Generalized thermodynamics and kinetic equations: Boltzmann, Landau, Kramers and Smoluchowski

We propose a formal extension of thermodynamics and kinetic theories to a larger class of entropy functionals. Kinetic equations associated to Boltzmann, Fermi, Bose and Tsallis entropies are recovered as a special case. This formalism first provides a unifying description of classical and quantum kinetic theories. On the other hand, a generalized thermodynamical framework is justified to describe complex systems exhibiting anomalous diffusion. Finally, a notion of generalized thermodynamics emerges in the context of the the violent relaxation of collisionless stellar systems and two-dimensional vortices due to the existence of Casimir invariants and incomplete relaxation. A thermodynamical analogy can also be developed to analyze the nonlinear dynamical stability of stationary solutions of the Vlasov and 2D Euler-Poisson systems. On general grounds, we suggest that generalized entropies arise due to the existence of “hidden constraints” that modify the form of entropy that we would naively expect. Generalized kinetic equations are therefore “effective” equations that are introduced heuristically to describe complex systems.