Asymptotic behaviour in the time synchronization model

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.

16 pages; first published in Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, AMS Translations - Series 2: Advances in the Mathematical Sciences (2006), Vol.217, pp.101-115; small corrections in text of the paper; bibliography updates