Pith. sign in

REVIEW 2 major objections 3 minor 105 references

Aperiodicity is sufficient for macroscopic thermalization

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single measurable quantity—the return probability—is enough to guarantee macroscopic thermalization.

desk verdict Real finite-time macroscopic thermalization result with a clean bound, but the printed Lemma 3.1 has a reciprocal error that must be fixed before the proof as written works. read the letter →

arxiv 2608.13462 v1 pith:UMAJX7PD submitted 2026-08-13 quant-ph cond-mat.stat-mechhep-thmath-phmath.MPnlin.CD

Amit Vikram This is my paper · ORCID
classification quant-phcond-mat.stat-mechhep-thmath-phmath.MPnlin.CD MSC 81Q5082C1037A60 PACS 05.30.-d
keywords macroscopicthermalizationaperiodicityreturnprobabilitypartialergodicityquantumequilibrationoperationalstatisticalmechanicsconcentratedobservableseffectivedimension
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that macroscopic thermalization—the settling of coarse-grained observables such as charge densities onto equilibrium values—follows from a single dynamical property of the initial state: aperiodicity, meaning the initial subspace of states has a small return probability over a finite time window. If that return probability is small enough, the subspace necessarily explores a correspondingly large fraction of the Hilbert space, and almost every state in any orthonormal basis of the subspace spends almost all times in the equilibrium subspaces of every sufficiently concentrated observable. This turns thermalization into a finitely computable, observable-independent prediction over finite time intervals, not just in the infinite-time limit. The result matters because eigenstate-based arguments cannot make finite-time predictions in the thermodynamic limit, while this mechanism can.

What carries the argument

The load-bearing object is the aperiodicity $A_{\mathcal{T}}[H_I]$, defined as the reciprocal of the double time-average of the subspace return probability $p_I(t,t')=\mathrm{Tr}[\hat{\Pi}_I(t)\hat{\Pi}_I(t')]/D_I$. Low return probability makes the time-averaged density operator $\hat{\rho}_w=\frac{1}{D_I}\int dt\, w(t)\hat{\Pi}_I(t)$ low-purity, so the nominal explored fraction $\mu_w(\hat{\Pi}_I)=1/(D\,\mathrm{Tr}[\hat{\rho}_w^2])$ is large (Lemma 3.1). Lemma 3.2 then uses Cauchy-Schwarz to show that any small subspace can capture only a small part of that history; Lemma 3.3 uses Markov's inequality to conclude that most basis states spend most weighted time in equilibrium; and Lemma 3.4 extends the bound from $[0,T]$ to arbitrarily long intervals by time-translation invariance.

What would settle it

Directly substitute Eq. (45) into the purity definition and compare with Eq. (46): the printed formula gives $\mu_w=(D_I/D)\int dt\, p_I(t)\int dt'\, w(t+t')w(t')$, while the substitution yields the reciprocal of that integral, so a numerical random-Hamiltonian check of $P_w$ distinguishes the two immediately. A physical check would measure the return probability of a prepared nonequilibrium subspace over $[0,T]$, compute $D_I A_{[0,T]}[H_I]$, and compare the predicted equilibrium fraction with the time spent by basis states in the equilibrium subspace of a concentrated observable.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 2.4: for an initial subspace $H_I$ of dimension $D_I$ and a time interval $[0,T]$, let $A_{[0,T]}[H_I]$ be the reciprocal of the time-averaged return probability of the subspace. Then every concentrated observable with nonequilibrium dimension $D_{\mathrm{neq}}$ satisfies $$f_{\mathrm{eq}} \ge 1-\frac{1}{\epsilon}\sqrt{\frac{D_{\mathrm{neq}}}{D_I A_{[0,T]}[H_I]}}$$ for the fraction of time spent within $\epsilon$ of equilibrium, uniformly over arbitrarily long intervals $[0,RT]$ including $R\to\infty$. In words, aperiodicity over a finite interval forces a partially ergodic exploration of the Hilbert space of size $D_I A_{[0,T]}[H_I]$, and that exploration forces almost all states in every orthonormal basis of $H_I$ to sit in the equilibrium subspace at almost all times, for every observable whose nonequilibrium subspace is small compared with the explored space. The same chain of lemmas also yields delocalization of the initial state over energy shells of width $\Delta E\sim 1/T$, and reduces to standard effective-dimension equilibration bounds in the infinite-time limit.

Load-bearing premise

The load-bearing premise is Lemma 3.1's bridge that turns a small return probability into a large explored Hilbert-space fraction; as printed, Eq. (46) states the reciprocal of the correct expression, so the written proof of Theorem 2.4(1) does not deliver its bound unless that formula is corrected.

