Kinetic theory of one-dimensional inhomogeneous long-range interacting N-body systems at order 1/N2 without collective effects

Long-range interacting systems irreversibly relax as a result of their finite number of particles, N. At order 1/N, this process is described by the inhomogeneous Balescu-Lenard equation. Yet, this equation exactly vanishes in one-dimensional inhomogeneous systems with a monotonic frequency profile and sustaining only 1:1 resonances. In the limit where collective effects can be neglected, we derive a closed and explicit 1/N2 collision operator for such systems. We detail its properties, highlighting in particular how it satisfies an H theorem for Boltzmann entropy. We also compare its predictions with direct N-body simulations. Finally, we exhibit a generic class of long-range interaction potentials for which this 1/N2 collision operator exactly vanishes.