American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020346

Orthogonality of fluxes in general nonlinear reaction networks

D. R. Michiel Renger 1, and  Johannes Zimmer 2,

1. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany
2. 

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Dedicated, in gratitude, to Alex Mielke on the occasion of his 60th birthday

Received  July 2019 Revised  November 2019 Published  May 2020

We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

Keywords: Chemical reactions, large deviations, orthogonality.
Mathematics Subject Classification: Primary: 82C35; Secondary: 82C31, 60J27.
Citation: D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020346
References:
[1]

S. Adams, N. Dirr, M. Peletier and J. Zimmer, Large deviations and gradient flows, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013), 17pp. doi: 10.1098/rsta.2012.0341.  Google Scholar

[2]

D. F. Anderson and T. G. Kurtz, Continuous time {M}arkov chain models for chemical reaction networks, in Design and Analysis of Biomolecular Circuits, Springer, NY, 2011, 3–42. doi: 10.1007/978-1-4419-6766-4_1.  Google Scholar

[3]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Macroscopic fluctuation theory, Rev. Modern Phys., 87 (2015), 593-636.  doi: 10.1103/RevModPhys.87.593.  Google Scholar

[4]

B. Hilder, M. A. Peletier, U. Sharma and O. Tse, An inequality connecting entropy distance, {F}isher Information and large deviations, Stochastic Process. Appl., 130 (2020), 2596-2638.  doi: 10.1016/j.spa.2019.07.012.  Google Scholar

[5]

M. Kaiser, R. L. Jack and J. Zimmer, Canonical structure and orthogonality of forces and currents in irreversible Markov chains, J. Stat. Phys., 170 (2018), 1019-1050.  doi: 10.1007/s10955-018-1986-0.  Google Scholar

[6]

T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probability, 7 (1970), 49-58.  doi: 10.2307/3212147.  Google Scholar

[7]

C. Maes, Frenetic bounds on the entropy production, Phys. Rev. Lett., 119 (2017). doi: 10.1103/PhysRevLett.119.160601.  Google Scholar

[8]

C. Maes and K. Netočný, Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states, Europhys. Lett. EPL, 82 (2008), 6pp. doi: 10.1209/0295-5075/82/30003.  Google Scholar

[9]

A. Mielke, R. I. A. Patterson, M. A. Peletier and D. R. M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM J. Appl. Math., 77 (2017), 1562-1585.  doi: 10.1137/16M1102240.  Google Scholar

[10]

A. Mielke, M. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., 41 (2014), 1293-1327.  doi: 10.1007/s11118-014-9418-5.  Google Scholar

[11]

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2), 91 (1953), 1505-1512.  doi: 10.1103/PhysRev.91.1505.  Google Scholar

[12]

R. I. A. Patterson and D. R. M. Renger, Large deviations of jump process fluxes, Math. Phys. Anal. Geom., 22 (2019), 32pp. doi: 10.1007/s11040-019-9318-4.  Google Scholar

[13]

D. R. M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, J. Stat. Phys., 172 (2018), 1291-1326.  doi: 10.1007/s10955-018-2083-0.  Google Scholar

[14]

D. R. M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy, 20 (2018). doi: 10.3390/e20080596.  Google Scholar

[15]

J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Modern Phys., 48 (1976), 571-585.  doi: 10.1103/RevModPhys.48.571.  Google Scholar

show all references

References:
[1]

S. Adams, N. Dirr, M. Peletier and J. Zimmer, Large deviations and gradient flows, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013), 17pp. doi: 10.1098/rsta.2012.0341.  Google Scholar

[2]

D. F. Anderson and T. G. Kurtz, Continuous time {M}arkov chain models for chemical reaction networks, in Design and Analysis of Biomolecular Circuits, Springer, NY, 2011, 3–42. doi: 10.1007/978-1-4419-6766-4_1.  Google Scholar

[3]

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, Macroscopic fluctuation theory, Rev. Modern Phys., 87 (2015), 593-636.  doi: 10.1103/RevModPhys.87.593.  Google Scholar

[4]

B. Hilder, M. A. Peletier, U. Sharma and O. Tse, An inequality connecting entropy distance, {F}isher Information and large deviations, Stochastic Process. Appl., 130 (2020), 2596-2638.  doi: 10.1016/j.spa.2019.07.012.  Google Scholar

[5]

