The covariant form of the Klein-Kramers equation and the associated moment equations

We provide a covariant, coordinate-free formulation of the many-dimensional Klein-Kramers equation for the phase space distribution of a Brownian particle. We construct a complete set of eigenfunctions of the collision operator adapted to the coordinate system, which involve covariant tensorial Hermite polynomials. The Klein-Kramers equation can then be reformulated as a system of coupled equations for the expansion coefficients with respect to this system. Truncation of this system of moment equations and application of a subsidiary condition yields a covariant generalization of Grad's thirteen-moment equations. As an application we give the explicit form of these equations for spherically symmetric, stationary solutions in spherical coordinates. We briefly comment on possible extensions of our treatment to slightly more complicated cases.