Editorial extensions

If this is right

  • A single finite-time return-probability measurement on an initial subspace certifies thermalization of every sufficiently concentrated observable, for almost all states in every orthonormal basis, over arbitrarily long intervals.
  • The same finite-time data imply the initial state is delocalized across energy shells of width $\Delta E\sim 1/T$, giving a finite-time version of effective-dimension equilibration bounds.
  • For a single pure state, certifying all concentrated observables requires an observation time $T\sim D_{\mathrm{neq}}/\sigma_E$, exponentially long for macroscopic observables, so useful finite-time predictions are statements about subspaces of initial states.
  • For any finite set of macroscopic observables, including noncommuting ones, joint thermalization holds for almost all basis states when $D_I=\Theta(D_{\mathrm{neq,max}})$.
  • The infinite-time limit recovers the standard effective-dimension and delocalization-based criteria for thermalization as a special case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the probe is a single subspace return probability, the criterion could be adapted to certify thermalization in classical dynamical systems by tracking the cycle length of a phase-space cell, a link the paper only offers as intuition.
  • The same forgetfulness measure may unify state-dependent macroscopic thermalization with observable-dependent autocorrelator and OTOC criteria, which would imply a single operational signature for all rows of the paper's Table 1; the paper raises this possibility but does not prove it.
  • A corrected version of Lemma 3.1 would change quantitative predictions for any application that uses Eq. (46) verbatim, so downstream numerical protocols should re-derive the explored-fraction formula from the purity rather than quoting the printed expression.
  • The bound in Theorem 2.4 is likely not tight: permutation caricatures give a linear dependence on return probability, while the quantum bound has a square root, so tighter quantum bounds may exist for special classes of initial subspaces.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper proposes that aperiodicity of an initial subspace, defined as the reciprocal of its time-averaged return probability over a finite interval, is sufficient for finite-time thermalization of macroscopic observables. The central result, Theorem 2.4, claims that aperiodicity implies partial ergodic exploration of Hilbert space and, for any concentrated observable, that almost all states in any orthonormal basis of the initial subspace spend almost all times in the equilibrium subspace over arbitrarily long intervals. The proof is built from a purity calculation for a weighted time-averaged density operator (Lemmas 3.1-3.4), and the corollaries connect the criterion to energy-shell delocalization, pure-state limits, and multiple-observable thermalization.

Significance. If established, the result would be a substantial advance: it gives an observable-independent, finite-time, finite-resolution criterion for macroscopic thermalization from a measurable return probability, with no fitted parameters and with the thermodynamic and infinite-time limits taken afterwards rather than assumed. The paper also recovers, as special cases, standard infinite-time effective-dimension bounds, and it is explicit about the limitations of the approach for individual pure states (Corollary 4.2) and about the operational assumptions needed for the preparation protocol (Section 5.1.2). The derivation is a chain of inequalities from definitions, and I found no circular reliance on the results being proven; the dependence on the author's prior works is citation of independent derivations. However, the printed proof contains a load-bearing algebraic error in Lemma 3.1 and a false quantitative claim in Theorem 2.4(1), which must be corrected before the central claim is fully supported.

major comments (2)
  1. [Sec. 3.1, Eq. (46)] Lemma 3.1 is stated with the reciprocal of the correct expression. Substituting Eq. (45) into Eq. (42) gives mu_w = (D_I/D) / [integral dt p_I(t) c_w(t)], where c_w(t) = integral dt' w(t+t')w(t'); for the uniform weight on [0,T] this is mu_w = (D_I/D) A_T[H_I]. The printed Eq. (46) instead has (D_I/D) times the integral, i.e. the reciprocal. Because Lemma 3.1 is the bridge from aperiodicity to the explored Hilbert-space fraction used in Lemmas 3.2-3.4 and in Theorem 2.4(2), the proof as written is invalid: applying the printed lemma would give D mu_w = D_I/A_T and would reverse the inequality in Lemma 3.3. The theorem statements and the corollaries consistently use the corrected direction, so I regard this as a typographical error rather than a conceptual failure, but it must be fixed and the subsequent uses of mu_w re-verified.
  2. [Theorem 2.4(1), Eq. (34)] The quantitative claim D_M >= p_M D_I A_T[H_I] is false as stated. Lemma 3.2(2), which is the stated basis for part (1), gives only D_M >= p_M^2 D_I A_T[H_I] (setting 1-epsilon = p_M). The stronger printed inequality fails for general density operators: e.g. for rho = diag(0.9, 0.05, 0.05) in D=3, one has D mu = 1/0.815 ~ 1.227; taking M to be the rank-one subspace spanned by the dominant eigenvector gives p_M = 0.9 and p_M D mu ~ 1.104 > 1 = D_M. The qualitative statement -- that any subspace containing most of the history must have size Omega(D_I A_T) -- survives, but Eq. (34) should be replaced by the p_M^2 bound or by a correctly stated weaker formulation. This does not affect Eq. (36), whose proof uses D_I A_T directly, but Theorem 2.4(1) as printed is a mathematical claim of the paper and must be corrected.
minor comments (3)
  1. [Sec. 4.3, Eq. (92)] The summand in Eq. (92) should carry the observable-subset index m, i.e. f_eq[BI, A_m](RT, epsilon), rather than the index-free f_eq[BI, M](RT, epsilon); as written the bound is not clearly the termwise union bound described in the text.
  2. [Sec. 5.1.2, Eqs. (103)-(106)] The existence of a dissipative preparation D with finite overlap p_0 and the state-wise bound p_k >= p_0/D_I is an additional operational assumption; the text should present it as an explicit assumption rather than as something 'taken for granted,' since Theorem 2.4 itself does not require this protocol.
  3. [Sec. 4.2 and Cor. 4.1] There are a few presentational slips: 'Dneq,,max' in the paragraph before Corollary 4.3 should read D_neq,max, and the notation mu_Theta(t in [0,T])/T in Corollary 4.1 is typeset in a way that makes the subscript hard to parse; a local definition of the subscript would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central bound is derived by explicit inequalities from the stated definitions.

