Collisional relaxation in the inhomogeneous Hamiltonian-Mean-Field model: diffusion coefficients

Abstract

Systems of particles with long range interactions present two important processes: first, the formation of out-of-equilibrium quasi-stationary states (QSS), and the collisional relaxation towards Maxwell-Boltzmann equilibrium in a much longer timescale. In this paper, we study the collisional relaxation in the Hamiltonian-Mean-Field model (HMF) using the appropriate kinetic equations for a system of $N$ particles at order $1/N$ : the Landau equation when collective effects are neglected and the Lenard-Balescu equation when they are taken into account. We derive explicit expressions for the diffusion coefficients using both equations for any magnetization, and we obtain analytic expressions for highly clustered configurations. An important conclusion is that in this system collective effects are crucial in order to describe the relaxation dynamics. We compare the diffusion calculated with the kinetic equations with simulations set up to simulate the system with or without collective effects, obtaining a very good agreement between theory and simulations.

15 pages, 10 figures. To be published in Physical Review E

References in corpus (4)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Relaxation of a 1-D gravitational system
• Secular resonant dressed orbital diffusion I : method and WKB limit for tepid discs
• Secular resonant dressed orbital diffusion II : application to an isolated self similar tepid galactic disc

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• Violent relaxation in one-dimensional self-gravitating system: deviation from the Vlasov limit due to finite-$N$ effect
• Classical Goldstone modes in Long-Range Interacting Systems
• The NewHorizon Simulation -- To Bar Or Not To Bar