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Simulating generalized fluids via interacting wave packet evolution

Andrew Urilyon, Leonardo Biagetti, Jitendra Kethepalli, and Jacopo De Nardis

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Phys. Rev. B 113, 014314 – Published 20 January, 2026

DOI: https://doi.org/10.1103/b587-8yyt

Abstract

One–dimensional integrable and quasi-integrable systems display, on macroscopic scales, a universal form of transport known as Generalized Hydrodynamics (GHD). In its standard Euler-scale formulation, GHD mirrors the equations of a two-dimensional compressible fluid but ignores fluctuations and becomes numerically unwieldy as soon as integrability-breaking perturbations are introduced. We show that GHD can be efficiently simulated as a gas of semiclassical wave packets—a natural generalization of hard-rod particles—whose trajectories are efficiently mapped onto those of point particles. This representation (i) provides a transparent route to incorporate integrability-breaking terms, and (ii) automatically embeds the exact fluctuating-hydrodynamics extension of GHD. The resulting framework enables fast, large-scale simulations of quasi-integrable systems even in the presence of complicated integrability-breaking perturbations. It also manifests the pivotal role of two-point correlations in systems confined by external potentials: we demonstrate that situations where local one-point observables appear thermalized can nevertheless sustain long-lived, far-from-equilibrium long-range correlations for arbitrarily long times, signaling that, differently from what was previously stated, true thermalization is not reached at diffusive time-scales.

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References (96)

