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Generalized Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy for Near-Integrable Dynamics

Leonardo Biagetti1, Maciej Łebek2, Miłosz Panfil2, and Jacopo De Nardis1

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Phys. Rev. X 16, 041021 – Published 21 October, 2026

DOI: https://doi.org/10.1103/sx1x-pyl1

Abstract

We study quantum and classical many-body Hamiltonian systems that combine integrable contact interactions with generic long-range two-body potentials. Starting from an ansatz for the state at time t, which we call the correlated fluid-cell ensemble, we show that the dynamics of local observables at macroscopic times and length scales can be cast into a generalized Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy formulated in terms of the quasiparticle densities of the underlying integrable model and their correlations. We derive this hierarchy and validate these predictions against microscopic molecular-dynamics simulations, finding perfect agreement. At late times, the one-particle distribution relaxes via a Boltzmann-type scattering integral encoding the interplay between integrable contact processes and long-range collisions, whereas higher-point correlations remain strongly nonthermal on thermalization timescales, indicative of a form of incomplete or generalized thermalization. Focusing on long-range dipolar quantum gases, where the relevant matrix elements can be obtained explicitly, we show that our collision integral reduces exactly to the Fermi golden rule result and provide a complete theoretical account of the experimental observations of Tang et al. [Phys. Rev. X 8, 021030 (2018)]. More broadly, our framework extends the BBGKY program to regimes with strong local interactions and applies to a wide class of experimentally relevant systems, from one-dimensional dipolar cold-atom gases to Lennard-Jones molecular fluids.

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Popular Summary

How do isolated many-body systems forget their past and equilibrate? For most physical systems, particle collisions gradually erase memory of the initial state, a process captured by the kinetic theory. Yet, in special one-dimensional systems (“integrable”), an extensive number of conserved quantities preserves this initial information and prevents the ordinary thermalization. Many real systems, such as low-dimensional cold-atomic gases or complex condensed-matter and optical systems, fall in between these two categories. How do such nearly integrable systems relax toward equilibrium? In this work, we develop a new framework—the generalized Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy—that extends the classic equations of nonequilibrium statistical mechanics to account for integrability and its weak breaking. It bridges the gap between idealized integrable dynamics and realistic many-body systems and provides a foundation for understanding their thermalization and prethermalization. It also establishes a systematic procedure to compute large-time kinetic equations in both classical and quantum systems.

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References (159)

  1. P. Calabrese, F. H. Essler, and G. Mussardo, Introduction to “quantum integrability in out of equilibrium systems”, J. Stat. Mech. (2016) 064001.
  2. A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  3. M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauss, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system, Nature (London) 481, 484 (2012).
  4. D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, and S. L. Sondhi, Many-body localization and thermalization in quantum statistical mechanics, Phys. Rev. B 88, 014206 (2013).
  5. M. Fagotti and P. Calabrese, Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field, Phys. Rev. A 78, 010306(R) (2008).
  6. V. Alba and P. Calabrese, Entanglement dynamics after quantum quenches in generic integrable systems, SciPost Phys. 4, 017 (2018).
  7. V. Alba, B. Bertini, and M. Fagotti, Entanglement evolution and generalized hydrodynamics: Interacting integrable systems, SciPost Phys. 7, 005 (2019).
  8. V. Alba and P. Calabrese, Quantum information scrambling after a quantum quench, Phys. Rev. B 100, 115150 (2019).
  9. J. Lopez-Piqueres, B. Ware, S. Gopalakrishnan, and R. Vasseur, Operator front broadening in chaotic and integrable quantum chains, Phys. Rev. B 104, 104307 (2021).
  10. P. Fendley, Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain, J. Phys. A 49, 30LT01 (2016).
  11. P. R. Pasnoori, Y. Tang, J. Lee, J. H. Pixley, N. Andrei, and P. Azaria, Spin fractionalization and zero modes in the spin-12 XXZ chain with boundary fields, Phys. Rev. B 112, 075121 (2025).
