Quasi-integrable systems are slow to thermalize but may be good scramblers

Jorge Kurchan; Tomer Goldfriend

REVIEW 3 major objections 6 minor 62 references

Quasi-integrable systems are slow to thermalize but may be good scramblers

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A weakly noise-driven integrable quantum system has a semiclassical Lyapunov exponent that matches the classical one and scales as $\epsilon^{1/3}$; quantum discreteness shuts off the exponential regime at small quantum numbers or small…

desk verdict Worth engaging: a genuinely new quantum tangent-space formalism with a clean ε^{1/3} check in the numerics, but the semiclassical claim as stated overreaches for γ<1/2 because the Magnus average breaks down. read the letter →

arxiv 1909.02145 v3 pith:E72MG5YZ submitted 2019-09-04 quant-ph cond-mat.stat-mechhep-th

Tomer Goldfriend , Jorge Kurchan This is my paper · ORCID

classification quant-phcond-mat.stat-mechhep-th MSC 81Q5070H08

keywords quantumLyapunovexponentout-of-time-ordercorrelatorquasi-integrablesystemsnoise-inducedchaostangentspaceprescramblingtimerandomlykickedrotorbound

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

This paper asks whether quantum versions of quasi-integrable systems—like a rotor weakly perturbed by random kicks—can scramble information quickly even though they thermalize slowly. It derives a linear superoperator equation, a quantum tangent space, that controls the growth of the out-of-time-order correlator (OTOC) for the Hamiltonian $H = H_{\rm int}(N) + \epsilon^{1/2}\eta(t)G(N,e^{i\Theta})$. In the semiclassical limit the resulting quantum Lyapunov exponent equals the classical one and scales as $\epsilon^{1/3}$, so prescrambling is fast while energy diffusion, which proceeds at a rate $\sim \epsilon$, remains slow. For small initial quantum numbers or very small perturbations the exponential Lyapunov regime disappears, a purely quantum effect. Because the dimensionless product $\beta\hbar\lambda_T$ can stay finite as the temperature $T$ goes to zero, the authors conclude that quasi-integrable systems are relatively good scramblers even at low temperature.

What carries the argument

The central object is the quantum tangent space, a pair of operators $C_\Theta=[e^{i\Theta},A_0]e^{-i\Theta}$ and $C_N=i[N,A_0]$ whose quadratic expectation values give the OTOC growth. Their evolution is governed by a linear superoperator equation whose noise-free part contains the integrable-drive superoperator $L$ and a purely quantum factor-reordering term $J\odot C = i(\omega_n-\omega_{n'})C_{nn'}$. After a Fokker-Planck step and a Magnus time-average, the closed set of equations reduces to the third-order ODE for $F^{\Theta\Theta}_{nn'}=|C^\Theta_{nn'}|^2$, namely $d^3F/dt^3 + 2l^2(n,n')(n-n')^2\,dF/dt - (\tilde\epsilon/4)\,2l^2(n,n')(V(n)+V(n'))^2(F_{n,n'+1}+F_{n+1,n'})=0$. The term linear in the first derivative, originating in $J$, obstructs the simple $\tilde\epsilon^{1/3}$ time rescaling and is responsible for the quantum cutoffs; dropping it self-consistently in the semiclassical limit yields the classical Lyapunov scaling.

What would settle it

For the randomly kicked rotor at fixed $n_0$, measure the OTOC growth rate $\lambda_Q$; the central claim fails if, as $\tilde\epsilon\to0$ with $n_0$ held large enough that $|\omega_{n_0}-\omega_{n_0+Z}|\ll\tilde\lambda_Q$, $\lambda_Q$ does not converge to $(1/2)2^{2/3}\tilde\epsilon^{1/3}(\gamma(\gamma-1)n_0^{\gamma-2})^{2/3}n_0^{2\mu/3}$. A second decisive test: fix $\tilde\epsilon$ and decrease $n_0$, checking that the exponential window disappears exactly when the level-spacing difference $|\omega_{n_0}-\omega_{n_0+Z}|$ becomes comparable to the classical Lyapunov rate, as Eq. (46) predicts.

