Phase transitions in the time synchronization model

Abstract

We continue the study of the time synchronization model from arXiv:1201.2141 . 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 find phase transitions in the clusterization (synchronization) behaviour of this system of particles on different time scales $t=t(N)$ relative to $N=N_1+N_2$.

10 pages; first published in Theory Probab. Appl. 50, pp. 134-141 (2006); small corrections in text of the paper; bibliography updates