  1. O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6, 041065 (2016).
  2. 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).
  3. M. Schemmer, I. Bouchoule, B. Doyon, and J. Dubail, Generalized hydrodynamics on an atom chip, Phys. Rev. Lett. 122, 090601 (2019).
  4. B. Doyon, J. Dubail, R. Konik, and T. Yoshimura, Large-scale description of interacting one-dimensional Bose gases: Generalized hydrodynamics supersedes conventional hydrodynamics, Phys. Rev. Lett. 119, 195301 (2017).
  5. J. De Nardis, D. Bernard, and B. Doyon, Hydrodynamic diffusion in integrable systems, Phys. Rev. Lett. 121, 160603 (2018).
  6. J. D. Nardis, D. Bernard, and B. Doyon, Diffusion in generalized hydrodynamics and quasiparticle scattering, SciPost Phys. 6, 049 (2019).
  7. B. Doyon, Lecture notes on generalised hydrodynamics, SciPost Phys. Lect. Notes 18 (2020).
  8. B. Doyon and J. Durnin, Free energy fluxes and the Kubo–Martin–Schwinger relation, J. Stat. Mech. (2021) 043206.
  9. B. Doyon and H. Spohn, Drude weight for the Lieb-Liniger Bose gas, SciPost Phys. 3, 039 (2017).
  10. B. Doyon, Diffusion and superdiffusion from hydrodynamic projections, J. Stat. Phys. 186, 25 (2022).
  11. J. Durnin, M. J. Bhaseen, and B. Doyon, Nonequilibrium dynamics and weakly broken integrability, Phys. Rev. Lett. 127, 130601 (2021).
  12. B. Doyon, H. Spohn, and T. Yoshimura, A geometric viewpoint on generalized hydrodynamics, Nucl. Phys. B 926, 570 (2018).
  13. A. Bastianello, B. Bertini, B. Doyon, and R. Vasseur, Introduction to the special issue on emergent hydrodynamics in integrable many-body systems, J. Stat. Mech. (2022) 014001.
  14. F. S. Møller, G. Perfetto, B. Doyon, and J. Schmiedmayer, Euler-scale dynamical correlations in integrable systems with fluid motion, SciPost Phys. Core 3, 016 (2020).
  15. A. Bastianello, B. Doyon, G. Watts, and T. Yoshimura, Generalized hydrodynamics of classical integrable field theory the sinh-Gordon model, SciPost Phys. 4, 045 (2018).
  16. B. Bertini, F. Heidrich-Meisner, C. Karrasch, T. Prosen, R. Steinigeweg, and M. Žnidarič, Finite-temperature transport in one-dimensional quantum lattice models, Rev. Mod. Phys. 93, 025003 (2021).
  17. A. Bastianello and A. De Luca, Integrability-protected adiabatic reversibility in quantum spin chains, Phys. Rev. Lett. 122, 240606 (2019).
  18. V. B. Bulchandani, S. Gopalakrishnan, and E. Ilievski, Superdiffusion in spin chains, J. Stat. Mech. (2021) 084001.
  19. B. Doyon and F. Hübner, Towards an ab initio derivation of generalised hydrodynamics from a gas of interacting wave packets, arXiv:2307.09307.
  20. F. Møller, P. Schüttelkopf, J. Schmiedmayer, and S. Erne, Whitham approach to generalized hydrodynamics, Phys. Rev. Res. 6, 013328 (2024).
  21. 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).
  22. F. S. Møller, F. Cataldini, and J. Schmiedmayer, Identifying diffusive length scales in one-dimensional Bose gases, SciPost Phys. Core 7, 025 (2024).
  23. P. Ruggiero, P. Calabrese, B. Doyon, and J. Dubail, Quantum generalized hydrodynamics, Phys. Rev. Lett. 124, 140603 (2020).
  24. F. Møller, N. Besse, I. E. Mazets, H.-P. Stimming, and N. J. Mauser, The dissipative Generalized Hydrodynamic equations and their numerical solution, J. Comput. Phys. 493, 112431 (2023).
  25. 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).
  26. F. Møller, S. Erne, N. J. Mauser, J. Schmiedmayer, and I. E. Mazets, Bridging effective field theories and generalized hydrodynamics, arXiv:2205.15871.
  27. A. Bastianello, V. Alba, and J.-S. Caux, Generalized hydrodynamics with space-time inhomogeneous interactions, Phys. Rev. Lett. 123, 130602 (2019).
  28. F. Møller, B. C. Nagy, M. Kormos, and G. Takács, Anomalous charge transport in the sine-Gordon model, arXiv:2411.11473.
  29. I. Bouchoule, B. Doyon, and J. Dubail, The effect of atom losses on the distribution of rapidities in the one-dimensional Bose gas, SciPost Phys. 9, 044 (2020).
  30. B. Doyon, S. Gopalakrishnan, F. Møller, J. Schmiedmayer, and R. Vasseur, Generalized hydrodynamics: A perspective, Phys. Rev. X 15, 010501 (2025).
  31. T. Bonnemain, B. Doyon, and G. El, Generalized hydrodynamics of the KdV soliton gas, J. Phys. A: Math. Theor. 55, 374004 (2022).
  32. A. Bastianello, Ž. Krajnik, and E. Ilievski, Landau-Lifschitz magnets: Exact thermodynamics and transport, Phys. Rev. Lett. 133, 107102 (2024).
