The Geometry of Thermodynamic Uncertainty Relations in Chemical Reaction Networks

TL;DR

This paper uses Hessian geometry to derive multivariate thermodynamic uncertainty relations for chemical reaction networks, linking geometric structures to thermodynamic and kinetic properties, and clarifying conditions for TUR equalities.

Contribution

It introduces a geometric framework based on Hessian geometry to derive and interpret multivariate thermodynamic uncertainty relations in chemical reaction networks.

Findings

TUR matrices are Riemannian metric tensors.

The current fluctuation tensor is a restriction of the entropy production tensor.

The error term in TUR is characterized as a norm of a flux subspace component.

Abstract

Recently, Hessian geometry - an extension of information geometry - has emerged as a framework to naturally connect the geometries appearing in the theory of chemical reaction networks (CRN) to their inherent thermodynamical and kinetic properties. This framework is used in this letter to derive multivariate thermodynamic uncertainty relations (TUR) for CRN. The matrices featured in the TUR are shown to be representations of Riemmanian metric tensors, whereby one tensor characterizes the pseudo entropy production rate and the other the current fluctuations. It is shown that the latter tensor is a restriction of the former one to a linear subspace in the flux tangent space. Therefore, in addition to clarifying the geometric origin of TUR in CRN, the Hessian geometric setup yields a characterization of the error term in the TUR as the norm of a linear subspace component of the flux vector and thus characterizes the fluxes where TUR become equalities.