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

## TL;DR

This paper derives a new kinetic equation for 1D homogeneous long-range systems that relax very slowly under 1/N^2 effects, explaining their long-term evolution and relaxation to equilibrium.

## Contribution

It introduces a closed, explicit kinetic equation for 1D homogeneous long-range systems that accounts for 1/N^2 effects and matches N-body simulations.

## Key findings

- Kinetic blocking prevents relaxation under 1/N effects in 1D homogeneous systems.
- The new kinetic equation satisfies an H-Theorem ensuring relaxation to equilibrium.
- Quantitative agreement between the kinetic equation and N-body simulations.

## 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.