Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the Fokker–Planck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N→+∞, a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1/N, the collision term of a homogeneous system has the form of the Lenard–Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard–Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker–Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N→+∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ→+∞, or for large times t⪢ξ-1, it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, 2D vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker–Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.