Advertisement

Journal of Statistical Physics

, Volume 174, Issue 2, pp 433–468 | Cite as

Fundamental Relation Between Entropy Production and Heat Current

  • Naoto ShiraishiEmail author
  • Keiji Saito
Article

Abstract

We investigate the fundamental relation between entropy production rate and the speed of energy exchange between a system and baths in classical Markov processes. We establish the fact that quick energy exchange inevitably induces large entropy production in a quantitative form. More specifically, we prove two inequalities on instantaneous quantities: one is applicable to general Markov processes induced by heat baths, and the other is applicable only to systems with the local detailed-balance condition but is stronger than the former one. We demonstrate the physical meaning of our result by applying to some specific setups. In particular, we show that our inequality is tight in the linear response regime.

Keywords

Heat engines Finite time thermodynamics Stochastic thermodynamics 

Notes

Acknowledgements

We are grateful to Hal Tasaki for fruitful discussion. He was a co-author in the joint work [41], and contributed to deriving several relations. NS was supported by Grant-in-Aid for JSPS Fellows JP17J00393. KS was supported by JSPS Grants-in-Aid for Scientific Research (No. JP25103003, JP16H02211 and JP17K05587).

References

  1. 1.
    Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401 (1993)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A Math. Gen. 31, 3719 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Jarzynski, C.: Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys. 98, 77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Mahan, G.D., Sofo, J.O.: The best thermoelectric. Proc. Natl Acad. Sci. U.S.A. 93, 7436 (1996)ADSCrossRefGoogle Scholar
  5. 5.
    Mahan, G.D., Sales, B., Sharp, J.: Thermoelectric materials: New approaches to an old problem. Phys. Today 50, 42 (1997)CrossRefGoogle Scholar
  6. 6.
    Majumdar, A.: Thermoelectricity in semiconductor nanostructures. Science 303, 777 (2004)CrossRefGoogle Scholar
  7. 7.
    Snyder, G.J., Toberer, E.R.: Complex thermoelectric materials. Nat. Mater. 7, 105 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Casati, G., Mejía-Monasterio, C., Prosen, T.: Incresing thermoelectric efficiency: a dynamic systems approach. Phys. Rev. Lett. 101, 016601 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    Shiraishi, N.: Attainability of Carnot efficiency with autonomous engines. Phys. Rev. E 92, 050101 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    Tajima, H., Hayashi, M.: Finite-size effect on optimal efficiency of heat engines. Phys. Rev. E 96, 012128 (2017)ADSCrossRefGoogle Scholar
  11. 11.
    Shiraishi, N.: Stationary engines in and beyond the linear response regime at the Carnot efficiency. Phys. Rev. E 95, 052128 (2017)ADSCrossRefGoogle Scholar
  12. 12.
    Benenti, G., Casati, G., Saito, K., Whitney, R.S.: Fundamental aspects of steady-state conversion of heat to work at the nanoscale. Phys. Rep. 694, 1 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Benenti, G., Saito, K., Casati, G.: Thermodynamic bounds on efficiency for systems with broken time-reversal symmetry. Phys. Rev. Lett. 106, 230602 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Allahverdyan, A.E., Hovhannisyan, K.V., Melkikh, A.V., Gevorkian, S.G.: Carnot cycle at finite power: attainability of maximal efficiency. Phys. Rev. Lett. 111, 050601 (2013)ADSCrossRefGoogle Scholar
  15. 15.
    Campisi, M., Fazio, R.: The power of a critical heat engine. Nat. Commun. 7, 11895 (2016)ADSCrossRefGoogle Scholar
  16. 16.
    Ponmurugan, M.: Attainability of maximum work and the reversible efficiency from minimally nonlinear irreversible heat engines. arXiv:1604.01912 (2016)Google Scholar
  17. 17.
