Particlelike structures and their interactions in spatiotemporal patterns generated by one-dimensional deterministic cellular-automaton rules

Abstract

Configurations generated by the evolution of some one-dimensional cellular automata may be viewed, after many time steps, as particlelike structures evolving in a regular background. A classification of the most frequently observed ``particles'' is proposed according to their specific behavior. The simplest---a straightforward generalization of the ``kinks'' in range-1 Rule 18 earlier studied by Grassberger [Phys. Rev. A 28, 3666 (1983)]---exhibit a diffusive motion and annihilate according to simple processes. Others have, in contrast, constant (positive, negative, or zero) velocities. The ``collision'' of particles with different velocities leads to some ``reactions'' in which some particles are annihilated and others are created. A detailed description of such ``reactions'' sheds new light on the large-time behavior of range-1 Rule 54 with a very slow decrease of the particle number, as ${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$ (\ensuremath{\gamma}\ensuremath{\simeq}0.15). More ``exotic'' behaviors are sometimes observed. Some particlelike structures radiate other ``particles.'' Some ``particles'' combine to generate a perturbation whose space extension increases with time and can be annihilated through the interactions with other ``particles.'' These different behaviors could lead to a more precise classification of cellular-automaton rules.