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Probabilistic cellular automata with deterministic updating are quantum systems. We employ the quantum formalism for an investigation of random probabilistic cellular automata, which start with a probability distribution over initial configurations. The properties of the deterministic updating are randomly distributed over space and time. We are interested in a possible continuum limit for a very large number of cells. As an example we consider bits with two colors, moving to the left or right on a linear chain. At randomly distributed scattering points, they change direction and color. A numerical simulation reveals the typical features of quantum systems. We find particular initial probability distributions which reemerge periodically after a certain number of time steps, as produced by the periodic evolution of energy eigenstates in quantum mechanics. Using a description in terms of wave functions allows to introduce statistical observables for momentum and energy. They characterize the probabilistic information without taking definite values for a given bit configuration, with a conceptual status similar to temperature in classical statistical thermal equilibrium. Conservation of energy and momentum are essential ingredients for the understanding of the evolution of our stochastic probabilistic automata. This evolution resembles in some aspects a single Dirac fermion in two dimensions with a random potential.
Kreuzkamp et al. (Thu,) studied this question.