Watch

Extended reading notes

Core claim

The paper establishes that in the semiclassical limit the quantum annealed Lyapunov exponent of a weakly noise-perturbed integrable system is given by the classical Lyapunov exponent of the same system, $2\tilde\lambda_Q = 2^{2/3}\tilde\epsilon^{1/3}\bigl(\gamma(\gamma-1)n_0^{\gamma-2}\bigr)^{2/3}n_0^{2\mu/3}$, obtained from the Bohr-Sommerfeld forms $\tilde H_{\rm int}(N)=N^\gamma$ and $\tilde q(N)\propto N^\mu$. Restoring units recovers exactly the classical Lyapunov exponent for a particle in a power-law potential with a $2\cos\Theta$ perturbation. The same calculation shows two purely quantum suppressions: for sufficiently small initial quantum number $n_0$, and for sufficiently small perturbation $\tilde\epsilon$, the exponential Lyapunov regime vanishes because the discreteness of the spectrum, encoded in a factor-reordering term in the tangent-space equations, prevents the Lyapunov rate from being much larger than the level-spacing differences. The paper further argues that, as $T\to0$, the combination $\beta\hbar\lambda_T$ can remain finite, so these systems do not violate the chaos bound but still count as relatively good scramblers.

Load-bearing premise

The whole calculation relies on treating the reference motion during the Lyapunov regime as the unperturbed integrable rotation, with the noise acting only on the tangent-space separation variables; if the noise appreciably changes the reference orbit, or if its typical frequency approaches the Lyapunov rate, the closed OTOC equations no longer follow.

Editorial extensions

If this is right

Quantum quasi-integrable systems have a prescrambling time much shorter than their energy-diffusion time, so information can spread over the quantum torus before approximate constants of motion relax.
Equation (48) gives a concrete, testable prediction for the randomly kicked rotor: at large $n_0$, $2\tilde\lambda_Q = 2^{2/3}\tilde\epsilon^{1/3}(\gamma(\gamma-1)n_0^{\gamma-2})^{2/3}n_0^{2\mu/3}$.
For fixed $n_0$, decreasing the noise strength eventually destroys the exponential OTOC growth, even though the classical formula would keep predicting a positive Lyapunov exponent; quantum discreteness imposes a floor.
At fixed noise, lowering $n_0$ shortens the prescrambling time measured in Lyapunov times, and below $n_0\sim O(1)$ no Lyapunov regime exists.
The product $\beta\hbar\lambda_T$ can stay finite as $T\to0$, so quasi-integrable systems can remain relatively good scramblers in the sense of the chaos bound while still thermalizing slowly.

Reading between the lines

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

Beyond the paper: for a chain of coupled rotors, the same quantum tangent-space formalism should yield a butterfly velocity, so scrambling would spread ballistically in space while growing exponentially in time; the paper sketches this as a future generalization but does not work it out.
Beyond the paper: the discreteness cutoff implies a sharp finite-size test—at fixed noise, the OTOC growth window should vanish as $n_0$ crosses the point where $|\omega_{n_0}-\omega_{n_0+Z}|\sim\tilde\lambda_Q$, a crossover that cold-atom experiments with tunable integrability breaking could search for.
Beyond the paper: because the semiclassical result depends on the perturbation mostly through its leading harmonic, the $\epsilon^{1/3}$ law should persist for colored noise correlated over times shorter than the Lyapunov time, extending the white-noise derivation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This one is worth a serious look. The authors construct a quantum tangent-space formalism for OTOC growth in weakly noise-perturbed integrable systems, and it's a genuine addition. The annealed Lyapunov exponent from the closed ODE matches the classical ε^{1/3} scaling for the potentials they test (γ=2, 3/2, 4/3), and the quantum suppression of the Lyapunov regime at small n0 or small ε is a new effect, argued clearly and backed by numerics. The low-temperature scrambling discussion is explicitly heuristic, and they say so.

The soft spots are mostly mechanical: Eq. (45) has a factor of 2 on the dF/dt term that direct algebra from Eqs. (41)–(43) doesn't support (I get l^2(n−n')^2, not 2l^2(n−n')^2). It's dropped in the semiclassical limit, so the exponent survives. The Fig. 6 caption says γ=5/2, which is outside the stated range 0<γ<2; it should be 3/2. The 'quenched' label in Fig. 5(a) contradicts the annealed averaging used throughout. These are typos, not the argument.

