# Mesoscopic Kinetic Basis of Macroscopic Chemical Thermodynamics: A Mathematical Theory

Source: [Mesoscopic Kinetic Basis of Macroscopic Chemical Thermodynamics: A Mathematical Theory](<https://www.sota2.com/research/paper/68fa34e31e209131ce89cdab>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T20:55:55.065Z

Authors: Hao Ge, Hong Qian

Published: 2016-01-13T08:05:37.000Z

## Abstract

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 $\\varphi^{ss}(\\vx)$ 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)\\varphi^{ss}\[\\vx(t)\]\\le 0$, and (2)$(\\rd/\\rd t)\\varphi^{ss}\[\\vx(t)\]=\\text{cmf}(\\vx)-\\sigma(\\vx)$ where entropy production rate $\\sigma\\ge 0$ 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,$\\varphi^{ss}(\\vx)$ is precisely the Gibbs' function $\\sum\_{i=1}^N x\_i\\big\[\\mu\_i^o+\\ln x\_i\\big\]$ 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 $\\varphi^{ss}(\\vx)$, 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1601.03159>)

[PDF](<https://arxiv.org/pdf/1601.03159.pdf>)