  33. L. Dubois, G. Thémèze, F. Nogrette, J. Dubail, and I. Bouchoule, Probing the local rapidity distribution of a one-dimensional Bose gas, Phys. Rev. Lett. 133, 113402 (2024).
  34. 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).
  35. 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).
  36. 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).
  37. F. Cataldini, F. Møller, M. Tajik, J. a. 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).
  38. 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, arXiv:2406.17569.
  39. D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, I. Bloch, and J. Zeiher, Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science 376, 716 (2022).
  40. M. Horvath, A. Bastianello, S. Dhar, R. Koch, Y. Guo, J.-S. Caux, M. Landini, and H.-C. Nägerl, Observing Bethe strings in an attractive Bose gas far from equilibrium, arXiv:2505.10550.
  41. L. Dubois, G. Thémèze, J. Dubail, and I. Bouchoule, Experimental investigation of a bipartite quench in a 1D Bose gas, SciPost Phys. 20, 008 (2026).
  42. B. Doyon, F. Hübner, and T. Yoshimura, New classical integrable systems from generalized TT¯-deformations, Phys. Rev. Lett. 132, 251602 (2024).
  43. B. Doyon, J. Durnin, and T. Yoshimura, The space of integrable systems from generalised TT¯-deformations, SciPost Phys. 13, 072 (2022).
  44. T. Bonnemain, V. Caudrelier, and B. Doyon, Hamiltonian formulation and aspects of integrability of generalised hydrodynamics, Ann. Henri Poincaré 27, 287 (2026).
  45. A. Aggarwal, Asymptotic scattering relation for the Toda lattice, arXiv:2503.08018.
  46. A. Bastianello, A. De Luca, B. Doyon, and J. De Nardis, Thermalization of a trapped one-dimensional Bose gas via diffusion, Phys. Rev. Lett. 125, 240604 (2020).
  47. L. Biagetti, G. Cecile, and J. De Nardis, Three-stage thermalization of a quasi-integrable system, Phys. Rev. Res. 6, 023083 (2024).
  48. B. Doyon, G. Perfetto, T. Sasamoto, and T. Yoshimura, Emergence of hydrodynamic spatial long-range correlations in nonequilibrium many-body systems, Phys. Rev. Lett. 131, 027101 (2023).
  49. B. Doyon, G. Perfetto, T. Sasamoto, and T. Yoshimura, Ballistic macroscopic fluctuation theory, SciPost Phys. 15, 136 (2023).
  50. T. Yoshimura and Ž. Krajnik, Anomalous current fluctuations from Euler hydrodynamics, Phys. Rev. E 111, 024141 (2025).
  51. L. Biagetti, M. Lebek, M. Panfil, and J. De Nardis, Generalised bbgky hierarchy for near-integrable dynamics (unpublished).
  52. F. Hübner, L. Biagetti, J. D. Nardis, and B. Doyon, Diffusive hydrodynamics of hard rods from microscopics, arXiv:2503.07794.
  53. S. Gopalakrishnan, A. Morningstar, R. Vasseur, and V. Khemani, Distinct universality classes of diffusive transport from full counting statistics, Phys. Rev. B 109, 024417 (2024).
  54. S. Gopalakrishnan, E. McCulloch, and R. Vasseur, Non-Gaussian diffusive fluctuations in Dirac fluids, Proc. Natl. Acad. Sci. USA 121, e2403327121 (2024).
  55. Ž. Krajnik, E. Ilievski, and T. Prosen, Absence of normal fluctuations in an integrable magnet, Phys. Rev. Lett. 128, 090604 (2022).
  56. E. McCulloch, R. Vasseur, and S. Gopalakrishnan, Ballistic modes as a source of anomalous charge noise, Phys. Rev. E 111, 015410 (2025).
  57. O. C. Eldridge and M. Feix, Numerical experiments with a plasma model, Phys. Fluids 6, 398 (1963).
  58. J. M. Dawson, Thermal relaxation in a one-species, one-dimensional plasma, Phys. Fluids 7, 419 (1964).
  59. J. L. Rouet and M. R. Feix, Relaxation for one-dimensional plasma: Test particles versus global distribution behavior, Phys. Fluids B 3, 1830 (1991).
  60. 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).
  61. 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).
  62. G. B. Rybicki, Exact statistical mechanics of a one-dimensional self-gravitating system, Astrophys. Space Sci. 14, 56 (1971).
  63. J. Reidl, J. Charles and B. N. Miller, Gravity in one dimension - quasi-equilibrium, Astrophys. J. 332, 619 (1988).
  64. L. Milanović, H. A. Posch, and W. Thirring, Statistical mechanics and computer simulation of systems with attractive positive power-law potentials, Phys. Rev. E 57, 2763 (1998).
  65. T. Tsuchiya, N. Gouda, and T. Konishi, Relaxation processes in one-dimensional self-gravitating many-body systems, Phys. Rev. E 53, 2210 (1996).
