Second order additive invariants in elementary cellular automata

TL;DR

This paper studies second order additive invariants in elementary cellular automata, revealing universal decay exponents and singularity behaviors similar to first-order invariant rules.

Contribution

It identifies the behavior of second order additive invariants, showing their fundamental diagrams are linear or singular, and proposes a universal decay exponent for invariant-related dynamics.

Findings

Rules with invariants show linear or singular fundamental diagrams.

Singularities occur only in rules with exactly one invariant.

Decay exponent near -1/2 suggests universality across invariant orders.

Abstract

We investigate second order additive invariants in elementary cellular automata rules. Fundamental diagrams of rules which possess additive invariants are either linear or exhibit singularities similar to singularities of rules with first-order invariant. Only rules which have exactly one invariants exhibit singularities. At the singularity, the current decays to its equilibrium value as a power law $t^{\alpha}$, and the value of the exponent $\alpha$ obtained from numerical simulations is very close to -1/2. This is in agreements with values previously reported for number-conserving rules, and leads to a conjecture that regardless of the order of the invariant, exponent $\alpha$ seems to have a universal value of 1/2.