Kibble-Zurek mechanism in driven underdamped Brownian motion

Pierre Naz\'e

REVIEW 4 major objections 4 minor 7 references

Kibble-Zurek mechanism in driven underdamped Brownian motion

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper argues that a single driven underdamped Brownian particle, in the double limit of vanishing friction and vanishing natural frequency taken at the same rate, exhibits the Kibble-Zurek mechanism: its relaxation time diverges, its…

desk verdict Interesting limit, but the central exponents are asserted without derivation, the response function is misnormalized, and the stated equations contradict the claimed scaling. read the letter →

arxiv 2412.01436 v1 pith:FZFJS5C3 submitted 2024-12-02 cond-mat.stat-mech

Pierre Nazé This is my paper · ORCID

classification cond-mat.stat-mech

keywords Kibble-ZurekmechanismunderdampedBrownianmotionlinearresponsetheoryrelaxationtimedivergenceirreversibleworkscalingsuddenprocessoptimalprotocolpausingeffect

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 argues that a single driven underdamped Brownian particle, when the friction and the natural frequency are made small at the same rate, behaves like a Kibble-Zurek system: its relaxation time diverges, its relaxation function freezes to a constant, and the irreversible work obeys a power law in the switching time. The paper derives the exponents $\eta_{KZ}=-2$ for sudden drives and $\eta_{KZ}=-1$ for slowly varying drives using linear response theory. This matters because it moves a paradigmatic nonequilibrium scaling from quantum chains at zero temperature to a minimal classical stochastic system, potentially making the mechanism easier to observe and control.

What carries the argument

The central object is the relaxation function $\Psi_0(t)/\Psi_0(0)$ of the underdamped Brownian motion, computed by linear response theory from the Langevin equation, together with the associated relaxation time $\tau_R=(\gamma^2+\omega_0^2)/(2\gamma\omega_0^2)$. The mechanism is the limit $\gamma,\omega_0\to 0^+$ at equal rates: this makes the relaxation time diverge while the relaxation function freezes at a constant, putting the dynamics in a critical-like frozen state. For the scaling, the key expression is Eq. (12), the linear-response formula for the impulse-part irreversible work, evaluated against the freezing point $\hat{t}=f(\tau)$ defined by Eq. (11).

What would settle it

Directly integrate the driven Langevin equation (1) for $\gamma=\omega_0=0.01$, $\delta\omega_0=0.001$, over switching times $\tau$ in the same range as Figures 1 and 2, computing the exact irreversible work from the full work distribution without the linear-response approximation, and compare the exponents of $W_{\rm irr}^I(\tau)$; if they are not $-2$ (sudden) and $-1$ (slowly varying), the central scaling claim fails.

Watch

Extended reading notes

Core claim

In the simultaneous limits $\gamma\to 0^+$ and $\omega_0\to 0^+$ with $\gamma/\omega_0=1$, the underdamped Langevin equation (1) has a relaxation time $\tau_R=(\gamma^2+\omega_0^2)/(2\gamma\omega_0^2)$ that diverges (Eq. 6), while the normalized relaxation function $\Psi_0(t)/\Psi_0(0)$ tends to the constant 3 (Eq. 7) and the initial susceptibility $\Psi_0(0)$ diverges (Eq. 8). These are the signatures of Kibble-Zurek behavior: the system freezes because the initial equilibrium temperature must scale as $T\propto 1/\gamma$, implying large thermal fluctuations that prevent equilibration. For a linear driving $\omega(t)=\omega_0-\delta\omega_0\,t/\tau$, the irreversible work in the impulse part (Eq. 12) scales as $\tau^{-2}$ for sudden processes and $\tau^{-1}$ for slowly varying processes, matching the known quantum-chain exponent in the second case and giving a new exponent in the first. The optimal protocol then reduces to a pause at half the driving with two jumps, which the author attributes to the generic Kibble-Zurek limit rather than to symmetry about a critical point.

Load-bearing premise

Everything rests on the assumption that the linear-response impulse-work formula, applied outside its formal range of validity (the paper states that $f(\tau)$ is always greater than the $\tau$ range used), still gives the correct Kibble-Zurek exponents because an earlier calculation with the same approximation worked.

