  A fundamental macroscopic description of a magnetized plasma is the Vlasov
equation supplemented by the nonlinear inverse-square force Fokker-Planck
collision operator [Rosenbluth et al., Phys. Rev., 107, 1957]. The Vlasov part
describes advection in a six-dimensional phase space whereas the collision
operator involves friction and diffusion coefficients that are weighted
velocity-space integrals of the particle distribution function. The
Fokker-Planck collision operator is an integro-differential, bilinear operator,
and numerical discretization of the operator is far from trivial. In this
letter, we describe a new approach to discretize the entire kinetic system
based on an expansion in Gaussian Radial Basis functions (RBFs). This approach
is particularly well-suited to treat the collision operator because the
friction and diffusion coefficients can be analytically calculated. Although
the RBF method is known to be a powerful scheme for the interpolation of
scattered multidimensional data, Gaussian RBFs also have a deep physical
interpretation as local thermodynamic equilibria. In this letter we outline the
general theory, highlight the connection to plasma fluid theories, and also
give 2D and 3D numerical solutions of the nonlinear Fokker-Planck equation. A
broad spectrum of applications for the new method is anticipated in both
astrophysical and laboratory plasmas.