M. Kaiser, R. L. Jack and J. Zimmer, Canonical structure and orthogonality of forces and currents in irreversible Markov chains, J. Stat. Phys., 170 (2018), 1019-1050.  doi: 10.1007/s10955-018-1986-0.  Google Scholar

[6]

T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probability, 7 (1970), 49-58.  doi: 10.2307/3212147.  Google Scholar

[7]

C. Maes, Frenetic bounds on the entropy production, Phys. Rev. Lett., 119 (2017). doi: 10.1103/PhysRevLett.119.160601.  Google Scholar

[8]

C. Maes and K. Netočný, Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states, Europhys. Lett. EPL, 82 (2008), 6pp. doi: 10.1209/0295-5075/82/30003.  Google Scholar

[9]

A. Mielke, R. I. A. Patterson, M. A. Peletier and D. R. M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM J. Appl. Math., 77 (2017), 1562-1585.  doi: 10.1137/16M1102240.  Google Scholar

[10]

A. Mielke, M. A. Peletier and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal., 41 (2014), 1293-1327.  doi: 10.1007/s11118-014-9418-5.  Google Scholar

[11]

L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. (2), 91 (1953), 1505-1512.  doi: 10.1103/PhysRev.91.1505.  Google Scholar

[12]

R. I. A. Patterson and D. R. M. Renger, Large deviations of jump process fluxes, Math. Phys. Anal. Geom., 22 (2019), 32pp. doi: 10.1007/s11040-019-9318-4.  Google Scholar

[13]

D. R. M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, J. Stat. Phys., 172 (2018), 1291-1326.  doi: 10.1007/s10955-018-2083-0.  Google Scholar

[14]

D. R. M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy, 20 (2018). doi: 10.3390/e20080596.  Google Scholar

[15]

J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Modern Phys., 48 (1976), 571-585.  doi: 10.1103/RevModPhys.48.571.  Google Scholar

[1]

Congming Li, Eric S. Wright. Modeling chemical reactions in rivers: A three component reaction. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 377-384. doi: 10.3934/dcds.2001.7.373
[2]

Arno F. Münster. Simulation of stationary chemical patterns and waves in ionic reactions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 35-46. doi: 10.3934/dcdsb.2002.2.35
[3]

Marzia Bisi, Maria Groppi, Giampiero Spiga. Flame structure from a kinetic model for chemical reactions. Kinetic & Related Models, 2010, 3 (1) : 17-34. doi: 10.3934/krm.2010.3.17
[4]

Marzia Bisi, Giampiero Spiga. On a kinetic BGK model for slow chemical reactions. Kinetic & Related Models, 2011, 4 (1) : 153-167. doi: 10.3934/krm.2011.4.153
[5]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
[6]

Wenxiong Chen, Congming Li, Eric S. Wright. On a nonlinear parabolic system-modeling chemical reactions in rivers. Communications on Pure & Applied Analysis, 2005, 4 (4) : 889-899. doi: 10.3934/cpaa.2005.4.889
[7]

Julien Coatléven, Claudio Altafini. A kinetic mechanism inducing oscillations in simple chemical reactions networks. Mathematical Biosciences & Engineering, 2010, 7 (2) : 301-312. doi: 10.3934/mbe.2010.7.301
[8]

Andrea Picco, Lamberto Rondoni. Boltzmann maps for networks of chemical reactions and the multi-stability problem. Networks & Heterogeneous Media, 2009, 4 (3) : 501-526. doi: 10.3934/nhm.2009.4.501
[9]

Steinar Evje, Aksel Hiorth, Merete V. Madland, Reidar I. Korsnes. A mathematical model relevant for weakening of chalk reservoirs due to chemical reactions. Networks & Heterogeneous Media, 2009, 4 (4) : 755-788. doi: 10.3934/nhm.2009.4.755
[10]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[11]

Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881
[12]

Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
[13]

Thomas Bogenschütz, Achim Doebler. Large deviations in expanding random dynamical systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 805-812. doi: 10.3934/dcds.1999.5.805
[14]

Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327
[15]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
[16]

Vesselin Petkov, Luchezar Stoyanov. Spectral estimates for Ruelle operators with two parameters and sharp large deviations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6391-6417. doi: 10.3934/dcds.2019277
[17]

Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245
[18]

Alexander Veretennikov. On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 523-549. doi: 10.3934/dcdsb.2013.18.523
[19]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
[20]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (39)
  • HTML views (212)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences