The stochastic process of equilibrium fluctuations, of a system with long range interactions

TL;DR

This paper investigates the equilibrium fluctuations in systems with long-range interactions by deriving an analytical Fokker-Planck equation and computing the diffusion coefficient, advancing understanding of their relaxation dynamics.

Contribution

It introduces an analytical approach to model the stochastic process of particles in long-range interacting systems, including explicit calculations of autocorrelation and diffusion.

Findings

Analytical autocorrelation function derived from microscopic dynamics.

Explicit expression for the diffusion coefficient.

Fokker-Planck equation describing particle impulsion in equilibrium.

Abstract

The relaxation towards equilibrium of systems with long range interactions is not yet fully understood. As a step towards such a comprehension, we propose the study of the dynamical equilibrium fluctuations in a model system with long range interaction. We compute analytically, from the microscopic dynamics, the autocorrelation function of the order parameter. From this result, we derive analytically a Fokker Planck equation which describes the stochastic process of the impulsion of a single particle in an equilibrium bath. The diffusion coefficient is explicitly computed.