Classical Fisher information for differentiable dynamical systems

TL;DR

This paper introduces a new classical information measure based on Lyapunov vectors to quantify uncertainty in deterministic dynamical systems, linking it to phase space geometry and flow dynamics.

Contribution

It defines a classical Fisher information analogue for deterministic systems using tangent space vectors, extending the concept beyond stochastic contexts.

Findings

Information depends on phase space curvature

Flow speed influences the information measure

Bounds relate to system stability and chaos

Abstract

Fisher information is a lower bound on the uncertainty in the statistical estimation of classical and quantum mechanical parameters. While some deterministic dynamical systems are not subject to random fluctuations, they do still have a form of uncertainty: Infinitesimal perturbations to the initial conditions can grow exponentially in time, a signature of deterministic chaos. As a measure of this uncertainty, we introduce another classical information, specifically for the deterministic dynamics of isolated, closed, or open classical systems not subject to noise. This classical measure of information is defined with Lyapunov vectors in tangent space, making it less akin to the classical Fisher information and more akin to the quantum Fisher information defined with wavevectors in Hilbert space. Our analysis of the local state space structure and linear stability lead to upper and lower bounds on this information, giving it an interpretation as the net stretching action of the flow. Numerical calculations of this information for illustrative mechanical examples show that it depends directly on the phase space curvature and speed of the flow.