Reimann's typicality framework (Nat. Commun. 7, 10821 (2016)) predicts that a closed many-body quantum system with macroscopically well-defined energy relaxes as ⟨A⟩ρ(t)≃⟨A⟩mc+F(t) (⟨A⟩ρ(0)−⟨A⟩mc)\langle A\rangle_{\rho(t)} \simeq \langle A\rangle_{\mathrm{mc}} + F(t)\,\big(\langle A\rangle_{\rho(0)} - \langle A\rangle_{\mathrm{mc}}\big), with the relaxation function F(t)F(t) depending only on the Hamiltonian spectrum and acting on Boltzmann-time scales, for Haar-typical observables and initial states. The paper itself declares the complementary problem open: most observables and initial conditions met in practice are 'special', namely exceptionally slow, almost-conserved quantities (local densities of conserved charges, transport-bottlenecked configurations, prethermal states). This problem asks for the missing general theory of those principally untypical but practically very common slow thermalization processes: (a) criteria that classify, for a concrete Hamiltonian, observable, and initial state, whether relaxation is typical or untypical, without appealing to an abstract Haar measure; (b) a quantitative account of how near-commutation with conserved or almost-conserved quantities modifies the relaxation law beyond the F(t)F(t) prediction and beyond the already-covered weakly or bandedly perturbed regimes; and (c) an analytic framework connecting the fast spectrum-controlled prethermal stage to the subsequent slow, often approximately exponential, global equilibration.

Background

A closed (isolated) many-body quantum system evolves unitarily, yet expectation values of many observables are observed to approach the values predicted by equilibrium statistical mechanics. The relevant equilibrium reference for a pure or mixed state with macroscopically sharp energy EE is the microcanonical ensemble ρmc\rho_{\mathrm{mc}}, the maximally mixed state on a narrow energy shell of Hilbert-space dimension DD around EE. The dominant microscopic explanation is the eigenstate thermalization hypothesis (ETH): each individual energy eigenstate within the shell already reproduces thermal expectation values of physical observables, so that dephasing over the populated eigenstates drives thermalization.

An independent, measure-concentration route is typicality. A unitary matrix drawn from the Haar measure (the unique rotation-invariant uniform distribution on unitary matrices) behaves, with overwhelming probability, like a 'generic' unitary; statements proved for Haar-random objects hold for all but an exponentially small fraction of cases. Von Neumann-type and canonical-typicality results, and the works this problem directly builds on — Malabarba, García-Pintos, Linden, Farrelly and Short (Phys. Rev. E 90, 012121 (2014)), showing that all two-outcome measurements equilibrate rapidly for typical Hamiltonians, and Goldstein, Hara and Tasaki (New J. Phys. 17, 045002 (2015)), showing extremely quick thermalization for typical nonequilibrium subspaces — establish that fast equilibration is generic in this Haar sense.

Reimann (Nat. Commun. 7, 10821 (2016)) sharpened this into a quantitative, equality-type prediction. Writing the Hamiltonian eigenbasis as ∣n⟩|n\rangle and the observable eigenbasis as ∣ψν⟩|\psi_\nu\rangle, the relative eigenbasis UU is the unitary connecting the two; the theory randomizes UU over the Haar measure and predicts, for the vast majority of UU and of times tt, the relaxation law ⟨A⟩ρ(t)≃⟨A⟩mc+F(t)(⟨A⟩ρ(0)−⟨A⟩mc)\langle A\rangle_{\rho(t)} \simeq \langle A\rangle_{\mathrm{mc}} + F(t)\big(\langle A\rangle_{\rho(0)} - \langle A\rangle_{\mathrm{mc}}\big), where F(t)F(t) depends only on the Hamiltonian level spectrum inside the populated energy window — not on AA or ρ(0)\rho(0) — and decays on extremely short scales such as the Boltzmann time tB=ℏ/kBTt_B = \hbar/k_B T (about 10−1310^{-13} s at room temperature), typically with non-exponential shape. The prediction quantitatively matched experimental and numerical relaxation curves including prethermalization of a coherently split Bose gas and pump-probe electron relaxation in solids.