  12. A. Burshtein and M. Goldstein, Inelastic decay from integrability, PRX Quantum 5, 020323 (2024).
  13. O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6, 041065 (2016).
  14. B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, Transport in out-of-equilibrium XXZ chains: Exact profiles of charges and currents, Phys. Rev. Lett. 117, 207201 (2016).
  15. S. Dubovsky, L. Hui, A. Nicolis, and D. T. Son, Effective field theory for hydrodynamics: Thermodynamics, and the derivative expansion, Phys. Rev. D 85, 085029 (2012).
  16. P. Ruggiero, P. Calabrese, B. Doyon, and J. Dubail, Quantum generalized hydrodynamics, Phys. Rev. Lett. 124, 140603 (2020).
  17. A. Lucas and K. C. Fong, Hydrodynamics of electrons in graphene, J. Phys. Condens. Matter 30, 053001 (2018).
  18. N. Malvania, Y. Zhang, Y. Le, J. Dubail, M. Rigol, and D. S. Weiss, Generalized hydrodynamics in strongly interacting 1d Bose gases, Science 373, 1129 (2021).
  19. M. Schemmer, I. Bouchoule, B. Doyon, and J. Dubail, Generalized hydrodynamics on an atom chip, Phys. Rev. Lett. 122, 090601 (2019).
  20. A. Scheie, N. E. Sherman, M. Dupont, S. E. Nagler, M. B. Stone, G. E. Granroth, J. E. Moore, and D. A. Tennant, Detection of Kardar–Parisi–Zhang hydrodynamics in a quantum Heisenberg spin-1/2 chain, Nat. Phys. 17, 726 (2021).
  21. F. Møller, C. Li, I. Mazets, H.-P. Stimming, T. Zhou, Z. Zhu, X. Chen, and J. Schmiedmayer, Extension of the generalized hydrodynamics to the dimensional crossover regime, Phys. Rev. Lett. 126, 090602 (2021).
  22. F. Cataldini, F. Møller, M. Tajik, J. Sabino, S.-C. Ji, I. Mazets, T. Schweigler, B. Rauer, and J. Schmiedmayer, Emergent Pauli blocking in a weakly interacting Bose gas, Phys. Rev. X 12, 041032 (2022).
  23. K.-Y. Li, Y. Zhang, K. Yang, K.-Y. Lin, S. Gopalakrishnan, M. Rigol, and B. L. Lev, Rapidity and momentum distributions of one-dimensional dipolar quantum gases, Phys. Rev. A 107, L061302 (2023).
  24. V. B. Bulchandani, R. Vasseur, C. Karrasch, and J. E. Moore, Bethe-Boltzmann hydrodynamics and spin transport in the XXZ chain, Phys. Rev. B 97, 045407 (2018).
  25. V. B. Bulchandani, C. Karrasch, and J. E. Moore, Superdiffusive transport of energy in one-dimensional metals, Proc. Natl. Acad. Sci. U.S.A. 117, 12713 (2020).
  26. L. Piroli, J. De Nardis, M. Collura, B. Bertini, and M. Fagotti, Transport in out-of-equilibrium XXZ chains: Nonballistic behavior and correlation functions, Phys. Rev. B 96, 115124 (2017).
  27. E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. 130, 1605 (1963).
  28. E. H. Lieb, Exact analysis of an interacting Bose gas. II. The excitation spectrum, Phys. Rev. 130, 1616 (1963).
  29. F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, and V. E. Korepin, The One-Dimensional Hubbard Model (Cambridge University Press, Cambridge, England, 2005).
  30. K. Mallayya, M. Rigol, and W. De Roeck, Prethermalization and thermalization in isolated quantum systems, Phys. Rev. X 9, 021027 (2019).
  31. J. Lopez-Piqueres and R. Vasseur, Integrability breaking from backscattering, Phys. Rev. Lett. 130, 247101 (2023).
  32. T. Goldfriend and J. Kurchan, Quasi-integrable systems are slow to thermalize but may be good scramblers, Phys. Rev. E 102, 022201 (2020).
