A note on the semi-Lagrangian formulation of the Vlasov equation

In a previous paper [Lewak (1969), see also Pflrsch (1966) for related treatment], it was shown that the Vlasov equation in the Semi-Lagrangian (S.L.) formulation, may be written in a form resembling the fluid equations.

plus Maxwell's equations with the source terms given by

where n is the determinant of the tensor T_ij = ∂g_i/∂ζ_j, and N is the constant mean number density of electrons. The averaging notation < > here is defined by

where f(σ) is the electron distribution function to be specified. The equations assume for simplicity a uniform fixed ion background, although this is not a necessary restriction and equations (1) and (2) need only an obvious modification to account for ions. The force fields in (1) are related to the electric field E and magnetic field B in the plasma by.