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

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.

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Cited by in corpus (16)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Thermodynamics and dynamics of systems with long-range interactions
• Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations
• Lynden-Bell and Tsallis distributions for the HMF model
• Kinetic equations for systems with long-range interactions: a unified description
• Kinetic theory with angle-action variables
• Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model
• Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction
• Kinetic theory of Onsager's vortices in two-dimensional hydrodynamics
• Self-gravitating Brownian particles in two dimensions: the case of N=2 particles
• Invariant measures of the 2D Euler and Vlasov equations
• Algebraic Correlation Function and Anomalous Diffusion in the HMF model
• Kinetic theory of two-dimensional point vortices with collective effects
• Curious behaviour of the diffusion coefficient and friction force for the strongly inhomogeneous HMF model
• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Solar Flare Heating with Turbulent Suppression of Thermal Conduction