The catch, stated in the paper's Discussion, is that typical predicted relaxation is far too fast for what experiments usually see, because 'most of the usually considered observables and initial conditions are actually quite special, namely exceptionally slow, almost conserved quantities'. An almost-conserved quantity is an operator that nearly commutes with the Hamiltonian, [A,H]≈0[A,H] \approx 0, so its expectation value can barely change under the unitary dynamics; prime examples are local descendants of exactly conserved quantities (the energy, charge, or particle number inside a subregion, which can only change via transport currents through the subregion boundary). Relaxation of such densities between macroscopically separated places is bottlenecked by the small surface-to-volume ratio, and the second, slow global equilibration step is often well approximated by a Markovian, roughly exponential decay with a model-dependent timescale. Related slow scenarios are prethermalization (a long-lived nonthermal plateau arising, e.g., when an integrable system — one with extensively many exact conservation laws — is subject to a weak integrability-breaking perturbation) and metastable quasi-equilibria. The 2016 framework deliberately covers only the first, rapid relaxation towards local equilibrium and explicitly excludes the slow stage; the paper's own closing sentence declares that 'a better understanding of those principally untypical but practically very common thermalization processes remains an open problem', citing the rapid-equilibration works of Malabarba et al. and Goldstein, Hara and Tasaki.

The subsequent literature resolved several restricted slices of this problem. Reimann and Dabelow proved prethermalization typical for weakly perturbed integrable-class systems (Phys. Rev. Lett. 122, 080603 (2019)); the Dabelow-Reimann program derived how weak-to-moderate time-independent perturbations H=H0+λVH = H_0 + \lambda V modify a known reference relaxation via a damping factor ∣g(t)∣2|g(t)|^2 fixed by the perturbation's strength and energy-band profile (Phys. Rev. Lett. 124, 120602 (2020); Phys. Rev. Research 2, 033210 (2020); J. Stat. Mech. 2021, 013106); and Mallayya, Rigol and De Roeck established two-step relaxation with a Fermi-golden-rule crossover for specific lattice-model classes (Phys. Rev. X 9, 021027 (2019)). Meanwhile Hamazaki and Ueda (Phys. Rev. Lett. 120, 080603 (2018)) showed that the Haar-typicality argument itself fails for most few-body observables when the energy-shell width shrinks only polynomially with system size, and the Osnabrück group exhibited concrete observables (particle current in the Hubbard chain, magnetization current in the XXZ chain) whose damping under perturbations is demonstrably non-simple, with the frequency and conditions of such counterexamples still unknown (Heitmann et al., Phys. Rev. E 104, 054145 (2021); Kempa et al., Phys. Rev. E 112, 024113 (2025)). What no later work provides is the general theory the source asked for: a classification of which (H,A,ρ(0))(H, A, \rho(0)) are untypical, a quantitative relaxation law for near-conserved cases, and a bridge between the spectrum-controlled fast stage and the transport-controlled slow stage.

Problem Statement

Develop a general theory of the 'principally untypical but practically very common' slow thermalization processes that the typical fast-thermalization framework of Reimann (Nat. Commun. 7, 10821 (2016)) excludes, for closed many-body quantum systems with time-independent Hamiltonians. Given a Hamiltonian HH, an observable AA, and an initial state ρ(0)\rho(0) with macroscopically well-defined energy populating a microcanonical shell of dimension DD, the theory should deliver three jointly linked results, all three of which are required; a partial answer to any one of them does not by itself resolve the task.

(1) Classification. Establish criteria, computable for a concrete (non-random) triple (H,A,ρ(0))(H, A, \rho(0)) without appealing to an abstract Haar measure over relative eigenbases, that decide whether the relaxation of ⟨A⟩ρ(t)\langle A\rangle_{\rho(t)} follows the typical fast law ⟨A⟩ρ(t)≃⟨A⟩mc+F(t)(⟨A⟩ρ(0)−⟨A⟩mc)\langle A\rangle_{\rho(t)} \simeq \langle A\rangle_{\mathrm{mc}} + F(t)(\langle A\rangle_{\rho(0)} - \langle A\rangle_{\mathrm{mc}}) with F(t)F(t) fixed by the Hamiltonian spectrum alone, or is untypical, i.e., parametrically slower. The criteria should capture the known untypical mechanisms — near-commutation of the observable or of the initial state with HH or with additional almost-conserved quantities, local densities of conserved charges that equilibrate only through transport, prethermal and metastable plateaus, weakly coupled subsystem bottlenecks — and should say how these mechanisms overlap and which dominates.

(2) Relaxation-law modification. Quantify how the degree of near-commutation (for instance commutator norms, perturbation band profiles, or overlaps with exactly or approximately conserved operators) modifies the relaxation law — its functional form, characteristic timescales, and their scaling with system size — beyond the regimes already covered by the existing perturbation theories for H=H0+λVH = H_0 + \lambda V with weak or banded λV\lambda V and for weak integrability breaking, and beyond the transportless-equilibration assumption (the regime in which relaxation is unaffected by transport currents, approximate constants of motion, or metastable states).

