Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction

TL;DR

This paper derives a kinetic equation for one-dimensional homogeneous long-range interacting systems with arbitrary potentials, explaining their slow relaxation dynamics driven by three-body correlations and confirming the theory with simulations.

Contribution

It provides the first explicit kinetic equation for such systems, capturing their long-term evolution beyond the standard ${1/N}$ effects, and demonstrates its consistency with simulations.

Findings

The kinetic equation satisfies an $H$-theorem and conserves particle number and energy.

Long-range interactions ensure the ${1/N^{2}}$ dynamics are always effective.

The theoretical predictions match direct $N$-body simulation results.

Abstract

Finite-$N$ effects unavoidably drive the long-term evolution of long-range interacting $N$-body systems. The Balescu-Lenard kinetic equation generically describes this process sourced by ${1/N}$ effects but this kinetic operator exactly vanishes by symmetry for one-dimensional homogeneous systems: such systems undergo a kinetic blocking and cannot relax as a whole at this order in ${1/N}$. It is therefore only through the much weaker ${1/N^{2}}$ effects, sourced by three-body correlations, that these systems can relax, leading to a much slower evolution. In the limit where collective effects can be neglected, but for an arbitrary pairwise interaction potential, we derive a closed and explicit kinetic equation describing this very long-term evolution. We show how this kinetic equation satisfies an $H$-theorem while conserving particle number and energy, ensuring the unavoidable relaxation of the system towards the Boltzmann equilibrium distribution. Provided that the interaction is long-range, we also show how this equation cannot suffer from further kinetic blocking, i.e., the ${1/N^{2}}$ dynamics is always effective. Finally, we illustrate how this equation quantitatively matches measurements from direct $N$-body simulations.