full rationale

The central claim (Theorem 2.4) is proved inside the paper as a chain of inequalities starting from explicit definitions: aperiodicity (Definition 2.3) is the reciprocal of the double time-averaged return probability; Eq. (45) computes the purity of the weighted Krylov ensemble from the return probability; Lemma 3.2 uses Cauchy-Schwarz to bound the dimension of any subspace carrying a large fraction of the history; Lemma 3.3 applies Markov's inequality to convert low nonequilibrium overlap into a large equilibrium-time fraction; Lemma 3.4 extends the bound to longer intervals by time-translation invariance. No parameter is fitted to data, and no prediction is used to fix a constant. The self-citations ([23,24,37,38]) supply terminology, intuition, and background results, but the aperiodicity-to-thermalization implication itself is not imported from those works: the crucial relation between the purity-derived nominal fraction and the return probability is written out in Eqs. (43)-(46), and the subsequent bounds are elementary. The printed Lemma 3.1 contains an algebraic direction error—substituting Eq. (45) into Eq. (42) gives mu_w = (D_I/D) A_T rather than the printed expression—but this is a proof-correctness defect, not circularity, and Theorem 2.4 uses the corrected direction. The paper also explicitly flags operational limitations (e.g., Sec. 4.2 on the long times needed for pure states and Sec. 5.1.2 on the difficulty of preparing the initial subspace), but these concern computability and time scales, not circular input. Overall, the derivation is self-contained and the result is not equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters appear in the derivation: the bound is expressed in terms of the measurable aperiodicity A_T, the subspace dimension D_I, and the nonequilibrium dimension D_neq. The axioms are standard background or domain assumptions. The only ad hoc element is the dissipative preparation protocol, which the paper explicitly flags as needing future work. No new physical entities are introduced.

assumptions (5)
  • domain assumption Finite-dimensional Hilbert space with autonomous unitary dynamics generated by a Hermitian Hamiltonian (Eq. 10).
    Standard setting for finite quantum systems; the paper explicitly restricts to D > 1 finite.
  • domain assumption Macroscopic observables are concentrated, meaning their equilibrium subspace occupies almost all of Hilbert space (Definition 2.1 and the spin-density example in Sec. 2.1).
    The theorem's bound is only nontrivial when D_neq is much smaller than D; this is the defining property of the observables the claim covers.
  • ad hoc to paper For the operational protocol, the existence of a dissipative process that prepares a proxy state rho_J with finite overlap p_0 with the initial subspace H_I (Sec. 5.1.2, Eq. 104).
    The paper itself leaves this to future work: 'pending the determination of such dissipative processes in different contexts of interest.' The theorem itself does not depend on this; the operational claim does.
  • standard math Cauchy-Schwarz inequality and Markov's inequality are applied in Lemmas 3.2 and 3.3.
    Standard inequalities; used correctly in the derivation.
  • domain assumption Time-translation invariance of the ergodicity measure (Lemma 3.4) relies on unitarity of the dynamics.
    Standard property of closed Hamiltonian evolution, required to extrapolate from the finite interval [0,T] to arbitrary intervals [0,RT].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Aperiodicity is sufficient for macroscopic thermalization." pith.science (2026). https://pith.science/paper/UMAJX7PD

@misc{pith2026260813462,
  author       = {Pith},
  title        = {Pith review of: Aperiodicity is sufficient for macroscopic thermalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMAJX7PD}},
  note         = {Machine review of arXiv:2608.13462}
}
read the original abstract

We identify a general mechanism for the finite-time thermalization of macroscopic observables, such as coarse-grained charge densities, in terms of elementary forms of the quantum dynamics of initial states: (1) aperiodicity, which provides a computable measure of (2) a dynamical partially ergodic exploration of the Hilbert space. Specifically, this mechanism predicts the equilibration of all (concentrated) macroscopic observables, in almost all states in an initial ensemble and almost all times within finite and longer intervals, given only the observable-independent information that the return probability of the ensemble of initial states is small over a finite time range. As a special case, it also accesses standard results on equilibration over infinitely long times in terms of (stronger versions of) the effective dimension of initial state delocalization in the energy eigenbasis. Our results incorporate macroscopic thermalization into the domain of operational quantum statistical mechanics, recently developed to provide finitely computable criteria for microscopic thermalization. We discuss an overall characterization of this approach as establishing connections between (1) the decay of a (theoretically or experimentally) computable probe indicating memorylessness, (2) a fundamental invariant mechanism in terms of the alignment of observables or states in the Hilbert space, and (3) predicting different natural forms of (classical and) quantum thermalization, most of which rigorously recover conventional eigenstate-based descriptions of infinite-time thermalization as a special case but provide stronger accessible predictions over finite observation times in the thermodynamic limit.

