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. 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. 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 f is a general distribution function of particles with momentum p at coordinates r and given time t. 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 f e ( r , p , t ) {displaystyle f_{e}(mathbf {r} ,mathbf {p} ,t)} and f i ( r , p , t ) {displaystyle f_{i}(mathbf {r} ,mathbf {p} ,t)} for electrons and (positive) plasma ions. The distribution function f α ( r , p , t ) {displaystyle f_{alpha }(mathbf {r} ,mathbf {p} ,t)} for species α describes the number of particles of the species α having approximately the momentum p {displaystyle mathbf {p} } near the position r {displaystyle mathbf {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): Here e is the electron charge, c is the speed of light, mi is the mass of the ion, E ( r , t ) {displaystyle mathbf {E} (mathbf {r} ,t)} and B ( r , t ) {displaystyle mathbf {B} (mathbf {r} ,t)} represent collective self-consistent electromagnetic field created in the point r {displaystyle mathbf {r} } at time moment t 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 f e ( r , p , t ) {displaystyle f_{e}(mathbf {r} ,mathbf {p} ,t)} and f i ( r , p , t ) {displaystyle f_{i}(mathbf {r} ,mathbf {p} ,t)} .