  33. A. J. Friedman, S. Gopalakrishnan, and R. Vasseur, Diffusive hydrodynamics from integrability breaking, Phys. Rev. B 101, 180302(R) (2020).
  34. A. Bastianello, A. De Luca, and R. Vasseur, Hydrodynamics of weak integrability breaking, J. Stat. Mech. (2021) 114003.
  35. P. Orlov, A. Tiutiakina, R. Sharipov, E. Petrova, V. Gritsev, and D. V. Kurlov, Adiabatic eigenstate deformations and weak integrability breaking of Heisenberg chain, Phys. Rev. B 107, 184312 (2023).
  36. D. Szász-Schagrin, B. Pozsgay, and G. Takács, Weak integrability breaking and level spacing distribution, SciPost Phys. 11, 037 (2021).
  37. B. Pozsgay, R. Sharipov, A. Tiutiakina, and I. Vona, Adiabatic gauge potential and integrability breaking with free fermions, SciPost Phys. 17, 075 (2024).
  38. B. Bertini, F. H. L. Essler, S. Groha, and N. J. Robinson, Prethermalization and thermalization in models with weak integrability breaking, Phys. Rev. Lett. 115, 180601 (2015).
  39. B. Bertini, F. H. L. Essler, S. Groha, and N. J. Robinson, Thermalization and light cones in a model with weak integrability breaking, Phys. Rev. B 94, 245117 (2016).
  40. F. H. L. Essler, S. Kehrein, S. R. Manmana, and N. J. Robinson, Quench dynamics in a model with tuneable integrability breaking, Phys. Rev. B 89, 165104 (2014).
  41. J. Lopez-Piqueres, B. Ware, S. Gopalakrishnan, and R. Vasseur, Hydrodynamics of nonintegrable systems from a relaxation-time approximation, Phys. Rev. B 103, L060302 (2021).
  42. F. M. Surace and O. Motrunich, Weak integrability breaking perturbations of integrable models, Phys. Rev. Res. 5, 043019 (2023).
  43. M. Łebek, M. Panfil, and R. M. Konik, Prethermalization in coupled one-dimensional quantum gases, SciPost Phys. 17, 007 (2024).
  44. J. A. Scaramazza, P. Smacchia, and E. A. Yuzbashyan, Consequences of integrability breaking in quench dynamics of pairing Hamiltonians, Phys. Rev. B 99, 054520 (2019).
  45. A. Bastianello, Lack of thermalization for integrability-breaking impurities, Europhys. Lett. 125, 20001 (2019).
  46. G. del Vecchio del Vecchio, A. D. Luca, and A. Bastianello, Transport through interacting defects and lack of thermalisation, SciPost Phys. 12, 060 (2022).
  47. J. Durnin, M. J. Bhaseen, and B. Doyon, Nonequilibrium dynamics and weakly broken integrability, Phys. Rev. Lett. 127, 130601 (2021).
  48. M. Panfil, S. Gopalakrishnan, and R. M. Konik, Thermalization of interacting quasi-one-dimensional systems, Phys. Rev. Lett. 130, 030401 (2023).
  49. M. Łebek and M. Panfil, Chapman-Enskog theory and crossover between diffusion and superdiffusion for nearly integrable quantum gases, SciPost Phys. 18, 186 (2025).
  50. S. Lepri, R. Livi, and A. Politi, Too close to integrable: Crossover from normal to anomalous heat diffusion, Phys. Rev. Lett. 125, 040604 (2020).
  51. L. Delacrétaz, A. L. Fitzpatrick, E. Katz, and M. Walters, Thermalization and hydrodynamics of two-dimensional quantum field theories, SciPost Phys. 12, 119 (2022).
