  It is well known that due to its divergence at large impact parameters, the
Boltzmann collision integral in the kinetic equation for 3D systems of
particles interacting through a $1/r$ potential must be replaced by a
Balescu-Lenard-like collision term. However, the latter diverges at small
impact parameters. This comes from the fact that only weak interactions are
considered while strong collisions between close particles are neglected in its
derivation. We show that a solution to this dilemma exists in the framework of
the BBGKY formulation of statistical mechanics. It is based on a separate
treatment of the contribution of the strong interactions from that of the weak
interactions. The strong interaction part leads to a new term that involves a
fractional Laplacian operator in velocity space while the weak interaction
component yields the Balescu-Lenard collision term with an explained lower
cut-off at the Landau length. For spatially uniform initial conditions, the
fractional Laplacian contribution leads to a long-tailed velocity distribution
as long as the spatial inhomogeneity remains small. We present results from
molecular dynamics simulations confirming the existence of such long tails.
