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Comparative Microscopic Study of Entropies and their Production (2403.09403v2)

Published 14 Mar 2024 in cond-mat.stat-mech, cond-mat.mes-hall, and quant-ph

Abstract: We study the time evolution of eleven microscopic entropy definitions (of Boltzmann-surface, Gibbs-volume, canonical, coarse-grained-observational, entanglement and diagonal type) and three microscopic temperature definitions (based on Boltzmann, Gibbs or canonical entropy). This is done for the archetypal nonequilibrium setup of two systems exchanging energy, modeled here with random matrix theory, based on numerical integration of the Schroedinger equation. We consider three types of pure initial states (local energy eigenstates, decorrelated and entangled microcanonical states) and three classes of systems: (A) two normal systems, (B) a normal and a negative temperature system and (C) a normal and a negative heat capacity system. We find: (1) All types of initial states give rise to the same macroscopic dynamics. (2) Entanglement and diagonal entropy sensitively depend on the microstate, in contrast to all other entropies. (3) For class B and C, Gibbs-volume entropies can violate the second law and the associated temperature becomes meaningless. (4) For class C, Boltzmann-surface entropies can violate the second law and the associated temperature becomes meaningless. (5) Canonical entropy has a tendency to remain almost constant. (6) For a Haar random initial state, entanglement or diagonal entropy behave similar or identical to coarse-grained-observational entropy.

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References (87)
  1. R. Clausius, Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Ann. Phys. 201, 353 (1865), https://doi.org/10.1002/andp.18652010702.
  2. M. Tribus and E. C. McIrvine, Energy and information, Sci. Am. 225(3), 179 (1971).
  3. J. von Neumann, Beweis des Ergodensatzes und des H𝐻{H}italic_H-Theorems in der neuen Mechanik, Z. Phys. 57, 30 (1929), 10.1007/BF01339852.
  4. J. von Neumann, Proof of the ergodic theorem and the H-theorem in quantum mechanics, European Phys. J. H 35, 201 (2010), 10.1140/epjh/e2010-00008-5.
  5. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer-Verlag, Berlin, 10.1007/978-3-642-61409-5 (1932).
  6. D. Ruelle, Statistical Mechanics, Benjamin, New York (1969).
  7. G. Gallavotti, Statistical Mechanics: A Short Treatise, Springer Science & Business Media, Berlin Heidelberg, 10.1007/978-3-662-03952-6 (1999).
  8. H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep. 478, 1 (2009), 10.1016/j.physrep.2009.05.002.
  9. A. Campa, T. Dauxois and S. Ruffo, Statistical mechanics and dynamics of solvable models with long-range interactions, Phys. Rep. 480, 57 (2009), 10.1016/j.physrep.2009.07.001.
  10. J. D. Bekenstein, Black‐hole thermodynamics, Phys. Today 33(1), 24 (1980), 10.1063/1.2913906.
  11. T. Padmanabhan, Statistical mechanics of gravitating systems, Phys. Rep. 188(5), 285 (1990), https://doi.org/10.1016/0370-1573(90)90051-3.
  12. R. Bousso, The holographic principle, Rev. Mod. Phys. 74, 825 (2002), 10.1103/RevModPhys.74.825.
  13. T. Padmanabhan, Thermodynamical aspects of gravity: new insights, Rep. Prog. Phys. 73(4), 046901 (2010), 10.1088/0034-4885/73/4/046901.
  14. Long-range interacting quantum systems, Rev. Mod. Phys. 95, 035002 (2023), 10.1103/RevModPhys.95.035002.
  15. H. Price, Time’s Arrow and Archimedes’ Point, Oxford University Press, New York (1996).
  16. D. Z. Albert, Time and Chance, Harvard University Press (2000).
  17. J. Earman, The ”Past Hypothesis”: Not even false, Stud. Hist. Phil. Sci. B 37, 399 (2006).
  18. J. Barbour, The Janus Point: A New Theory of Time, Penguin Random House, London (2020).
  19. I. Bloch, J. Dalibard and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008), 10.1103/RevModPhys.80.885.
  20. M. Lewenstein, A. Sanpera and V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems, Oxford University Press, Oxford, 10.1093/acprof:oso/9780199573127.001.0001 (2012).
  21. R. Blatt and C. Roos, Quantum simulations with trapped ions, Nature Phys. 8, 277–284 (2012), 10.1038/nphys2252.
  22. Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions, Springer Nature Switzerland, Cham, https://doi.org/10.1007/978-3-319-99046-0 (2018).
  23. S. Deffner and S. Campbell, Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information, Morgan & Claypool, San Rafael, CA, 10.1088/2053-2571/ab21c6 (2019).
  24. P. Strasberg, Quantum Stochastic Thermodynamics: Foundations and Selected Applications, Oxford University Press, Oxford, 10.1093/oso/9780192895585.001.0001 (2022).
  25. Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas, Nat. Phys. 8, 325 (2012), 10.1038/nphys2232.
  26. Experimental verification of Landauer’s principle linking information and thermodynamics, Nature (London) 483, 187 (2012), 10.1038/nature10872.
