  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.

Statistical Mechanics