Thermodynamic uncertainty relations constrain non-equilibrium fluctuations

Abstract

In equilibrium thermodynamics, there exists a well-established connection between dynamical fluctuations of a physical system and the dissipation of its energy into an environment. However, few similarly quantitative tools are available for the description of physical systems out of equilibrium. Here, we offer our perspective on the recent development of a new class of inequalities known as thermodynamic uncertainty relations, which have revealed that dissipation constrains current fluctuations in steady states arbitrarily far from equilibrium. We discuss the stochastic thermodynamic origin of these inequalities, and highlight recent efforts to expand their applicability, which have focused on connections between current fluctuations and the fluctuation theorems.

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Fig. 1: Thermodynamic constraints on fluctuations in particle currents.

References

  1. 1.

    Van den Broeck, C. & Esposito, M. Ensemble and trajectory thermodynamics: a brief introduction. Physica A 418, 6–16 (2015).

  2. 2.

    Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75, 126001 (2012).

  3. 3.

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

  4. 4.

    Barato, A. C. & Seifert, U. Thermodynamic uncertainty relation for biomolecular processes. Phys. Rev. Lett. 114, 158101 (2015).

  5. 5.

    Gingrich, T. R., Horowitz, J. M., Perunov, N. & England, J. L. Dissipation bounds all steady-state current fluctuations. Phys. Rev. Lett. 116, 120601 (2016).

  6. 6.

    Touchette, H. & Harris, R. J. in Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond (eds Klages, R. et al.) 335–360 (Wiley-VCH, 2013).

  7. 7.

    Pietzonka, P., Ritort, F. & Seifert, U. Finite-time generalization of the thermodynamic uncertainty relation. Phys. Rev. E 96, 012101 (2017).

  8. 8.

    Pietzonka, P. & Seifert, U. Universal trade-off between power, efficiency, and constancy in steady-state heat engines. Phys. Rev. Lett. 120, 190602 (2018).

  9. 9.

    Gingrich, T. R., Rotskoff, G. M. & Horowitz, J. M. Inferring dissipation from current fluctuations. J. Phys. A 50, 184004 (2017).

  10. 10.

    Horowitz, J. M. & Gingrich, T. R. Proof of the finite-time thermodynamic uncertainty relation for steady-state currents. Phys. Rev. E 96, 020103 (2017).

  11. 11.

    Yan, J. Achievability of thermodynamic uncertainty relations. Preprint at https://arxiv.org/abs/1905.00929 (2019).

  12. 12.

    Polettini, M., Lazarescu, A. & Esposito, M. Tightening the uncertainty principle for stochastic currents. Phys. Rev. E 94, 052104 (2016).

  13. 13.

    Nardini, C. & Touchette, H. Process interpretation of current entropic bounds. Eur. Phys. J. B 91, 16 (2018).

  14. 14.

    Dechant, A. & Sasa, S.-I. Current fluctuations and transport efficiency for general langevin systems. J. Stat. Mech. Theor. Exp. 2018, 063209 (2018).

  15. 15.

    Dechant, A. & Sasa, S.-I. Fluctuation-response inequality out of equilibrium. Preprint at https://arxiv.org/abs/1804.08250 (2018).

  16. 16.

    Dechant, A. Multidimensional thermodynamic uncertainty relations. J. Phys. A 52 (2018).

  17. 17.

    Hasegawa, Y. & Van, Vu,T. Uncertainty relations in stochastic processes: an information inequality approach. Phys. Rev. E 99, 062126 (2019).

  18. 18.

    Pigolotti, S., Neri, I., Roldán, E. & Jülicher, F. Generic properties of stochastic entropy production. Phys. Rev. Lett. 119, 140604 (2017).

  19. 19.

    Pietzonka, P., Barato, A. C. & Seifert, U. Universal bounds on current fluctuations. Phys. Rev. E 93, 052145 (2016).

  20. 20.