The substantive problem is exactly the one flagged in the stress-test, and it holds up on reading. The derivation of Eq. (45) relies on the Magnus time-average, valid when λ_Q ≪ ω_{n0}. With H~(N)=N^γ, ω_n ~ γ n^{γ−1}, and Eq. (48) gives λ_Q ~ ε^{1/3} n^{(γ−2)/3}, so λ_Q/ω_n ~ ε^{1/3} n^{(1−2γ)/3}. For every γ<1/2 this diverges as n→∞. The paper explicitly claims 0<γ<2, so the central semiclassical result is not derived in that region. The numerics don't cover it. That doesn't kill the paper, but it means the claim as stated is too broad. A referee should ask for a restriction of the range or a separate treatment of γ<1/2.

The citation pattern is clean: the classical benchmark is the authors' own Ref. [28], but it's a published, independent derivation, and the quantum calculation doesn't assume the result. No code or data, but the methods are specified well enough to reimplement.

Who gets value from this: people working on OTOCs in near-integrable systems, prethermalization, and the quantum bound on chaos. It deserves a proper refereeing round, and with the range fixed it will be a useful contribution.

Recommend sending to peer review, with a request to address the γ<1/2 gap and fix the small errors.

Referee Report

3 major / 6 minor

Summary. The manuscript studies a quantum integrable system weakly driven by Gaussian white noise, written in action-angle variables as H = ω0ℏ[H̃_int(N) + ε̃^{1/2} q̃(N,e^{iΘ}) η(t)]. It develops a 'quantum tangent space' superoperator formalism for the growth of the commutator (OTOC), derives a closed third-order ODE for squared commutator matrix elements (Eq. (45)), and in the Bohr-Sommerfeld semiclassical limit obtains a quantum annealed Lyapunov exponent (Eq. (48)) that scales as ε̃^{1/3} and matches the classical noise-induced Lyapunov exponent of Eq. (15). Numerical simulations for kicked power-law wells are presented for γ = 2, 3/2, and 4/3, supporting the ε̃^{1/3} scaling and showing that the Lyapunov regime is suppressed for sufficiently small n0 or sufficiently small ε̃. The paper also argues, in Sec. IX, that quasi-integrable systems may be relatively good scramblers in the sense that the combination βℏλ_T may stay finite at low temperature.

Significance. If the central result is correct, the paper supplies one of the few analytically tractable bridges between classical quasi-integrable chaos and quantum OTOC growth, with a concrete, falsifiable prediction: the annealed quantum Lyapunov exponent follows the classical ε^{1/3} law in the semiclassical regime and deviates from it in a quantal or weak-noise regime. The strengths are its self-contained derivation of the tangent-space equations, the explicit semiclassical formula, and numerics that check the predicted exponent rather than extracting it from a fit. The formalism of a quantum tangent space is likely to be useful beyond this specific model. However, the derivation as written overclaims the parameter range 0<γ<2, and the key ODE contains an algebraic error whose correction is needed even if the final exponent is unaffected. The low-temperature 'good scrambler' conclusion is heuristic and should be labeled as such.

major comments (3)

[Sec. VII D and Appendix D2, Eq. (48)] The Magnus time-average that removes oscillating components of F(t)F(t) is justified in Appendix D2 by the condition λ̃_Q ≪ ω_{n0}. With the Bohr-Sommerfeld data H̃(N)=N^γ one has ω_{n0}≈γ n0^{γ-1}, while Eq. (48) gives 2λ̃_Q = O(ε̃^{1/3} n0^{(γ-2)/3}); hence λ̃_Q/ω_{n0} ∼ ε̃^{1/3} n0^{(1-2γ)/3}. For every γ<1/2, which is explicitly allowed by the stated range 0<γ<2 in Appendix C, this ratio diverges as n0→∞ at fixed ε̃, so the time-averaging step leading to Eq. (45) and to Eq. (48) is uncontrolled in precisely the semiclassical limit. The same condition affects the classical formula (15), so agreement with the classical result does not repair the problem. The numerics in Sec. VIII cover only γ=2, 3/2, and 4/3. The claim should be restricted to γ>1/2 with an explicit condition, or supplemented by a controlled treatment of γ<1/2.
[Eq. (45)] Direct algebra from Eqs. (41)-(44) gives for the first-derivative term the coefficient l²(n,n')(n-n')², not 2l²(n,n')(n-n')². Differentiating Eq. (41), using Eqs. (42)-(43), and substituting j²=(n-n')²l² yields the third-order equation with a single factor (n-n')²l² in the first-derivative term. The printed factor of 2 is therefore spurious. This discrepancy does not by itself change Eq. (48), because that term is discarded in the semiclassical reduction, but Eq. (45) is presented as the key outcome of the derivation and the crossover criterion in Eq. (46) is tied to this term; the equation and criterion should be corrected.
[Sec. IX and Abstract] The conclusion that βℏλ_T may remain finite as T→0 is stated in the abstract as a finding, but the supporting discussion is an extrapolation. Equation (58) is a semiclassical expression, and the argument that quantization cuts off the growth at n_T=O(1) is not converted into a quantitative estimate; in the model as written the thermal average n_T→0 as T→0, which is below the regime where λ̃_Q is applicable. The manuscript itself flags this reasoning as an argument, but because the claim is part of the advertised results, the abstract and Sec. IX should either present a controlled calculation of the crossover or clearly state that this part is heuristic.

