Stochastic Synchronization in a Large System of Identical Particles

We consider a basic stochastic particle system consisting of $N$ identical particles with isotropic $k$-particle synchronization, ${k\ge 2}$. In the limit when both the number of particles $N$ and the time $t=t(N)$ grow to infinity we study an asymptotic behavior of a coordinate spread of the particle system. We describe three time stages of $t(N)$ for which a qualitative behavior of the system is completely different. Moreover, we discuss the case when a spread of the initial configuration depends on $N$ and increases to infinity as $N\to\infty$.