Editorial extensions

If this is right

If the claimed limit is realized, the driven underdamped Brownian particle becomes a minimal classical system in which the relaxation time diverges, so the Kibble-Zurek mechanism is not restricted to quantum chains at zero temperature.
For sudden switching, the irreversible work in the impulse part decays as $\tau^{-2}$, a new scaling distinct from the usual $\tau^{-1}$ of the transverse-field Ising chain; the author attributes the difference to the different nature of the relaxation time.
For slowly varying processes, the scaling exponent is $\eta_{KZ}=-1$, the same as in the quantum Ising chain.
The optimal protocol that minimizes average work and its fluctuations becomes approximately constant at $1/2$, with two jumps, meaning the driving pauses in the middle; the author notes this is a coincidence with the critical-point pause, since the driving here is not symmetric about a critical point.
The initial equilibrium temperature must be high ($T\propto 1/\gamma$), so the frozen state is associated with strong thermal fluctuations rather than low-temperature criticality.

Reading between the lines

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

A natural extension not pursued in the paper is that the same equal-rate vanishing limit may transfer the $\eta_{KZ}=-2$ sudden-process exponent to other underdamped linear models, such as trapped ions or levitated nanoparticles, where both $\gamma$ and $\omega_0$ can be tuned toward zero simultaneously.
If the scaling survives beyond linear response, a direct integration of the exact Langevin dynamics would show a collapse of $W_{\rm irr}^I(\tau)$ onto the predicted power laws; checking that collapse would separate the linear-response extrapolation from genuine Kibble-Zurek physics.
The pausing optimal protocol suggests a practical control strategy: to avoid large irreversible work, one should hold the trap frequency fixed in the middle of a fast drive rather than sweeping it through the whole interval.
The paper fixes the ratio $\gamma/\omega_0=1$; exploring nearby ratios could reveal a crossover between exponents or a family of scaling curves, a question the paper leaves open.

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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

The paper has a genuinely interesting observation buried under a pile of unsubstantiated claims. The limit γ,ω0→0 with γ/ω0=1 does make the relaxation time diverge, and that is a legitimate curiosity. But the two new exponents, ηKZ=-2 and -1, are asserted, not derived. The only justification given is a self-citation (ref. [4]) after the paper itself concedes that the linear-response regime is not reached. That alone would worry me; what worries me more is that the equations as written do not support the plotted slopes. If f(τ)>τ, the lower limit in Eq. (12) is negative; clipping it to zero gives an O(1) result, not τ^{-2}. And Eq. (3) fails a basic check: for γ/ω0=1, the right-hand side at t=0 equals 3, not 1, so the frozen value in Eq. (7) is at least partly a normalization artifact. These are load-bearing problems, not minor typos. The high-temperature condition T∝1/γ is stated without derivation and looks ad hoc. What the paper does well is identify a corner of the problem worth thinking about: a classical Brownian system whose relaxation time diverges in a particular limit, with a possible analogy to Kibble-Zurek. The idea is not crazy, but the execution is a sketch, not a paper. I would not send this to a referee in its current form. The author needs to supply a derivation of the exponents, fix the response function normalization, and show that the plotted scaling actually follows from the equations. If that can be done, this might become a solid contribution. Right now, it is not. Recommendation: reject, but do not close the door; a rewritten version with the missing steps could deserve a serious look.

Referee Report

4 major / 4 minor

Summary. The manuscript claims that a driven underdamped Brownian motion exhibits the Kibble-Zurek mechanism in the simultaneous limit γ,ω₀→0⁺ with γ/ω₀=1. Using linear response theory, it derives a divergent relaxation time (Eq. 6), a frozen relaxation function (Eq. 7), a high-temperature initial condition (Eq. 9), new scaling exponents η_KZ=-2 and -1 for the impulse part of the irreversible work (Eqs. 13-14), and a pausing effect in the optimal protocol (Eq. 15). The results are presented in a very brief form, with several key steps asserted rather than derived.