    Chiuchiù, C. & Pigolotti, S. Mapping of uncertainty relations between continuous and discrete time. Phys. Rev. E 97, 032109 (2018).

  21. 21.

    Di Terlizzi, I. & Baiesi, M. Kinetic uncertainty relation. J. Phys. A 52, 02LT03 (2018).

  22. 22.

    Shiraishi, N. Finite-time thermodynamic uncertainty relation do not hold for discrete-time Markov process. Preprint at https://arxiv.org/abs/1706.00892 (2017).

  23. 23.

    Proesmans, K. & Van den Broeck, C. Discrete-time thermodynamic uncertainty relation. Europhys. Lett. 119, 20001 (2017).

  24. 24.

    Falasco, G., Pfaller, R., Bregulla, A. P. & Kroy, K. Exact symmetries in the velocity fluctuations of a hot Brownian swimmer. Phys. Rev. E 94, 030620(R) (2016).

  25. 25.

    Hyeon, C. & Hwang, W. Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials. Phys. Rev. E 96, 012156 (2017).

  26. 26.

    Barato, A. C. & Seifert, U. Coherence of biochemical oscillations is bounded by driving force and network topology. Phys. Rev. E 95, 062409 (2017).

  27. 27.

    Proesmans, K., Peliti, L. & Lacoste, D. in Chemical Kinetics: Beyond the Textbook (eds Lindenberg, K. et al.) Ch. 17 (World Scientific, 2019).

  28. 28.

    Wierenga, H., ten Wolde, P. R. & Beck, N. B. Quantifying fluctautions in reversible enzymatic cycles and clocks. Phys. Rev. E 97, 042404 (2018).

  29. 29.

    Marsland, R., Cui, W. & Horowitz, J. M. The thermodynamic uncertainty relation in biochemcial oscillations. J. R. Soc. Interface 16, (2019).

  30. 30.

    Brown, A. I. & Sivak, D. A. Pulling cargo increases the precision of molecular motor progress. Europhy. Lett. 126, 40004 (2019).

  31. 31.

    Shankar, S. & Marchetti, M. C. Hidden entropy production and work fluctuations in an ideal active gas. Phys. Rev. E 98, 020604(R) (2018).

  32. 32.

    Lee, S., Hyeon, C. & Jo, J. Thermodynamic uncertainty relation of interacting oscillators in synchrony. Phys. Rev. E 98, 032119 (2018).

  33. 33.

    Li, J., Horowitz, J. M., Gingrich, T. R. & Fakhri, N. Quantifying dissipation using fluctauting currents. Nat. Commun. 10, 1666 (2019).

  34. 34.

    Pietzonka, P., Barato, A. C. & Seifert, U. Universal bound on the efficiency of molecular motors. J. Stat. Mech. Theor. Exp. 2016, 124004 (2016).

  35. 35.

    Seifert, U. Stochastic thermodynamics: from principles to the cost of precision. Physica A 504, 176–191 (2018).

  36. 36.

    Barato, A. C. & Seifert, U. Cost and precision of Brownian clocks. Phys. Rev. X 6, 041053 (2016).

  37. 37.

    Ptaszynński, K. Coherence-enhanced constancy of a quantum thermoelectric generator. Phys. Rev. B 98, 085425 (2018).

  38. 38.

    Agarwalla, B. K. & Segal, D. Assessing the validity of the thermodynamic uncertainty relation in quantum systems. Phys. Rev. B 98, 155438 (2018).

  39. 39.

    Liu, J. & Segal, D. Thermodynamic uncertainty relation in quantum thermoelectric junctions. Phys. Rev. E 99, 062141 (2019).

  40. 40.

    Brandner, K., Hanazato, T. & Saito, K. Thermodynamic bounds on precision in ballistic multiterminal transport. Phys. Rev. Lett. 120, 090601 (2018).

  41. 41.

    Macieszczak, K., Brandner, K. & Garrahan, J. P. Unified thermodynamic uncertainty relations in linear response. Phys. Rev. Lett. 121, 130601 (2018).

