Dynamics and thermodynamics of systems with long‐range interactions: interpretation of the different functionals

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.