  The long-term dynamics of long-range interacting $N$-body systems can
generically be described by the Balescu-Lenard kinetic equation. However, for
${1D}$ 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 new,
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.
