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  • Letter
  • Open Access

Asymmetric thermal relaxation in driven systems: Rotations go opposite ways

Cai Dieball, Gerrit Wellecke*, and Aljaž Godec†

  • Mathematical bioPhysics group, Max Planck Institute for Multidisciplinary Sciences, 37077 Göttingen, Germany

  • *Present address: Theory of Biological Fluids, Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany.
  • †agodec@mpinat.mpg.de
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Phys. Rev. Research 5, L042030 – Published 27 November, 2023

DOI: https://doi.org/10.1103/PhysRevResearch.5.L042030

Abstract

It was predicted and recently experimentally confirmed that systems with microscopically reversible dynamics in quadratic potentials warm up faster than they cool down. This thermal relaxation asymmetry challenged our understanding of relaxation far from equilibrium. Because the intuition and proof hinged on the dynamics obeying detailed balance, it was not clear whether the asymmetry persists in systems with irreversible dynamics. To fill this gap, we here prove the relaxation asymmetry for systems driven out of equilibrium by a general linear drift. The asymmetry persists due to a nontrivial isomorphism between driven and reversible processes. Moreover, rotations of level sets of probability densities emerge that, strikingly, occur in opposite directions during heating and cooling. This highlights that noisy systems do not relax by passing through local equilibria.

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References (108)

