Butterfly Effect and Spatial Structure of Information Spreading in a Chaotic Cellular Automaton

TL;DR

This paper introduces a classical measure of local information spreading in a chaotic cellular automaton, revealing scaling laws, butterfly velocity, and Lyapunov exponents, providing new insights into classical chaos and information dynamics.

Contribution

It develops an analytic framework for the decorrelator in a stochastic cellular automaton, including explicit formulas for butterfly velocity and Lyapunov exponents, bridging classical and quantum chaos concepts.

Findings

Derived scaling form of the decorrelator with velocity-dependent Lyapunov exponent.

Obtained analytic expressions for butterfly velocity and scaling exponent.

Established the decorrelator as a unifying diagnostic for information spreading.

Abstract

Inspired by recent developments in the study of chaos in many-body systems, we construct a measure of local information spreading for a stochastic Cellular Automaton in the form of a spatiotemporally resolved Hamming distance. This decorrelator is a classical version of an Out-of-Time-Order Correlator studied in the context of quantum many-body systems. Focusing on the one-dimensional Kauffman Cellular Automaton, we extract the scaling form of our decorrelator with an associated butterfly velocity $v_b$ and a velocity-dependent Lyapunov exponent $\lambda(v)$. The existence of the latter is not a given in a discrete classical system. Second, we account for the behaviour of the decorrelator in a framework based solely on the boundary of the information spreading, including an effective boundary random walk model yielding the full functional form of the decorrelator. In particular, we obtain analytic results for $v_b$ and the exponent $\beta$ in the scaling ansatz $\lambda(v) \sim \mu (v - v_b)^\beta$, which is usually only obtained numerically. Finally, a full scaling collapse establishes the decorrelator as a unifying diagnostic of information spreading.