Violent Relaxation in Galaxies and Stellar Systems

The statistical mechanics of violently relaxing stellar systems is derived directly from the 6-N dimensional Liouville theorem. This is in contrast to previous derivations by Lynden-Bell and Saslaw which used qualitative assumptions about the distribution of particles in phase space. In particular, the present approach suggests a quantitative definition of violent relaxation, namely $$\langle{\frac{\partial U_{[f]}}{\partial \text{r}}\,\frac{\partial f}{\partial \text{p}}}\rangle\,=\frac{\partial\langle U[f]\rangle}{\partial \text{r}} \frac{\partial\langle f\rangle}{\partial \text{p}}$$ . Here $${U}_{[f]}$$ is the mean field gravitational potential, which is a functional of the N -particle distribution function f , r and p are position and momentum coordinates, and brackets denote time averages.