  We study the formation and the evolution of velocity distribution tails for
systems with long-range interactions. In the thermal bath approximation, the
evolution of the distribution function of a test particle is governed by a
Fokker-Planck equation where the diffusion coefficient depends on the velocity.
We extend the theory of Potapenko et al. [Phys. Rev. E, {\bf 56}, 7159 (1997)]
developed for power law potentials to the case of an arbitrary potential of
interaction. We study how the structure and the progression of the front depend
on the behavior of the diffusion coefficient for large velocities. Particular
emphasis is given to the case where the velocity dependence of the diffusion
coefficient is Gaussian. This situation arises in Fokker-Planck equations
associated with one dimensional systems with long-range interactions such as
the Hamiltonian Mean Field (HMF) model and in the kinetic theory of
two-dimensional point vortices in hydrodynamics. We show that the progression
of the front is extremely slow (logarithmic) in that case so that the
convergence towards the equilibrium state is peculiar.
