Classical Fisher information for differentiable dynamical systems

Area of Science:

• Classical mechanics
• Information theory
• Dynamical systems theory

Background:

• Fisher information provides a lower bound for uncertainty in parameter estimation for classical and quantum systems.
• Deterministic dynamical systems, even without noise, exhibit uncertainty due to exponential growth of initial condition perturbations, a hallmark of chaos.

Purpose of the Study:

• Introduce a novel classical information measure for deterministic dynamics in isolated, closed, or open systems.
• Develop a measure of uncertainty distinct from classical Fisher information, analogous to quantum Fisher information.

Main Methods:

• Define the new classical information measure using Lyapunov vectors in tangent space.
• Analyze local state space structure and linear stability to derive bounds for the information measure.
• Perform numerical calculations on illustrative mechanical systems.

Main Results:

• The new information measure quantifies uncertainty in deterministic chaotic systems.
• It is defined using Lyapunov vectors, similar to how quantum Fisher information uses wavevectors.
• Derived upper and lower bounds interpret the measure as the net stretching action of the flow.
• Numerical results show dependence on phase space curvature and flow speed.

Conclusions:

• The proposed classical information measure offers a new perspective on uncertainty in deterministic dynamics.
• This measure provides insights into the stretching dynamics within the phase space.
• The findings connect information theory concepts to the study of chaotic mechanical systems.