Figures

Figures reproduced from arXiv: 2608.13462 by the authors.

Figure 1
Figure 1. A depiction of the role of ergodic dynamics in macroscopic thermalization in a schematic state [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

105 extracted references · 52 canonical work pages

  1. [1]

    R. C. Tolman,The principles of statistical mechanics, Vol. 51 (Dover Publications, 2010)

    2010

  2. [2]

    A. Y . Khinchin,Mathematical foundations of statistical mechanics(Courier Corporation, 1949),ISBN: 9780486601472. 12Beyond eigenstate structure conclusions that follow from the assumedexacttime-translation invariance stemming from isolated Hamiltonian dynamics, which itself may be restricted to only early times (without persisting long enough to carry imp...

    1949

  3. [3]

    L. D. Landau and E. M. Lifshitz,Statistical physics, Vol. 5 (Elsevier, 1980),ISBN: 9780080570464

    1980

  4. [4]

    P . R. Halmos,Lectures on ergodic theory(Dover Publications, 2017),ISBN: 9780486814896

    2017

  5. [5]

    Y. G. Sinai,Introduction to ergodic theory, Vol. 18 (Princeton University Press, 1977), ISBN: 978-0691081823

    1977

  6. [6]

    I. P . Cornfeld, S. V . Fomin, and Y. G. Sinai,Ergodic theory(Springer-Verlag New York, 1982),ISBN: 978-1-4615-6927-5

    1982

  7. [7]

    Many-body localization and thermalization in quantum statistical mechanics

    R. Nandkishore and D. A. Huse, “Many-body localization and thermalization in quantum statistical mechanics”, Annu. Rev. Condens. Matter Phys.6, 15–38 (2015)

    2015

  8. [8]

    Equilibration, thermalisation, and the emergence of statis- tical mechanics in closed quantum systems

    C. Gogolin and J. Eisert, “Equilibration, thermalisation, and the emergence of statis- tical mechanics in closed quantum systems”, Rep. Prog. Phys.79, 056001 (2016)

    2016

Show all 105 references
  1. [9]

    Thermalization and prethermaliza- tion in isolated quantum systems: a theoretical overview

    T . Mori, T . N. Ikeda, E. Kaminishi, and M. Ueda, “Thermalization and prethermaliza- tion in isolated quantum systems: a theoretical overview”, J. Phys. B.: At. Mol. Opt. Phys.51, 112001 (2018)

    2018

  2. [10]

    Black holes as mirrors: quantum information in random subsystems

    P . Hayden and J. Preskill, “Black holes as mirrors: quantum information in random subsystems”, J. High Energy Phys.2007, 120 (2007)

    2007

  3. [11]

    Chaos, complexity , and random matrices

    J. Cotler, N. Hunter-Jones, J. Liu, and B. Yoshida, “Chaos, complexity , and random matrices”, J. High Energy Phys.2017, 1–60 (2017)

    2017

  4. [12]

    Quantum many-body simulations on digital quantum computers: state-of-the-art and future challenges

    B. Fauseweh, “Quantum many-body simulations on digital quantum computers: state-of-the-art and future challenges”, Nat. Comm.15, 2123 (2024)

    2024

  5. [13]

    Thermalization and criti- cality on an analogue–digital quantum simulator

    T . I. Andersen, N. Astrakhantsev, A. H. Karamlou, J. Berndtsson, J. Motruk, A. Szasz, J. A. Gross, A. Schuckert, T . Westerhout, Y. Zhang, et al., “Thermalization and criti- cality on an analogue–digital quantum simulator”, Nature638, 79–85 (2025)

    2025

  6. [14]

    Digital quantum magnetism on a trapped- ion quantum computer

    R. Haghshenas, E. Chertkov, M. Mills, W . Kadow, S.-H. Lin, Y.-H. Chen, C. Cade, I. Niesen, T . Beguši´c, M. S. Rudolph, et al., “Digital quantum magnetism on a trapped- ion quantum computer”, Nature653, 56–62 (2026)

    2026

  7. [15]

    M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information (Cambridge University Press, 2010)

    2010

  8. [16]

    CODATA recommended values of the fundamental physical constants: 2022

    P . J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022”, J. Phys. Chem. Ref. Data54, 033105 (2025)

    2025

  9. [17]

    O’Raifeartaigh,Group structure of gauge theories(Cambridge University Press, 1986),ISBN: 9780511564031

    L. O’Raifeartaigh,Group structure of gauge theories(Cambridge University Press, 1986),ISBN: 9780511564031

    1986

  10. [18]

    Boucheron, G

    S. Boucheron, G. Lugosi, and P . Massart,Concentration inequalities: a nonasymptotic theory of independence(Oxford Univ. Press, Oxford, 2013),ISBN: 9780199535255

    2013

  11. [19]

    S. M. Ross,A first course in probability(Pearson Harlow, UK, 2020),ISBN: 9780138076719

    2020

  12. [20]

    Gallavotti,Statistical mechanics: a short treatise(Springer Science & Business Media, 1999)

    G. Gallavotti,Statistical mechanics: a short treatise(Springer Science & Business Media, 1999)

    1999

  13. [21]

    Typicality of thermal equilibrium and thermalization in isolated macro- scopic quantum systems

