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Kinetic Theory

Theory & Computation

Fundamentally, a plasma is an N-body system of mobile charged particles and electromagnetic fields. The basic equations which describe the system classically are the Lorentz-force equation and the (microscopic) Maxwell equations. Unfortunately, for a macroscopic amount of plasma, a complete global simulation of such a large N-body system by direct integration of the Maxwell-Lorentz equations is impractical, even with the most powerful computers, and even if we could solve the system exactly we would have far more information than we would typically require. For these reasons a variety of statistical models of plasma dynamics have been developed.

In any macroscopic physical system containing many individual particles, there are basically three levels of description:

  • the exact microscopic description
  • kinetic theory
  • macroscopic or fluid description.

In a microscopic description, one imagines writing down Newton second law for a large number N of particles and solving for their interacting trajectories. Such a description is in principle exact, classically. It is still unimaginable today, even by the most advanced computers. Even the initial data itself is beyond the magnitude of imaginable storage devices. Moreover, if solutions were known, they would be mostly irrelevant information requiring another unimaginably advanced computers to distill into useful form. When the sensitivity of the exact solution to minuscule initial condition errors is considered — the modern study of chaos — the situation becomes even more absurd. Nonetheless, the microscopic description is useful as a formal starting point from which to derive soluble, practical descriptions.

The microscopic theory passed to kinetic theory by the application of statistical, probability concepts. Since one is not interested in all the microscopic particle data, one considers statistical ensembles of systems. By averaging out the microscopic information in the exact theory, one obtains statistical, kinetic equations. When possible, these are reduced further to give equations for the one-particle (i.e., electron or ion) distribution functions. Examples of kinetic equations are the Vlasov equation and the Boltzmann equation. Although the precise locations of individual particles are lost in kinetic theory, detailed knowledge of particle motion is required. In this sense kinetic theory is still microscopic, even though statistical averages have been employed.

Finally, in some cases, it is possible to reduce kinetic theory even further. Here, one has only macroscopic quantities such as density, temperature, and fluid velocity, and closed equations giving their evolution in space and time. No knowledge of individual particle motion is required to describe observable phenomena.

The modern  plasma kinetic theory starts from first principles with the fundamental microscopic equations and then systematically derives the major equations in current use in plasma physics research, such as kinetic equations named after Klimontovich, Landau, Liouville, BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon), Balescu-Lenard, Fokker-Planck, Vlasov, ...