Asymptotic behaviour in the time synchronization model

TL;DR

This paper analyzes the long-term behavior of a particle system with two types, focusing on how initial desynchronization evolves as the total number of particles grows large, with fixed ratios and time.

Contribution

It provides a detailed study of the asymptotic behavior of a two-type particle system under fixed ratios and large particle numbers, highlighting initial desynchronization dynamics.

Findings

Behavior of the system as total particles tend to infinity

Impact of fixed ratios on system dynamics

Insights into initial desynchronization process

Abstract

There are two types $i=1,2$ of particles on the line $R$, with $N_{i}$ particles of type $i$. Each particle of type $i$ moves with constant velocity $v_{i}$. Moreover, any particle of type $i=1,2$ jumps to any particle of type $j=1,2$ with rates $N_{j}^{-1}\alpha_{ij}$. We discuss in details the initial desynchronization of this particle system, namely, we are interested in behaviour of the process when the total number of particles $N_1+N_2$ tends to infinity, the ratio $N_1/N_2$ is constant and the time $t>0$ is fixed.