Field theory of reaction-diffusion: Law of mass action with an energetic variational approach.

We extend the energetic variational approach so it can be applied to a chemical reaction system with general mass action kinetics. Our approach starts with an energy-dissipation law. We show that the chemical equilibrium is determined by the choice of the free energy and the dynamics of the chemical reaction is determined by the choice of the dissipation. This approach enables us to couple chemical reactions with other effects, such as diffusion and drift in an electric field. As an illustration, we apply our approach to a nonequilibrium reaction-diffusion system in a specific but canonical setup. We show by numerical simulations that the input-output relation of such a system depends on the choice of the dissipation.

[1]  Alexander Mielke,et al.  Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions , 2012 .

[2]  R. Toupin.  ELASTIC MATERIALS WITH COUPLE STRESSES, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS , 1962 .

[3]  R. Jackson,et al.  General mass action kinetics , 1972 .

[4]  Martin Feinberg,et al.  Foundations of Chemical Reaction Network Theory , 2019, Applied Mathematical Sciences.

[5]  Thomas G. Kurtz,et al.  Stochastic Analysis of Biochemical Systems , 2015 .

[6]  I. Prigogine,et al.  Book Review: Modern Thermodynamics: From Heat Engines to Dissipative Structures , 1998 .

[7]  Wen-An Yong.  Conservation-dissipation structure of chemical reaction systems. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  T. Kurtz.  The Relationship between Stochastic and Deterministic Models for Chemical Reactions , 1972 .

[9]  L. Ropolyi,et al.  Analogies between point mechanics and chemical reaction kinetics , 1984 .

[10]  Simo Wu,et al.  Non-Isothermal Electrokinetics: Energetic Variational Approach , 2017, 1710.08031.

[11]  Anna Scotti,et al.  Positivity and Conservation Properties of Some Integration Schemes for Mass Action Kinetics , 2011, SIAM J. Numer. Anal..

[12]  H. Qian.  Stochastic Population Kinetics and Its Underlying Mathematicothermodynamics , 2019, The Dynamics of Biological Systems.

[13]  Mi-Ho Giga,et al.  Variational Modeling and Complex Fluids , 2017 .

[14]  B. L. Clarke.  Stability of Complex Reaction Networks , 2007 .

[15]  J. Wei,et al.  Axiomatic Treatment of Chemical Reaction Systems , 1962 .

[16]  Miroslav Grmela,et al.  GENERIC guide to the multiscale dynamics and thermodynamics , 2018 .

[17]  Oliver Junge,et al.  A Fully Discrete Variational Scheme for Solving Nonlinear Fokker-Planck Equations in Multiple Space Dimensions , 2017, SIAM J. Numer. Anal..

[18]  Stein Shiromoto,et al.  Lyapunov functions , 2012 .

[19]  Chun Liu,et al.  An Introduction of Elastic Complex Fluids: An Energetic Variational Approach , 2009 .

[20]  John William Strutt,et al.  Some General Theorems relating to Vibrations , 1871 .

[21]  James P. Keener,et al.  Mathematical physiology , 1998 .

[22]  Chun Liu,et al.  On Lagrangian schemes for the multidimensional porous medium equations by a discrete energetic variational approach , 2019, 1905.12225.

[23]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[24]  Mark A. Peletier,et al.  Non-equilibrium Thermodynamical Principles for Chemical Reactions with Mass-Action Kinetics , 2015, SIAM J. Appl. Math..

[25]  D Shear,et al.  An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions. , 1967, Journal of theoretical biology.

[26]  Brian J. Edwards,et al.  Thermodynamics of flowing systems : with internal microstructure , 1994 .

[27]  Chun Liu,et al.  On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach , 2019, J. Comput. Phys..

[28]  Yoshikazu Giga,et al.  Handbook of Mathematical Analysis in Mechanics of Viscous Fluids , 2017 .

[29]  Adrian Sandu.  Positive numerical integration methods for chemical kinetic systems , 2001 .

[30]  B. Eisenberg.  Channels as Enzymes: Oxymoron and Tautology , 2011, 1112.2363.

[31]  Carsten Wiuf,et al.  Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks , 2015, Bulletin of mathematical biology.

[32]  Mark A. Peletier,et al.  Variational modelling : energies, gradient flows, and large deviations , 2014, 1402.1990.

[33]  Hong Qian,et al.  Mesoscopic kinetic basis of macroscopic chemical thermodynamics: A mathematical theory. , 2016, Physical review. E.

[34]  J. Carrillo,et al.  A blob method for diffusion , 2017, Calculus of Variations and Partial Differential Equations.

[35]  L. Desvillettes,et al.  Exponential decay toward equilibrium via entropy methods for reaction–diffusion equations , 2006 .

[36]  H. Qian,et al.  Mathematical Formalism of Nonequilibrium Thermodynamics for Nonlinear Chemical Reaction Systems with General Rate Law , 2016, 1604.07115.

[37]  Masao Doi,et al.  Onsager’s variational principle in soft matter , 2011, Journal of physics. Condensed matter : an Institute of Physics journal.

[38]  José A. Carrillo,et al.  A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes , 2017, J. Sci. Comput..

[39]  L. Onsager.  Reciprocal Relations in Irreversible Processes. II. , 1931 .

[40]  D. Bedeaux,et al.  Entropy production in mesoscopic stochastic thermodynamics: nonequilibrium kinetic cycles driven by chemical potentials, temperatures, and mechanical forces , 2016, Journal of physics. Condensed matter : an Institute of Physics journal.

[41]  M. Grmela,et al.  Multiscale Thermo-Dynamics , 2018 .

[42]  B. Perthame.  Parabolic Equations in Biology , 2015 .

[43]  P. Waage,et al.  Studies concerning affinity , 1986 .

[44]  YunKyong Hyon,et al.  Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids. , 2010, The Journal of chemical physics.

[45]  Alexander Mielke,et al.  A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems , 2011 .

[46]  David Jou,et al.  Understanding Non-equilibrium Thermodynamics , 2008 .