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 (for example, Coulomb) interaction. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938 (see also ) 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 ${\displaystyle f_{e}(\mathbf {r} ,\mathbf {p} ,t)}$ and ${\displaystyle f_{i}(\mathbf {r} ,\mathbf {p} ,t)}$ for electrons and (positive) plasma ions. The distribution function ${\displaystyle f_{\alpha }(\mathbf {r} ,\mathbf {p} ,t)}$ for species α describes the number of particles of the species α having approximately the momentum ${\displaystyle \mathbf {p} }$ near the position ${\displaystyle \mathbf {r} }$ 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):

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