minor comments (6)

[Sec. VIII B and Fig. 6] The text states that the simulations are for γ=4/3 and γ=3/2, while the Fig. 6 caption says γ=4/3 and γ=5/2; these should be reconciled.
[Figs. 5 and 6 captions] The captions label the data as 'quenched', but Sec. III explicitly states that the paper computes the annealed average (the average of the squared commutator); the labels should be corrected to avoid confusion.
[Sec. V A, Eq. (16)] The sentence accompanying Eq. (16) says the diffusion time is shorter than the Lyapunov time, but the inequality λ^{-1}_cl ≪ I0²/(ε̄q²) expresses the opposite, namely that the Lyapunov time is much shorter than the diffusion time; the wording should be fixed.
[Title and Introduction] The title has a spurious space in 'g ood scramblers', and the Introduction contains 'prethermalizad'; these typos should be corrected.
[Sec. VII C, initial conditions for Eq. (45)] The three initial conditions needed to integrate the third-order ODE (45) are only described by reference to Eqs. (41)-(44); explicit initial data for FΘΘ, FNN, and the symmetric/antisymmetric combinations would make the ODE reproducible.
[Eq. (33)] The approximation of evaluating F(N,e^{iΘ}) on the unperturbed trajectory is stated with O(ε̃) correction, but the precise smallness condition under which this O(ε̃) backreaction cannot affect the leading Lyapunov exponent should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the semiclassical quantum Lyapunov exponent is derived independently and only compared with the classical benchmark.

full rationale

The paper's central prediction, Eq. (48), is obtained by solving the closed ODE (45) derived from the commutator chain rule and Fokker-Planck equations for the quantum tangent space (Sec. VII A-C). The Bohr-Sommerfeld limits l(n,n+Z) ≈ γ(γ−1)n^{γ−2} and V(n) ≈ n^μ are inserted into Eq. (45), yielding the third-order equation whose exponential solution gives 2λ_Q = 2^{2/3} ε^{1/3}(γ(γ−1)n^{γ−2})^{2/3} n^{2μ/3}. No step in this derivation imports Eq. (48) or the classical exponent (15) as an input. The classical result from Ref. [28] (co-authored by Kurchan) is cited only as a benchmark in Sec. V and then reproduced by restoring units from the quantum formula; the quantum derivation stands alone. The numerical simulations (Sec. VIII) verify the prediction rather than fit it. The Magnus time-averaging validity condition λ_Q ≪ ω_{n0} is stated in Appendix D2; whether it is satisfied for all 0<γ<2 is a correctness/scoping question, not a circularity. Self-citations (Refs. 27, 28) are for background and methodology, not load-bearing for the target result. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation assumes the standard Bohr-Sommerfeld quantization, canonical-invariance of Poisson brackets, and the Magnus/Fokker-Planck machinery used in the classical treatment. The key structural inputs are the unperturbed-trajectory approximation (Eq. 33) and the time-averaging condition lambda << omega_n (Appendix D2), plus the specific cosine-coupling model. No constants are fitted to data; epsilon_tilde and n0 are control parameters.