  52. A. Miron, J. Cividini, A. Kundu, and D. Mukamel, Derivation of fluctuating hydrodynamics and crossover from diffusive to anomalous transport in a hard-particle gas, Phys. Rev. E 99, 012124 (2019).
  53. S. Chen, J. Wang, G. Casati, and G. Benenti, Nonintegrability and the Fourier heat conduction law, Phys. Rev. E 90, 032134 (2014).
  54. H. Zhao and W. g. Wang, Fourier heat conduction as a strong kinetic effect in one-dimensional hard-core gases, Phys. Rev. E 97, 010103(R) (2018).
  55. S. Lepri, R. Livi, and A. Politi, Too close to integrable: Crossover from normal to anomalous heat diffusion, Phys. Rev. Lett. 125, 040604 (2020).
  56. M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science 337, 1318 (2012).
  57. F. Møller, C. Li, I. Mazets, H.-P. Stimming, T. Zhou, Z. Zhu, X. Chen, and J. Schmiedmayer, Extension of the generalized hydrodynamics to the dimensional crossover regime, Phys. Rev. Lett. 126, 090602 (2021).
  58. Y. Tang, W. Kao, K.-Y. Li, S. Seo, K. Mallayya, M. Rigol, S. Gopalakrishnan, and B. L. Lev, Thermalization near integrability in a dipolar quantum Newton’s cradle, Phys. Rev. X 8, 021030 (2018).
  59. C. Colléaux, J. Skipp, J. Laurie, and S. Nazarenko, Semi-local one-dimensional optical wave turbulence, arXiv:2412.14153.
  60. S. Lepri, P. Sandri, and A. Politi, The one-dimensional Lennard-Jones system: Collective fluctuations and breakdown of hydrodynamics, Eur. Phys. J. B 47, 549 (2005).
  61. M. Mareschal and A. Amellal, Thermal conductivity in a one-dimensional Lennard-Jones chain by molecular dynamics, Phys. Rev. A 37, 2189 (1988).
  62. M. Eltohfa, X. Wang, C. M. Griffin, and F. Robicheaux, Simulations of classical three-body thermalization in one dimension, Phys. Rev. E 110, 014114 (2024).
  63. A. Parameshwaran and B. S. Gupta, Kinetics of vapor-liquid transition of active matter system in confined geometry, Phys. Rev. E 111, 025405 (2025).
  64. G. Benettin, G. Orsatti, and A. Ponno, On the role of the integrable toda model in one-dimensional molecular dynamics, J. Stat. Phys. 190, 131 (2023).
  65. O. C. Eldridge and M. Feix, Numerical experiments with a plasma model, Phys. Fluids 6, 398 (1963).
  66. J. M. Dawson, Thermal relaxation in a one-species, one-dimensional plasma, Phys. Fluids 7, 419 (1964).
  67. J. L. Rouet and M. R. Feix, Relaxation for one-dimensional plasma: Test particles versus global distribution behavior, Phys. Fluids B 3, 1830 (1991).
  68. J.-B. Fouvry, B. Bar-Or, and P.-H. Chavanis, Kinetic theory of one-dimensional homogeneous long-range interacting systems sourced by 1/N2 effects, Phys. Rev. E 100, 052142 (2019).
  69. J.-B. Fouvry, P.-H. Chavanis, and C. Pichon, Kinetic theory of one-dimensional homogeneous long-range interacting systems with an arbitrary potential of interaction, Phys. Rev. E 102, 052110 (2020).
  70. J. Berges, G. S. Denicol, M. P. Heller, and T. Preis, Far from equilibrium hydrodynamics of nonthermal fixed points, arXiv:2504.18754.
  71. J.-B. Fouvry and M. Roule, Kinetic blockings in long-range interacting inhomogeneous systems, Phys. Rev. E 108, 054108 (2023).
  72. G. B. Rybicki, Exact statistical mechanics of a one-dimensional self-gravitating system, Astrophys. Space Sci. 14, 56 (1971).