  42. 42.

    Fischer, L. P., Pietzonka, P. & Seifert, U. Large deviation function for a driven underdamped particle in a periodic potential. Phys. Rev. E 97, 022143 (2018).

  43. 43.

    Chun, H.-M., Fischer, L. P. & Seifert, U. Effect of a magnetic field on the thermodynamic uncertainty relation. Phys. Rev. E 99, 042128 (2019).

  44. 44.

    Hasegawa, Y. & Van Vu, T. Fluctuation theorem uncertainty relation. Phys. Rev. Lett. 123, 110602 (2019).

  45. 45.

    Timpanaro, A. M., Guarnieri, G., Goold, J. & Landi, G. T. Thermodynamic uncertainty relations from exchange fluctuation theorems. Phys. Rev. Lett. 123, 090604 (2019).

  46. 46.

    Seifert, U. From stochastic thermodynamics to thermodynamic inference. Annu. Rev. Condens. Matter Phys. 10, 171–192 (2019).

  47. 47.

    Proesmans, K. & Horowitz, J. M. Hysteretic thermodynamic uncertainty relation for systems with broken time-reversal symmetry. J. Stat. Mech. Theor. Exp. 2019, 054005 (2019).

  48. 48.

    Potts, P. P. & Samuelsson, P. Thermodynamic uncertainty relations including measurement and feedback. Preprint at https://arxiv.org/abs/1904.04913 (2019).

  49. 49.

    Garrahan, J. P. Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables. Phys. Rev. E 95, 032134 (2017).

  50. 50.

    Gingrich, T. R. & Horowitz, J. M. Fundamental bounds on first passage time fluctuations for currents. Phys. Rev. Lett. 119, 170601 (2017).

  51. 51.

    Guioth, J. & Lacoste, D. Thermodynamic bounds on equilibrium fluctuations of a global or local order parameter. Europhys. Lett. 115, 60007 (2016).

  52. 52.

    Barato, A. C., Chetrite, R., Faggionato, A. & Gabrielli, D. Bounds on current fluctuations in periodically driven systems. New J. Phys. 20, 103023 (2018).

  53. 53.

    Barato, A. C., Chetrite, R., Faggionato, A. & Gabrielli, D. A unifying picture of generalized thermodynamic uncertainty relations. J. Stat. Mech. Theor. Exp. 2019, 084017 (2019).

  54. 54.

    Koyuk, T., Seifert, U. & Pietzonka, P. A generalization of the thermodynamic uncertainty relation to periodically driven systems. J. Phys. A 52, 02LT02 (2018).

  55. 55.

    Pietzonka, P., Barato, A. C. & Seifert, U. Affinity-and topology-dependent bound on current fluctuations. J. Phys. A 49, 34LT01 (2016).

  56. 56.

    Koyuk, T. & Seifert, U. Operationally accessible bounds on fluctuations and entropy production in periodically driven systems. Phys. Rev. Lett. 122, 230601 (2019).

  57. 57.

    Owen, J. A., Gingrich, T. R. & Horowitz, J. M. Universal thermodynamic bounds on nonequilibrium response with biochemical applications. Preprint at https://arxiv.org/abs/1905.07449 (2019).

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Acknowledgements

We gratefully acknowledge our collaborators on this work, J. England, N. Perunov, G. M. Rostkoff, K. Proesmans, H. Vroylandt, J. Li, N. Fakhri, R. Marsland III and W. Cui.

Author information

Correspondence to Jordan M. Horowitz or Todd R. Gingrich.

Additional information

Peer review information Nature Physics thanks Shin-ichi Sasa and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Horowitz, J.M., Gingrich, T.R. Thermodynamic uncertainty relations constrain non-equilibrium fluctuations. Nat. Phys. 16, 15–20 (2020). https://doi.org/10.1038/s41567-019-0702-6

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