assumptions (6)

domain assumption Bohr-Sommerfeld quantization: H_int = N^gamma and q_tilde proportional to N^mu for power-law potentials (Appendix C)
Used to evaluate l(n,n') in the semiclassical limit, Sec. VII D.
domain assumption Unperturbed-trajectory approximation: F(N,e^{iTheta}) evaluated along integrable evolution, Eq. (33)
Basis of the closed Fokker-Planck equations; load-bearing for the derivation.
domain assumption Time-averaging (Magnus) approximation valid when lambda_Q << omega_n, Appendix D2
Used to drop oscillating terms in Eq. (40) and obtain Eq. (44).
ad hoc to paper The specific perturbation form q_tilde = V(N) cos(Theta) + cos(Theta) V(N) captures generic behavior (Sec. VII C)
Simplification to close the equations; higher harmonics are dropped.
domain assumption Classical noise as a model for integrability breaking (Sec. IV, Conclusion)
Paper explicitly notes this may not capture quantum-origin noise effects.
domain assumption The kicked system with tau << omega_n^{-1} << t_Lyp approximates white noise (Sec. VIII)
Used in numerics to realize the noise model.

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Pith. "Pith review of Quasi-integrable systems are slow to thermalize but may be good scramblers." pith.science (2026). https://pith.science/paper/E72MG5YZ

@misc{pith2026190902145,
  author       = {Pith},
  title        = {Pith review of: Quasi-integrable systems are slow to thermalize but may be good scramblers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E72MG5YZ}},
  note         = {Machine review of arXiv:1909.02145}
}

abstract

Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time -- the most clear example being the Solar System -- but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent, defined by the evolution of the 4-point out-of-time-order correlator (OTOC), of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. We find that i) in the semi-classical limit the quantum Lyapunov exponent is given by the classical one: it scales as $\epsilon^{1/3}$, with $\epsilon$ being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is $\sim \epsilon^{-1}$). ii) in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and iii) for sufficiently small perturbations the $\epsilon^{1/3}$ dependence is also suppressed -- another purely quantum effect which we explain. These essential features of the problem are already present in a rotor that is kicked weakly but randomly. Concerning quantum limits on chaos, we find that quasi-integrable systems are relatively good scramblers in the sense that the ratio between the Lyapunov exponent and $kT/\hbar$ may stay finite at a low temperature $T$.

Figures

Figures reproduced from arXiv: 1909.02145 by the authors.

**Figure 2.** FIG. 2. The separation of two initially close by trajectorie [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3. The OTOC growth for the quantum randomly kicked [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. The spreading of the initial wave-function [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5. (a) The (quenched) growth rate of the OTOC for a fixed re [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: FIG. 6. The (quenched) growth rate of the OTOC under the evolu [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

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(32) The above dynamics cannot yield an exponential growth for the OTOC

Integrable case When there is no external noise, ˜ ǫ = 0, the dynamics of the OTOC follows ( ˙CN ˙CΘ ) = ( 0 0 L⊙ iJ ⊙ )( CN CΘ ) . (32) The above dynamics cannot yield an exponential growth for the OTOC. Only CΘ may grow exponentially, but since iJ = i[Ω(N ), ·] we get CΘ (t) = e−iΩtCΘ 0 eiΩt, an oscillatory term. B. Annealed Lyapunov exponent The deriva...

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Classical As a reference, and demonstration of the theoretical description presented in Sec. V, we study the analog clas- sical problem Hcl(I, Θ) = I 2 2 + 2η(t)˜ǫ1/ 2 cos Θ. (52) The Hamilton equation yields the random map It+dt =It + 2rt sin Θt, (53) Θ t+dt = Θ t +Itdt (mod 2π), (54) wherert is taken from a normal distribution of zero mean and variance ...

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In Fig- ure 3 we show the evolution of the OTOC in Eq

Quantum Let us now move to the quantum problem. In Fig- ure 3 we show the evolution of the OTOC in Eq. (51) for 11 101 103 105 107 -40 -30 -20 -10 0 (a) 0 50 100 150 200 -40 -30 -20 -10 0 0 100 200 -40 -20 0 (b) FIG. 2. The separation of two initially close by trajectorie s for the classical randomly kicked rotor (averaged over 1000 pairs of trajectories)...

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Canonical variables In relation to the problem studied in the paper, we consider action-a ngle variables (I, Θ). The derivations holds for any canonical variables, e.g., coordinates and momentum ( q,p ), and for many-degrees of freedom. Since the Poisson brackets act as derivatives ∂ ∂I = −{·, Θ }, ∂ ∂Θ = {·,I }, (A1) they also have a corresponding chain ...

