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Asymptotics of work distributions in a stochastically driven system

  • Sreekanth K. Manikandan
  • Supriya Krishnamurthy
Open Access
Regular Article
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Abstract

We determine the asymptotic forms of work distributions at arbitrary times T, in a class of driven stochastic systems using a theory developed by Nickelsen and Engel (EN theory) [D. Nickelsen and A. Engel, Eur. Phys. J. B 82, 207 (2011)], which is based on the contraction principle of large deviation theory. In this paper, we extend the theory, previously applied in the context of deterministically driven systems, to a model in which the driving is stochastic. The models we study are described by overdamped Langevin equations and the work distributions in path integral form, are characterised by having quadratic augmented actions. We first illustrate EN theory, for a deterministically driven system – the breathing parabola model, and show that within its framework, the Crooks fluctuation theorem manifests itself as a reflection symmetry property of a certain characteristic polynomial, which also determines the exact moment-generating-function at arbitrary times. We then extend our analysis to a stochastically driven system, studied in references [S. Sabhapandit, EPL 89, 60003 (2010); A. Pal, S. Sabhapandit, Phys. Rev. E 87, 022138 (2013); G. Verley, C. Van den Broeck, M. Esposito, New J. Phys. 16, 095001 (2014)], for both equilibrium and non-equilibrium steady state initial distributions. In both cases we obtain new analytic solutions for the asymptotic forms of (dissipated) work distributions at arbitrary T. For dissipated work in the steady state, we compare the large T asymptotic behaviour of our solution to the functional form obtained in reference [New J. Phys. 16, 095001 (2014)]. In all cases, special emphasis is placed on the computation of the pre-exponential factor and the results show excellent agreement with numerical simulations. Our solutions are exact in the low noise (β → ∞) limit.

Keywords

Statistical and Nonlinear Physics 
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References

