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(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.