A Hessian Geometric Structure of Chemical Thermodynamic Systems with Stoichiometric Constraints

TL;DR

This paper introduces a Hessian geometric framework for chemical thermodynamics that characterizes equilibrium states and entropy production without relying on ideal gas assumptions or mass action kinetics, using duality and divergence minimization.

Contribution

It develops a novel Hessian geometric structure for CRNs, deriving equilibrium conditions and entropy measures without traditional assumptions, unifying thermodynamics and information geometry.

Findings

Established a Hessian geometric structure for CRNs.

Derived existence and uniqueness conditions for equilibrium states.

Linked entropy production to Bregman divergence and generalized Kullback-Leibler divergence.

Abstract

We establish a Hessian geometric structure in chemical thermodynamics which describes chemical reaction networks (CRNs) with equilibrium states. In our setup, the ideal gas assumption and mass action kinetics are not required. The existence and uniqueness condition of the equilibrium state is derived by using the Legendre duality inherent to the Hessian structure. The entropy production during a relaxation to the equilibrium state can be evaluated by the Bregman divergence. Furthermore, the equilibrium state is characterized by four distinct minimization problems of the divergence, which are obtained from the generalized Pythagorean theorem originating in the dual flatness. For the ideal gas case, we confirm that our existence and uniqueness condition implies Birch's theorem, and that the entropy production represented by the divergence coincides with the generalized Kullback-Leibler divergence. In addition, under mass action kinetics, our general framework reproduces the local detailed balance condition.