We address the generalized thermodynamics and the collapse of a system of
self-gravitating Langevin particles exhibiting anomalous diffusion in a space
of dimension D. The equilibrium states correspond to polytropic distributions.
The index n of the polytrope is related to the exponent of anomalous diffusion.
We consider a high-friction limit and reduce the problem to the study of the
nonlinear Smoluchowski-Poisson system. We show that the associated Lyapunov
functional is the Tsallis free energy. We discuss in detail the equilibrium
phase diagram of self-gravitating polytropes as a function of D and n and
determine their stability by using turning points arguments and analytical
methods. When no equilibrium state exists, we investigate self-similar
solutions describing the collapse. These results can be relevant for
astrophysical systems, two-dimensional vortices and for the chemotaxis of
bacterial populations. Above all, this model constitutes a prototypical
dynamical model of systems with long-range interactions which possesses a rich
structure and which can be studied in great detail.Comment: Submitted to Phys. Rev.