  73. J. Reidl, Charles J., and B. N. Miller, Gravity in one dimension—quasi-equilibrium, Astrophys. J. 332, 619 (1988).
  74. L. Milanović, H. Posch, and W. Thirring, Statistical mechanics and computer simulation of systems with attractive positive power-law potentials, Phys. Rev. E 57, 2763 (1998).
  75. T. Tsuchiya, N. Gouda, and T. Konishi, Relaxation processes in one-dimensional self-gravitating many-body systems, Phys. Rev. E 53, 2210 (1996).
  76. B. N. Miller, K. R. Yawn, and V. P. Youngkins, Recent progress in studies of one-dimensional gravitating systems, in Dynamical Systems, Plasmas and Gravitation (Springer, Berlin, Heidelberg, 1996), pp. 164–175.
  77. M. Łebek and M. Panfil, Navier-Stokes equations for nearly integrable quantum gases, Phys. Rev. Lett. 134, 010405 (2025).
  78. R. Balescu, Equilibrium and Non-Equilibrium Statistical Mechanics (Wiley-Interscience, New York, 1975).
  79. J. G. Kirkwood, The statistical mechanical theory of transport processes II. Transport in gases, J. Chem. Phys. 15, 72 (1947).
  80. J. G. Kirkwood, The statistical mechanical theory of transport processes I. General theory, J. Chem. Phys. 14, 180 (1946).
  81. J. Yvon, La théorie statistique des fluides et l’équation d’état, Actual. Sci. & Indust. No. 203 (Hermann, Paris, 1935).
  82. N. N. Bogoliubov and K. P. Gurov, Kinetic equations in quantum mechanics, J. Exp. Theor. Phys. 17, 614 (1947).
  83. V. I. Gerasimenko, Kinetic equations and hierarchies of evolution equations of quantum systems, arXiv:2107.10872.
  84. L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar physics: A review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2023).
  85. C. Chen, G. Bornet, M. Bintz, G. Emperauger, L. Leclerc, V. S. Liu, P. Scholl, D. Barredo, J. Hauschild, S. Chatterjee, M. Schuler, A. M. Läuchli, M. P. Zaletel, T. Lahaye, N. Y. Yao, and A. Browaeys, Continuous symmetry breaking in a two-dimensional Rydberg array, Nature (London) 616, 691 (2023).
  86. M. Płodzień, G. Lochead, J. de Hond, N. J. van Druten, and S. Kokkelmans, Rydberg dressing of a one-dimensional Bose-Einstein condensate, Phys. Rev. A 95, 043606 (2017).
  87. J. Rincón, M. Ganahl, and G. Vidal, Lieb-Liniger model with exponentially decaying interactions: A continuous matrix product state study, Phys. Rev. B 92, 115107 (2015).
  88. E. Orignac, S. De Palo, L. Salasnich, and R. Citro, Breathing mode of a quantum droplet in a quasi-one-dimensional dipolar Bose gas, Phys. Rev. A 109, 043316 (2024).
  89. I. Frérot, P. Naldesi, and T. Roscilde, Entanglement and fluctuations in the XXZ model with power-law interactions, Phys. Rev. B 95, 245111 (2017).
  90. L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-fluctuation-driven crossover from a dilute Bose-Einstein condensate to a macrodroplet in a dipolar quantum fluid, Phys. Rev. X 6, 041039 (2016).
  91. A. Recati and S. Stringari, Supersolidity in ultracold dipolar gases, Nat. Rev. Phys. 5, 735 (2023).
  92. G. Emperauger, M. Qiao, C. Chen, F. Caleca, S. Bocini, M. Bintz, G. Bornet, R. Martin, B. Gély, L. Klein, D. Barredo, S. Chatterjee, N. Yao, F. Mezzacapo, T. Lahaye, T. Roscilde, and A. Browaeys, Tomonaga-Luttinger liquid behavior in a Rydberg-encoded spin chain, Phys. Rev. X 15, 031021 (2025).
