Relaxation of a test particle in systems with long-range interactions: diffusion coefficient and dynamical friction

Abstract

We study the relaxation of a test particle immersed in a bath of field particles interacting via weak long-range forces. To order 1/N in the $N\to +\infty$ limit, the velocity distribution of the test particle satisfies a Fokker-Planck equation whose form is related to the Landau and Lenard-Balescu equations in plasma physics. We provide explict expressions for the diffusion coefficient and friction force in the case where the velocity distribution of the field particles is isotropic. We consider (i) various dimensions of space $d=3,2$ and 1 (ii) a discret spectrum of masses among the particles (iii) different distributions of the bath including the Maxwell distribution of statistical equilibrium (thermal bath) and the step function (water bag). Specific applications are given for self-gravitating systems in three dimensions, Coulombian systems in two dimensions and for the HMF model in one dimension.

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Cited by in corpus (15)

• Statistical mechanics and dynamics of solvable models 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
• Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems
• Kinetic theory with angle-action variables
• Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory
• Statistical theory of quasi stationary states beyond the single water-bag case study
• Statistics of the gravitational force in various dimensions of space: from Gaussian to Levy laws
• Algebraic Correlation Function and Anomalous Diffusion in the HMF model
• Two-dimensional Brownian vortices
• Phase separation of bacterial colonies in a limit of high degradation. Analogy with Jupiter's great red spot
• Escape of stars from gravitational clusters in the Chandrasekhar model
• Collisional relaxation of two-dimensional self-gravitating systems
• Emergence of a collective crystal in a classical system with long-range interactions