    Polettini, M., Esposito, M.: Carnot efficiency at divergent power output. Europhys. Lett. 118, 40003 (2017)ADSCrossRefGoogle Scholar
  18. 18.
    Johnson, C.V.: Approaching the Carnot limit at finite power: an exact solution. Phys. Rev. D 98, 026008 (2018)ADSCrossRefGoogle Scholar
  19. 19.
    Curzon, F.L., Ahlborn, B.: Efficiency of a Carnot engine at maximum power output. Am. J. Phys. 43, 22 (1975)ADSCrossRefGoogle Scholar
  20. 20.
    Andresen, B., Berry, R.S., Ondrechen, M.J., Salamon, P.: Thermodynamics for processes in finite time. Acc. Chem. Res. 17, 266 (1984)CrossRefGoogle Scholar
  21. 21.
    Sothmann, B., Büttiker, M.: Magnon-driven quantum-dot heat engine. Europhys. Lett. 99, 27001 (2012)ADSCrossRefGoogle Scholar
  22. 22.
    Brandner, K., Saito, K., Seifert, U.: Strong bounds on Onsager coefficients and efficiency for three-terminal thermoelectric transport in a magnetic field. Phys. Rev. Lett. 110, 070603 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    Brandner, K., Seifert, U.: Multi-terminal thermoelectric transport in a magnetic field: bounds on Onsager coefficients and efficiency. New J. Phys. 15, 105003 (2013)ADSCrossRefGoogle Scholar
  24. 24.
    Balachandran, V., Benenti, G., Casati, G.: Efficiency of three-terminal thermoelectric transport under broken time-reversal symmetry. Phys. Rev. B 87, 165419 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    Brandner, K., Seifert, U.: Bound on thermoelectric power in a magnetic field within linear response. Phys. Rev. E 91, 012121 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    Yamamoto, K., Entin-Wohlman, O., Aharony, A., Hatano, N.: Efficiency bounds on thermoelectric transport in magnetic fields: the role of inelastic processes. Phys. Rev. B 94, 121402 (2016)ADSCrossRefGoogle Scholar
  27. 27.
    Brandner, K., Saito, K., Seifert, U.: Thermodynamics of micro-and nano-systems driven by periodic temperature variations. Phys. Rev. X 5, 031019 (2015)Google Scholar
  28. 28.
    Proesmans, K., Van den Broeck, C.: Onsager coefficients in periodically driven systems. Phys. Rev. Lett. 115, 090601 (2015)ADSCrossRefGoogle Scholar
  29. 29.
    Proesmans, K., Cleuren, B., Van den Broeck, C.: Linear stochastic thermodynamics for periodically driven systems. J. Stat. Mech. P023202 (2016)Google Scholar
  30. 30.
    Pietzonka, P., Seifert, U.: Universal trade-off between power, efficiency and constancy in steady-state heat engines. Phys. Rev. Lett. 120, 190602 (2018)ADSCrossRefGoogle Scholar
  31. 31.
    Sekimoto, K., Sasa, S.-I.: Complementarity relation for irreversible process derived from stochastic energetics. J. Phys. Soc. Jpn. 66, 3326 (1997)ADSCrossRefGoogle Scholar
  32. 32.
    Aurell, E., Gawȩdzki, K., Mejía-Monasterio, C., Mohayaee, R., Muratore-Ginanneschi, P.: Refined second law of thermodynamics for fast random processes. J. Stat. Phys. 147, 487 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Raz, O., Subaşı, Y., Pugatch, R.: Geometric heat engines featuring power that grows with efficiency. Phys. Rev. Lett. 116, 160601 (2016)ADSCrossRefGoogle Scholar
  34. 34.
    Barato, A.C., Seifert, U.: Thermodynamic uncertainty relation for biomolecular processes. Phys. Rev. Lett. 114, 158101 (2015)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Gingrich, T.R., Horowitz, J.M., Perunov, N., England, J.L.: Dissipation bounds all steady-state current fluctuations. Phys. Rev. Lett. 116, 120601 (2016)ADSCrossRefGoogle Scholar
  36. 36.
    Gingrich, T.R., Rotskoff, G.M., Horowitz, J.M.: Inferring dissipation from current fluctuations. J. Phys. A Math. Theor. 50, 184004 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pietzonka, P., Ritort, F., Seifert, U.: Finite-time generalization of the thermodynamic uncertainty relation. Phys. Rev. E 96, 012101 (2017)ADSCrossRefGoogle Scholar
  38. 38.
