Mesoscopic kinetic basis of macroscopic chemical thermodynamics: A mathematical theory

Gibbs' macroscopic chemical thermodynamics is one of the most important theories in chemistry. Generalizing it to mesoscaled nonequilibrium systems is essential to biophysics. The nonequilibrium stochastic thermodynamics of chemical reaction kinetics suggested a free energy balance equation $d{F}^{\left(\text{meso}\right)}/dt=E_{\text{in}}-e_p$ in which the free energy input rate $E_{\text{in}}$ and dissipation rate $e_p$ are both non-negative, and $E_{\text{in}}\le e_p$. We prove that in the macroscopic limit by merely allowing the molecular numbers to be infinite, the generalized mesoscopic free energy $F^{\left(\text{meso}\right)}$ converges to ${\phi }^{\text{ss}}$, the large deviation rate function for the stationary distributions. This generalized macroscopic free energy ${\phi }^{\text{ss}}$ now satisfies a balance equation $d{\phi }^{\text{ss}}\left(\mathbf{x}\right)/dt=\text{cmf}\left(\mathbf{x}\right)-\sigma \left(\mathbf{x}\right)$, in which $\mathbf{x}$ represents chemical concentration. The chemical motive force $\text{cmf}\left(\mathbf{x}\right)$ and entropy production rate $\sigma \left(\mathbf{x}\right)$ are both non-negative, and $\text{cmf}\left(\mathbf{x}\right)\le \sigma \left(\mathbf{x}\right)$. The balance equation is valid generally in isothermal driven systems and is different from mechanical energy conservation and the first law; it is actually an unknown form of the second law. Consequences of the emergent thermodynamic quantities and equalities are further discussed. The emergent “law” is independent of underlying kinetic details. Our theory provides an example showing how a macroscopic law emerges from a level below.