  27. A Thermoelectric Heat Engine with Ultracold Atoms, Science 342, 713 (2013), 10.1126/science.1242308.
  28. Negative Absolute Temperature for Motional Degrees of Freedom, Science 339(6115), 52 (2013), 10.1126/science.1227831.
  29. M. Gavrilov, R. Chétrite and J. Bechhoefer, Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs-Shannon form, Proc. Natl. Acad. Sci. USA 114, 11097 (2017), https://doi.org/10.1073/pnas.1708689114.
  30. Reaching the ultimate energy resolution of a quantum detector, Nat. Comm. 11, 367 (2020), 10.1038/s41467-019-14247-2.
  31. P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (04), P04010 (2005), 10.1088/1742-5468/2005/04/P04010.
  32. A. Polkovnikov, Microscopic diagonal entropy and its connection to basic thermodynamic relations, Ann. Phys. 326(2), 486 (2011), https://doi.org/10.1016/j.aop.2010.08.004.
  33. Entropy of Isolated Quantum Systems after a Quench, Phys. Rev. Lett. 107, 040601 (2011), 10.1103/PhysRevLett.107.040601.
  34. V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Nat. Acad. Sci. 114, 7947 (2017), 10.1073/pnas.1703516114.
  35. K. Ptaszyński and M. Esposito, Entropy production in open systems: The predominant role of intraenvironment correlations, Phys. Rev. Lett. 123, 200603 (2019), 10.1103/PhysRevLett.123.200603.
  36. Quantum coarse-grained entropy and thermodynamics, Phys. Rev. A 99, 010101 (2019), 10.1103/PhysRevA.99.010101.
  37. Quantum coarse-grained entropy and thermalization in closed systems, Phys. Rev. A 99, 012103 (2019), 10.1103/PhysRevA.99.012103.
  38. Typical and extreme entropies of long-lived isolated quantum systems, Phys. Rev. A 101, 052101 (2020), 10.1103/PhysRevA.101.052101.
  39. Classical dynamical coarse-grained entropy and comparison with the quantum version, Phys. Rev. E 102, 032106 (2020), 10.1103/PhysRevE.102.032106.
  40. Entropy growth during free expansion of an ideal gas, J, Phys. A: Math. Theor. 55(39), 394002 (2022), 10.1088/1751-8121/ac8a7e.
  41. Boltzmann Entropy of a Freely Expanding Quantum Ideal Gas, J. Stat. Phys. 190, 142 (2023), 10.1007/s10955-023-03154-y.
  42. K. Ptaszyński and M. Esposito, Ensemble dependence of information-theoretic contributions to the entropy production, Phys. Rev. E 107, L052102 (2023), 10.1103/PhysRevE.107.L052102.
  43. K. Ptaszyński and M. Esposito, Quantum and Classical Contributions to Entropy Production in Fermionic and Bosonic Gaussian Systems, PRX Quantum 4, 020353 (2023), 10.1103/PRXQuantum.4.020353.
  44. Microscopic contributions to the entropy production at all times: From nonequilibrium steady states to global thermalization, arXiv: 2309.11812 (2023).
  45. C. Bartsch, R. Steinigeweg and J. Gemmer, Occurrence of exponential relaxation in closed quantum systems, Phys. Rev. E 77, 011119 (2008), 10.1103/PhysRevE.77.011119.
  46. P. Strasberg, T. E. Reinhard and J. C. Schindler, Everything Everywhere All At Once: A First Principles Numerical Demonstration of Emergent Decoherent Histories, arXiv: 2304.10258 (2023).
  47. E. Wigner, Random matrices in physics, SIAM Reviews 9, 1 (1967), 10.1137/1009001.
  48. Random-matrix physics: spectrum and strength fluctuations, Rev. Mod. Phys. 53, 385 (1981), 10.1103/RevModPhys.53.385.
  49. C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69, 731 (1997), 10.1103/RevModPhys.69.731.
  50. Random-matrix theories in quantum physics: common concepts, Phys. Rep. 299, 189 (1998), 10.1016/S0370-1573(97)00088-4.
  51. F. Haake, Quantum Signatures of Chaos, Springer-Verlag, Berlin Heidelberg (2010).
  52. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016), 10.1080/00018732.2016.1198134.
  53. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018), 10.1088/1361-6633/aac9f1.
  54. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991), 10.1103/PhysRevA.43.2046.
  55. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994), 10.1103/PhysRevE.50.888.
  56. M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J. Phys. A 32(7), 1163 (1999), 10.1088/0305-4470/32/7/007.
  57. P. Reimann and L. Dabelow, Refining Deutsch’s approach to thermalization, Phys. Rev. E 103, 022119 (2021), 10.1103/PhysRevE.103.022119.
  58. A. Askar and A. S. Cakmak, Explicit integration method for the time‐dependent Schrodinger equation for collision problems, J. Chem. Phys. 68(6), 2794 (1978), 10.1063/1.436072.
