- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Thermalization is typical in large classical and quantum harmonic systems

Marco Cattaneo1,*, Marco Baldovin2, Dario Lucente3, Paolo Muratore-Ginanneschi4, and Angelo Vulpiani5

  • *Contact author: marco.cattaneo@helsinki.fi
PDF

Phys. Rev. Research 7, L032002 – Published 2 July, 2025

DOI: https://doi.org/10.1103/h9fj-ylgm

Abstract

We establish an analytical criterion for dynamical thermalization within harmonic systems, applicable to both classical and quantum models. Specifically, we prove that thermalization of various observables—such as particle energies in physically relevant random quadratic Hamiltonians—is typical for large systems (N≫1) with initial conditions drawn from the microcanonical distribution. Moreover, we show that thermalization can also arise from nontypical initial conditions, where only a finite fraction of the normal modes is excited. A different choice of initial conditions, such as all the initial energy localized in a single particle, instead leads to energy equipartition without thermalization. Since the models we consider are integrable, our findings provide a general dynamical basis for an approach to thermalization that bypasses chaos and ergodicity, focusing instead on the physical requirement that thermodynamic observables depend on a large number of normal modes, and they build a bridge between the classical and quantum theories of thermalization.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (83)

  1. L. Peliti, Statistical Mechanics in a Nutshell (Princeton University Press, Princeton, NJ, 2011).
  2. K. Huang, Statistical Mechanics (Wiley, Hoboken, New Jersey, 2008).
  3. L. Landau and E. Lifshitz, Statistical Physics. Part 1 (Pergamon, Oxford, 1980).
  4. S.-K. Ma, Statistical Mechanics (World Scientific, Singapore, 1985).
  5. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Interdisciplinary Applied Mathematics Vol. 1 (Springer, New York, 1990).
  6. P. Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge Nonlinear Science Series (Cambridge University Press, Cambridge, 1998).
  7. Edited by J. Bricmont, D. Dürr, M. C. Galavotti, G. Ghirardi, F. Petruccione, and N. Zanghi, Chance in Physics–Foundations and Perspectives (Springer-Verlag, Berlin/Heidelberg, 2001).
  8. P. Castiglione, M. Falcioni, A. Lesne, and A. Vulpiani, Chaos and Coarse Graining in Statistical Mechanics (Cambridge University Press, Cambridge, 2008).
  9. L. Sklar, Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics (Cambridge University Press, Cambridge, 1993).
  10. J. Bricmont, Making Sense of Statistical Mechanics, Undergraduate Lecture Notes in Physics (Springer Nature, Switzerland, 2022).
  11. Edited by A. Bassi, S. Goldstein, R. Tumulka, and N. Zanghi, Physics and the Nature of Reality (Springer, Berlin, 2024).
  12. M. Baldovin, G. Gradenigo, A. Vulpiani, and N. Zanghì, On the foundations of statistical mechanics, Phys. Rep. 1132, 1 (2025).
  13. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  14. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  15. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  16. L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  17. J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys. 81, 082001 (2018).
  18. H. Tasaki, From quantum dynamics to the canonical distribution: General picture and a rigorous example, Phys. Rev. Lett. 80, 1373 (1998).
  19. J. Gemmer and G. Mahler, Distribution of local entropy in the Hilbert space of bi-partite quantum systems: Origin of Jaynes' principle, Eur. Phys. J. B 31, 249 (2003).
  20. S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Canonical typicality, Phys. Rev. Lett. 96, 050403 (2006).
  21. S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nat. Phys. 2, 754 (2006).
  22. P. Reimann, Foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. Lett. 101, 190403 (2008).
  23. N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantum mechanical evolution towards thermal equilibrium, Phys. Rev. E 79, 061103 (2009).