    H. Tasaki, “Typicality of thermal equilibrium and thermalization in isolated macro- scopic quantum systems”, J. Stat. Phys.163, 937–997 (2016)

    2016

  14. [22]

    Big omicron and big omega and big theta

    D. E. Knuth, “Big omicron and big omega and big theta”, ACM Sigact News8, 18–24 (1976). 34

    1976

  15. [23]

    Bypassing eigenstate thermalization with experimentally accessible quantum dynamics

    A. Vikram, “Bypassing eigenstate thermalization with experimentally accessible quantum dynamics”, SciPost Phys.21, 029 (2026)

    2026

  16. [24]

    Provable quantum thermalization without statistical averages

    A. Vikram, “Provable quantum thermalization without statistical averages”, arXiv preprint arXiv:2604.02417 (2026)

    2026 arXiv

  17. [25]

    A tale of many H0

    L. Verde, N. Schöneberg, and H. Gil-Marín, “A tale of many H0”, Annu. Rev. Astron. Astrophys.62, 287–331 (2024)

    2024

  18. [26]

    Proof of the ergodic theorem and the H-theorem in quantum mechanics. Translation of: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik

    J. von Neumann, “Proof of the ergodic theorem and the H-theorem in quantum mechanics. Translation of: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik”, Eur. Phys. J. H35, Original (in German): Zeit. für Phys. 57, 30 (1929), 201–237 (2010)

    1929

  19. [27]

    Approach to thermal equilibrium of macroscopic quantum systems

    S. Goldstein, J. L. Lebowitz, C. Mastrodonato, R. Tumulka, and N. Zanghi, “Approach to thermal equilibrium of macroscopic quantum systems”, Phys. Rev. E81, 011109 (2010)

    2010

  20. [28]

    Thermal equilibrium of a macroscopic quantum system in a pure state

    S. Goldstein, D. A. Huse, J. L. Lebowitz, and R. Tumulka, “Thermal equilibrium of a macroscopic quantum system in a pure state”, Phys. Rev. Lett.115, 100402 (2015)

    2015

  21. [29]

    Macroscopic and micro- scopic thermal equilibrium

    S. Goldstein, D. A. Huse, J. L. Lebowitz, and R. Tumulka, “Macroscopic and micro- scopic thermal equilibrium”, Annalen der Physik529, 1600301 (2017)

    2017

  22. [30]

    Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace

    S. Goldstein, T . Hara, and H. Tasaki, “Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace”, New J. Phys.17, 045002 (2015)

    2015

  23. [31]

    The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace

    S. Goldstein, T . Hara, and H. Tasaki, “The approach to equilibrium in a macroscopic quantum system for a typical nonequilibrium subspace”, arXiv preprint arXiv:1402.3380 (2014)

    2014 arXiv

  24. [32]

    Time scales in the approach to equilibrium of macroscopic quantum systems

    S. Goldstein, T . Hara, and H. Tasaki, “Time scales in the approach to equilibrium of macroscopic quantum systems”, Phys. Rev. Lett.111, 140401 (2013)

    2013

  25. [33]

    Macroscopic thermalization for highly degenerate Hamiltonians

    B. Roos, S. Teufel, R. Tumulka, and C. Vogel, “Macroscopic thermalization for highly degenerate Hamiltonians”, J. Stat. Phys.192, 109 (2025)

    2025

  26. [34]

    Undecidability in quantum thermalization

    N. Shiraishi and K. Matsumoto, “Undecidability in quantum thermalization”, Nat. Comm.12, 5084 (2021)

    2021

  27. [35]

    The complexity of thermalization in finite quantum systems

    D. Devulapalli, T . Mooney, and J. D. Watson, “The complexity of thermalization in finite quantum systems”, arXiv preprint arXiv:2507.00405 (2025)

    2025

  28. [36]

    F . W . Byron and R. W . Fuller,Mathematics of classical and quantum physics(Dover Publications, 2012),ISBN: 9780486135069

    2012

  29. [37]

    Enhanced entanglement from quantum ergodicity

    A. Vikram, “Enhanced entanglement from quantum ergodicity”, arXiv preprint arXiv:2507.08067 (2025)

    2025 arXiv

  30. [38]

    Dynamical quantum ergodicity from energy level statistics

    A. Vikram and V . Galitski, “Dynamical quantum ergodicity from energy level statistics”, Phys. Rev. Res.5, 033126 (2023)

    2023

  31. [39]

    Canonical typicality

    S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghì, “Canonical typicality”, Phys. Rev. Lett.96, 050403 (2006)

    2006

  32. [40]

    Entanglement and the foundations of statisti- cal mechanics

    S. Popescu, A. J. Short, and A. Winter, “Entanglement and the foundations of statisti- cal mechanics”, Nat. Phys.2, 754–758 (2006). 35

    2006

  33. [41]

    Normal typicality and von Neumann’s quantum ergodic theorem

    S. Goldstein, J. L. Lebowitz, C. Mastrodonato, R. Tumulka, and N. Zanghì, “Normal typicality and von Neumann’s quantum ergodic theorem”, Proc. Roy. Soc. Lond. A 466, 3203–3224 (2010)

