Covariant currents and a thermodynamic uncertainty relation on curved manifolds

TL;DR

This paper develops a covariant framework for stochastic currents on curved manifolds, extending thermodynamic uncertainty relations to account for geometry and coordinate invariance in diffusion processes.

Contribution

It introduces a covariant Stratonovich-Langevin equation and extends thermodynamic uncertainty relations to curved spaces with arbitrary coordinates.

Findings

Covariant formulation of stochastic currents on Riemannian manifolds.

Extension of thermodynamic uncertainty relation to curved geometries.

Applicable to diffusion processes in complex geometries.

Abstract

A framework for defining stochastic currents associated with diffusion processes on curved Riemannian manifolds is presented. This is achieved by introducing an overdamped Stratonovich-Langevin equation that remains fully covariant under non-linear transformations of state variables. The approach leads to a covariant extension of the thermodynamic uncertainty relation, describing a trade-off between the total entropy production rate and thermodynamic precision associated with short-time currents in curved spaces and arbitrary coordinate systems.