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Pith tools

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

[1] [28]

Bohrdt, C

A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap, Scramblin g and thermalization in a diﬀusive quantum many-body system, New J. Phys. 19, 063001 (2017)

work page 2017

[2] [1]

(32) The above dynamics cannot yield an exponential growth for the OTOC

Integrable case When there is no external noise, ˜ ǫ = 0, the dynamics of the OTOC follows ( ˙CN ˙CΘ ) = ( 0 0 L⊙ iJ ⊙ )( CN CΘ ) . (32) The above dynamics cannot yield an exponential growth for the OTOC. Only CΘ may grow exponentially, but since iJ = i[Ω(N ), ·] we get CΘ (t) = e−iΩtCΘ 0 eiΩt, an oscillatory term. B. Annealed Lyapunov exponent The deriva...

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[3] [2]

(23) concerns the commutation of a time-evolving operator with some initial Hermitian operator A0

Initial conditions The deﬁnition of CΘ and CN in Eq. (23) concerns the commutation of a time-evolving operator with some initial Hermitian operator A0. We may choose A0 = |ψ0⟩⟨ψ0| as a projection on an initial wavepacket concen- trated around an eigenstate |n0⟩, or in the extreme case, just as |n0⟩⟨n0|. The corresponding commutators at time t = 0 then rea...

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[4] [3]

V, we study the analog clas- sical problem Hcl(I, Θ) = I 2 2 + 2η(t)˜ǫ1/ 2 cos Θ

Classical As a reference, and demonstration of the theoretical description presented in Sec. V, we study the analog clas- sical problem Hcl(I, Θ) = I 2 2 + 2η(t)˜ǫ1/ 2 cos Θ. (52) The Hamilton equation yields the random map It+dt =It + 2rt sin Θt, (53) Θ t+dt = Θ t +Itdt (mod 2π), (54) wherert is taken from a normal distribution of zero mean and variance ...

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[5] [4]

In Fig- ure 3 we show the evolution of the OTOC in Eq

Quantum Let us now move to the quantum problem. In Fig- ure 3 we show the evolution of the OTOC in Eq. (51) for 11 101 103 105 107 -40 -30 -20 -10 0 (a) 0 50 100 150 200 -40 -30 -20 -10 0 0 100 200 -40 -20 0 (b) FIG. 2. The separation of two initially close by trajectorie s for the classical randomly kicked rotor (averaged over 1000 pairs of trajectories)...

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[6] [5]

The derivations holds for any canonical variables, e.g., coordinates and momentum ( q,p ), and for many-degrees of freedom

Canonical variables In relation to the problem studied in the paper, we consider action-a ngle variables (I, Θ). The derivations holds for any canonical variables, e.g., coordinates and momentum ( q,p ), and for many-degrees of freedom. Since the Poisson brackets act as derivatives ∂ ∂I = −{·, Θ }, ∂ ∂Θ = {·,I }, (A1) they also have a corresponding chain ...

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[7] [6]

Instead of working in the action-angle space (I, Θ), we change coordinates to ( I,g (Θ))

Non-canonical variables We now consider the case when the pair of variables are not canonica lly conjugate. Instead of working in the action-angle space (I, Θ), we change coordinates to ( I,g (Θ)). Then, the Poisson brackets are related to the derivatives according to − {·, Θ } = ∂ ∂I = − 1 g′ {·,g }, (A6) ∂ ∂g = 1 g′ {·,I }, (A7) where g′ ≡∂g (Θ)/∂Θ. The...

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[8] [7]

(B1) Proof: for an integer power g(x) = xk we have the known formula (readily proven by induction) [ A,Bk] =∑ k r=1Br−1[A,B ]Bk−r, which is equivalent to the expression in Eq

General relation We prove the following general statement: for an analytic function (at some domain) g, and operators A andB we have [A,g (B)] = lim s→0 g(B +s[A,B ]) −g(B) s . (B1) Proof: for an integer power g(x) = xk we have the known formula (readily proven by induction) [ A,Bk] =∑ k r=1Br−1[A,B ]Bk−r, which is equivalent to the expression in Eq. (B1)...

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