    2010

  34. [42]

    Statistical behavior in deterministic quantum systems with few degrees of freedom

    R. V . Jensen and R. Shankar, “Statistical behavior in deterministic quantum systems with few degrees of freedom”, Phys. Rev. Lett.54, 1879 (1985)

    1985

  35. [43]

    Quantum statistical mechanics in a closed system

    J. M. Deutsch, “Quantum statistical mechanics in a closed system”, Phys. Rev. A43, 2046–2049 (1991)

    1991

  36. [44]

    Chaos and quantum thermalization

    M. Srednicki, “Chaos and quantum thermalization”, Phys. Rev. E50, 888–901 (1994)

    1994

  37. [45]

    The approach to thermal equilibrium in quantized chaotic systems

    M. Srednicki, “The approach to thermal equilibrium in quantized chaotic systems”, J. Phys. A: Math. Gen.32, 1163–1175 (1999)

    1999

  38. [46]

    Thermalization and its mechanism for generic isolated quantum systems

    M. Rigol, V . Dunjko, and M. Olshanii, “Thermalization and its mechanism for generic isolated quantum systems”, Nature452, 854–858 (2008)

    2008

  39. [47]

    From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, “From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics”, Adv. Phys. 65, 239–362 (2016)

    2016

  40. [48]

    Eigenstate thermalization hypothesis

    J. M. Deutsch, “Eigenstate thermalization hypothesis”, Rep. Prog. Phys.81, 082001 (2018)

    2018

  41. [49]

    From quantum dynamics to the canonical distribution: general picture and a rigorous example

    H. Tasaki, “From quantum dynamics to the canonical distribution: general picture and a rigorous example”, Phys. Rev. Lett.80, 1373 (1998)

    1998

  42. [50]

    Foundation of statistical mechanics under experimentally realistic conditions

    P . Reimann, “Foundation of statistical mechanics under experimentally realistic conditions”, Phys. Rev. Lett.101, 190403 (2008)

    2008

  43. [51]

    Quantum mechanical evolution towards thermal equilibrium

    N. Linden, S. Popescu, A. J. Short, and A. Winter, “Quantum mechanical evolution towards thermal equilibrium”, Phys. Rev. E79, 061103 (2009)

    2009

  44. [52]

    Equilibration of quantum systems and subsystems

    A. J. Short, “Equilibration of quantum systems and subsystems”, New J. Phys.13, 053009 (2011)

    2011

  45. [53]

    Quantum equilibration in finite time

    A. J. Short and T . C. Farrelly, “Quantum equilibration in finite time”, New J. Phys. 14, 013063 (2012)

    2012

  46. [54]

    The uncertainty relation between energy and time in non-relativistic quantum mechanics

    L. Mandelstam and I. Tamm, “The uncertainty relation between energy and time in non-relativistic quantum mechanics”, inSelected papers(Springer, 1991), pp. 115– 123

    1991

  47. [55]

    Geometry of quantum evolution

    J. Anandan and Y. Aharonov, “Geometry of quantum evolution”, Phys. Rev. Lett.65, 1697 (1990)

    1990

  48. [56]

    The maximum speed of dynamical evolution

    N. Margolus and L. B. Levitin, “The maximum speed of dynamical evolution”, Physica D120, 188–195 (1998)

    1998

  49. [57]

    Fundamental limit on the rate of quantum dynamics: the unified bound is tight

    L. B. Levitin and T . Toffoli, “Fundamental limit on the rate of quantum dynamics: the unified bound is tight”, Phys. Rev. Lett.103, 160502 (2009)

    2009

  50. [58]

    Bounds in nonequilibrium quantum dynamics

    Z. Gong and R. Hamazaki, “Bounds in nonequilibrium quantum dynamics”, Int. J. Mod. Phys. B.36(2022)

    2022

  51. [59]

    Equi- libration time scales of physically relevant observables

    L. P . García-Pintos, N. Linden, A. S. Malabarba, A. J. Short, and A. Winter, “Equi- libration time scales of physically relevant observables”, Phys. Rev. X7, 031027 (2017). 36

    2017

  52. [60]

    P . A. M. Dirac,The principles of quantum mechanics, 27 (Oxford university press, 1981)

    1981

  53. [61]

    Shankar,Principles of quantum mechanics(Springer Science & Business Media, 2012),ISBN: 978-1-4757-0576-8

    R. Shankar,Principles of quantum mechanics(Springer Science & Business Media, 2012),ISBN: 978-1-4757-0576-8

    2012

  54. [62]

    Reif,Fundamentals of statistical and thermal physics(Waveland Press, 2009),ISBN: 978-1-57766-612-7

    F . Reif,Fundamentals of statistical and thermal physics(Waveland Press, 2009),ISBN: 978-1-57766-612-7

    2009

  55. [63]

    L. E. Reichl,A modern course in statistical physics(John Wiley & Sons, 2016),ISBN: 978-3-527-41349-2

    2016

  56. [64]

    Science of chaos or chaos in science?

