A model of particles movement on a discrete contour

We study an exclusive process on a circle. In this paper, we study a discrete closed contour, containing N cells and M particles. Each particle occupies a cell at every time. There is not more than one particle in each cell at every moment. At every time t=0,1, 2, … , each particle tries to move onto a cell forward with probability p, this particle tries to move back with probability q, and the particle does not try to move with probability s, p+q+s=1. Under assumptions that q=0, the system of this type was considered by M. Kanai et. al. As it follows from results of these authors, in the case q=0, the process is time reversible, i.e., in the stationary state, the behavior of process does not change if the direction of time-axis is changed. The ergodic properties of some more general exclusive process were studied by M. Blank but, in the case 0<p, q<1, values of steady probabilities have not been found. Under the assumptions that M=2, N=2, s=0, we have obtained a formula for the average velocity of particles and the particle transitions intensity. In this paper, under assumptions that M=2, it has been proved the following. The process is time-reversible if M=2, N is even and s=0. The process is not time-reversible if M=2, s>0, or N is odd and s=0. We have proved that the process can be non-reversible if M ≥ 3, s=0.