Classical Geometric Fluctuation Relations

TL;DR

This paper introduces geometric fluctuation relations based on stochastic Fisher Information, revealing new insights into nonequilibrium systems through a geometric and trajectory-dependent framework, supported by tests on two models.

Contribution

It uncovers geometric fluctuation relations using stochastic Fisher Information and explores trajectory-dependent uncertainty relations in nonequilibrium systems.

Findings

Derived fluctuation relations with geometric interpretation.

Established a stochastic length in entropy space.

Validated relations with two nonequilibrium models.

Abstract

Fisher Information (FI) is a quantity ubiquitously measured in such varied areas like metrology, machine learning, and biological complexity. Mathematically, it represents a lower bound in the variance of unknown parameters that are related to the distributions one has access, and a metric for probability manifolds. A stochastic analogous of the Fisher Information, dubbed stochastic Fisher Information was recently introduced in the literature by some of us. By exploring the probability distributions of the Stochastic Fisher Information (SFI), we uncover two fluctuation relations with an inherent geometric nature, as the SFI acts as a single nonequilibrium trajectory metric. The geometric nature of these relations is expressed through a stochastic length in entropy space derived from the system entropy associated with a nonequilibrium trajectory. We also explore the possibility of trajectory-dependent uncertainty relations linked to the SFI with time as a parameter. Finally, we test our geometric fluctuation relations using two nonequilibrium models.