasma in its self-consistent electric field. The Vlasov-Poisson equations are a combination of the Vlasov equation for each species $alpha$ (we consider the nonrelativistic zero-magnetic field limit):

:$frac\{partial\; f\_\{alpha\{partial\; t\}\; +\; vec\{v\}\; cdot\; frac\{partial\; f\_\{alpha\{partial\; vec\{x\; +\; frac\{q\_\{alpha\}vec\{E\{m\_\{alpha\; cdot\; frac\{partial\; f\_\{alpha\{partial\; vec\{v\; =\; 0,$

and Poisson’s equation for self-consistent electric field:

:$abla\; cdot\; vec\{E\}\; =\; -frac\{partial^2phi\}\{partial\; x^2\}\; =\; 4\; pi\; ho.$

Here $q\_\{alpha\}$ is the particle’s electric charge, $m\_\{alpha\}$ is the particle’s mass, $vec\{E\}(vec\{x\},t)$ is the self-consistent electric field, $phi(vec\{x\},\; t)$ the self-consistent electric potential and $ho$ is the electric charge density.

Vlasov-Poisson equations are used to describe various phenomena in plasma, in particular to study the distributions in a double layer plasma, where they are necessarily strongly non-Maxwellian, and therefore inaccessible to fluid models.

References

ee also

*A. A. Vlasov, "Many-Particle Theory and Its Application to Plasma", Gordon and Breach, 1961.