  24. P. Reimann and M. Evstigneev, Quantum versus classical foundation of statistical mechanics under experimentally realistic conditions, Phys. Rev. E 88, 052114 (2013).
  25. M. P. Müller, E. Adlam, L. Masanes, and N. Wiebe, Thermalization and canonical typicality in translation-invariant quantum lattice systems, Commun. Math. Phys. 340, 499 (2015).
  26. P. Correia, G. Dias Carvalho, and T. R. de Oliveira, Is entanglement necessary for the typicality argument in statistical mechanics? arXiv:2504.21090.
  27. Note that the definition of “integrable system” is problematic in QM [51], while in CM we refer to systems with a maximal number of conserved quantities.
  28. M. Rigol, A. Muramatsu, and M. Olshanii, Hard-core bosons on optical superlattices: Dynamics and relaxation in the superfluid and insulating regimes, Phys. Rev. A 74, 053616 (2006).
  29. M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett. 98, 050405 (2007).
  30. L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. (2016) 064007.
  31. H. Spohn, Generalized Gibbs ensembles of the classical toda chain, J. Stat. Phys. 180, 4 (2020).
  32. V. Alba, Eigenstate thermalization hypothesis and integrability in quantum spin chains, Phys. Rev. B 91, 155123 (2015).
  33. J. M. Magán, Random free fermions: An analytical example of eigenstate thermalization, Phys. Rev. Lett. 116, 030401 (2016).
  34. S. Nandy, A. Sen, A. Das, and A. Dhar, Eigenstate Gibbs ensemble in integrable quantum systems, Phys. Rev. B 94, 245131 (2016).
  35. T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, Entanglement and matrix elements of observables in interacting integrable systems, Phys. Rev. E 100, 062134 (2019).
  36. M. Gluza, J. Eisert, and T. Farrelly, Equilibration towards generalized Gibbs ensembles in non-interacting theories, SciPost Phys. 7, 038 (2019).
  37. P. Łydżba, R. Świetek, M. Mierzejewski, M. Rigol, and L. Vidmar, Normal weak eigenstate thermalization, Phys. Rev. B 110, 104202 (2024).
  38. H. Tasaki, Macroscopic irreversibility in quantum systems: Free expansion in a fermionic chain, Phys. Rev. E 111, L062103 (2025).
  39. K. Ptaszyński and M. Esposito, Open-system eigenstate thermalization in a noninteracting integrable model, Phys. Rev. E 111, 014129 (2025).
  40. A. Y. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York, 1949).
  41. P. Mazur and J. Van Der Linden, Asymptotic form of the structure function for real systems, J. Math. Phys. 4, 271 (1963).
  42. M. Kac, On the distribution of values of sums of the type ∑f(2kt), Ann. Math. 47, 33 (1946).
  43. “Observables” in the language of quantum theory; we will use these terms interchangeably throughout the work.
  44. P. Mazur and E. Montroll, Poincaré cycles, ergodicity, and irreversibility in assemblies of coupled harmonic oscillators, J. Math. Phys. 1, 70 (1960).
  45. G. W. Ford, M. Kac, and P. Mazur, Statistical mechanics of assemblies of coupled oscillators, J. Math. Phys. 6, 504 (1965).
  46. M. Baldovin, R. Marino, and A. Vulpiani, Ergodic observables in non-ergodic systems: The example of the harmonic chain, Physica A 630, 129273 (2023).
  47. F. Jin, T. Neuhaus, K. Michielsen, S. Miyashita, M. A. Novotny, M. I. Katsnelson, and H. De Raedt, Equilibration and thermalization of classical systems, New J. Phys. 15, 033009 (2013).
  48. N. Cocciaglia, A. Vulpiani, and G. Gradenigo, Thermalization without chaos in harmonic systems, Physica A 601, 127581 (2022).
  49. M. Baldovin, A. Vulpiani, and G. Gradenigo, Statistical mechanics of an integrable system, J. Stat. Phys. 183, 41 (2021).
  50. M. C. Bañuls, J. I. Cirac, and M. B. Hastings, Strong and weak thermalization of infinite nonintegrable quantum systems, Phys. Rev. Lett. 106, 050405 (2011).
  51. C. Gogolin and J. Eisert, Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems, Rep. Prog. Phys. 79, 056001 (2016).
  52. A. Giorgilli, Notes on Hamiltonian Dynamical Systems (Cambridge University Press, Cambridge, 2022).
  53. F. Strocchi, Complex coordinates and quantum mechanics, Rev. Mod. Phys. 38, 36 (1966).
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/h9fj-ylgm for the proofs of all results presented in the main text, additional technical details, a discussion of the relation between our work and Khinchin's approach, and the example of the linear chain, which includes Refs. [78, 79, 80, 81, 82, 83].
