Kinetic Derivation of the Hessian Geometric Structure in Chemical Reaction Systems

TL;DR

This paper reveals the Hessian geometric structure underlying chemical reaction systems, connecting kinetics, thermodynamics, and complex balanced states, providing a unified geometric framework for analyzing chemical dynamics.

Contribution

It introduces a Hessian geometric framework derived from chemical kinetics assumptions, linking thermodynamics and nonequilibrium steady states in chemical systems.

Findings

Hessian geometry underpins chemical reaction systems.

Equilibrium and complex balanced states form toric varieties.

The framework extends to nonequilibrium steady states.

Abstract

The theory of chemical kinetics form the basis to describe the dynamics of chemical systems. Owing to physical and thermodynamic constraints, chemical reaction systems possess various structures, which can be utilized to characterize important physical properties of the systems. In this work, we reveal the Hessian geometry which underlies chemical reaction systems and demonstrate how it originates from the interplay of stoichiometric and thermodynamic constraints. Our derivation is based on kinetics, we assume the law of mass action and characterize the equilibrium states by the detailed balance condition. The obtained geometric structure is then related to thermodynamics via the Hessian geometry appearing in a pure thermodynamic derivation. We demonstrate, based on the fact that both equilibrium and complex balanced states form toric varieties, how the Hessian geometric framework can be extended to nonequilibrium complex balanced steady states. We conclude that Hessian geometry provides a natural framework to capture the thermodynamic aspects of chemical reaction kinetics.