  1. S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, 2nd ed. (North-Holland, Amsterdam, 1962).
  2. L. Onsager, Reciprocal relations in irreversible processes. I., Phys. Rev. 37, 405 (1931).
  3. L. Onsager, Reciprocal relations in irreversible processes. II., Phys. Rev. 38, 2265 (1931).
  4. R. Kubo, M. Yokota, and S. Nakajima, Statistical-mechanical theory of irreversible processes. II. Response to thermal disturbance, J. Phys. Soc. Jpn. 12, 1203 (1957).
  5. M. Baiesi and C. Maes, An update on the nonequilibrium linear response, New J. Phys. 15, 013004 (2013).
  6. C. Maes, K. Netočný, and B. Wynants, Monotonic return to steady nonequilibrium, Phys. Rev. Lett. 107, 010601 (2011).
  7. M. Polettini and M. Esposito, Nonconvexity of the relative entropy for Markov dynamics: A Fisher information approach, Phys. Rev. E 88, 012112 (2013).
  8. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Minimum dissipation principle in stationary non-equilibrium states, J. Stat. Phys. 116, 831 (2004).
  9. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory, Rev. Mod. Phys. 87, 593 (2015).
  10. R. S. Maier and D. L. Stein, Escape problem for irreversible systems, Phys. Rev. E 48, 931 (1993).
  11. F. Bouchet and J. Reygner, Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes, Ann. Henri Poincaré 17, 3499 (2016).
  12. J. Casademunt, R. Mannella, P. V. E. McClintock, F. E. Moss, and J. M. Sancho, Relaxation times of non-Markovian processes, Phys. Rev. A 35, 5183 (1987).
  13. I. M. Sokolov, Solutions of a class of non-Markovian Fokker-Planck equations, Phys. Rev. E 66, 041101 (2002).
  14. I. M. Sokolov, Cyclization of a polymer: First-passage problem for a Non-Markovian process, Phys. Rev. Lett. 90, 080601 (2003).
  15. B. Dybiec, I. M. Sokolov, and A. V. Chechkin, Relaxation to stationary states for anomalous diffusion, Commun. Nonlinear Sci. Numer. Simul. 16, 4549 (2011).
  16. D. E. Makarov, Interplay of non-Markov and internal friction effects in the barrier crossing kinetics of biopolymers: Insights from an analytically solvable model, J. Chem. Phys. 138, 014102 (2013).
  17. R. Satija and D. E. Makarov, Generalized Langevin equation as a model for barrier crossing dynamics in biomolecular folding, J. Phys. Chem. B 123, 802 (2019).
  18. A. Lapolla and A. Godec, Manifestations of projection-induced memory: General theory and the tilted single file, Front. Phys. 7, 182 (2019).
  19. A. Lapolla and A. Godec, Single-file diffusion in a bi-stable potential: Signatures of memory in the barrier-crossing of a tagged-particle, J. Chem. Phys. 153, 194104 (2020).
  20. P. Talkner and P. Hänggi, Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical, Rev. Mod. Phys. 92, 041002 (2020).
  21. C. Ayaz, L. Scalfi, B. A. Dalton, and R. R. Netz, Generalized Langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme, Phys. Rev. E 105, 054138 (2022).
  22. L. F. Cugliandolo, D. S. Dean, and J. Kurchan, Fluctuation-dissipation theorems and entropy production in relaxational systems, Phys. Rev. Lett. 79, 2168 (1997).
  23. E. Lippiello, M. Baiesi, and A. Sarracino, Nonequilibrium fluctuation-dissipation theorem and heat production, Phys. Rev. Lett. 112, 140602 (2014).
  24. A. Gal and O. Raz, Precooling strategy allows exponentially faster heating, Phys. Rev. Lett. 124, 060602 (2020).
  25. Z. Lu and O. Raz, Nonequilibrium thermodynamics of the Markovian Mpemba effect and its inverse, Proc. Natl. Acad. Sci. USA 114, 5083 (2017).
  26. A. Lasanta, F. Vega Reyes, A. Prados, and A. Santos, When the hotter cools more quickly: Mpemba effect in granular fluids, Phys. Rev. Lett. 119, 148001 (2017).
  27. M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvión, A. Gordillo-Guerrero, D. Iñiguez, A. Lasanta, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. M. Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane et al., The Mpemba effect in spin glasses is a persistent memory effect, Proc. Natl. Acad. Sci. USA 116, 15350 (2019).
