Nonergodicity and growth are compatible for 1D local interaction

Abstract

We present results of Monte Carlo simulation and chaos approximation of a class of Markov processes with a countable or continuous set of states. Each of these states can be written as a finite (finite case) or infinite in both directions (infinite case) sequence of pluses and minuses denoted by ⊕ and ⊖. As continuous time goes on, our sequence undergoes the following three types of local transformations: the first one, called flip, changes any minus into plus and any plus into minus with a rate β; the second, called annihilation, eliminates two neighbor components with a rate α whenever they are in differents states; and the third, called mitosis, doubles any component with a rate γ. All of them occur at any place of the sequence independently. Our simulations and approximations suggest that with appropriate positive values of α, β and γ this process has the following two properties. Growth: In the finite case, as the process goes on, the length of the sequence tends to infinity with a probability which tends to 1 when the length of the initial sequence tends to ∞. Nonergodicity: The infinite process is nonergodic and the finite process keeps most of the time at two extremes, occasionally swinging from one to the other.