Dynamic aspects of the flip-annihilation process

• Published in: Journal of mathematical physics 2020-05, Vol.61 (5)
• Main Authors: Costa, L. T., Ramos, A. D.
• Format: Article
• Language: English
• Subjects: Computer simulation; Ergodic processes; Physics; Upper bounds
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.5119080

A one-dimensional interacting particle system is revisited. It has discrete time, and its components are located in the set of integers. These components can disappear in the functioning process. Each component assumes two possible states, called plus and minus, and interacts at every time step only with its nearest neighbors. The following two transformations happen: The first one is called flip, under its action, a component in state minus turns into a plus with probability β. The second one is called annihilation, under its action, whenever a component in state plus is a left neighbor of a component in state minus, both components disappear with probability α. Let us consider a set of initial measures to the process. For these measures, we show the upper bound for the mean time of convergence, which is a function of the initial measure. Moreover, we obtain the upper bound to the mean quantity of minuses on the process in each time step. Considering the initial measure concentrated at the configuration whose components are in the state minus, we improved a well-known result that the process is non-ergodic when β < α2/250. Now, we are able to offer non-ergodicity when β < 9α2/1000. We also established new conditions to the ergodicity of the process. Finally, we performed some Monte Carlo simulations for this process.