Relaxation of the distribution function tails for systems described by Fokker-Planck equations

TL;DR

This paper investigates how velocity distribution tails evolve in systems with long-range interactions, focusing on the effects of velocity-dependent diffusion coefficients, especially Gaussian ones, on the convergence to equilibrium.

Contribution

It extends existing theory to arbitrary potentials and analyzes the slow, logarithmic progression of distribution fronts in specific long-range interacting systems.

Findings

Front progression is logarithmically slow for Gaussian diffusion coefficients.

Velocity distribution tails evolve differently depending on the diffusion coefficient behavior.

Convergence to equilibrium exhibits peculiar, slow dynamics in these systems.

Abstract

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.