Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

Abstract

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N tending to infinity, it reduces to the Vlasov equation describing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasi stationary state (QSS) on the coarse-grained scale. We interprete the physical nature of the QSS in relation to Lynden-Bell's statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.

References in corpus (6)

• A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model
• Lynden-Bell and Tsallis distributions for the HMF model
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Relaxation of a 1-D gravitational system
• Kinetic theory of two dimensional point vortices from a BBGKY-like hierarchy

Cited by in corpus (5)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Ergodicity and Central Limit Theorem in Systems with Long-Range Interactions
• Two-dimensional Brownian vortices