Soliton turbulence in one-dimensional cellular automata

Statistical properties of cellular automata which support simple solitary waves are numerically studied. Under these rules, spatio-temporal patterns are sensitively dependent on the collision processes of solitons, and other elementary excitations such as breathers, kinks, and nuclei. Patterns typically become randomized as collisions proceed. Turbulent states are realized after many collisions. The resulting global patterns can be classified by the types of elementary excitations they contain. In many turbulent states, the spectra reveal a long-range order with a k^-_v anomaly for k ≪ 1. The irreversible process leading to turbulent equilibrium is characterized by the transient spectrum together with the Allan variance. The mean free motion of a soliton in the turbulent states is measured by the mutual information flow, and the information loss is shown to obey an inverse power law. The transient time to reach an attractor grows exponentially with system size, which suggests that in the thermodynamic limit soliton turbulence is not an attractor but rather a transient state.