  66. 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.
  67. 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).
  68. J. Durnin, A. De Luca, J. De Nardis, and B. Doyon, Diffusive hydrodynamics of inhomogenous Hamiltonians, J. Phys. A: Math. Theor. 54, 494001 (2021).
  69. F. Hübner, L. Biagetti, J. De Nardis, and B. Doyon, Diffusive hydrodynamics from long-range correlations, Phys. Rev. Lett. 134, 187101 (2025).
  70. B. Doyon and T. Yoshimura, A note on generalized hydrodynamics: inhomogeneous fields and other concepts, SciPost Phys. 2, 014 (2017).
  71. X. Cao, V. B. Bulchandani, and J. E. Moore, Incomplete thermalization from trap-induced integrability breaking: Lessons from classical hard rods, Phys. Rev. Lett. 120, 164101 (2018).
  72. D. Bagchi, J. Kethepalli, V. B. Bulchandani, A. Dhar, D. A. Huse, M. Kulkarni, and A. Kundu, Unusual ergodic and chaotic properties of trapped hard rods, Phys. Rev. E 108, 064130 (2023).
  73. B. Doyon, T. Yoshimura, and J.-S. Caux, Soliton gases and generalized hydrodynamics, Phys. Rev. Lett. 120, 045301 (2018).
  74. J. L. Lebowitz, J. K. Percus, and J. Sykes, Time evolution of the total distribution function of a one-dimensional system of hard rods, Phys. Rev. 171, 224 (1968).
  75. J. Percus, Exact solution of kinetics of a model classical fluid, Phys. Fluids 12, 1560 (1969).
  76. C. Boldrighini, R. L. Dobrushin, and Y. M. Sukhov, One-dimensional hard rod caricature of hydrodynamics, J. Stat. Phys. 31, 577 (1983).
  77. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, Heidelberg, 1991).
  78. C. Boldrighini and Y. M. Sukhov, One-dimensional hard rod caricature of hydrodynamics: Navier-Stokes correction for locally-equilibrium initial states, Commun. Math. Phys. 189, 577 (1997).
  79. B. Pozsgay, T. Gombor, and A. Hutsalyuk, Integrable hard-rod deformation of the Heisenberg spin chains, Phys. Rev. E 104, 064124 (2021).
  80. 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).
  81. M. A. Cazalilla, Bosonizing one-dimensional cold atomic gases, J. Phys. B: At. Mol. Opt. Phys. 37, S1 (2004).
  82. M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, One dimensional bosons: From condensed matter systems to ultracold gases, Rev. Mod. Phys. 83, 1405 (2011).
  83. I. Bouchoule and J. Dubail, Generalized hydrodynamics in the one-dimensional Bose gas: theory and experiments, J. Stat. Mech. (2022) 014003.
  84. J. Ford, S. D. Stoddard, and J. S. Turner, On the integrability of the Toda lattice, Prog. Theor. Phys. 50, 1547 (1973).
  85. M. Toda, Vibration of a chain with nonlinear interaction, J. Phys. Soc. Jpn. 22, 431 (1967).
  86. S. Lepri, R. Livi, and A. Politi, Too close to integrable: Crossover from normal to anomalous heat diffusion, Phys. Rev. Lett. 125, 040604 (2020).
  87. T. Goldfriend and J. Kurchan, Equilibration of quasi-integrable systems, Phys. Rev. E 99, 022146 (2019).
  88. M. Mestyán and V. Alba, Molecular dynamics simulation of entanglement spreading in generalized hydrodynamics, SciPost Phys. 8, 055 (2020).
  89. J.-S. Caux, B. Doyon, J. Dubail, R. Konik, and T. Yoshimura, Hydrodynamics of the interacting Bose gas in the Quantum Newton Cradle setup, SciPost Phys. 6, 070 (2019).
  90. F. Hohl and J. W. Campbell, Statistical mechanics of a collisionless self-gravitating system, Astron. J. 73, 611 (1968).
  91. H. L. Wright, B. N. Miller, and W. E. Stein, The relaxation time of a one-dimensional self-gravitating system, Astrophys. Space Sci. 84, 421 (1982).
  92. M. Joyce and T. Worrakitpoonpon, Relaxation to thermal equilibrium in the self-gravitating sheet model, J. Stat. Mech. (2010) P10012.
  93. T. N. Teles, Y. Levin, and R. Pakter, Statistical mechanics of 1d self-gravitating systems: the core–halo distribution, Mon. Not. R. Astron. Soc. 417, L21 (2011).
  94. M. Roule, J.-B. Fouvry, C. Pichon, and P.-H. Chavanis, Long-term relaxation of one-dimensional self-gravitating systems, Phys. Rev. E 106, 044118 (2022).
  95. 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).
  96. M. Medenjak, J. D. Nardis, and T. Yoshimura, Diffusion from convection, SciPost Phys. 9, 075 (2020).

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Phys. Rev. B 113, 014314– Published 20 January, 2026
  • Received 23 July 2025
  • Revised 8 October 2025
  • Accepted 14 October 2025
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