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We establish a Hessian geometric structure in chemical thermodynamics whichdescribes chemical reaction networks (CRNs) with equilibrium states. In oursetup, the ideal gas assumption and mass action kinetics are not required. Theexistence and uniqueness condition of the equilibrium state is derived by usingthe Legendre duality inherent to the Hessian structure. The entropy productionduring a relaxation to the equilibrium state can be evaluated by the Bregmandivergence. Furthermore, the equilibrium state is characterized by fourdistinct minimization problems of the divergence, which are obtained from thegeneralized Pythagorean theorem originating in the dual flatness. For the idealgas case, we confirm that our existence and uniqueness condition impliesBirch's theorem, and that the entropy production represented by the divergencecoincides with the generalized Kullback-Leibler divergence. In addition, undermass action kinetics, our general framework reproduces the local detailedbalance condition.

A Hessian Geometric Structure of Chemical Thermodynamic Systems with Stoichiometric Constraints