  55. For simplicity, we provide all the results for a real C; generalizations to a complex unitary are trivial.
  56. S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, Any orthonormal basis in high dimension is uniformly distributed over the sphere, Ann. Inst. H. Poincaré Probab. Statist. 53, 701 (2017).
  57. A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  58. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature (London) 618, 500 (2023).
  59. J. P. Pekola, B. Karimi, M. Cattaneo, and S. Maniscalco, Long-time relaxation of a finite spin bath linearly coupled to a qubit, Open Syst. Inf. Dyn. 30, 2350009 (2023).
  60. S. O'Rourke, V. Vu, and K. Wang, Eigenvectors of random matrices: A survey, J. Comb. Theor. Ser. A 144, 361 (2016).
  61. G. W. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices (Cambridge University Press, Cambridge, 2009).
  62. A. Altland, M. Janssen, and B. Shapiro, Perturbation theory for the Rosenzweig-Porter matrix model, Phys. Rev. E 56, 1471 (1997).
  63. D. Venturelli, L. F. Cugliandolo, G. Schehr, and M. Tarzia, Replica approach to the generalized Rosenzweig-Porter model, SciPost Phys. 14, 110 (2023).
  64. L. Erdős, B. Schlein, and H.-T. Yau, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann. Probab. 37, 815 (2009).
  65. L. Erdős, B. Schlein, and H.-T. Yau, Local semicircle law and complete delocalization for wigner random matrices, Commun. Math. Phys. 287, 641 (2009).
  66. M. Rudelson and R. Vershynin, Delocalization of eigenvectors of random matrices with independent entries, Duke Math. J. 164, 2507 (2015).
  67. L. Benigni and P. Lopatto, Optimal delocalization for generalized Wigner matrices, Adv. Math. 396, 108109 (2022).
  68. M. Rudelson, Delocalization of eigenvectors of random matrices. Lecture notes, arXiv:1707.08461.
  69. S. Pappalardi, Quantum (chaotic) dynamics. Lecture notes (2024).
  70. M. Rigol and M. Fitzpatrick, Initial-state dependence of the quench dynamics in integrable quantum systems, Phys. Rev. A 84, 033640 (2011).
  71. P. Łydżba, M. Rigol, and L. Vidmar, Eigenstate entanglement entropy in random quadratic Hamiltonians, Phys. Rev. Lett. 125, 180604 (2020).
  72. P. Łydżba, M. Mierzejewski, M. Rigol, and L. Vidmar, Generalized thermalization in quantum-chaotic quadratic Hamiltonians, Phys. Rev. Lett. 131, 060401 (2023).
  73. J. P. Pekola and B. Karimi, Quantum thermalization via multiwave mixing, Phys. Rev. Res. 6, L042023 (2024).
  74. G. P. Berman and F. M. Izrailev, The Fermi–Pasta–Ulam problem: Fifty years of progress, Chaos 15, 015104 (2005).
  75. Edited by G. Gallavotti, The Fermi-Pasta-Ulam Problem (Springer, Berlin, 2008).
  76. S. Goldstein, J. L. Lebowitz, C. Mastrodonato, R. Tumulka, and N. Zanghi, Approach to thermal equilibrium of macroscopic quantum systems, Phys. Rev. E 81, 011109 (2010).
  77. G. Livan, M. Novaes, and P. Vivo, Introduction to Random Matrices, SpringerBriefs in Mathematical Physics Vol. 26 (Springer International, Cham, 2018).
  78. G. Marsaglia, Choosing a Point from the Surface of a Sphere, Ann. Math. Stat. 43, 645 (1972).
  79. J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford Graduate Texts (Oxford University Press, Oxford, UK, 2010).
  80. W. Greiner, L. Neise, and H. Stöcker, Thermodynamics and Statistical Mechanics (Springer, New York, 1995).
  81. O. E. Lanford and J. L. Lebowitz, Time evolution and ergodic properties of harmonic systems, in Dynamical Systems, Theory and Applications (1975), pp. 144–177.
  82. S. Graffi and A. Martinez, Ergodic properties of infinite quantum harmonic crystals: An analytic approach, J. Math. Phys. 37, 5111 (1996).
  83. G. Jona-Lasinio and C. Presilla, Chaotic properties of quantum many-body systems in the thermodynamic limit, Phys. Rev. Lett. 77, 4322 (1996).

Outline

Information

Phys. Rev. Research 7, L032002– Published 2 July, 2025
  • Received 11 September 2024
  • Revised 18 March 2025
  • Accepted 12 June 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation