In statistical mechanics he had developed in his thesis new methods for the solution of the Boltzmann equation.
The Boltzmann equation for gas dynamics is a typical example of this activity.
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The model for the plasma bulk is based on 2d-fluid model (zero and first order moments of Boltzmann equation) and the full set of the Maxwellian equations leading to the Helmholtz equation for the magnetic field.
A kinetic description is achieved by solving the Boltzmann equation or, when the correct description of long-range Coulomb interaction is necessary, by the Vlasov equation which contains self-consistent collective electromagnetic field, or by the Fokker-Planck equation, in which approximations have been used to derive manageable collision terms.
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.
Finally, in the 1970s E.G.D. Cohen and J.R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible.
Mark Kac, Probability and related topics in the physical sciences. 1959 (with contributions by Uhlenbeck on the Boltzmann equation, Hibbs on quantum mechanics, and van der Pol on finite difference analogues of the wave and potential equations, Boulder Seminar 1957).
The Poisson–Boltzmann equation plays a role in the development of the Debye–Hückel theory of dilute electrolyte solutions.