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Reversibility and entropy production of inhomogeneous Markov chains

Part of: Markov processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  14 July 2016

Hao Ge ,
Da-Quan Jiang  and
Min Qian
Hao Ge*
Affiliation:
Peking University
Da-Quan Jiang*
Affiliation:
Peking University
Min Qian*
Affiliation:
Peking University
*
∗Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
∗Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
∗Postal address: LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China.
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Abstract

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In this paper we introduce the concepts of instantaneous reversibility and instantaneous entropy production rate for inhomogeneous Markov chains with denumerable state spaces. The following statements are proved to be equivalent: the inhomogeneous Markov chain is instantaneously reversible; it is in detailed balance; its entropy production rate vanishes. In particular, for a time-periodic birth-death chain, which can be regarded as a simple version of a physical model (Brownian motors), we prove that its rotation number is 0 when it is instantaneously reversible or periodically reversible. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors can occur only in nonequilibrium and irreversible systems.

Keywords

Brownian motorinhomogeneous Markov chaininstantaneous reversibilityinstantaneous entropy productionrotation number

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) 82C70: Transport processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1028 - 1043
Copyright
© Applied Probability Trust 2006 

References

Arizmendi, C. M. and Family, F. (1999). Algorithmic complexity and efficiency of a ratchet. Physica A 269, 285–292.CrossRefGoogle Scholar
Astumian, R. D. (1997). Thermodynamics and kinetics of a Brownian motor. Science 276, 917–922.CrossRefGoogle ScholarPubMed
Astumian, R. D. and Bier, M. (1994). Fluctuation driven ratchets: molecular motors. Phys. Rev. Lett. 72, 1766–1769.CrossRefGoogle ScholarPubMed
Berdichevsky, V. and Gitterman, M. (1998). Stochastic resonance and ratchets - new manifestations. Physica A 249, 88–95.CrossRefGoogle Scholar
Bier, M. (1997). Brownian ratchets in physics and biology. Contemp. Phys. 38, 371–379.CrossRefGoogle Scholar
Dollard, J. D. and Friedman, C. N. (1979). Product Integration with Applications to Differential Equations (Encyclopaedia Math. Appl. 10). Addison-Wesley, Reading, MA.Google Scholar
Elston, T. C. and Doering, C. R. (1996). Numerical and analytical studies of nonequilibrium fluctuation-induced transport processes. J. Statist. Phys. 83, 359–383.CrossRefGoogle Scholar
Gammaitoni, L., Hänggi, P., Jung, P. and Marchesoni, F. (1998). Stochastic resonance. Rev. Mod. Phys. 70, 223–287.CrossRefGoogle Scholar
Ge, H., Jiang, D. Q. and Qian, M. (2006). A simple discrete model of Brownian motors: time-periodic Markov chains. J. Statist. Phys. 123, 831–859.CrossRefGoogle Scholar
Hasegawa, H. (1976). On the construction of a time-reversed Markoff process. Progress Theoret. Phys. 55, 90–105.CrossRefGoogle Scholar
Hasegawa, H. (1976). Variational principle for non-equilibrium states and the Onsager–Machlup formula. Progress Theoret. Phys. 56, 44–60.CrossRefGoogle Scholar
Hasegawa, H. (1977). Thermodynamic properties of non-equilibrium states subject to Fokker–Planck equations. Progress Theoret. Phys. 57, 1523–1537.CrossRefGoogle Scholar
Hasegawa, H. (1977). Variational approach in studies with Fokker–Planck equations. Progress Theoret. Phys. 58, 128–146.CrossRefGoogle Scholar
Hu, D. H. (1983). Markov Process with Countable State Space. Wuhan University Press (in Chinese).Google Scholar
Jiang, D. Q., Qian, M. and Qian, M. P. (2004). Mathematical Theory of Nonequilibrium Steady States. On the Frontier of Probability and Dynamical Systems (Lecture Notes Math. 1833). Springer, Berlin.CrossRefGoogle Scholar
Jülicher, F., Ajdari, A. and Prost, J. (1997). Modeling molecular motors. Rev. Mod. Phys. 69, 1269–1281.CrossRefGoogle Scholar
Lin, X., Zhang, H. J. and Hou, Z. T. (2000). The property of transition matrix of nonhomogeneous Markov chains. J. Changsha Railway Univ. 18, 86–90 (in Chinese).Google Scholar
Magnasco, M. O. (1993). Forced thermal ratchets. Phys. Rev. Lett. 71, 1477–1481.CrossRefGoogle ScholarPubMed
Magnasco, M. O. (1994). Molecular combustion motors. Phys. Rev. Lett. 72, 2656–2659.CrossRefGoogle ScholarPubMed
Nicolis, G. and Prigogine, I. (1997). Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations. John Wiley, New York.Google Scholar
Qian, M. P. and Qian, M. (1982). Circulation for recurrent Markov chains. Z. Wahrscheinlichkeitsth. 59, 203–210.Google Scholar
Qian, M. P. and Qian, M. (1985). The entropy production and reversibility of Markov processes. Sci. Bull. 30, 165–167.Google Scholar
Qian, M. P., Qian, M. and Gong, G. L. (1991). The reversibility and the entropy production of Markov processes. Contemp. Math. 118, 255–261.CrossRefGoogle Scholar
Qian, M. P., Qian, C. and Qian, M. (1984). Circulations of Markov chains with continuous time and the probability interpretation of some determinants. Sci. Sinica A 27, 470–481.Google Scholar
Reimann, P. (2002). Brownian motors: noisy transport far from equilibrium. Phys. Rep. 361, 57–265.CrossRefGoogle Scholar
Schnakenberg, J. (1976). Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys. 48, 571–585.CrossRefGoogle Scholar
Walters, P. (1982). An Introduction to Ergodic Theory (Grad. Texts Math. 79). Springer, New York.CrossRefGoogle Scholar
Wang, H., Peskin, C. S. and Elston, T. C. (2003). A robust numerical algorithm for studying biomolecular transport processes. J. Theoret. Biol. 221, 491–511.CrossRefGoogle ScholarPubMed
Xing, J., Wang, H. Y. and Oster, G. (2005). From continuum Fokker–Planck models to discrete kinetic models. Biophys. J. 89, 1551–1563.CrossRefGoogle ScholarPubMed
Yin, G. G. and Zhang, Q. (1998). Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach (Appl. Math. 37). Springer, Berlin.Google Scholar
Zhang, F. X. (2005). Exclusion processes on groups: entropy production density and reversibility. Physica A 348, 131–139.CrossRefGoogle Scholar