  28. A. Kumar and J. Bechhoefer, Exponentially faster cooling in a colloidal system, Nature (London) 584, 64 (2020).
  29. F. Carollo, A. Lasanta, and I. Lesanovsky, Exponentially accelerated approach to stationarity in Markovian open quantum systems through the Mpemba effect, Phys. Rev. Lett. 127, 060401 (2021).
  30. A. Kumar, R. Chétrite, and J. Bechhoefer, Anomalous heating in a colloidal system, Proc. Natl. Acad. Sci. USA 119, e2118484119 (2022).
  31. I. Klich, O. Raz, O. Hirschberg, and M. Vucelja, Mpemba index and anomalous relaxation, Phys. Rev. X 9, 021060 (2019).
  32. R. Holtzman and O. Raz, Landau theory for the Mpemba effect through phase transitions, Commun. Phys. 5, 280 (2022).
  33. J. Degünther and U. Seifert, Anomalous relaxation from a non-equilibrium steady state: An isothermal analog of the Mpemba effect, Europhys. Lett. 139, 41002 (2022).
  34. I. L. Morgan, R. Avinery, G. Rahamim, R. Beck, and O. A. Saleh, Glassy dynamics and memory effects in an intrinsically disordered protein construct, Phys. Rev. Lett. 125, 058001 (2020).
  35. A. Militaru, A. Lasanta, M. Frimmer, L. L. Bonilla, L. Novotny, and R. A. Rica, Kovacs memory effect with an optically levitated nanoparticle, Phys. Rev. Lett. 127, 130603 (2021).
  36. F. Bouchet, K. Gawedzki, and C. Nardini, Perturbative calculation of quasi-potential in non-equilibrium diffusions: A mean-field example, J. Stat. Phys. 163, 1157 (2016).
  37. Y. Baek and Y. Kafri, Singularities in large deviation functions, J. Stat. Mech. (2015) P08026.
  38. J. P. Garrahan, R. L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, and F. van Wijland, Dynamical first-order phase transition in kinetically constrained models of glasses, Phys. Rev. Lett. 98, 195702 (2007).
  39. T. Speck, A. Engel, and U. Seifert, The large deviation function for entropy production: the optimal trajectory and the role of fluctuations, J. Stat. Mech. (2012) P12001.
  40. P. Tsobgni Nyawo and H. Touchette, A minimal model of dynamical phase transition, Europhys. Lett. 116, 50009 (2016).
  41. V. Ermolaev and C. Külske, Low-temperature dynamics of the curie-weiss model: Periodic orbits, multiple histories, and loss of gibbsianness, J. Stat. Phys. 141, 727 (2010).
  42. J. Meibohm and M. Esposito, Finite-time dynamical phase transition in nonequilibrium relaxation, Phys. Rev. Lett. 128, 110603 (2022).
  43. J. Meibohm and M. Esposito, Landau theory for finite-time dynamical phase transitions, New J. Phys. 25, 023034 (2023).
  44. K. Blom and A. Godec, Global speed limit for finite-time dynamical phase transition and nonequilibrium relaxation, arXiv:2209.14287.
  45. P. Pietzonka, F. Ritort, and U. Seifert, Finite-time generalization of the thermodynamic uncertainty relation, Phys. Rev. E 96, 012101 (2017).
  46. A. Dechant and S.-i. Sasa, Current fluctuations and transport efficiency for general Langevin systems, J. Stat. Mech. (2018) 063209.
  47. K. Liu, Z. Gong, and M. Ueda, Thermodynamic uncertainty relation for arbitrary initial states, Phys. Rev. Lett. 125, 140602 (2020).
  48. T. Koyuk and U. Seifert, Operationally accessible bounds on fluctuations and entropy production in periodically driven systems, Phys. Rev. Lett. 122, 230601 (2019).
  49. T. Koyuk and U. Seifert, Thermodynamic uncertainty relation for time-dependent driving, Phys. Rev. Lett. 125, 260604 (2020).
  50. C. Dieball and A. Godec, Direct route to thermodynamic uncertainty relations and their saturation, Phys. Rev. Lett. 130, 087101 (2023).
  51. N. Shiraishi, K. Funo, and K. Saito, Speed limit for classical stochastic processes, Phys. Rev. Lett. 121, 070601 (2018).
  52. E. Aurell, C. Mejía-Monasterio, and P. Muratore-Ginanneschi, Optimal protocols and optimal transport in stochastic thermodynamics, Phys. Rev. Lett. 106, 250601 (2011).
  53. K. Yoshimura and S. Ito, Thermodynamic uncertainty relation and thermodynamic speed limit in deterministic chemical reaction networks, Phys. Rev. Lett. 127, 160601 (2021).
  54. S. Ito and A. Dechant, Stochastic time evolution, information geometry, and the Cramér-Rao bound, Phys. Rev. X 10, 021056 (2020).
