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 $1/{N}^{2}$ 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 $1/{N}^{2}$ collision operator
exactly vanishes.