Abstract
On networks representing probability currents between states of a system, we generalize Schnakenberg’s theory of nonequilibrium observables to nonsteady states, with the introduction of a new set of macroscopic observables that, for planar graphs, are related by a duality. We apply this duality to the linear regime, obtaining a dual proposition for the minimum entropy production principle, and to discrete electromagnetism, finding that it exchanges fields with sources. We interpret duality as reversing the role of system and environment, and discuss generalization to nonplanar graphs. The results are based on two theorems regarding the representation of bilinear and quadratic forms over the edge vector space of an oriented graph in terms of observables associated to cycles and cocycles.
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References
Schnakenberg, J.: Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys. 48, 571 (1976)
Hill, T.L.: Free Energy Transduction and Biochemical Cycle Kinetics. Dover, New York (2005)
Schnakenberg, J.: Thermodynamic Network Analysis of Biological Systems. Springer, Berlin (1977)
Polettini, M., Esposito, M.: Irreversible thermodynamics of open chemical networks I: emergent cycles and broken conservation laws. J. Chem. Phys. 141, 024117 (2014)
Andrieux, D., Gaspard, P.: Fluctuation theorem and Onsager reciprocity relations. J. Chem. Phys. 121, 6167 (2004)
Andrieux, D., Gaspard, P.: Fluctuation theorem and mesoscopic chemical clocks. J. Chem. Phys. 128, 154506 (2008)
Liepelt, S., Lipowsky, R.: Steady-state balance conditions for molecular motor cycles and stochastic nonequilibrium processes. Europhys. Lett. 77, 50002 (2007)
Liepelt, S., Lipowsky, R.: Kinesin’s network of chemomechanical motor cycles. Phys. Rev. Lett. 98, 258102 (2007)
Faggionato, A., Di Pietro, D.: Gallavotti-cohen-type symmetry related to cycle decompositions for Markov chains and biochemical applications. J. Stat. Phys. 143, 11 (2011)
Andrieux, D., Gaspard, P.: A fluctuation theorem for currents and non-linear response coefficients. J. Stat. Mech. P01011 (2006)
Andrieux, D., Gaspard, P.: Fluctuation theorem for currents and Schnakenberg network theory. J. Stat. Phys. 127, 107 (2007)
Qian, H., Qian, M.: Pumped biochemical reactions, nonequilibrium circulation, and stochastic resonance. Phys. Rev. Lett. 84, 2271 (2000)
Qian, H.: Cycle kinetics, steady state thermodynamics and motors—a paradigm for living matter physics. J. Phys. 17, S3783 (2005)
Jiang, D.-J., Qian, M., Qian, M.-P.: Mathematical Theory of Nonequilibrium Steady States. Springer, Berlin (2004)
Kalpazidou, S.: Cycle Representations of Markov Processes. Springer, New York (1995)
Polettini, M.: Nonequilibrium thermodynamics as a gauge theory. Eur. Phys. Lett. 97, 30003 (2012)
Zia, R.K.P., Schmittmann, B.: A possible classification of nonequilibrium steady states. J. Phys. A 39, L407 (2006)
Seifert, U.: Stochastic thermodynamics: principles and perspectives. Eur. Phys. J. B 64, 423 (2008)
Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1974)
Nakanishi, N.: Graph Theory and Feynman Integrals. Gordon and Breach, New York (1971)
Brunnemann, J., Rideout, D.: Oriented matroids-combinatorial structures underlying loop quantum gravity. Classical and Quantum Gravity 27, 205008 (2010)
Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Surveys in Combinatorics 2005, London Math. Soc. Lecture Note Ser. vol. 327, p. 173. Cambridge University Press, Cambridge (2005)
Iyer, T.S.K.V.: Circuit Theory. Tata Mac Graw-Hill, New Dehli (2006)
Bjorken, J., Drell, S.: Relativistic Quantum Fields. McGraw-Hill, New York (1965)
David, F., Tuan, R.H.: A duality property of planar Feynman diagrams. Phys. Lett. B 158, 435 (1985)
Wu, F.Y.: The potts model. Rev. Mod. Phys. 54, 235 (1982)
Polettini, M.: Macroscopic constraints for the minimum entropy production principle. Phys. Rev. E 84, 051117 (2011)
Klein, M., Meijer, P.H.E.: Principle of minimum entropy production. Phys. Rev. 96, 250 (1954)
Timm, C.: Gauge theory for the rate equations: electrodynamics on a network. Phys. Rev. Lett. 98, 070604 (2007)
McKee, T.A.: The logic of graph-theoretic duality. Am. Math. Mon. 92(7), 457 (1985)
Esposito, M., van den Broeck, C.: The three faces of the second law: I. Master equation formulation. Phys. Rev. E 82, 011143 (2010)
Hinrichsen, H., Gogolin, C., Janotta, P.: Non-equilibrium dynamics, thermalization and entropy production. J. Phys. 297, 012011 (2011)
Desburn, M., Kanso, E., Tong, Y.: Discrete differential forms for computational modeling. In: Grinspun, E., Schroder, P., Desbrun, M. (eds.) Discrete Differential Geometry. ACM SIGGRAPH, Boston (2006)
Shai, O., Pennock, G.R.: Extension of graph theory to the duality between static systems and mechanisms. J. Mech. Des. 128(1), 179 (2006)
Shai, O., Pennock, G.R.: A study of the duality between planar kinematics and statics. J. Mech. Des. 128(3), 587 (2006)
Acknowledgements
The author warmly thanks A. Maritan and D. Andrieux for discussion, M. Esposito and M. Dalmonte for helping out with the first drafts. The research was partly supported by the National Research Fund Luxembourg in the frame of the AFR Postdoc Grant 5856127.
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Polettini, M. (2015). System/Environment Duality of Nonequilibrium Network Observables. In: Mugnolo, D. (eds) Mathematical Technology of Networks. Springer Proceedings in Mathematics & Statistics, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-319-16619-3_13
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DOI: https://doi.org/10.1007/978-3-319-16619-3_13
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