Equilibrium and Non-Equilibrium in a Reversible and Conservative Cellular Automaton

Abstract

The main goal of this thesis, relies the dynami s of a reversible and on- servative ellular automaton Q2R model. Q2R is a automaton that runs on a two-dimensional grid of nite size and is reversible in a physi al sense, that is, not only is the automaton rule invertible, but the ba kward rule reads exa tly the same as the forward one. This model is a dynami al variation of the Ising model for ferromagnetism that possesses quite a ri h and omplex dynami s. As expe ted, the Q2R automaton only possesses xed points and periodi orbits and it has been shown that possesses an energy like quantity, and, at least an extra onserved quantity. Although, the dynami s in ludes only xed points and periodi orbits, numeri al simulations show that the system ex- hibits a ferromagneti phase transition in the large system size limit for a well de ned riti al energy. In the present work, we hara terize the on guration spa e, that is om- posed of a huge number of y les with exponentially long periods. More pre- isely, we quantify the probability distribution fun tions of states in terms of the aforementioned invariants. We show that the dynami s of the system in the phase spa e appears to be, depending on the energy, a random walk or a Levy ight. The main ontribution of the present thesis is the appli ation of a oarse- graining approa h that allows to write a oarse-grained master equation, whi h hara terizes equilibrium and non equilibrium statisti al properties, for the Q2R model. Following Ni olis and ollaborators, a oarse-graining approa h is applied to the time series of the total magnetization, leading to a onsistent master equation that governs the ma ros opi irreversible dynami s of the Q2R automata. The methodology is repli ated for various latti e sizes. In the ase of small systems, we show that the master equation leads to a tra table probability transfer matrix of moderate size, whi h provides a master equation for a oarse-grained probability distribution. The method is validated and some expli it examples are dis ussed.