  93. K. Yang, Y. Zhang, K.-Y. Li, K.-Y. Lin, S. Gopalakrishnan, M. Rigol, and B. L. Lev, Phantom energy in the nonlinear response of a quantum many-body scar state, Science 385, 1063 (2024).
  94. D. Bernard and B. Doyon, Conformal field theory out of equilibrium: A review, J. Stat. Mech. (2016) 064005.
  95. B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, and R. Vasseur, Generalized hydrodynamics: A perspective, Phys. Rev. X 15, 010501 (2025).
  96. P. Schüttelkopf, M. Tajik, N. Bazhan, F. Cataldini, S.-C. Ji, J. Schmiedmayer, and F. Møller, Characterising transport in a quantum gas by measuring Drude weights, Science 391, 290 (2025).
  97. R. Vasseur and J. E. Moore, Nonequilibrium quantum dynamics and transport: From integrability to many-body localization, J. Stat. Mech. (2016) 064010.
  98. M. L. Kerr and K. V. Kheruntsyan, The theory of generalised hydrodynamics for the one-dimensional Bose gas, AAPPS Bull. 33, 25 (2023).
  99. V. Alba, B. Bertini, M. Fagotti, L. Piroli, and P. Ruggiero, Generalized-hydrodynamic approach to inhomogeneous quenches: Correlations, entanglement and quantum effects, J. Stat. Mech. (2021) 114004.
  100. F. H. Essler, A short introduction to generalized hydrodynamics, Physica (Amsterdam) 631A, 127572 (2023), lecture Notes of the 15th International Summer School of Fundamental Problems in Statistical Physics.
  101. S. Scopa, P. Ruggiero, P. Calabrese, and J. Dubail, One-particle density matrix and momentum distribution of the out-of-equilibrium one-dimensional Tonks-Girardeau gas: Analytical results at large n, Phys. Rev. A 108, 013324 (2023).
  102. P. Suret, S. Randoux, A. Gelash, D. Agafontsev, B. Doyon, and G. El, Soliton gas: Theory, numerics, and experiments, Phys. Rev. E 109, 061001 (2024).
  103. A. Bastianello, A. Tikan, F. Copie, S. Randoux, and P. Suret, Observation of a generalized Gibbs ensemble in photonics, Phys. Rev. A 113, 013514 (2026).
  104. T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton’s cradle, Nature (London) 440, 900 (2006).
  105. M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of interacting fermions in a quasirandom optical lattice, Science 349, 842 (2015).
  106. W. Kao, K.-Y. Li, K.-Y. Lin, S. Gopalakrishnan, and B. L. Lev, Topological pumping of a 1d dipolar gas into strongly correlated prethermal states, Science 371, 296 (2021).
  107. W. Morong, F. Liu, P. Becker, K. S. Collins, L. Feng, A. Kyprianidis, G. Pagano, T. You, A. V. Gorshkov, and C. Monroe, Publisher correction: Observation of stark many-body localization without disorder, Nature (London) 601, E13 (2022).
  108. S. Birnkammer, A. Bastianello, and M. Knap, Prethermalization in one-dimensional quantum many-body systems with confinement, Nat. Commun. 13, 7663 (2022).
  109. Y. Le, Y. Zhang, S. Gopalakrishnan, M. Rigol, and D. S. Weiss, Observation of hydrodynamization and local prethermalization in 1d Bose gases, Nature (London) 618, 494 (2023).
  110. T. Young, J. LLoyd, and C. von Keyserlingk, Breakdown of Fermi’s golden rule in 1d systems at non-zero temperature, arXiv:2508.00254.
  111. P. A. E. M. Janssen, Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr. 33, 863 (2003).
  112. J. Garnier, M. Lisak, and A. Picozzi, Toward a wave turbulence formulation of statistical nonlinear optics, J. Opt. Soc. Am. B 29, 2229 (2012).
  113. F. Hübner, L. Biagetti, J. De Nardis, and B. Doyon, Diffusive hydrodynamics from long-range correlations, Phys. Rev. Lett. 134, 187101 (2025).
