Stochastic dynamics in systems with unidirectional delay coupling: two-state description.

We study stochastic dynamics of two-state particles coupled unidirectionally with delay. We give exact results for the stationary distribution function p st and the time correlation function (TCF) when the system consists of two (N=2) and three (N=3) particles. Based on these results, effects of delay are discussed and compared with the N=1 case, studied by Tsimring and Pikovsky [Phys. Rev. Lett. 87, 250602 (2001)]. Next, we consider the general N-particle system, for which we give exact expressions for p st and the TCF, which are inferred based on the N=2 and N=3 solutions and then confirmed via detailed arguments. It is pointed out that the stationary state is mapped to Ising spin model with ferro(antiferro)magnetic interaction when delay feedback is positive (negative).