Journal of Statistical Physics

, Volume 123, Issue 4, pp 831–859 | Cite as

A Simple Discrete Model of Brownian Motors: Time-periodic Markov Chains

  • Hao Ge
  • Da-Quan Jiang
  • Min Qian


In this paper, we consider periodically inhomogeneous Markov chains, which can be regarded as a simple version of physical model—Brownian motors. We introduce for them the concepts of periodical reversibility, detailed balance, entropy production rate and circulation distribution. We prove the equivalence of the following statements: The time-periodic Markov chain is periodically reversible; It is in detailed balance; Kolmogorov's cycle condition is satisfied; Its entropy production rate vanishes; Every circuit and its reversed circuit have the same circulation weight. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors, i.e. the existence of net circulation, can occur only in nonequilibrium and irreversible systems. Moreover, we verify the large deviation property and the Gallavotti-Cohen fluctuation theorem of sample entropy production rates of the Markov chain.


Brownian motor time-periodic Markov chain periodical reversibility detailed balance entropy production circulation, fluctuation theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. R. D. Astumian, Thermodynamics and kinetics of a Brownian motor. Science 276:917–922 (1997).CrossRefGoogle Scholar
  2. R. D. Astumian and M. Bier, Fluctuation driven ratchets: Molecular motors. Phys. Rev. Lett. 72:1766–1769 (1994).CrossRefADSGoogle Scholar
  3. R. Benzi, A. Sutera and A. Vulpiani, The mechanism of stochastic resonance. J. Phys. A: Math. Gen. 14:L453–L457 (1981).CrossRefADSMathSciNetGoogle Scholar
  4. V. Berdichevsky, and M. Gitterman, Stochastic resonance and ratchets–new manifestations. Physica A 249:88–95 (1998).CrossRefGoogle Scholar
  5. R.N. Bhattacharya and E. C. Waymire, Stochastic processes with applications. New York: John Wiley & Sons, Inc. 1990.Google Scholar
  6. M. Bier, Brownian ratchets in physics and biology. Contemporary Phys. 38(6), 371–379 (1997).CrossRefADSGoogle Scholar
  7. R. Durrett, Probability: theory and examples. (2nd ed.) Belmont: Duxbury Press 1996.Google Scholar
  8. D. J. Evans, and D. J. Searles, Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50(2), 1645–1648 (1994).CrossRefADSGoogle Scholar
  9. D. J. Evans and D. J. Searles, The fluctuation theorem. Adv. Phys. 51(7), 1529–1585 (2002).CrossRefADSGoogle Scholar
  10. G. Gallavotti and E. G. D. Cohen, Dynamical ensembles in stationary states. J. Statist. Phys. 80: 931–970 (1995).CrossRefADSMathSciNetMATHGoogle Scholar
  11. L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance. Rev. Mod. Phys. 70(1), 223–287 (1998).CrossRefADSGoogle Scholar
  12. G. P. Harmer and D. Abbott, Game theory: Losing strategies can win by Parrondo's paradox. Nature 402:864 (1999).CrossRefADSGoogle Scholar
  13. G. P. Harmer, D. Abbott, P. G. Taylor and J. M. Parrondo, Parrondo's paradoxical games and the discrete Brownian ratchet. In Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations (eds Abbott, D. and Kish, L. B.), 189–200; Melville, N.Y, American Institute of Physics 2000.Google Scholar
  14. H. Hasegawa, On the construction of a time-reversed Markoff process, Prog. Theor. Phys. 55:90–105 (1976); Variational principle for non-equilibrium states and the Onsager-Machlup formula, ibid. 56, 44–60 (1976); Thermodynamic properties of non-equilibrium states subject to Fokker-Planck equations, ibid. 57:1523–1537 (1977); Variational approach in studies with Fokker-Planck equations, ibid. 58, 128–146 (1977).CrossRefADSMathSciNetMATHGoogle Scholar
