Kinetic blockings in long-range interacting inhomogeneous systems

Long-range interacting systems unavoidably relax through Poisson shot noise fluctuations generated by their finite number of particles, $N$. When driven by two-body correlations, i.e. ${1/N}$ effects, this long-term evolution is described by the inhomogeneous Balescu-Lenard equation. Yet, in one-dimensional systems with a monotonic frequency profile and only subject to 1:1 resonances, this kinetic equation exactly vanishes: this is a first-order full kinetic blocking. These systems' long-term evolution is then driven by three-body correlations, i.e. ${1/N^2}$ effects. In the limit of dynamically hot systems, this is described by the inhomogeneous ${1/N^2}$ Landau equation. We investigate numerically the long-term evolution of systems for which this second kinetic equation also exactly vanishes: this a second-order bare kinetic blocking. We demonstrate that these systems relax through the "leaking" contributions of dressed three-body interactions that are neglected in the inhomogeneous ${1/N^2}$ Landau equation. Finally, we argue that these never-vanishing contributions prevent four-body correlations, i.e. ${1/N^{3}}$ effects, from ever being the main driver of relaxation.