Ornstein-Uhlenbeck limit for the velocity process of an $N$-particle system interacting stochastically

TL;DR

This paper studies a stochastic N-particle system with energy conservation, showing that as N grows large, some velocity components behave independently following an Ornstein-Uhlenbeck process, revealing a limit behavior.

Contribution

It demonstrates the convergence of finite velocity components to Ornstein-Uhlenbeck processes in the large N limit for a conserved-energy stochastic particle system.

Findings

Velocity components become independent in the limit

Finite-dimensional distributions converge to Ornstein-Uhlenbeck processes

System evolution described as diffusion on a sphere

Abstract

An $N$-particle system with stochastic interactions is considered. Interactions are driven by a Brownian noise term and total energy conservation is imposed. The evolution of the system, in velocity space, is a diffusion on a $(3N-1)$-dimensional sphere with radius fixed by the total energy. In the $N\rightarrow\infty$ limit, a finite number of velocity components are shown to evolve independently and according to an Ornstein-Uhlenbeck process.