    Horowitz, J.M., Gingrich, T.R.: Proof of the finite-time thermodynamic uncertainty relation for steady-state currents. Phys. Rev. E 96, 020103 (2017)ADSCrossRefGoogle Scholar
  39. 39.
    Dechant, A., Sasa, S.: Current fluctuations and transport efficiency for general Langevin systems. J. Stat. Mech. 063209 (2018)Google Scholar
  40. 40.
    Dechant, A., Sasa, S.: Fluctuation-response inequality out of equilibrium. arXiv:1804.08250 (2018)Google Scholar
  41. 41.
    Shiraishi, N., Saito, K., Tasaki, H.: Universal trade-off relation between power and efficiency for heat engines. Phys. Rev. Lett. 117, 190601 (2016)ADSCrossRefGoogle Scholar
  42. 42.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1991)CrossRefzbMATHGoogle Scholar
  43. 43.
    Taneja, I.J.: Bounds on triangular discrimination, harmonic mean and symmetric chi-square divergences. arXiv:math/0505238 (2005)Google Scholar
  44. 44.
    Shiraishi, N., Sagawa, T.: Fluctuation theorem for partially masked nonequilibrium dynamics. Phys. Rev. E 91, 012130 (2015)ADSCrossRefGoogle Scholar
  45. 45.
    Shiraishi, N., Ito, S., Kawaguchi, K., Sagawa, T.: Role of measurement-feedback separation in autonomous Maxwell’s demons. New J. Phys. 17, 045012 (2015)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Shiraishi, N., Matsumoto, T., Sagawa, T.: Measurement-feedback formalism meets information reservoirs. New J. Phys. 18, 013044 (2016)ADSCrossRefGoogle Scholar
  47. 47.
    Shiraishi, N., Saito, K.: Incompatibility between Carnot efficiency and finite power in Markovian dynamics. arXiv:1602.03645 (2016)Google Scholar
  48. 48.
    Van Kampen, N.G.: Stochastic Process in Physics and Chemistry, 3rd edn. Elsevier, Amsterdam (2007)Google Scholar
  49. 49.
    Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)zbMATHGoogle Scholar
  50. 50.
    Shiraishi, N., Tajima, H.: Efficiency versus speed in quantum heat engines: rigorous constraint from Lieb–Robinson bound. Phys. Rev. E 96, 022138 (2017)ADSCrossRefGoogle Scholar
  51. 51.
    Perarnau-Llobet, M., Wilming, H., Riera, A., Gallego, R., Eisert, J.: Strong coupling corrections in quantum thermodynamics. Phys. Rev. Lett. 120, 120602 (2018)ADSCrossRefGoogle Scholar
  52. 52.
    Funo, K., Shiraishi, N., Saito, K.: Speed limit for open quantum systems. arXiv:1810.03011 (2018)Google Scholar
  53. 53.
    Brandner, K., Hanazato, T., Saito, K.: Thermodynamic bounds on precision in ballistic multi-terminal transport. Phys. Rev. Lett. 120, 090601 (2018)ADSCrossRefGoogle Scholar
  54. 54.
    Macieszczak, K., Brandner, K., Garrahan, J.P.: Unified thermodynamic uncertainty relations in linear response. Phys. Rev. Lett. 121, 130601 (2018)ADSCrossRefGoogle Scholar
  55. 55.
    Shiraishi, N.: Finite-time thermodynamic uncertainty relation do not hold for discrete-time Markov process. arXiv:1706.00892 (2017)Google Scholar
  56. 56.
    Proesmans, K., Van den Broeck, C.: Discrete-time thermodynamic uncertainty relation. Europhys. Lett. 119, 20001 (2017)ADSCrossRefGoogle Scholar
  57. 57.
    Shiraishi, N., Funo, K., Saito, K.: Speed limit for classical stochastic processes. Phys. Rev. Lett. 121, 070601 (2018)ADSCrossRefGoogle Scholar
  58. 58.
    Siegel, A.: Differential-operator approximations to the linear Boltzmann equation. J. Am. Phys. 1, 378 (1960)ADSMathSciNetzbMATHGoogle Scholar
  59. 59.
    Van den Broeck, C., Kawai, R., Meurs, P.: Microscopic analysis of a thermal Brownian motor. Phys. Rev. Lett. 93, 090601 (2004)CrossRefGoogle Scholar
  60. 60.
    Fruleux, A., Kawai, R., Sekimoto, K.: Momentum transfer in nonequilibrium steady states. Phys. Rev. Lett. 108, 160601 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    Seifert, U.: Stochastic thermodynamics, fluctuation theorems, and molecular machines. Rep. Prog. Phys. 75, 126001 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsKeio UniversityYokohamaJapan

Personalised recommendations