Time scales in large systems of Brownian particles with stochastic synchronization

TL;DR

This paper investigates the asymptotic behavior of large systems of Brownian particles with stochastic synchronization, revealing three distinct time scales as both the number of particles and time tend to infinity.

Contribution

It introduces a new analysis of large Brownian particle systems with random synchronization times, identifying three different asymptotic time scales.

Findings

Identification of three distinct time scales for system behavior

Asymptotic properties characterized as system size and time grow

Independence assumption between Brownian motions and synchronization times

Abstract

We consider a system $x(t)=(x_{1}(t),...,x_{N}(t))$ consisting of $N$ Brownian particles with synchronizing interaction between them occurring at random time moments $\{\tau_{n}\}_{n=1}^{\infty}$. Under assumption that the free Brownian motions and the sequence $\{\tau_{n}\}_{n=1}^{\infty}$ are independent we study asymptotic properties of the system when both the dimension~$N$ and the time~$t$ go to infinity. We find three time scales $t=t(N)$ of qualitatively different behavior of the system.