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Refined Second Law of Thermodynamics for Fast Random Processes

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Abstract

We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport and becomes relevant for processes far from quasi-stationary regime. General discussion is illustrated by numerical analysis of the optimal memory erasure protocol for a model for micron-size particle manipulated by optical tweezers.

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References

  1. Andresen, B.: Current trends in finite-time thermodynamics. Angew. Chem., Int. Ed. Engl. 50, 2690–2704 (2011)

    Article  Google Scholar 

  2. Andrieux, D., Gaspard, P.: Dynamical randomness, information, and Landauer’s principle. Europhys. Lett. 81, 28004 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  3. Aurell, E., Mejía-Monasterio, C., Muratore-Ginanneschi, P.: Optimal protocols and optimal transport in stochastic thermodynamics. Phys. Rev. Lett. 106, 250601 (2011)

    Article  ADS  Google Scholar 

  4. Aurell, E., Mejía-Monasterio, C., Muratore-Ginanneschi, P.: Boundary layers in stochastic thermodynamics. Phys. Rev. E 85, 020103(R) (2012)

    ADS  Google Scholar 

  5. Bellman, R.E.: Dynamic Programming. Princeton University Press, Princeton (1957)

    MATH  Google Scholar 

  6. Benamou, J.-D., Brenier, Y.: A numerical method for the optimal time-continuous mass transport problem and related problems. In: Monge Ampère Equation: Applications to Geometry and Optimization. Deerfield Beach, FL, 1997, pp. 1–11. Am. Math. Soc., Providence (1999)

    Chapter  Google Scholar 

  7. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (1999)

    Article  MathSciNet  Google Scholar 

  8. Bennett, Ch.H.: The thermodynamics of computation—a review. Int. J. Theor. Phys. 21, 905–940 (1982)

    Article  Google Scholar 

  9. Bertsekas, D.P.: Network Optimization: Continuous and Discrete Models. Athena Scientific, Belmont (1998)

    MATH  Google Scholar 

  10. Bérut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., Lutz, E.: Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483, 187–189 (2012)

    Article  ADS  Google Scholar 

  11. Brenier, Y., Frisch, U., Hénon, M., Loeper, G., Matarrese, S., Mohayaee, R., Sobolevski, A.: Reconstruction of the early Universe as a convex optimization problem. Mon. Not. R. Astron. Soc. 346, 501–524 (2003)

    Article  ADS  Google Scholar 

  12. Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Clarendon Press, Oxford (2006)

    Google Scholar 

  13. Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974)

    MATH  Google Scholar 

  14. Chen, L., Sun, F. (eds.): Advances in Finite Time Thermodynamics: Analysis and Optimization. Nova Science Publ., Hauppauge (2004)

    Google Scholar 

  15. Chernyak, V., Chertkov, M., Jarzynski, C.: Path-integral analysis of fluctuation theorems for general Langevin processes. J. Stat. Mech. 08, P08001 (2006)

    Article  Google Scholar 

  16. Chetrite, R., Gawȩdzki, K.: Fluctuation relations for diffusion processes. Commun. Math. Phys. 282, 469–518 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Dillenschneider, R., Lutz, E.: Memory erasure in small systems. Phys. Rev. Lett. 102, 210601 (2009)

    Article  ADS  Google Scholar 

  18. Esposito, M., Van den Broeck, C.: Second law and Landauer principle far from equilibrium. Europhys. Lett. 95, 40004 (2011)

    Article  ADS  Google Scholar 

  19. Esposito, M., Kawai, R., Lindenberg, K., Van den Broeck, C.: Finite-time thermodynamics for a single-level quantum dot. Europhys. Lett. 89, 20003 (2010)

    Article  ADS  Google Scholar 

  20. Esposito, M., Kawai, R., Lindenberg, K., Van den Broeck, C.: Efficiency at maximal power of low-dissipation Carnot engines. Phys. Rev. Lett. 105, 150603 (2010)

    Article  ADS  Google Scholar 

  21. Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 51, 1529–1585 (2002)

    Article  ADS  Google Scholar 

  22. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer, New York (2005)

    Google Scholar 

  23. Gallavotti, G.: Fluctuation Theorem and chaos. Eur. Phys. J. B 64, 315–320 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  24. Gallavotti, G.: On thermostats: Isokinetic or Hamiltonian? Finite or infinite? Chaos 19, 013101 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  25. Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177, 113–161 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ge, H., Jiang, D.-Q., Qian, M.: Reversibility and entropy production of inhomogeneous Markov chains. J. Appl. Probab. 43, 1028–1043 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Glansdorff, P., Prigogine, I.: Thermodynamic Theory of Structure, Stability and Fluctuations. Wiley, London (1971)

    MATH  Google Scholar 

  28. Gomez-Marin, A., Schmiedl, T., Seifert, U.: Optimal protocols for minimal work processes in underdamped stochastic thermodynamics. J. Chem. Phys. 129, 024114 (2008)

    Article  ADS  Google Scholar 

  29. Guerra, F., Morato, L.: Quantization of dynamical systems and stochastic control theory. Phys. Rev. D 27, 1774–1786 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  30. Hatano, T., Sasa, S.: Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463–3466 (2001)

    Article  ADS  Google Scholar 

  31. Hongler, M.-O., Soner, H.M., Streit, L.: Stochastic control for a class of random evolution models. Appl. Math. Optim. 49, 113–121 (2004)

    MathSciNet  MATH  Google Scholar 

  32. Jarzynski, C.: Equilibrium free energy differences from nonequilibrium measurements: a master equation approach. Phys. Rev. E 56, 5018–5035 (1997)

