The largest Lyapunov exponent is the standard diagnostic for chaos. It measures the exponential rate at which nearby trajectories diverge — if it's positive, the system is chaotic. Trani, Di Cintio, and Ginolfi tested whether Shannon entropy agrees.
In simple systems, it does. The Hénon-Heiles potential — a textbook two-dimensional chaotic system — produces the same verdict from both measures. Lyapunov says chaotic; entropy says chaotic. The agreement has reinforced the assumption that the Lyapunov exponent is sufficient.
In gravitational N-body systems, the measures diverge. As the number of particles increases, the largest Lyapunov exponent stays constant. Shannon entropy decreases monotonically. By one metric, chaos is unchanged. By the other, the system becomes progressively more ordered.
The mechanism: the Lyapunov exponent is a local measure — it tracks what happens to nearby trajectories in a small patch of phase space. Shannon entropy is a global measure — it quantifies the mixing of the entire phase-space distribution. In gravitational systems with many bodies, local chaos persists (nearby orbits still diverge at the same rate) while global mixing changes character. The phase space develops large-scale structure — filaments, voids, correlated motions — that suppresses global entropy even as local divergence continues.
This means the Lyapunov exponent is blind to a structural feature of many-body gravity. It reports unchanged chaos while the system quietly organizes itself at scales the exponent cannot see. For astrophysicists modeling galaxy formation, stellar clusters, or planetary systems, the local chaos metric may be systematically misleading about the global dynamics.
The through-claim: when two meters disagree, the problem isn't that one is wrong. It's that they're measuring at different scales, and the system has structure at both. The danger is using the local meter and assuming it speaks for the whole — because in a system that organizes at one scale while churning at another, the meter that reports no change is the one missing the story.