Speed limits on classical chaos

TL;DR

This paper derives a classical uncertainty relation that limits the speed of divergence in chaotic systems, linking phase space geometry with information theory through Fisher information and Lyapunov vectors.

Contribution

It introduces a classical speed limit based on Fisher information and Lyapunov vectors, extending quantum uncertainty concepts to classical chaos.

Findings

Establishes a classical speed limit for local observables in chaotic systems.

Connects phase space geometry with information measures in classical dynamics.

Applicable to open, closed, conservative, and dissipative systems.

Abstract

Uncertainty in the initial conditions of dynamical systems can cause exponentially fast divergence of trajectories, a signature of deterministic chaos. Here, we derive a classical uncertainty relation that sets a speed limit on the rates of local observables underlying this behavior. For systems with a time-invariant stability matrix, this general speed limit simplifies to classical analogues of the Mandelstam-Tamm versions of the time-energy uncertainty relation. This classical bound derives from our definition of Fisher information in terms of Lyapunov vectors on tangent space, analogous to the quantum Fisher information defined in terms of wavevectors on Hilbert space. This information measures fluctuations in local stability of the state space and sets a lower bound on the time of classical, dynamical systems to evolve between two distinguishable states. The bounds it sets apply to systems that are open or closed, conservative or dissipative, actively driven or passively evolving, and directly connect the geometries of phase space and information.