  55. N. Shiraishi and K. Saito, Information-theoretical bound of the irreversibility in thermal relaxation processes, Phys. Rev. Lett. 123, 110603 (2019).
  56. M. Ibáñez, C. Dieball, A. Lasanta, A. Godec, and R. A. Rica, Heating and cooling are fundamentally asymmetric and evolve along distinct pathways, arXiv:2302.09061 [Nat. Phys. (to be published)].
  57. A. Lapolla and A. Godec, Faster uphill relaxation in thermodynamically equidistant temperature quenches, Phys. Rev. Lett. 125, 110602 (2020).
  58. T. V. Vu and Y. Hasegawa, Toward relaxation asymmetry: Heating is faster than cooling, Phys. Rev. Res. 3, 043160 (2021).
  59. S. K. Manikandan, Equidistant quenches in few-level quantum systems, Phys. Rev. Res. 3, 043108 (2021).
  60. J. Meibohm, D. Forastiere, T. Adeleke-Larodo, and K. Proesmans, Relaxation-speed crossover in anharmonic potentials, Phys. Rev. E 104, L032105 (2021).
  61. A. Lapolla and A. Godec, Erratum: Faster uphill relaxation in thermodynamically equidistant temperature quenches [Phys. Rev. Lett. 125, 110602 (2020)], Phys. Rev. Lett. 128, 229901(E) (2022).
  62. C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Accelerating diffusions, Ann. Appl. Probab. 15, 1433 (2005).
  63. K. S. Turitsyn, M. Chertkov, and M. Vucelja, Irreversible Monte Carlo algorithms for efficient sampling, Phys. D: Nonlinear Phenom. 240, 410 (2011).
  64. S. C. Kapfer and W. Krauth, Irreversible local Markov chains with rapid convergence towards equilibrium, Phys. Rev. Lett. 119, 240603 (2017).
  65. L. Rey-Bellet and K. Spiliopoulos, Irreversible Langevin samplers and variance reduction: a large deviations approach, Nonlinearity 28, 2081 (2015).
  66. F. Coghi, R. Chetrite, and H. Touchette, Role of current fluctuations in nonreversible samplers, Phys. Rev. E 103, 062142 (2021).
  67. H. Qian, A decomposition of irreversible diffusion processes without detailed balance, J. Math. Phys. 54, 053302 (2013).
  68. C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Accelerating Gaussian diffusions, Ann. Appl. Probab. 3, 897 (1993).
  69. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.5.L042030 for further details, derivations, and figures.
  70. J. L. Lebowitz and P. G. Bergmann, Irreversible Gibbsian ensembles, Ann. Phys. (NY) 1, 1 (1957).
  71. M. C. Mackey, The dynamic origin of increasing entropy, Rev. Mod. Phys. 61, 981 (1989).
  72. S. Vaikuntanathan and C. Jarzynski, Dissipation and lag in irreversible processes, Europhys. Lett. 87, 60005 (2009).
  73. M. Esposito and C. Van den Broeck, Three detailed fluctuation theorems, Phys. Rev. Lett. 104, 090601 (2010).
  74. C. Van den Broeck and M. Esposito, Three faces of the second law. II. Fokker-Planck formulation, Phys. Rev. E 82, 011144 (2010).
  75. S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statist. 22, 79 (1951).
  76. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 1st ed. (North Holland, Amsterdam, 1981).
  77. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences (Springer-Verlag, Berlin, 1985).
  78. C. Dieball and A. Godec, Coarse graining empirical densities and currents in continuous-space steady states, Phys. Rev. Res. 4, 033243 (2022).
  79. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  80. B. H. Zimm, Dynamics of polymer molecules in dilute solution: Viscoelasticity, flow birefringence and dielectric loss, J. Chem. Phys. 24, 269 (1956).
  81. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, International Series of Monographs on Physics (Clarendon Press, Oxford, 1988).
  82. R. R. Cheng, A. T. Hawk, and D. E. Makarov, Exploring the role of internal friction in the dynamics of unfolded proteins using simple polymer models, J. Chem. Phys. 138, 074112 (2013).
  83. A. Soranno, B. Buchli, D. Nettels, R. R. Cheng, S. Müller-Späth, S. H. Pfeil, A. Hoffmann, E. A. Lipman, D. E. Makarov, and B. Schuler, Quantifying internal friction in unfolded and intrinsically disordered proteins with single-molecule spectroscopy, Proc. Natl. Acad. Sci. USA 109, 17800 (2012).
