Kinetic theory of $1D$ homogeneous long-range interacting systems sourced by $1/N^2$ effects

Abstract

The long-term dynamics of long-range interacting $N$-body systems can generically be described by the Balescu-Lenard kinetic equation. However, for $1D$ homogeneous systems, this collision operator exactly vanishes by symmetry. These systems undergo a kinetic blocking, and cannot relax as a whole under $1/N$ resonant effects. As a result, these systems can only relax under $1/N^2$ effects, and their relaxation is drastically slowed down. In the context of the homogeneous Hamiltonian Mean Field model, we present a new, closed and explicit kinetic equation describing self-consistently the very long-term evolution of such systems, in the limit where collective effects can be neglected, i.e. for dynamically hot initial conditions. We show in particular how that kinetic equation satisfies an $H$-Theorem that guarantees the unavoidable relaxation to the Boltzmann equilibrium distribution. Finally, we illustrate how that kinetic equation quantitatively matches with the measurements from direct $N$-body simulations.

15 pages, 4 figures, submitted to PRE