Significance. If established, the work would provide a simple classical analog of the Kibble-Zurek mechanism and could be of interest to the stochastic thermodynamics community. However, the paper does not provide a self-contained derivation of its scaling claims; the only support for the central exponents is the author's own prior work, and the manuscript explicitly acknowledges that the linear response calculation is used outside its range of validity. The normalization error in the response function further undermines the quantitative claims. The significance is therefore contingent on a substantial revision that the present manuscript does not provide.

major comments (4)

[I.A, Eq. (3)] Equation (3) is not a normalized response function: at t=0 its right-hand side equals ω²/ω₀² = 3 when γ/ω₀=1, rather than 1 as required by the definition Ψ₀(0)/Ψ₀(0)=1. This error propagates into the 'frozen' value 3 reported in Eq. (7) and into the impulse-work integrand in Eq. (12), since all subsequent results use Ψ₀(t)/Ψ₀(0). Please re-derive Eq. (3) from the Langevin equation (1) and correct the normalization before any scaling claims can be assessed.
[I.C, Eq. (12)] The paper states that for the plotted ranges f(τ)>τ, so the lower integration limit τ-f(τ) in Eq. (12) is negative and the integral effectively runs from 0 to τ. In that regime the normalized response (Eq. (3) with γ,ω₀ small and γ/ω₀=1) is approximately constant (≈3) over the integration domain, so the double integral is O(τ²) and W_irr^I is O(δω₀²), i.e., independent of τ. This contradicts the claimed η_KZ=-2 scaling in Eq. (13) and the slope in Fig. 1. Please clarify how the numerical results were obtained and whether Eq. (12) is the correct expression in the regime of the figures.
[I.C, Eqs. (13)-(14)] The scaling exponents are asserted without derivation. No closed form for f(τ) is given, no asymptotic analysis of Eq. (12) is shown, and no direct numerical integration of the original Langevin dynamics is provided. The only justification offered is a citation to the author's previous work [4]. Given that the paper itself admits that the linear response validity condition is violated in the plotted regime, the exponents cannot be considered established unless they are derived within this manuscript or verified by a calculation that does not rely on the invalid approximation.
[I.B, Eq. (9)] The 'high-temperature condition' T∝1/γ is stated without derivation or a clear argument from the fluctuation-dissipation theorem. As presented, it is an additional assumption that is singular in the γ→0 limit and places the system in a regime where the validity of linear response around the initial equilibrium state must be re-examined. Please provide a derivation or explicitly label this as a condition that defines the KZ limit rather than a consequence of linear response.

minor comments (4)

[Abstract/Introduction] The paper never defines the Kibble-Zurek exponent for the quantum Ising chain or states the comparison quantitatively. A few sentences defining the standard KZ scaling and how it is generalized here would place the results in context.
[Fig. 1 and Fig. 2] The axis labels are unclear (e.g., 'τ-2 Wirr I' in Fig. 1). Please use standard notation, e.g., W_irr^I on the y-axis, and state explicitly whether the figures show W_irr^I versus τ on a log-log scale or some rescaled quantity.
[I.D, Eq. (15)] The optimal protocol result g*(t)≈1/2 is taken directly from reference [5] and is not derived in this manuscript. Since the connection to the Kibble-Zurek mechanism is the point of this section, a more explicit explanation of how the pausing effect follows from the preceding analysis would be helpful.
[References] Reference [1] is missing its article number/page (likely J. Chem. Phys. 129, 024114 (2008)). Please verify all references for completeness.

Circularity Check

3 steps flagged · score 8.0 of 10

The claimed frozen value 3 is built into the paper's own unnormalized relaxation function, and the new KZ exponents are validated only by the author's prior work after the paper admits the linear-response range is not respected.

self definitional [Section I.A, Eqs. (3) and (7)]
"lim γ,ω0→0+ γ/ω0=1 Ψ0(t)/Ψ0(0) = 3, with Eq. (3): Ψ0(t)/Ψ0(0) = e−γ|t| [ 2 + (ω2/ω02 − 2) cos ωt + γω/ω02 sin ω|t| ]."

