  We discuss the dynamics and thermodynamics of systems with weak long-range
interactions. Generically, these systems experience a violent collisionless
relaxation in the Vlasov regime leading to a (usually) non-Boltzmannian quasi
stationary state (QSS), followed by a slow collisional relaxation leading to
the Boltzmann statistical equilibrium state. These two regimes can be explained
by a kinetic theory, using an expansion of the BBGKY hierarchy in powers of
1/N, where N is the number of particles. We discuss the physical meaning of the
different functionals appearing in the analysis: the Boltzmann entropy, the
Lynden-Bell entropy, the "generalized" entropies arising in the reduced space
of coarse-grained distribution functions, the Tsallis entropy, the generalized
H-functions increasing during violent relaxation (not necessarily
monotonically) and the convex Casimir functionals used to settle the formal
nonlinear dynamical stability of steady states of the Vlasov equation. We show
the connection between the different variational problems associated with these
functionals. We also introduce a general class of nonlinear mean field
Fokker-Planck (NFP) equations that can be used as numerical algorithms to solve
these constrained optimization problems.
