Kinetic theory of a two-dimensional magnetized plasma

Abstract

Several features of the equilibrium and non-equilibrium statistical mechanics of a two-dimensional plasma in a uniform d.c. magnetic field are investigated. The calculations have been motivated by the recent derivation of Bohm's diffusion coefficient given by Taylor & McNamara for this system. The charges interact only through electrostatic (logarithmic) potentials. The problem is considered both with and without the guidiiig-centre approximation. With the guiding centre approximation, an appropriate Liouville equation and BBGKV hierarchy predict no approach to thermal equilibrium for the spatially uniform case. For the spatially non-uniform situation, a guiding-centre ‘Vlasov’ equation is discussed, and is solved in special cases. The most interesting features of thermal equilibrium theory (with and without the guiding-centre approximation) are (i) a collapse of the system above a critical value of the plasma parameter, and (ii) a divergence in the electric field fluctuation spectrum (minus the selfenergy terms) for small plasma parameter and very large systems. For the nonequilibrium, non-guiding-centre case, a Boltzmann equation and a Fokker—Planck equation are derived in the appropriate limits. The latter is more tractable than the former, and can be shown to obey conservation laws and an H-theorem, but contains a divergent integral, which must be cut off on physical grounds. Several unsolved problems are posed.