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Mesoscopic Kinetic Basis of Macroscopic Chemical Thermodynamics: A Mathematical Theory

From a mathematical model that describes a complex chemical kinetic system of $N$ species and $M$ elementrary reactions in a rapidly stirred vessel of size $V$ as a Markov process, we show that a macroscopic chemical thermodynamics emerges as $V\rightarrow\infty$. The theory is applicable to linear and nonlinear reactions, closed systems reaching chemical equilibrium, or open, driven systems approaching to nonequilibrium steady states. A generalized mesoscopic free energy gives rise to a macroscopic chemical energy function $φ^{ss}(\vx)$ where $\vx=(x_1,\cdots,x_N)$ are the concentrations of the $N$ chemical species. The macroscopic chemical dynamics $\vx(t)$ satisfies two emergent laws: (1) $(\rd/\rd t)φ^{ss}[\vx(t)]\le 0$, and (2)$(\rd/\rd t)φ^{ss}[\vx(t)]=\text{cmf}(\vx)-σ(\vx)$ where entropy production rate $σ\ge 0$ represents the sink for the chemical energy, and chemical motive force $\text{cmf}\ge 0$ is non-zero if the system is driven under a sustained nonequilibrium chemostat. For systems with detailed balance $\text{cmf}=0$, and if one assumes the law of mass action,$φ^{ss}(\vx)$ is precisely the Gibbs' function $\sum_{i=1}^N x_i\big[μ_i^o+\ln x_i\big]$ for ideal solutions. For a class of kinetic systems called complex balanced, which include many nonlinear systems as well as many simple open, driven chemical systems, the $φ^{ss}(\vx)$, with global minimum at $\vx^*$, has the generic form $\sum_{i=1}^N x_i\big[\ln(x_i/x_i^*)-x_i+x_i^*\big]$,which has been known in chemical kinetic literature.Macroscopic emergent "laws" are independent of the details of the underlying kinetics. This theory provides a concrete example from chemistry showing how a dynamic macroscopic law can emerge from the kinetics at a level below.