    J. Bricmont, “Science of chaos or chaos in science?”, arXiv preprint chao-dyn/9603009 (1996)

    1996 arXiv

  57. [65]

    Boltzmann’s approach to statistical mechanics

    S. Goldstein, “Boltzmann’s approach to statistical mechanics”, arXiv preprint cond- mat/0105242 (2001)

    2001

  58. [66]

    Microscopic origins of macroscopic behavior

    J. L. Lebowitz, “Microscopic origins of macroscopic behavior”, arXiv preprint arXiv:2105.03470 (2021)

    2021 arXiv

  59. [67]

    Hamiltonian systems and transformation in Hilbert space

    B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space”, Proc. Natl. Acad. Sci. U.S.A.17, 315–318 (1931)

    1931

  60. [68]

    Zur Operatorenmethode in der klassischen Mechanik

    J. von Neumann, “Zur Operatorenmethode in der klassischen Mechanik”, Ann. Math. 33, 587–642 (1932)

    1932

  61. [69]

    Zusätze Zur Arbeit, Zur Operatorenmethode

    J. von Neumann, “Zusätze Zur Arbeit, Zur Operatorenmethode...”, Ann. Math.33, 789–791 (1932)

    1932

  62. [70]

    Approximation of ergodic dynamic systems by periodic transformations

    A. B. Katok and A. M. Stepin, “Approximation of ergodic dynamic systems by periodic transformations”, Dokl. Akad. Nauk SSSR171, 1268–1271 (1966)

    1966

  63. [71]

    Approximations in ergodic theory

    A. B. Katok and A. M. Stepin, “Approximations in ergodic theory”, Russ. Math. Surv. 22, 77 (1967)

    1967

  64. [72]

    Theory of dynamical systems and general transformation groups with invariant measure

    A. B. Katok, Y . G. Sinai, and A. M. Stepin, “Theory of dynamical systems and general transformation groups with invariant measure”, J. Math. Sci.7, 974–1065 (1977)

    1977

  65. [73]

    M. G. Nadkarni,Spectral theory of dynamical systems, 2nd edition(Springer, Singapore, 2020),ISBN: 978-981-15-6225-9

    2020

  66. [74]

    E. S. Meckes,The random matrix theory of the classical compact groups, Vol. 218, Cambridge Tracts in Mathematics (Cambridge University Press, 2019)

    2019

  67. [75]

    Vershynin,High-dimensional probability: an introduction with applications in data science, Vol

    R. Vershynin,High-dimensional probability: an introduction with applications in data science, Vol. 47 (Cambridge university press, 2018)

    2018

  68. [76]

    Level clustering in the regular spectrum

    M. V . Berry and M. Tabor, “Level clustering in the regular spectrum”, Proc. Roy . Soc. Lond. A356, 375–394 (1977)

    1977

  69. [77]

    Spectrum and eigenfunctions for a hamiltonian with stochastic trajectories

    S. W . McDonald and A. N. Kaufman, “Spectrum and eigenfunctions for a hamiltonian with stochastic trajectories”, Phys. Rev. Lett.42, 1189 (1979)

    1979

  70. [78]

    On the connection between quantization of nonintegrable systems and statistical theory of spectra

    G. Casati, F . Valz-Gris, and I. Guarnieri, “On the connection between quantization of nonintegrable systems and statistical theory of spectra”, Lett. Nuovo Cimento28, 279–282 (1980)

    1980

  71. [79]

    Quantizing a classically ergodic system: Sinai’s billiard and the KKR method

    M. V . Berry, “Quantizing a classically ergodic system: Sinai’s billiard and the KKR method”, Ann. Phys.131, 163–216 (1981). 37

    1981

  72. [80]

    Characterization of chaotic quantum spectra and universality of level fluctuation laws

    O. Bohigas, M.-J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws”, Phys. Rev. Lett.52, 1 (1984). [81]F . Haake,Quantum signatures of chaos(Springer, Berlin, Heidelberg, 2001)

    1984

  73. [82]

    Quantum ergodicity and a quantum measure algebra

    E. B. Stechel, “Quantum ergodicity and a quantum measure algebra”, J. Chem. Phys. 82, 364 (1985)

    1985

  74. [83]

    Quantum ergodicity and spectral chaos

    E. Stechel and E. Heller, “Quantum ergodicity and spectral chaos”, Annu. Rev. Phys. Chem.35, 563–589 (1984)

    1984

  75. [84]

    Quantum ergodicity and mixing

    S. Zelditch, “Quantum ergodicity and mixing”, arXiv preprint math-ph/0503026 (2005)

    2005 arXiv

  76. [85]

    Out-of-equilibrium eigenstate thermaliza- tion hypothesis

    L. Foini, A. Dymarsky , and S. Pappalardi, “Out-of-equilibrium eigenstate thermaliza- tion hypothesis”, SciPost Phys.18, 136 (2025)

    2025

  77. [86]

    Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control

    S. Deffner and S. Campbell, “Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control”, J. Phys. A: Math. Theor.50, 453001 (2017)

    2017

  78. [87]

    Geometric speed limit of accessible many- body state preparation

    M. Bukov, D. Sels, and A. Polkovnikov, “Geometric speed limit of accessible many- body state preparation”, Phys. Rev. X9, 011034 (2019)

    2019

  79. [88]