  84. B. Schuler, A. Soranno, H. Hofmann, and D. Nettels, Single-molecule FRET spectroscopy and the polymer physics of unfolded and intrinsically disordered proteins, Annu. Rev. Biophys. 45, 207 (2016).
  85. J. O. Daldrop, J. Kappler, F. N. Brünig, and R. R. Netz, Butane dihedral angle dynamics in water is dominated by internal friction, Proc. Natl. Acad. Sci. USA 115, 5169 (2018).
  86. D. E. Smith, H. P. Babcock, and S. Chu, Single-polymer dynamics in steady shear flow, Science 283, 1724 (1999).
  87. S. Gerashchenko and V. Steinberg, Statistics of tumbling of a single polymer molecule in shear flow, Phys. Rev. Lett. 96, 038304 (2006).
  88. P. S. Doyle, B. Ladoux, and J.-L. Viovy, Dynamics of a tethered polymer in shear flow, Phys. Rev. Lett. 84, 4769 (2000).
  89. T. T. Perkins, D. E. Smith, R. G. Larson, and S. Chu, Stretching of a single tethered polymer in a uniform flow, Science 268, 83 (1995).
  90. C. M. Schroeder, R. E. Teixeira, E. S. G. Shaqfeh, and S. Chu, Characteristic periodic motion of polymers in shear flow, Phys. Rev. Lett. 95, 018301 (2005).
  91. R. E. Teixeira, H. P. Babcock, E. S. G. Shaqfeh, and S. Chu, Shear thinning and tumbling dynamics of single polymers in the flow-gradient plane, Macromolecules 38, 581 (2005).
  92. M. Harasim, B. Wunderlich, O. Peleg, M. Kröger, and A. R. Bausch, Direct observation of the dynamics of semiflexible polymers in shear flow, Phys. Rev. Lett. 110, 108302 (2013).
  93. A. Alexander-Katz, M. F. Schneider, S. W. Schneider, A. Wixforth, and R. R. Netz, Shear-flow-induced unfolding of polymeric globules, Phys. Rev. Lett. 97, 138101 (2006).
  94. T. T. Perkins, D. E. Smith, and S. Chu, Single polymer dynamics in an elongational flow, Science 276, 2016 (1997).
  95. Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: The fountain of probability, Phys. Rev. Lett. 101, 128301 (2008).
  96. G. Metafune, Lp-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30, 97 (2001).
  97. G. Metafune, D. Pallara, and E. Priola, Spectrum of Ornstein-Uhlenbeck operators in lp spaces with respect to invariant measures, J. Funct. Anal. 196, 40 (2002).
  98. H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10, 545 (1959).
  99. T. Hatano and S.-i. Sasa, Steady-state thermodynamics of Langevin systems, Phys. Rev. Lett. 86, 3463 (2001).
  100. A. Dechant and S.-i. Sasa, Continuous time reversal and equality in the thermodynamic uncertainty relation, Phys. Rev. Res. 3, L042012 (2021).
  101. C. Dieball and A. Godec, Mathematical, thermodynamical, and experimental necessity for coarse graining empirical densities and currents in continuous space, Phys. Rev. Lett. 129, 140601 (2022).
  102. G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations (Almqvist & Wiksell, Uppsala, 1958).
  103. V. Blickle and C. Bechinger, Realization of a micrometre-sized stochastic heat engine, Nature Phys. 8, 143 (2012).
  104. I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, and R. A. Rica, Adiabatic processes realized with a trapped Brownian particle, Phys. Rev. Lett. 114, 120601 (2015).
  105. I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, Brownian Carnot engine, Nature Phys. 12, 67 (2016).
  106. S. Krishnamurthy, S. Ghosh, D. Chatterji, R. Ganapathy, and A. K. Sood, A micrometre-sized heat engine operating between bacterial reservoirs, Nature Phys. 12, 1134 (2016).
  107. M. Rademacher, M. Konopik, M. Debiossac, D. Grass, E. Lutz, and N. Kiesel, Nonequilibrium control of thermal and mechanical changes in a levitated system, Phys. Rev. Lett. 128, 070601 (2022).
  108. J. Gladrow, M. Ribezzi-Crivellari, F. Ritort, and U. F. Keyser, Experimental evidence of symmetry breaking of transition-path times, Nature Commun. 10, 55 (2019).

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Phys. Rev. Research 5, L042030– Published 27 November, 2023
  • Received 13 April 2023
  • Revised 12 June 2023
  • Accepted 24 October 2023
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