Vlasov equation

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938 and later discussed by him in detail in a monograph.
Difficulties of the standard kinetic approach.
First, Vlasov argues that the standard kinetic approach based on the Boltzmann equation has difficulties when applied to a description of the plasma with long-range Coulomb interaction. He mentions the following problems arising when applying the kinetic theory based on pair collisions to plasma dynamics:
Vlasov suggests that these difficulties originate from the long-range character of Coulomb interaction. He starts with the collisionless Boltzmann equation (sometimes called the Vlasov equation, anachronistically in this context), in generalized coordinates:
explicitly a PDE:
and adapted it to the case of a plasma, leading to the systems of equations shown below. Here is a general distribution function of particles with momentum at coordinates and given time .
The Vlasov–Maxwell system of equations (gaussian units).
Instead of collision-based kinetic description for interaction of charged particles in plasma, Vlasov utilizes a self-consistent collective field created by the charged plasma particles. Such a description uses distribution functions formula_3 and formula_4 for electrons and (positive) plasma ions. The distribution function formula_5 for species describes the number of particles of the species having approximately the momentum formula_6 near the position formula_7 at time . Instead of the Boltzmann equation, the following system of equations was proposed for description of charged components of plasma (electrons and positive ions):
Here is the elementary charge (formula_10), is the speed of light, is the mass of the ion, formula_11 and formula_12 represent collective self-consistent electromagnetic field created in the point formula_7 at time moment by all plasma particles. The essential difference of this system of equations from equations for particles in an external electromagnetic field is that the self-consistent electromagnetic field depends in a complex way on the distribution functions of electrons and ions formula_3 and formula_4.
The Vlasov–Poisson equation.
The Vlasov–Poisson equations are an approximation of the Vlasov–Maxwell equations in the nonrelativistic zero-magnetic field limit:
and Poisson's equation for self-consistent electric field:
Here is the particle's electric charge, is the particle's mass, formula_18 is the self-consistent electric field, formula_19 the self-consistent electric potential and is the electric charge density.
Vlasov–Poisson equations are used to describe various phenomena in plasma, in particular Landau damping and the distributions in a double layer plasma, where they are necessarily strongly non-Maxwellian, and therefore inaccessible to fluid models.
Moment equations.
In fluid descriptions of plasmas (see plasma modeling and magnetohydrodynamics (MHD)) one does not consider the velocity distribution. This is achieved by replacing formula_20 with plasma moments such as number density , flow velocity and pressure . They are named plasma moments because the -th moment of formula_21 can be found by integrating formula_22 over velocity. These variables are only functions of position and time, which means that some information is lost. In multifluid theory, the different particle species are treated as different fluids with different pressures, densities and flow velocities. The equations governing the plasma moments are called the moment or fluid equations.
Below the two most used moment equations are presented (in SI units). Deriving the moment equations from the Vlasov equation requires no assumptions about the distribution function.
Continuity equation.
The continuity equation describes how the density changes with time. It can be found by integration of the Vlasov equation over the entire velocity space.
After some calculations, one ends up with
The number density , and the momentum density , are zeroth and first order moments:
Momentum equation.
The rate of change of momentum of a particle is given by the Lorentz equation:
By using this equation and the Vlasov Equation, the momentum equation for each fluid becomes
where formula_29 is the pressure tensor. The material derivative is
The pressure tensor is defined as the particle mass times the covariance matrix of the velocity:
The frozen-in approximation.
 As for ideal MHD, the plasma can be considered as tied to the magnetic field lines when certain conditions are fulfilled. One often says that the magnetic field lines are frozen into the plasma. The frozen-in conditions can be derived from Vlasov equation.
We introduce the scales and for time, distance and speed respectively. They represent magnitudes of the different parameters which give large changes in formula_21. By large we mean that
We then write
Vlasov equation can now be written
So far no approximations have been done. To be able to proceed we set formula_36, where formula_37 is the gyro frequency and is the gyroradius. By dividing by , we get
If formula_39 and formula_40, the two first terms will be much less than formula_21 since formula_42 and formula_43 due to the definitions of and above. Since the last term is of the order of formula_21, we can neglect the two first terms and write
This equation can be decomposed into a field aligned and a perpendicular part:
The next step is to write formula_47, where
It will soon be clear why this is done. With this substitution, we get
If the parallel electric field is small,
This equation means that the distribution is gyrotropic. The mean velocity of a gyrotropic distribution is zero. Hence, formula_51 is identical with the mean velocity, , and we have
To summarize, the gyro period and the gyro radius must be much smaller than the typical times and lengths which give large changes in the distribution function. The gyro radius is often estimated by replacing with the thermal velocity or the Alfvén velocity. In the latter case is often called the inertial length. The frozen-in conditions must be evaluated for each particle species separately. Because electrons have much smaller gyro period and gyro radius than ions, the frozen-in conditions will more often be satisfied.