Dynamical theory of collisionless relaxation

In his theory of violent relaxation, Lynden-Bell gave a rigorous derivation of the equilibrium distribution, but only a qualitative discussion of the manner in which equilibrium is attained Here we present a fully explicit dynamical theory of collisionless relaxation towards Lynden-Bel equilibrium.The analysis proceeds from the coarse-graining in phase space of the collisionless Boltzmann equation the mesh size being determined by the precision of the observational data. The theoretical developmen leads to a kinetic equation generalizing that obtained by Kadomtsev and Pogutse in the rather differen context of homogeneous plasma turbulence. The collision integral differs from the classical Fokker Planck type essentially by the appearance of products of three distribution functions. It drives th systems towards the Lynden-Bell equilibrium state, on a time-scale which is inversely proportional to th coarse-graining mesh and, in the non-degenerate limit, to the fine-grained phase density. Owing to th various approximations introduced, the theory does not, however, describe the violent relaxation proces itself, but rather its late quiescent phases.