Information Geometric Inequalities of Chemical Thermodynamics

TL;DR

This paper establishes a novel connection between chemical thermodynamics and information geometry, deriving inequalities that serve as bounds on chemical fluctuations and reaction rates, applicable to various chemical systems.

Contribution

It introduces information-geometric inequalities as generalizations of the Cramér-Rao bound for chemical reaction networks, including non-normalized and oscillatory systems.

Findings

Derived inequalities provide bounds on the rate of change of Gibbs free energy.

Established a trade-off relation between speed and time on concentration distribution manifolds.

Applicable to both closed and open chemical reaction networks.

Abstract

We study a connection between chemical thermodynamics and information geometry. We clarify a relation between the Gibbs free energy of an ideal dilute solution and an information-geometric quantity called an $f$-divergence. From this relation, we derive information-geometric inequalities that give a speed limit for a changing rate of the Gibbs free energy and a general bound of chemical fluctuations. These information-geometric inequalities can be regarded as generalizations of the Cram\'{e}r--Rao inequality for chemical reaction networks described by rate equations, where unnormalized concentration distributions are of importance rather than probability distributions. They hold true for damped oscillatory reaction networks and systems where the total concentration is not conserved so that the distribution cannot be normalized. We also formulate a trade-off relation between speed and time on a manifold of concentration distribution by using the geometrical structure induced by the $f$-divergence. Our results apply to both closed and open chemical reaction networks, thus they are widely useful for thermodynamic analysis of chemical systems from the viewpoint of information geometry.