At t=0, Eq. (3) gives Ψ0(0)/Ψ0(0)=ω2/ω02. For γ/ω0=1, ω2/ω02=3, so the claimed frozen value 3 is exactly the t=0 value of the quotient as defined by Eq. (3). The 'frozen state' is thus a constant already present in the definition of the relaxation function, not a dynamically derived consequence of the Kibble-Zurek mechanism. A properly normalized relaxation function would equal 1 at t=0 and would not produce the same freeze.
self citation load bearing [Section C, after Eqs. (12)-(14)]
"Observe that it is not guaranteed that the values of the irreversible work in the impulse part correspond to the exact value. Indeed the solution f (τ) is always greater than the range of τ used in these graphics, which indicates that the range of validity of linear response is not respected. However, previous studies illustrate that the Kibble-Zurek exponent is calculated correctly [4]."

The exponents ηKZ=-2 and ηKZ=-1 in Eqs. (13)-(14) are stated without a derivation from Eq. (12). The paper itself says f(τ) is always greater than the plotted τ, so Eq. (12) is evaluated outside its linear-response validity and cannot by itself yield the claimed scaling. The only evidence offered that the exponent is nevertheless correct is reference [4], a prior paper by the same author. The central new scaling prediction is therefore not independently derived; it is imported from a self-citation.

1 more flagged steps

self citation load bearing [Section D, Eq. (15)]
"the optimal protocol for the average work and its fluctuations, in linear response theory, will be [5] g∗(t) ≈ 1/2"

The pausing effect is presented as a consequence of the Kibble-Zurek limit, but the optimal-protocol form is taken entirely from reference [5], by the same author, with no derivation shown in the present paper. This makes the 'pausing effect' claim a self-citation rather than a derived result. It is less central than the scaling exponents, but it follows the same pattern of load-bearing self-reference.

full rationale

The paper is not uniformly circular: the divergence of the relaxation time in Eq. (6) follows directly from Eq. (5), and the high-temperature condition in Eq. (9) is a stated consequence of the fluctuation-dissipation theorem. However, the two headline 'Kibble-Zurek consequences' do not survive a derivation-chain check. The frozen constant 3 in Eq. (7) is simply the t=0 value of the relaxation-function quotient defined in Eq. (3) when γ/ω0=1; the quotient by construction equals ω2/ω02 at t=0, so the 'freeze' is definitional. More importantly, the new scaling exponents in Sec. C are not derived from Eq. (12); the paper admits f(τ)>τ in the plotted ranges, putting the calculation outside the linear-response validity, and it then invokes previous work by the same author (ref. [4]) to assert that the exponent is correct. The central claim of a new KZ scaling is thus supported by a self-citation rather than by the equations in this manuscript. The optimal-protocol 'pausing effect' is likewise imported from the author's own ref. [5]. Because the central new results are either definitional or justified by self-citation, the circularity score is 8.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The paper's central scaling claims rely on a chosen limit path and a singular high-temperature scaling, plus an assumption that the linear response calculation remains valid outside its range. No new physical entities are introduced.

free parameters (2)

Limit path ratio γ/ω_0 = 1 = 1
Chosen by hand in Section A to define the double limit. Any ratio γ/ω_0 < 2 would give a diverging relaxation time, so this specific value is an ad hoc example, not a derived constraint.
High-temperature scaling T ∝ 1/γ
Introduced in Section B (Eq. 9) to keep the noise amplitude finite as γ→0. This scaling is presented as a requirement from the fluctuation-dissipation theorem, but it is effectively an ad hoc modeling choice for a singular limit.

assumptions (7)

domain assumption Langevin equation with white noise (Eqs. 1-2)
The equation of motion and noise properties are taken as the starting model.
standard math Linear response theory gives the relaxation function (Eq. 3)
The relaxation function is cited from refs. [1,2] without derivation; it is a standard result.
standard math Fluctuation-dissipation theorem holds (Eq. 2)
The noise strength is tied to temperature and friction, a standard equilibrium condition.
domain assumption Initial thermal equilibrium
Linear response requires the system to start in equilibrium; the paper invokes this in Section B.
domain assumption Underdamped condition γ < 2ω_0
Used to define ω = sqrt(4ω_0^2 - γ^2) as real in Eq. (3).
ad hoc to paper High-temperature scaling T ∝ 1/γ
Needed to maintain finite noise in the γ→0 limit; not a standard equilibrium limit.
ad hoc to paper Kibble-Zurek exponents valid despite linear response violation
Section C states that the calculation is outside the linear response range of validity, but the exponents are asserted to be correct based on the author's earlier work (ref. [4]).

