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.

arXiv comment15 pages, 10 figures. To be published in Physical Review E

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• Formation and relaxation of quasi-stationary states in particle systems with power law interactions
• Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction
• Long-term relaxation of $1D$ self-gravitating systems
• The secular evolution of discrete quasi-Keplerian systems. II. Application to a multi-mass axisymmetric disc around a supermassive black hole
• On the long-term evolution of razor-thin galactic discs: Balescu-Lenard prediction and perspectives
• Classical Goldstone modes in Long-Range Interacting Systems
• Violent relaxation in one-dimensional self-gravitating system: deviation from the Vlasov limit due to finite-$N$ effect
• The NewHorizon Simulation -- To Bar Or Not To Bar