(3) Unified two-stage framework. Connect the fast, spectrum-controlled stage F(t)F(t) to the subsequent slow global equilibration (transport- or bottleneck-limited, often approximately exponential) within one analytic framework that predicts, or rigorously bounds, both stages and the crossover between them for a given (H,A,ρ(0))(H, A, \rho(0)).

The common scientific objective is a predictive relaxation theory for the actually-occurring slow cases that completes the typical fast-thermalization framework; the three components jointly serve this objective and must be consistent with the documented exemplar classes — prethermalization plateaus, golden-rule crossover, the nontrivial damping of current autocorrelations in the XXZ and Hubbard chains, and the fast relaxation data the original framework already explains. These three components are complementary rather than nested — the computable criteria of (1) and the general near-conservation law of (2) are not supplied by a two-stage framework that predicts the two stages only for one particular triple, and (3) is in turn not supplied by (1) and (2) — and a complete resolution must establish all three for arbitrary triples (H,A,ρ(0))(H, A, \rho(0)) in the stated setting (closed systems with time-independent Hamiltonians and macroscopically well-defined energy); reproducing the documented exemplar classes is required validation of such a theory but does not by itself constitute the intended scope.

References

  1. P. Reimann, 'Typical fast thermalization processes in closed many-body systems', Nat. Commun. 7, 10821 (2016), DOI 10.1038/ncomms10821, https://doi.org/10.1038/ncomms10821 (arXiv:1603.00669, https://arxiv.org/abs/1603.00669)
  2. A. S. L. Malabarba, L. P. García-Pintos, N. Linden, T. C. Farrelly, A. J. Short, 'Quantum systems equilibrate rapidly for most observables', Phys. Rev. E 90, 012121 (2014), DOI 10.1103/PhysRevE.90.012121, https://doi.org/10.1103/PhysRevE.90.012121
  3. S. Goldstein, T. Hara, H. Tasaki, 'Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace', New J. Phys. 17, 045002 (2015), DOI 10.1088/1367-2630/17/4/045002, https://doi.org/10.1088/1367-2630/17/4/045002
  4. B. N. Balz, P. Reimann, 'Typical relaxation of isolated many-body systems which do not thermalize', Phys. Rev. Lett. 118, 190601 (2017), DOI 10.1103/PhysRevLett.118.190601, https://doi.org/10.1103/PhysRevLett.118.190601
  5. L. P. García-Pintos, N. Linden, A. S. L. Malabarba, A. J. Short, A. Winter, 'Equilibration time scales of physically relevant observables', Phys. Rev. X 7, 031027 (2017), DOI 10.1103/PhysRevX.7.031027, https://doi.org/10.1103/PhysRevX.7.031027
  6. R. Gallego, H. Wilming, J. Eisert, C. Gogolin, 'What it takes to avoid equilibration', Phys. Rev. A 98, 022135 (2018), DOI 10.1103/PhysRevA.98.022135, https://doi.org/10.1103/PhysRevA.98.022135
  7. R. Hamazaki, M. Ueda, 'Atypicality of most few-body observables', Phys. Rev. Lett. 120, 080603 (2018), DOI 10.1103/PhysRevLett.120.080603, https://doi.org/10.1103/PhysRevLett.120.080603
  8. J. Richter, F. Jin, H. De Raedt, K. Michielsen, J. Gemmer, R. Steinigeweg, 'Real-time dynamics of typical and untypical states in nonintegrable systems', Phys. Rev. B 97, 174430 (2018), DOI 10.1103/PhysRevB.97.174430, https://doi.org/10.1103/PhysRevB.97.174430
  9. L. Knipschild, J. Gemmer, 'Stability of quantum dynamics under constant Hamiltonian perturbations', Phys. Rev. E 98, 062103 (2018), DOI 10.1103/PhysRevE.98.062103, https://doi.org/10.1103/PhysRevE.98.062103
  10. T. Mori, T. N. Ikeda, E. Kaminishi, M. Ueda, 'Thermalization and prethermalization in isolated quantum systems: a theoretical overview', J. Phys. B 51, 192001 (2018), DOI 10.1088/1361-6455/aabcdf, https://doi.org/10.1088/1361-6455/aabcdf
