A Certain Discontinuous Markov Process in Stellar Dynamics

Abstract

The basic assumption in Chandrasekhar’s approach of statistical stellar dynamics (Chandrasekhar, 1942) is the postulate that a test star within a stellar system being stationary in the sense of collisionless continuum theory suffers random displacements in velocity space generated by the fluctuating part of the gravitational field in a manner that can be described in terms of a random walk. This is equivalent to the assertion that the increments of velocity are regarded as stochastically independent in disjoint time intervals. From this Chandrasekhar derived a diffusion process in velocity space. The equation of motion of the probability density W(r, u, t) in the whole 6-dimensional phase space is then written in the form of a Fokker-Planck-type equation:

$$\frac{{\partial W}}{{\partial t}}+u\cdot{\nabla_r}W+{\nabla_r}\Phi\cdot{\nabla_u}W={\nabla_u}\left({q{\nabla_u}W+\eta{W_u}}\right)$$

(1)

r, u = position, velocity vector; Φ = gravitational potential of the ‘smoothed out’ distribution of matter; q = diffusion coefficient; and η = coefficient of dynamical friction.

Now at Universitäts Sternwarte, Göttingen, West Germany.