Kinetic theory of one-dimensional homogeneous long-range interacting systems sourced by 1/N^{2} effects

The long-term dynamics of long-range interacting N-body systems can generically be described by the Balescu-Lenard kinetic equation. However, for one-dimensional 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 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.