text: 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:
  # Theory of pair collisions disagrees with the discovery by Rayleigh, Irving Langmuir and Lewi Tonks of natural vibrations in electron plasma.
  # Theory of pair collisions is formally not applicable to Coulomb interaction due to the divergence of the kinetic terms.
  # 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 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  is a general distribution function of particles with momentum  at coordinates  and given time . Note that the term  is the force  acting on the particle.