The Vlasov Equation and Related Systems

Abstract

AbstractThere are many situations in which the forces acting among the particles are long-range and thus the effect of collisions (short-range interactions) can be neglected with respect to the averaged action of several particles at distance. One well-known case arises with charged particles in a plasma or in a semiconductor; in the second case the short-range interactions are with the impurities (including thermal vibrations) of the lattice and usually cannot be neglected, but there are several situations in plasma physics where the mean electromagnetic field plays a much more important role than the interparticle collisions. In the non-relativistic case this is dealt with via the force term in the left-hand side of the collisionless Boltzmann equation; the electric field is assumed to possess a potential which is computed via the Poisson equation, where the charge density is determined by the integral of the distribution function with respect to momentum. This produces a nonlinear system which is called the Vlasov—Poisson system. If the electric field is expressed as an integral with respect to space coordinates via the Green’s function and substituted into the collisionless Boltzmann equation, we obtain a non-linear equation, usually called the Vlasov equation. We shall call the Vlasov equation, generally speaking, the collisionless Boltzmann equation in which the force term depends on fields which in turn are determined by field equations containing the distribution function in the source term. When both the mean field and the collisions are taken into account (as in semiconductors) one talks of the Boltzmann-Vlasov equation.KeywordsBlack HoleSpherical SymmetryGravitational CollapseNaked SingularityVlasov EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.