Kinetic theory of inhomogeneous systems with long-range interactions and fluctuation-dissipation theorem

TL;DR

This paper advances the kinetic theory of inhomogeneous systems with long-range interactions by deriving response functions, kinetic equations, and fluctuation-dissipation relations, including for multi-species and stochastic systems, with applications to various physical models.

Contribution

It introduces a simplified formalism for the kinetic theory of inhomogeneous long-range systems, deriving new equations and relations, including a generalized fluctuation-dissipation theorem and stochastic kinetic equations.

Findings

Derived the diffusion tensor and friction by polarization.

Established the fluctuation-dissipation theorem for long-range systems.

Reduced the inhomogeneous Lenard-Balescu equation to the Landau equation when collective effects are neglected.

Abstract

We complete the kinetic theory of inhomogeneous systems with long-range interactions initiated in previous works. We use a simpler and more physical formalism. We consider a system of particles submitted to a small external stochastic perturbation and determine the response of the system to the perturbation. We derive the diffusion tensor and the friction by polarization of a test particle. We introduce a general Fokker-Planck equation involving a diffusion term and a friction term. When the friction by polarization can be neglected, we obtain a secular dressed diffusion (SDD) equation sourced by the external noise. When the external perturbation is created by a discrete collection of $N$ field particles, we obtain the inhomogeneous Lenard-Balescu kinetic equation reducing to the inhomogeneous Landau kinetic equation when collective effects are neglected. We consider a multi-species system of particles. When the field particles are at statistical equilibrium (thermal bath), we establish the proper expression of the fluctuation-dissipation theorem for systems with long-range interactions relating the power spectrum of the fluctuations to the response function of the system. In that case, the friction and diffusion coefficients satisfy the Einstein relation and the Fokker-Planck equation reduces to the inhomogeneous Kramers equation. We also consider a gas of Brownian particles with long-range interactions described by $N$ coupled stochastic Langevin equations and determine its mean and mesoscopic evolution. We discuss the notion of stochastic kinetic equations and the role of fluctuations possibly triggering random transitions from one equilibrium state to the other. Our presentation parallels the one given for two-dimensional point vortices in a previous paper [P.H. Chavanis, Eur. Phys. J. Plus 138, 136 (2023)].