    Article  ADS  Google Scholar 

  33. Jarzynski, C.: Equalities and inequalities: irreversibility and the second law of thermodynamics at the nanoscale. Ann. Rev. Condens. Matter Phys. 2, 329–351 (2011)

    Article  ADS  Google Scholar 

  34. Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker Planck equation. SIAM J. Math. Anal. 29, 1–17 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jun, Y., Bechhoefer, J.: Experimental study of memory erasure in a double-well potential, abstract of a talk at APS March Meeting 2011. Bull. Am. Phys. Soc. 56(1) (2011)

  36. Kantorovich, L.: On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37, 199–201 (1942)

    MathSciNet  Google Scholar 

  37. Kawai, R., Parrondo, J.M.R., Van den Broeck, C.: Dissipation: The phase-space perspective. Phys. Rev. Lett. 98, 080602 (2007)

    Article  ADS  Google Scholar 

  38. Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5 (3), 183–191 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  39. Maes, C.: The fluctuation theorem as a Gibbs property. J. Stat. Phys. 95, 367–392 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. Maes, C.: On the origin and the use of fluctuation relations for the entropy. Séminaire Poincaré 2, 29–62 (2003)

    Google Scholar 

  41. Maes, C., Netočný, K., Wynants, B.: Steady state statistics of driven diffusions. Physica A 387, 2675–2689 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  42. Maes, C., Netočný, K., Wynants, B.: On and beyond entropy production: the case of Markov jump processes. Markov Process. Relat. Fields 14, 445–464 (2008)

    MATH  Google Scholar 

  43. Maes, C., Redig, F., Van Moffaert, A.: On definition of entropy production, via examples. J. Math. Phys. 41, 1528–1554 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Monge, G.: Mémoir sur la thórie des déblais et des remblais. In: Histoire de l’Académie Royale des Sciences, Année 1781, pp. 666–704. Imprimerie Royale, Paris (1784)

    Google Scholar 

  45. Maroney, O.J.E.: Landauer’s erasure principle in non-equilibrium systems. arXiv:1112.0898v1 [cond-mat.stat-mech]

  46. Muratore-Ginanneschi, P., Mejía-Monasterio, C., Peliti, L.: Heat release by controlled continuous-time Markov jump processes. arXiv:1203.4062v1 [cond-mat.stat-mech]

  47. Nelson, E.: Dynamic Theories of Brownian Motion. Princeton University Press, Princeton (1967)

    Google Scholar 

  48. Schmiedl, T., Seifert, U.: Optimal finite-time processes in stochastic thermodynamics. Phys. Rev. Lett. 98, 108301 (2007)

    Article  ADS  Google Scholar 

  49. Schmiedl, T., Seifert, U.: Efficiency at maximum power: An analytically solvable model for stochastic heat engines. Europhys. Lett. 81, 20003 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  50. Schnakenberg, J.: Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Mod. Phys. 48, 571–585 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  51. Seifert, U.: Entropy production along a stochastic trajectory and an integral fluctuation theorem. Phys. Rev. Lett. 95, 040602 (2005)

    Article  ADS  Google Scholar 

  52. Seifert, U.: Stochastic thermodynamics: Principles and perspectives. Eur. Phys. J. B 64, 423–431 (2008)

    Article  ADS  MATH  Google Scholar 

  53. Sekimoto, K.: Stochastic Energetics. Lecture Notes in Physics, vol. 799. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  54. Szilard, L.: Über die Ausdehnung der phänomenologischen Thermodynamik auf die Schwankungserscheinungen. Z. Phys. 32, 753–788 (1925)

    Article  ADS  Google Scholar 

  55. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 38. Am. Math. Soc., Providence (2003)

    MATH  Google Scholar 

  56. Zander, C., Plastino, A.R., Plastino, A., Casas, M., Curilef, S.: Landauer’s Principle and divergenceless dynamical systems. Entropy 11, 586–597 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

E.A. acknowledges support from the Academy of Finland as part of its Finland Distinguished Professor program, Project No. 129024. K.G. thanks S. Ciliberto, U. Seifert and C. Van den Broeck for inspiring discussions. His work was partly done within the framework of the STOSYMAP project ANR-11-BS01-015-02. R.M.’s work was partly supported by the OTARIE project ANR-07-BLAN-0235. P.M.-G. acknowledges support of the Center of Excellence “Analysis and Dynamics” of the Academy of Finland.

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Authors and Affiliations

  1. ACCESS Linnaeus Center, KTH, 10044, Stockholm, Sweden

    Erik Aurell

  2. Computational Biology Department, AlbaNova University Center, KTH, 10691, Stockholm, Sweden

    Erik Aurell

  3. Department of Information and Computer Science, Aalto University, PO Box 15400, 00076, Aalto, Finland

    Erik Aurell

  4. CNRS, Laboratoire de Physique, ENS Lyon, Université de Lyon, 46 Allée d’Italie, 69364, Lyon, France

    Krzysztof Gawȩdzki

  5. Laboratory of Physical Properties, Department of Rural Engineering, Technical University of Madrid, Avenida Complutense s/n, 28040, Madrid, Spain

    Carlos Mejía-Monasterio

  6. CNRS, Institut d’Astrophysique de Paris, Université Pierre et Marie Curie, 8 bis boulevard Arago, 75014, Paris, France

    Roya Mohayaee

  7. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 FIN-00014, Helsinki, Finland

    Paolo Muratore-Ginanneschi

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  1. Erik Aurell
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Correspondence to Krzysztof Gawȩdzki.

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Aurell, E., Gawȩdzki, K., Mejía-Monasterio, C. et al. Refined Second Law of Thermodynamics for Fast Random Processes. J Stat Phys 147, 487–505 (2012). https://doi.org/10.1007/s10955-012-0478-x

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