A graphic formulation of non-isothermal chemical reaction systems and the analysis of detailed balanced networks

In this paper, we provide a graphic formulation of non-isothermal reaction systems and show that a non-isothermal detailed balanced network system converges (locally) asymptotically to the unique equilibrium within the invariant manifold determined by the initial condition. To model thermal effects, the proposed modeling approach extends the classical chemical reaction network by adding two parameters to each direct (reaction) edge, depicting, respectively, the instantaneous internal energy change after the firing of the reaction and the variation of the reaction rate with respect to the temperature. For systems possessing thermodynamic equilibria, our modeling approach provides a compact formulation of the dynamics where reaction topology and thermodynamic information are presented simultaneously. Finally, using this formulation and the Legendre transformation, we show that non-isothermal detailed balanced network systems admit some fundamental properties: dissipativeness, the detailed balancing of each equilibrium, the existence and uniqueness of the equilibrium, and the asymptotic stability of each equilibrium. In general, the analysis and results of this work provide insights into the research of non-isothermal chemical reaction systems.