  114. See Supplemental Material at http://link.aps.org/supplemental/10.1103/sx1x-pyl1 for additional computations supporting the results presented in the main text.
  115. B. Doyon and T. Yoshimura, A note on generalized hydrodynamics: Inhomogeneous fields and other concepts, SciPost Phys. 2, 014 (2017).
  116. J. Durnin, A. D. Luca, J. De Nardis, and B. Doyon, Diffusive hydrodynamics of inhomogenous Hamiltonians, J. Phys. A 54, 494001 (2021).
  117. A. A. Vlasov, The vibrational properties of an electron gas, Sov. Phys. Usp. 10, 721 (1968).
  118. W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/n limit of interacting classical particles, Commun. Math. Phys. 56, 101 (1977).
  119. P. H. Chavanis, J. Sommeria, and R. Robert, Statistical mechanics of two-dimensional vortices and collisionless stellar systems, Astrophys. J. 471, 385 (1996).
  120. J. De Nardis, D. Bernard, and B. Doyon, Diffusion in generalized hydrodynamics and quasiparticle scattering, SciPost Phys. 6, 049 (2019).
  121. M. Medenjak, J. De Nardis, and T. Yoshimura, Diffusion from convection, SciPost Phys. 9, 075 (2020).
  122. S. Gopalakrishnan, D. A. Huse, V. Khemani, and R. Vasseur, Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems, Phys. Rev. B 98, 220303(R) (2018).
  123. B. Doyon, G. Perfetto, T. Sasamoto, and T. Yoshimura, Ballistic macroscopic fluctuation theory, SciPost Phys. 15, 136 (2023).
  124. P. Lisiak, M. Łebek, and M. Panfil, Burgers equation from non-thermal stationary states in nearly-integrable gases, arXiv:2512.09726.
  125. Z. Krajnik, J. Schmidt, V. Pasquier, E. Ilievski, and T. Prosen, Exact anomalous current fluctuations in a deterministic interacting model, Phys. Rev. Lett. 128, 160601 (2022).
  126. S. Gopalakrishnan, E. McCulloch, and R. Vasseur, Non-Gaussian diffusive fluctuations in Dirac fluids, Proc. Natl. Acad. Sci. U.S.A. 121, e2403327121 (2024).
  127. E. McCulloch, R. Vasseur, and S. Gopalakrishnan, Ballistic modes as a source of anomalous charge noise, Phys. Rev. E 111, 015410 (2025).
  128. T. Yoshimura and Žiga Krajnik, Anomalous current fluctuations from Euler hydrodynamics, Phys. Rev. E 111, 024141 (2025).
  129. B. Doyon, Hydrodynamic noise in one dimension: Projected kubo formula and its vanishing in integrable models, SciPost Phys. 20, 111 (2026).
  130. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer-Verlag, Berlin, 1991).
  131. M. Flicker, Pair distribution function of a one-dimensional hard rod gas, J. Math. Phys. (N.Y.) 9, 171 (1968).
  132. A. Robledo and J. Rowlinson, The distribution of hard rods on a line of finite length, Mol. Phys. 58, 711 (1986).
  133. B. Doyon and H. Spohn, Dynamics of hard rods with initial domain wall state, J. Stat. Mech. (2017) 073210.
  134. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991).
  135. C. Boldrighini, R. L. Dobrushin, and Y. M. Sukhov, One-dimensional hard rod caricature of hydrodynamics, J. Stat. Phys. 31, 577 (1983).
  136. C. Boldrighini and Y. M. Suhov, One-dimensional hard-rod caricature of hydrodynamics: “Navier–Stokes correction” for local equilibrium initial states, Commun. Math. Phys. 189, 577 (1997).
  137. B. Doyon, Lecture notes on generalised hydrodynamics, SciPost Physics Lecture Notes 18, (2020), 10.21468/scipostphyslectnotes.18.
