On the coarse-grained evolution of collisionless stellar systems

We describe the dynamical evolution of collisionless stellar systems on a coarse-grained scale. We first discuss the statistical theory of violent relaxation, following the seminal paper of Lynden-Bell ( 1967). Consistently with this statistical approach, we present kinetic equations for the coarse-grained distribution function (f) over bar (r, v, t) based on a Maximum Entropy Production Principle or on a quasi-linear theory of the Vlasov-Poisson system. Then, we develop a deterministic approach where the coarse-grained distribution function is defined as a convolution of the fine-grained distribution function f (r, v, t) by a Gaussian window. We derive the dynamical equation satisfied by f (r, v, t) and show that its stationary states are different from those predicted by the statistical theory of violent relaxation. This implies that the notion of coarse-graining must be defined with care. We apply these results to the HMF (Hamiltonian Mean Field) model and find that the spatial density is similar to a Tsallis q-distribution where the q parameter is related to the resolution length.