Vlasov Equation - Difficulties of The Standard Kinetic Approach

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:

1. Theory of pair collisions disagree with the discovery by Rayleigh, Irving Langmuir and Lewi Tonks of natural vibrations in electron plasma.
2. Theory of pair collisions formally not applicable to Coulomb interaction due to the divergence of the kinetic terms.
3. Theory of pair collisions cannot explain experiments by Harrison Merrill and Harold Webb on anomalous electron scattering in gaseous plasma.

Vlasov suggests that these difficulties originate from the long-range character of Coulomb interaction. He starts with the collisionless Boltzmann equation (sometimes anachronistically called the Vlasov equation in this context),

and adapted it to the case of a plasma, leading to the systems of equations shown below.