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Cite this review

Pith. "Pith review of Kibble-Zurek mechanism in driven underdamped Brownian motion." pith.science (2026). https://pith.science/paper/FZFJS5C3

@misc{pith2026241201436,
  author       = {Pith},
  title        = {Pith review of: Kibble-Zurek mechanism in driven underdamped Brownian motion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FZFJS5C3}},
  note         = {Machine review of arXiv:2412.01436}
}

read the original abstract

Kibble-Zurek mechanism is widely known to appear in the transverse-field quantum Ising chain in the thermodynamic limit at zero temperature, having notorious characteristics, like the divergence of its relaxation time. In this work, I present the same effect in a simple system, the driven underdamped Brownian motion. Using linear response theory, I show the appropriate limits where the Kibble-Zurek mechanism happens. The divergence of the relaxation time, the high-temperature condition in the initial thermal equilibrium state, a new Kibble-Zurek scaling at sudden processes, and the pausing effect in the optimal protocol are presented as consequences.

Figures

Figures reproduced from arXiv: 2412.01436 by the authors.

**Figure 2.** FIG. 2. Kibble-Zurek scaling for slowly-varying processes [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages

[4]

Deﬀner, Physical Review E 96, 052125 (2017)

S. Deﬀner, Physical Review E 96, 052125 (2017)

work page 2017
[1]

× 10-6 Wirr I FIG. 2. Kibble-Zurek scaling for slowly-varying processes . It is given by ηKZ = − 1, considering γ = 0 . 01, ω 0 = 0 . 01, δω 0 = 0. 001, meaning that the relaxation time of the system is τR = 100. 3 fact: divergence of its relaxation time, the initial high- temperature heat bath, a new Kibble-Zurek scaling for sudden processes, and the pau...

work page
[2]

Gomez-Marin, T

A. Gomez-Marin, T. Schmiedl, and U. Seifert, The Journal of chemical physics 129 (2008)

work page 2008
[3]

R. Kubo, M. Toda, and N. Hashitsume, Statistical physics II: nonequilibrium statistical mechanics , Vol. 31 (Springer Science & Business Media, 2012)

2012
[5]

Naz´ e, M

P. Naz´ e, M. V. Bonan¸ ca, and S. Deﬀner, Entropy 24, 666 (2022)

work page 2022
[6]

Naz´ e, Journal of Statistical Mechanics: Theory and E x- periment 2024, 073205 (2024)

P. Naz´ e, Journal of Statistical Mechanics: Theory and E x- periment 2024, 073205 (2024)

work page 2024
[7]

Chen and D

H. Chen and D. A. Lidar, Physical Review Applied 14, 014100 (2020)

work page 2020

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [4]

Deﬀner, Physical Review E 96, 052125 (2017)

S. Deﬀner, Physical Review E 96, 052125 (2017)

work page 2017

[2] [1]

× 10-6 Wirr I FIG. 2. Kibble-Zurek scaling for slowly-varying processes . It is given by ηKZ = − 1, considering γ = 0 . 01, ω 0 = 0 . 01, δω 0 = 0. 001, meaning that the relaxation time of the system is τR = 100. 3 fact: divergence of its relaxation time, the initial high- temperature heat bath, a new Kibble-Zurek scaling for sudden processes, and the pau...

work page

[3] [2]

Gomez-Marin, T

A. Gomez-Marin, T. Schmiedl, and U. Seifert, The Journal of chemical physics 129 (2008)

work page 2008

[4] [3]

R. Kubo, M. Toda, and N. Hashitsume, Statistical physics II: nonequilibrium statistical mechanics , Vol. 31 (Springer Science & Business Media, 2012)

2012

[5] [5]

Naz´ e, M

P. Naz´ e, M. V. Bonan¸ ca, and S. Deﬀner, Entropy 24, 666 (2022)

work page 2022

[6] [6]

Naz´ e, Journal of Statistical Mechanics: Theory and E x- periment 2024, 073205 (2024)

P. Naz´ e, Journal of Statistical Mechanics: Theory and E x- periment 2024, 073205 (2024)

work page 2024

[7] [7]

Chen and D

H. Chen and D. A. Lidar, Physical Review Applied 14, 014100 (2020)

work page 2020