  11. P. Reimann, 'Transportless equilibration in isolated many-body quantum systems', New J. Phys. 21, 053014 (2019), DOI 10.1088/1367-2630/ab1a63, https://doi.org/10.1088/1367-2630/ab1a63 (arXiv:1905.03114)
  12. P. Reimann, L. Dabelow, 'Typicality of prethermalization', Phys. Rev. Lett. 122, 080603 (2019), DOI 10.1103/PhysRevLett.122.080603, https://doi.org/10.1103/PhysRevLett.122.080603
  13. K. Mallayya, M. Rigol, W. De Roeck, 'Prethermalization and thermalization in isolated quantum systems', Phys. Rev. X 9, 021027 (2019), DOI 10.1103/PhysRevX.9.021027, https://doi.org/10.1103/PhysRevX.9.021027
  14. J. Richter, F. Jin, L. Knipschild, H. De Raedt, K. Michielsen, J. Gemmer, R. Steinigeweg, 'Exponential damping induced by random and realistic perturbations', Phys. Rev. E 101, 062133 (2020), DOI 10.1103/PhysRevE.101.062133, https://doi.org/10.1103/PhysRevE.101.062133
  15. L. Dabelow, P. Reimann, 'Relaxation theory for perturbed many-body quantum systems versus numerics and experiment', Phys. Rev. Lett. 124, 120602 (2020), DOI 10.1103/PhysRevLett.124.120602, https://doi.org/10.1103/PhysRevLett.124.120602 (arXiv:1903.11881)
  16. L. Dabelow, P. Vorndamme, P. Reimann, 'Modification of quantum many-body relaxation by perturbations exhibiting a banded matrix structure', Phys. Rev. Research 2, 033210 (2020), DOI 10.1103/PhysRevResearch.2.033210, https://doi.org/10.1103/PhysRevResearch.2.033210
  17. T. Heitmann, J. Richter, J. Gemmer, R. Steinigeweg, 'Nontrivial damping of quantum many-body dynamics', Phys. Rev. E 104, 054145 (2021), DOI 10.1103/PhysRevE.104.054145, https://doi.org/10.1103/PhysRevE.104.054145
  18. L. Dabelow, P. Reimann, 'Typical relaxation of perturbed quantum many-body systems', J. Stat. Mech. 2021, 013106 (2021), DOI 10.1088/1742-5468/abd026, https://doi.org/10.1088/1742-5468/abd026
  19. P. Reimann, L. Dabelow, 'Refining Deutsch's approach to thermalization', Phys. Rev. E 103, 022119 (2021), DOI 10.1103/PhysRevE.103.022119, https://doi.org/10.1103/PhysRevE.103.022119
  20. K. Mallayya, M. Rigol, 'Prethermalization, thermalization, and Fermi's golden rule in quantum many-body systems', Phys. Rev. B 104, 184302 (2021), DOI 10.1103/PhysRevB.104.184302, https://doi.org/10.1103/PhysRevB.104.184302
  21. L. Dabelow, P. Vorndamme, P. Reimann, 'Thermalization of locally perturbed many-body quantum systems', Phys. Rev. B 105, 024310 (2022), DOI 10.1103/PhysRevB.105.024310, https://doi.org/10.1103/PhysRevB.105.024310 (arXiv:2202.00271, https://arxiv.org/abs/2202.00271)
  22. L. Dabelow, P. Reimann, 'Random-matrix approach to time-dependent forcing in many-body quantum systems', Phys. Rev. B 110, 144308 (2024), DOI 10.1103/PhysRevB.110.144308, https://doi.org/10.1103/PhysRevB.110.144308
  23. C. Bertoni, C. Wassner, G. Guarnieri, J. Eisert, 'Typical thermalization of low-entanglement states', Commun. Phys. 8, 301 (2025), DOI 10.1038/s42005-025-02161-7, https://doi.org/10.1038/s42005-025-02161-7
  24. M. Kempa, M. Kraft, J. Wang, R. Steinigeweg, 'Nontrivial damping of magnetization currents in perturbed spin chains', Phys. Rev. E 112, 024113 (2025), DOI 10.1103/92fv-ndt1, https://doi.org/10.1103/92fv-ndt1 (arXiv:2502.14537)
  25. K. Çeven, L. Peinemann, F. Heidrich-Meisner, 'Hierarchy of timescales in a disordered spin-1/2 XX ladder', Phys. Rev. B 113, 045126 (2026), DOI 10.1103/85kv-972r, https://doi.org/10.1103/85kv-972r (arXiv:2509.20078, https://arxiv.org/abs/2509.20078)
  26. L. Dabelow, C. Eidecker-Dunkel, P. Reimann, 'Basis dependence of eigenstate thermalization', preprint (2026), arXiv:2603.23058, https://arxiv.org/abs/2603.23058