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 α describes the number of particles of the species α 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}\cdot\nabla f_e - e\Bigl(\vec{E}+\frac{1}{c}(\vec{v}\times\vec{B})\Bigr)\cdot\frac{\partial f_e}{\partial\vec{p}} = 0
\frac{\partial f_i}{\partial t} + \vec{v}\cdot\nabla f_i + e\Bigl(\vec{E}+\frac{1}{c}(\vec{v}\times\vec{B})\Bigr)\cdot\frac{\partial f_i}{\partial\vec{p}} = 0
\nabla\times\vec{B}=\frac{4\pi\vec{j}}{c}+\frac{1}{c}\frac{\partial\vec{E}}{\partial t},\quad \nabla\times\vec{E}=-\frac{1}{c}\frac{\partial\vec{B}}{\partial t}
\nabla\cdot\vec{E}=4\pi\rho,\quad \nabla\cdot\vec{B}=0
\rho=e\int(f_i-f_e)d^3\vec{p},\quad \vec{j}=e\int(f_i-f_e)\vec{v}d^3\vec{p},\quad \vec{v} = \frac{\vec{p}/m_\alpha}{(1+p^2/(m_\alpha c)^2)^{1/2}}
Here e is the electron charge, c is the speed of light, mα the mass of the electron and ion respectively, \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).
[edit] The Vlasov-Poisson equation
The Vlasov-Poisson equations are an approximation of the Vlasov-Maxwell equations in 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 (in CGS units):
\nabla \cdot \vec{E} = -\nabla^2 \phi = 4 \pi \rho.
Here qα is the particle’s electric charge, mα 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 ρ 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.