    Quantum computation and quantum- state engineering driven by dissipation

    F . Verstraete, M. M. Wolf, and J. Ignacio Cirac, “Quantum computation and quantum- state engineering driven by dissipation”, Nat. Phys.5, 633–636 (2009)

    2009

  80. [89]

    Kamenev,Field theory of non-equilibrium systems(Cambridge University Press, 2023)

    A. Kamenev,Field theory of non-equilibrium systems(Cambridge University Press, 2023)

    2023

  81. [90]

    Introduction to nonequilibrium quantum field theory

    J. Berges, “Introduction to nonequilibrium quantum field theory”, in AIP Conf. Proc. Vol. 739, 1 (American Institute of Physics, 2004), pp. 3–62

    2004

  82. [91]

    Characteristic vectors of bordered matrices with infinite dimensions. JSTOR: Annals of Mathematics, Second Series, Vol. 62, No. 3 (Nov., 1955), pp. 548-564

    E. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions. JSTOR: Annals of Mathematics, Second Series, Vol. 62, No. 3 (Nov., 1955), pp. 548-564”, Ann. Math.62, 548–564 (1955)

    1955

  83. [92]

    Statistical Theory of the Energy Levels of Complex Systems. I

    F . J. Dyson, “Statistical Theory of the Energy Levels of Complex Systems. I”, J. Math. Phys.3, 140–156,ISSN: 0022-2488 (1962)

    1962

  84. [93]

    Statistical theory of the energy levels of complex systems. IV

    F . J. Dyson and M. L. Mehta, “Statistical theory of the energy levels of complex systems. IV”, J. Math. Phys.4, 701–712 (1963). [94]M. L. Mehta,Random matrices(Elsevier, 2004),ISBN: 978-0-12-088409-4

    1963

  85. [95]

    Periodic orbits and a correlation function for the semiclassical density of states

    J. H. Hannay and A. M. Ozorio de Almeida, “Periodic orbits and a correlation function for the semiclassical density of states”, J. Phys. A: Math. Gen.17, 3429 (1984)

    1984

  86. [96]

    Semiclassical theory of spectral rigidity

    M. V . Berry, “Semiclassical theory of spectral rigidity”, Proc. Roy . Soc. Lond. A400, 229–251 (1985)

    1985

  87. [97]

    Some quantum-to-classical asymptotics

    M. V . Berry, “Some quantum-to-classical asymptotics”, inLes houches lecture series LII (1989), edited by M. J. Giannoni, A. Voros, and J. Zinn-Justin (Elsevier Science Publishers B.V ., 1991), pp. 251–304,ISBN: 978-0444892775

    1989

  88. [98]

    Semiclassical foundation of universality in quantum chaos

    S. Müller, S. Heusler, P . Braun, F . Haake, and A. Altland, “Semiclassical foundation of universality in quantum chaos”, Phys. Rev. Lett.93, 014103 (2004)

    2004

  89. [99]

    Periodic-orbit theory of universality in quantum chaos

    S. Müller, S. Heusler, P . Braun, F . Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos”, Phys. Rev. E72, 046207 (2005)

    2005

  90. [100]

    Arithmetical chaos

    E. Bogomolny, B. Georgeot, M.-J. Giannoni, and C. Schmit, “Arithmetical chaos”, Phys. Rep.291, 219–324 (1997). 38

    1997

  91. [101]

    Level statistics in arithmetical and pseudo-arithmetical chaos

    P . Braun and F . Haake, “Level statistics in arithmetical and pseudo-arithmetical chaos”, J. Phys. A: Math. Theor.43, 262001 (2010)

    2010

  92. [102]

    Spectral anomalies and broken symmetries in maximally chaotic quantum maps

    L. Shou, A. Vikram, and V . Galitski, “Spectral anomalies and broken symmetries in maximally chaotic quantum maps”, SciPost Phys.18, 151 (2025)

    2025

  93. [103]

    Two types of quantum chaos: testing the limits of the Bohigas-Giannoni-Schmit conjecture

    J. M. Magan and Q. Wu, “Two types of quantum chaos: testing the limits of the Bohigas-Giannoni-Schmit conjecture”, arXiv preprint arXiv:2411.08186 (2024)

    2024 arXiv

  94. [104]

    Quantum thermalization and the route to ergodicity

    A. Vardi and D. Cohen, “Quantum thermalization and the route to ergodicity”, Phys. Rev. Res.7, 043293 (2025)

    2025

  95. [105]

    On the reconstruction map in JT gravity

    C. Akers, A. Lucas, and A. Vikram, “On the reconstruction map in JT gravity”, J. High Energy Phys.12, 045 (2025)

    2025

  96. [106]

    Exact universal bounds on quantum dynamics and fast scrambling

    A. Vikram and V . Galitski, “Exact universal bounds on quantum dynamics and fast scrambling”, Phys. Rev. Lett.132, 040402 (2024)

    2024

  97. [107]

    Proof of a universal speed limit on fast scrambling in quantum systems

    A. Vikram, L. Shou, and V . Galitski, “Proof of a universal speed limit on fast scrambling in quantum systems”, Phys. Rev. Lett.136, 150401 (2026). 39

    2026

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.