  15. P. Imkeller and I. Pavlyukevich, Stochastic resonance in two-state Markov chains. Arch. Math. 77:107–115 (2001).CrossRefMathSciNetMATHGoogle Scholar
  16. D. Q. Jiang, M. Qian and M. P. Qian, Entropy production and information gain in Axiom-A systems. Commun. Math. Phys. 214(2), 389–409 (2000).CrossRefADSMathSciNetMATHGoogle Scholar
  17. D. Q. Jiang, M. Qian and M. P. Qian, Mathematical theory of nonequilibrium steady states - On the frontier of probability and dynamical systems. (Lect. Notes Math. 1833) Berlin: Springer-Verlag 2004.MATHGoogle Scholar
  18. D. Q. Jiang, M. Qian and F. X. Zhang, Entropy production fluctuations of finite Markov chains. J. Math. Phys. 44(9), 4176–4188 (2003).CrossRefADSMathSciNetMATHGoogle Scholar
  19. F. Jülicher, A. Ajdari and J. Prost, Modeling molecular motors. Rev. Mod. Phys. 69(4), 1269–1281 (1997).CrossRefADSGoogle Scholar
  20. Kurchan, J, Fluctuation theorem for stochastic dynamics. J. Phys. A: Math. Gen. 31: 3719–3729 (1998).CrossRefADSMathSciNetMATHGoogle Scholar
  21. Lan, Y.Z, Concise lectures of advanced algebra. Peking University Press 2002Google Scholar
  22. J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Statist. Phys. 95(1–2), 333–365 (1999).CrossRefMathSciNetMATHGoogle Scholar
  23. M. O. Magnasco, Forced thermal ratchets. Phys. Rev. Lett. 71:1477–1481 (1993).CrossRefADSGoogle Scholar
  24. M. O. Magnasco, Molecular combustion motors. Phys. Rev. Lett. 72:2656–2659 (1994).CrossRefADSGoogle Scholar
  25. C. Nicolis, Stochastic aspects of climatic transitions—response to a periodic forcing. Tellus 34:1–9 (1982).ADSMathSciNetCrossRefGoogle Scholar
  26. G. Nicolis and I. Prigogine, Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. New York: Wiley 1977.Google Scholar
  27. Qian, M.P. and Gong, G.L, Applied Stochastic Processes. Peking University Press 1998Google Scholar
  28. M. P. Qian and M, Qian, Circulation for recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 59:203–210 (1982).CrossRefMathSciNetMATHGoogle Scholar
  29. M. P. Qian and M. Qian, The entropy production and reversibility of Markov processes. Scinence Bulletin 30(3), 165–167 (1985).Google Scholar
  30. M. P. Qian, M. Qian and G. L. Gong, The reversibility and the entropy production of Markov processes. Contemp. Math. 118:255–261 (1991).MathSciNetGoogle Scholar
  31. M. P. Qian, C. Qian and M. Qian, Circulations of Markov chains with continuous time and the probability interpretation of some determinants. Sci. Sinica (Series A) 27(5), 470–481 (1984).MATHGoogle Scholar
  32. P. Reimann, Brownian motors: noisy transport far from equilibrium. Physics Reports 361:57–265 (2002).CrossRefADSMathSciNetMATHGoogle Scholar
  33. D. Ruelle, Positivity of entropy production in nonequilibrium statistical mechanics. J. Statist. Phys. 85(1–2), 1–23 (1996).CrossRefADSMathSciNetMATHGoogle Scholar
  34. J. Schnakenberg, Network theory of microscopic and macroscopic behaviour of master equation systems. Rev. Modern Phys. 48(4), 571–585 (1976).CrossRefADSMathSciNetGoogle Scholar
  35. S. R. S. Varadhan, Large deviations and applications. Philadelphia: Society for Industrial and Applied Mathematics 1984.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingP. R. China
  2. 2.LMAM, School of Mathematical SciencesPeking UniversityBeijingP. R. China
  3. 3.Institute of Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations