Riemannian Geometry of Optimal Driving and Thermodynamic Length and its Application to Chemical Reaction Networks

TL;DR

This paper extends the Riemannian geometric framework for optimal driving of chemical reaction networks to nonequilibrium conditions, providing new bounds on dissipation and insights into thermodynamic length without restrictive assumptions.

Contribution

It generalizes the geometric approach to nonequilibrium systems, introduces a weighted Fisher information metric for chemical concentrations, and derives dissipation bounds along steady state manifolds.

Findings

Dissipation can be bounded using the embedded steady state manifold.

The weighted Fisher information metric characterizes dissipation due to diffusive driving.

Analytic expressions for geodesics and dissipation bounds are provided.

Abstract

It is known that the trajectory of an endoreversibly driven system with minimal dissipation is a geodesic on the equilibrium state space. Thereby, the state space is equipped with the Riemannian metric given by the Hessian of the free energy function, known as Fisher information metric. However, the derivations given until now require both the system and the driving reservoir to be in local equilibrium. In the present work, we rederive the framework for chemical reaction networks and thereby enhance its scope of applicability to the nonequilibrium situation. Moreover, because our results are derived without restrictive assumptions, we are able to discuss phenomena that could not been seen previously. We introduce a suitable weighted Fisher information metric on the space of chemical concentrations and show that it characterizes the dissipation caused by diffusive driving, with arbitrary diffusion rate constants. This allows us to consider driving far from equilibrium. As the main result, we show that the isometric embedding of a steady state manifold into the concentration space yields a lower bound for the dissipation when the system is driven along the manifold. We give an analytic expression for this bound and for the corresponding geodesic, and thereby are able to dissect the contributions from the driving kinetics and from thermodynamics. Finally, we discuss in detail the application to quasi-thermostatic steady states.