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

TL;DR

This paper investigates the collisional relaxation process in the Hamiltonian-Mean-Field model, deriving diffusion coefficients using kinetic equations and highlighting the importance of collective effects in the relaxation dynamics.

Contribution

It provides explicit expressions for diffusion coefficients in the HMF model, including collective effects, and compares theoretical predictions with simulations for validation.

Findings

Collective effects are crucial for accurate relaxation description.

Derived explicit diffusion coefficients for various configurations.

Good agreement between kinetic theory and simulations.

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.