In plasma physics, the Vlasov equation is a differential equation describing time evolution of the distribution function of collisionless plasma consisting of charged particles with long-range interaction, such as the Coulomb interaction.
The equation was first suggested for the description of plasma by Anatoly Vlasov in 1938[1][2] and later discussed by him in detail in a monograph.
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:
and adapted it to the case of a plasma, leading to the systems of equations shown below.
[5] Here f is a general distribution function of particles with momentum p at coordinates r and given time t. Note that the term
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.
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):
), c is the speed of light, mi is the mass of the ion,
represent collective self-consistent electromagnetic field created in the point
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
and Poisson's equation for self-consistent electric field:
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.
In fluid descriptions of plasmas (see plasma modeling and magnetohydrodynamics (MHD)) one does not consider the velocity distribution.
These variables are only functions of position and time, which means that some information is lost.
Below the two most used moment equations are presented (in SI units).
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.
The rate of change of momentum of a particle is given by the Lorentz equation:
The pressure tensor is defined as the particle mass times the covariance matrix of the velocity:
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 T, L, and V for time, distance and speed respectively.
They represent magnitudes of the different parameters which give large changes in
This equation can be decomposed into a field aligned and a perpendicular part:
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 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.