Fisher information and Hamilton's canonical equations

TL;DR

The paper demonstrates that Fisher information in Gibbs' canonical distributions reflects the structure of classical mechanics and relates to temperature and degrees of freedom, revealing a universal form involving forces and accelerations.

Contribution

It establishes a connection between Fisher information and Hamiltonian mechanics, showing a universal form valid for all systems with Hamiltonian T+V and exploring its implications.

Findings

Fisher information incorporates features of classical mechanics.

Fisher information per degree of freedom is proportional to inverse temperature.

Equipartition of Fisher information holds in both linear and certain nonlinear systems.

Abstract

We show that the mathematical form of the information measure of Fisher's I for a Gibbs' canonical probability distribution (the most important one in statistical mechanics) incorporates important features of the intrinsic structure of classical mechanics and has a universal form in terms of "forces" and "accelerations", i.e., one that is valid for all Hamiltonian of the form T+V. If the system of differential equations associated to Hamilton's canonical equations of motion is linear, one can easily ascertain that the Fisher information per degree of freedom is proportional to the inverse temperature and to the number of these degrees. This equipartition of I is also seen to hold in a simple example involving a non-linear system of differential equations.