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Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law

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Abstract

This paper studies a mathematical formalism of nonequilibrium thermodynamics for chemical reaction models with N species, M reactions, and general rate law. We establish a mathematical basis for J. W. Gibbs’ macroscopic chemical thermodynamics under G. N. Lewis’ kinetic law of entire equilibrium (detailed balance in nonlinear chemical kinetics). In doing so, the equilibrium thermodynamics is then naturally generalized to nonequilibrium settings without detailed balance. The kinetic models are represented by a Markovian jumping process. A generalized macroscopic chemical free energy function and its associated balance equation with nonnegative source and sink are the major discoveries. The proof is based on the large deviation principle of this type of Markov processes. A general fluctuation dissipation theorem for stochastic reaction kinetics is also proved. The mathematical theory illustrates how a novel macroscopic dynamic law can emerges from the mesoscopic kinetics in a multi-scale system.

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Acknowledgements

The authors would like to thank Xiao Jin, Tiejun Li for comments and helpful discussions. H. Ge is supported by NSFC (Nos. 21373021 and 11622101), and the 863 program (No. 2015AA020406).

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Authors and Affiliations

  1. Beijing International Center for Mathematical Research (BICMR) and Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing, 100871, People’s Republic of China

    Hao Ge

  2. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195-3925, USA

    Hong Qian

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  1. Hao Ge
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Ge, H., Qian, H. Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law. J Stat Phys 166, 190–209 (2017). https://doi.org/10.1007/s10955-016-1678-6

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