Dispersion of ensembles of non-interacting particles

The dynamics of an ensemble of particles emanating from a common point with a distribution of velocities is modeled as a continuum of particles described by a phase space distribution function. A general solution for the distribution function and the associated spatial density function is obtained for a general dynamical system. The special cases of linear dynamical systems and slow dispersion from a circular orbit are treated in detail. A transcendental equation is derived, the roots of which determine the time since initial dispersion from knowledge of the spatial density function at later times.