  1. 1.
    U. Seifert, Rep. Progr. Phys. 75, 126001 (2012) ADSCrossRefGoogle Scholar
  2. 2.
    G.E. Crooks, Phys. Rev. E 61, 2361 (2000) ADSCrossRefGoogle Scholar
  3. 3.
    C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997) ADSCrossRefGoogle Scholar
  4. 4.
    C. Jarzynski, Phys. Rev. E 56, 5018 (1997) ADSCrossRefGoogle Scholar
  5. 5.
    R. van Zon, E.G.D. Cohen, Phys. Rev. E 67, 046102 (2003) ADSCrossRefGoogle Scholar
  6. 6.
    R. van Zon, E.G.D. Cohen, Phys. Rev. E 69, 056121 (2004) ADSCrossRefGoogle Scholar
  7. 7.
    R. van Zon, E.G.D. Cohen, Phys. Rev. Lett. 91, 110601 (2003) ADSCrossRefGoogle Scholar
  8. 8.
    R. van Zon, S. Ciliberto, E.G.D. Cohen, Phys. Rev. Lett. 92, 130601 (2004) ADSCrossRefGoogle Scholar
  9. 9.
    D. Nickelsen, A. Engel, Eur. Phys. J. B 82, 207 (2011) ADSCrossRefGoogle Scholar
  10. 10.
    T. Speck, J. Phys. A: Math. Theor. 44, 305001 (2011) ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Jarzynski, Phys. Rev. E 56, 5018 (1997) ADSCrossRefGoogle Scholar
  12. 12.
    D.M. Carberry, J.C. Reid, G.M. Wang, E.M. Sevick, D.J. Searles, D.J. Evans, Phys. Rev. Lett. 92, 140601 (2004) ADSCrossRefGoogle Scholar
  13. 13.
    C. Kwon, J.D. Noh, H. Park, Phys. Rev. E 88, 062102 (2013) ADSCrossRefGoogle Scholar
  14. 14.
    A. Ryabov, M. Dierl, P. Chvosta, M. Einax, P. Maass, J. Phys. A: Math. Theor. 46, 075002 (2013) ADSCrossRefGoogle Scholar
  15. 15.
    B. Saha, S. Mukherji, Eur. Phys. J. B 88, 146 (2015) ADSCrossRefGoogle Scholar
  16. 16.
    H. Touchette, Phys. Rep. 478, 1 (2009) ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    A. Pal, S. Sabhapandit, Phys. Rev. E 87, 022138 (2013) ADSCrossRefGoogle Scholar
  18. 18.
    S. Sabhapandit, EPL 96, 20005 (2011) ADSCrossRefGoogle Scholar
  19. 19.
    L.F. Shampine, M.W. Reichelt, J. Kierzenka, Solving Boundary Value Problems for Ordinary Differential Equations in MATLAB with bvp4c, Available at www.mathworks.com/bvp_tutorial
  20. 20.
    V. Holubec, M. Dierl, M. Einax, P. Maass, P. Chvosta, A. Ryabov, Phys. Scr. 2015, 014024 (2015) CrossRefGoogle Scholar
  21. 21.
    D. Nickelsen, A. Engel, Phys. Scr. 86, 058503 (2012) CrossRefGoogle Scholar
  22. 22.
    G. Verley, C. Van den Broeck, M. Esposito, New J. Phys. 16, 095001 (2014) ADSCrossRefGoogle Scholar
  23. 23.
    P. Pietzonka, F. Ritort, U. Seifert, Phys. Rev. E 96, 012101 (2017) ADSCrossRefGoogle Scholar
  24. 24.
    R.P. Feynman, A.R. Hibbs, Quantum mechanics and path integrals. International series in pure and applied physics (McGraw-Hill, Maidenhead, Berkshire, 1965) Google Scholar
  25. 25.
    L.S. Schulman, Techniques and applications of path integration (Wiley, New York, 1996) Google Scholar
  26. 26.
    K. Kirsten, A.J. McKane, Ann. Phys. 308, 502 (2003) ADSCrossRefGoogle Scholar
  27. 27.
    G.M. Falco, A.A. Fedorenko, I.A. Gruzberg, J. Phys. A: Math. Theor. 50, 485201 (2017) CrossRefGoogle Scholar
  28. 28.
    A. Engel, Phys. Rev. E 80, 021120 (2009) ADSCrossRefGoogle Scholar
  29. 29.
    K. Sekimoto, J. Phys. Soc. Jpn. 66, 1234 (1997) ADSCrossRefGoogle Scholar
  30. 30.
    K. Sekimoto, Progr. Theor. Phys. Suppl. 130, 17 (1998) ADSCrossRefGoogle Scholar
  31. 31.
    S. Machlup, L. Onsager, Phys. Rev. 91, 1512 (1953) ADSCrossRefGoogle Scholar
  32. 32.
    L. Onsager, S. Machlup, Phys. Rev. 91, 1505 (1953) ADSCrossRefGoogle Scholar
  33. 33.
    R. Courant, D. Hilbert, in Methods of mathematical physics (Wiley, New York, 1953), Vol. 1 Google Scholar
  34. 34.
    K. Kirsten, P. Loya, Am. J. Phys. 76, 60 (2008) ADSCrossRefGoogle Scholar
  35. 35.
    C. Kwon, J.D. Noh, H. Park, Phys. Rev. E 83, 061145 (2011) ADSCrossRefGoogle Scholar
  36. 36.
    J.R. Gomez-Solano, L. Bellon, A. Petrosyan, S. Ciliberto, EPL 89, 60003 (2010) ADSCrossRefGoogle Scholar
  37. 37.
    V.Y. Chernyak, M. Chertkov, C. Jarzynski, J. Stat. Mech.: Theory Exp. 2006, P08001 (2006) CrossRefGoogle Scholar
  38. 38.
    M. Chaichian, A. Demichev, Path integrals in physics: volume I: Stochastic processes and quantum mechanics (Institute of Physics Publishing, Bristol and Philadelphia, 2001) Google Scholar
  39. 39.
    M. Doi, J. Phys. A: Math. Gen. 9, 1465 (1976) ADSCrossRefGoogle Scholar
  40. 40.
    M. Doi, J. Phys. A: Math. Gen. 9, 1479 (1976) ADSCrossRefGoogle Scholar
  41. 41.
    L. Peliti, J. Phys. 46, 1469 (1985) MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017. This article is published with open access at Springerlink.com 2017

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Sreekanth K. Manikandan
    • 1
  • Supriya Krishnamurthy
    • 1
  1. 1.Department of PhysicsStockholm UniversityStockholmSweden

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