  59. A brief introduction to observational entropy, Found. Phys. 51, 101 (2021), 10.1007/s10701-021-00498-x.
  60. P. Strasberg and A. Winter, First and Second Law of Quantum Thermodynamics: A Consistent Derivation Based on a Microscopic Definition of Entropy, PRX Quantum 2, 030202 (2021), 10.1103/PRXQuantum.2.030202.
  61. J. W. Gibbs, Elementary principles in statistical mechanics, Charles Scribner’s Sons, New-York (1902).
  62. J. Dunkel and S. Hilbert, Consistent thermostatistics forbids negative absolute temperatures, Nat. Phys. 10, 67 (2014).
  63. S. Hilbert, P. Hänggi and J. Dunkel, Thermodynamic laws in isolated systems, Phys. Rev. E 90, 062116 (2014), 10.1103/PhysRevE.90.062116.
  64. D. Frenkel and P. B. Warren, Gibbs, Boltzmann, and negative temperatures, Am. J. Phys. 83(2), 163 (2015), 10.1119/1.4895828.
  65. M. Campisi, Construction of microcanonical entropy on thermodynamic pillars, Phys. Rev. E 91, 052147 (2015), 10.1103/PhysRevE.91.052147.
  66. E. Abraham and O. Penrose, Physics of negative absolute temperatures, Phys. Rev. E 95, 012125 (2017), 10.1103/PhysRevE.95.012125.
  67. R. H. Swendsen, Thermodynamics of finite systems: a key issues review, Rep. Prog. Phys. 81, 072001 (2018), 10.1088/1361-6633/aac18c.
  68. Statistical mechanics of systems with negative temperature, Phys. Rep. 923, 1 (2021), https://doi.org/10.1016/j.physrep.2021.03.007.
  69. R. H. Swendsen, Continuity of the entropy of macroscopic quantum systems, Phys. Rev. E 92, 052110, 10.1103/PhysRevE.92.052110.
  70. Comparison of canonical and microcanonical definitions of entropy, Physica A 467, 474 (2017), https://doi.org/10.1016/j.physa.2016.10.030.
  71. U. Seifert, Entropy and the second law for driven, or quenched, thermally isolated systems, Physica A 552, 121822 (2020), 10.1016/j.physa.2019.121822.
  72. Clausius inequality for finite baths reveals universal efficiency improvements, Phys. Rev. E 104, L022103 (2021), 10.1103/PhysRevE.104.L022103.
  73. Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010), 10.1103/RevModPhys.82.277.
  74. Volume-law entanglement entropy of typical pure quantum states, PRX Quantum 3, 030201 (2022), 10.1103/PRXQuantum.3.030201.
  75. W. Muschik, Empirical foundation and axiomatic treatment of non-equilibrium temperature, Arch. Ration. Mech. Anal. 66, 379 (1977), 10.1007/BF00248902.
  76. W. Muschik and G. Brunk, A concept of non-equilibrium temperature, Int. J. Eng. Sci. 15, 377 (1977), 10.1016/0020-7225(77)90047-7.
  77. H. Tasaki, Jarzynski relations for quantum systems and some applications, arXiv: cond-mat/0009244 (2000), 10.48550/arXiv.cond-mat/0009244.
  78. J. Casas-Vázquez and D. Jou, Temperature in non-equilibrium states: a review of open problems and current proposals, Rep. Prog. Phys. 66, 1937 (2003), 10.1088/0034-4885/66/11/R03.
  79. A. Puglisi, A. Sarracino and A. Vulpiani, Temperature in and out of equilibrium: A review of concepts, tools and attempts, Physics Reports 709-710, 1 (2017), https://doi.org/10.1016/j.physrep.2017.09.001.
  80. C. Bartsch and J. Gemmer, Dynamical Typicality of Quantum Expectation Values, Phys. Rev. Lett. 102, 110403 (2009), 10.1103/PhysRevLett.102.110403.
  81. P. Reimann, Dynamical typicality of isolated many-body quantum systems, Phys. Rev. E 97, 062129 (2018), 10.1103/PhysRevE.97.062129.
  82. P. Reimann and J. Gemmer, Why are macroscopic experiments reproducible? Imitating the behavior of an ensemble by single pure states, Physica A 552, 121840 (2020), https://doi.org/10.1016/j.physa.2019.121840.
  83. X. Xu, C. Guo and D. Poletti, Typicality of nonequilibrium quasi-steady currents, Phys. Rev. A 105, L040203 (2022), 10.1103/PhysRevA.105.L040203.
  84. S. Teufel, R. Tumulka and C. Vogel, Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace, J. Stat. Phys. 190, 69 (2023), 10.1007/s10955-023-03074-x.
  85. Normal typicality and von Neumann’s quantum ergodic theorem, Proc. R. Soc. A 466, 3203 (2010).
  86. M. Rigol and M. Srednicki, Alternatives to Eigenstate Thermalization, Phys. Rev. Lett. 108, 110601 (2012), 10.1103/PhysRevLett.108.110601.
  87. P. Reimann, Generalization of von Neumann’s Approach to Thermalization, Phys. Rev. Lett. 115, 010403 (2015), 10.1103/PhysRevLett.115.010403.
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