Coarse-grained distributions and superstatistics

We show an interesting connection between non-standard (non-Boltzmannian) distribution functions arising in the theory of violent relaxation for collisionless stellar systems [D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136 (1967) 101.] and the notion of superstatistics recently introduced by [Beck and Cohen Physica A 322 (2003) 267]. The common link between these two theories is the emergence of coarse-grained distributions arising out of fine-grained distributions. The coarse-grained distribution functions are written as a superposition of Boltzmann factors weighted by a non-universal function. Even more general distributions can arise in case of incomplete violent relaxation (non-ergodicity). They are stable stationary solutions of the Vlasov equation. We also discuss analogies and differences between the statistical equilibrium state of a multi-components self-gravitating system and the metaequilibrium (or quasi-equilibrium) states of a collisionless stellar system. Finally, we stress the important distinction between entropies, generalized entropies, relative entropies and H-functions. We discuss applications of these ideas in two-dimensional turbulence and for other systems with long-range interactions.