Dynamics of the Cellular Automaton Rule 142

Abstract

We investigate dynamics of the cellular automaton Rule 142. This rule possesses additive invariant of the second order, namely it conserves the number of blocks "10." Rule 142 can be alternatively described as an operation on a binary string in which we simultaneously flip all symbols which have dissenting right neighbors. We show that the probability of having a dissenting neighbor can be computed exactly using the fact that the surjective Rule 60 transforms Rule 142 into Rule 226. We also demonstrate that the conservation of the number of 10 blocks implies that these blocks move with speed -1 or stay in the same place, depending on the state of the preceding site. At the density of blocks 10 equal to 0.25, Rule 142 exhibits a phenomenon similar to the jamming transitions occurring in discrete models of traffic flow.