  138. M. Olshanii, Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons, Phys. Rev. Lett. 81, 938 (1998).
  139. F. Deuretzbacher, J. C. Cremon, and S. M. Reimann, Ground-state properties of few dipolar bosons in a quasi-one-dimensional harmonic trap, Phys. Rev. A 81, 063616 (2010).
  140. F. Deuretzbacher, J. C. Cremon, and S. M. Reimann, Erratum: Ground-state properties of few dipolar bosons in a quasi-one-dimensional harmonic trap [Phys. Rev. A 81, 063616 (2010)]; 87, 039903(E) (2013).
  141. A. Urilyon, L. Biagetti, J. Kethepalli, and J. De Nardis, Simulating generalized fluids via interacting wave packet evolution, Phys. Rev. B 113, 014314 (2026).
  142. K. Mallayya and M. Rigol, Prethermalization, thermalization, and Fermi’s golden rule in quantum many-body systems, Phys. Rev. B 104, 184302 (2021).
  143. J. De Nardis and M. Panfil, Density form factors of the 1d Bose gas for finite entropy states, J. Stat. Mech. (2015) P02019.
  144. J. De Nardis and M. Panfil, Exact correlations in the Lieb-Liniger model and detailed balance out-of-equilibrium, SciPost Phys. 1, 015 (2016).
  145. J. De Nardis and M. Panfil, Particle-hole pairs and density–density correlations in the Lieb–Liniger model, J. Stat. Mech. (2018) 033102.
  146. A. C. Cubero and M. Panfil, Thermodynamic bootstrap program for integrable QFT’s: Form factors and correlation functions at finite energy density, J. High Energy Phys. 01 (2019) 104.
  147. M. Panfil, The two particle–hole pairs contribution to the dynamic correlation functions of quantum integrable models, J. Stat. Mech. (2021) 013108.

  148. Note the additional missing factor of 2π with respect to Ref. [47].

  149. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, England, 1993).
  150. M. Panfil and R. M. Konik, Extending the thermodynamic form factor bootstrap program: Multiple particle-hole excitations, crossing symmetry, and reparameterization invariance, J. High Energy Phys. 10 (2023) 072.
  151. C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021).
  152. P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Quasiparticle engineering and entanglement propagation in a quantum many-body system, Nature (London) 511, 202 (2014).
  153. B. Kloss and Y. Bar Lev, Spin transport in a long-range-interacting spin chain, Phys. Rev. A 99, 032114 (2019).
  154. J. T. Schneider, S. J. Thomson, and L. Sanchez-Palencia, Entanglement spectrum and quantum phase diagram of the long-range XXZ chain, Phys. Rev. B 106, 014306 (2022).
  155. S. Nazarenko, Wave Turbulence, Lecture Notes in Physics Vol. 825 (Springer, Berlin, Heidelberg, 2011).
  156. L. Fache, H. Damart, F. Copie, T. Bonnemain, T. Congy, G. Roberti, P. Suret, G. El, and S. Randoux, Dissipation-driven emergence of a soliton condensate in a nonlinear electrical transmission line, Phys. Rev. Lett. 134, 147201 (2025).
  157. A. Mucci, P. Suret, F. Copie, S. Randoux, R. Mullyadzhanov, and A. Gelash, Manipulation of strongly interacting solitons in optical fiber experiments, Phys. Rev. Lett. 134, 193804 (2025).
  158. L. Fache, F. Copie, P. Suret, and S. Randoux, Perturbed nonlinear evolution of optical soliton gases: Growth and decay in integrable turbulence, Phys. Rev. Lett. 135, 157201 (2025).
  159. L. Biagetti, G. Cecile, and J. De Nardis, Three-stage thermalization of a quasi-integrable system, Phys. Rev. Res. 6, 023083 (2024).

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Phys. Rev. X 16, 041021– Published 21 October, 2026
  • Received 28 August 2025
  • Accepted 20 May 2026
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