Invariant Measures and Convergence Properties for Cellular Automaton 184 and Related Processes

Abstract

Our results concern long time limit properties of a deterministic dynamics that is common for a wide class of processes that have been studied so far during at least last two decades. The most widely known process from this class is a cellular automaton that acquired number 184 in the classification of S. Wolfram. This CA 184 is being intensively used to model vehicular traffic. However, our results are mainly derived with help of another process that offers a helpful insight into the studied dynamics, it is a so-called Ballistic Annihilation Model (abbreviated by BA). BA is a model for chemical reaction A+B → inert. In BA, A and B-type particles move in opposite directions with velocities 1 and -1, respectively, and annihilate upon collisions. Certain results concerning BA and CA 184 are also formulated in terms of another process known as a Model of Surface Growth (SG, for short); the surface shape in this process behaves as the integrated profile of particle distribution in CA 184.

Our results are as follows. First, we characterize the invariant measures of the dynamics in interest. The bulk of our effort is devoted to the characterization of those of them that are not translation invariant; we call them phase separating invariant measures. In the case of BA, such measures are concentrated on the configurations consisting of two converging infinite blocks of (not necessarily adjacent) particles. In the case of CA 184, a phase separating measure describes the transition from free traffic phase to jammed phase. We also analyze domains of attraction of invariant measures and rates of convergence to them. This analysis then allows us to express the long time limit of particle current in CA 184 as a function of certain characteristics of its initial distribution, when it is translation invariant. This expression has been used in a companion paper (V. Belitsky, J. Krug, E. J. Neves and G. Schütz, A cellular automaton model for two-lane traffic, J. Stat. phys.103(5/6):945--971 (2001)) to show the enhancement of cars’ current caused by the possibility of lane changes in a model of traffic on a two-lane highway that was created by putting two CA 184’s in parallel. Our other results concern hydrodynamic limits of BA and CA 184. We prove that if the integrated profile of initial particle configuration of BA or CA 184 converges, as n → ∞, to some stochastic process W(x), x ∈ R, when being re-scaled by n ^-1 along x-axis and by c _n ^-1 along y-axis for some sequence c _n , then the integrated profile of particle configuration at time n under the same re-scaling, will converge, as time → ∞, to the local moving minimum of the process W(·), that is, to the process W ^min(·) defined by W ^min(x):=min{W(y) :x-1≤ y≤ x+1}. This hydrodynamic limit is then interpreted in terms of the limiting shape of surface in SG.