Rare events for low energy domain in bouncing ball model

Diego F. M. Oliveira; Edson D. Leonel

REVIEW 4 major objections 4 minor 26 references

Rare events for low energy domain in bouncing ball model

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

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Pith's one-line read In the bouncing ball model's chaotic low-energy sea, successive-collision bursts follow a universal power law with exponent -4.

desk verdict Plausible numerical scaling law for rare collision bursts in the Fermi-Ulam model, but the exponent and collapse need convergence checks and a theory that actually implies a power law. read the letter →

arxiv 2411.16945 v1 pith:2BLVUZB4 submitted 2024-11-25 nlin.CD

Edson D. Leonel , Diego F. M. Oliveira This is my paper · ORCID

classification nlin.CD MSC 37D45 PACS 05.45.-a

keywords Fermi-Ulammodelbouncingballchaosrareeventsscalinginvariancesuccessivecollisionspowerlawdecaylowenergydomain

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 tries to establish that rare events in the chaotic low-energy domain of the bouncing ball model—bursts of many successive collisions with the moving wall—are described by a universal statistical law. The central claim is that the probability of observing a burst of size $n$ decays as a power law $F(n) \propto n^{\gamma}$ with $\gamma = -4$, and that the cumulative distributions for different control parameters collapse onto one curve under the rescaling $H \to H/\epsilon$. If true, the burst-size statistics in this regime are fixed by a single exponent and a single scaling function, with no free dependence on the wall amplitude.

What carries the argument

The load-bearing object is the collision zone $x\in[-\epsilon,\epsilon]$ and the probability $p=q=(\epsilon - V^*)/(2\epsilon)$ that a particle entering it with velocity $V^*$ moves right or left. This turns the number of successive collisions into a binomial count $P_N = \frac{N!}{n_1!(N-n_1)!} p^{n_1} q^{N-n_1}$, whose cumulative form $H(n)$ is the quantity measured numerically. The collapse is carried by the rescaling $H \to H/\epsilon$, which removes the control-parameter dependence and exposes the single power-law exponent.

What would settle it

A direct test would repeat the measurement with several different initial conditions in the chaotic sea and with ensembles of many shorter orbits; if the fitted exponent moves away from $-4$ or the $H/\epsilon$ collapse fails for some starting regions, the claimed universality is not supported.

Watch

Extended reading notes

Core claim

The paper's central discovery is that successive collisions—repeated impacts with the moving wall that happen before the particle leaves the collision zone $x\in[-\epsilon,\epsilon]$—are governed by a scaling-invariant probability distribution. Using a long orbit of $10^{11}$ collisions, the authors measure the cumulative distribution $H(n)$ for the number of right-moving and left-moving successive collisions, fit a power law $F(n)\propto n^{\gamma}$ in both cases, and obtain $\gamma=-3.98(3)$ and $\gamma=-3.99(5)$, which they summarize as $\gamma=-4$. They then show that the transformation $H(n)\to H(n)/\epsilon$ superimposes the curves for several control parameters onto a single universal plot, establishing scaling invariance with respect to the control parameter in the chaotic low-energy regime.

Load-bearing premise

The numerical claim rests on the assumption that one very long orbit of $10^{11}$ collisions samples the chaotic low-energy sea uniformly enough that the measured histogram of burst sizes has converged and is independent of the starting point.

Editorial extensions

If this is right

The burst-size distribution for successive collisions in the low-energy chaotic domain is controlled by a single exponent, $\gamma = -4$.
Distributions from different control parameters $\epsilon$ collapse onto one universal curve when $H$ is rescaled by $\epsilon$.
Large bursts are rare but not exponentially rare: their probability decays as a power law rather than a tail.
The known scaling invariance of the Fermi-Ulam chaotic sea extends to the statistics of successive collisions at very low energy.
The fitted exponents for right-moving and left-moving collision counts agree within uncertainty, indicating one underlying process.

Reading between the lines

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

If the exponent $-4$ holds up, the same scaling collapse should appear in other Hamiltonian impact systems whose collision-zone dynamics are linear in the wall velocity; this is an extrapolation the paper does not make.
A testable extension is to measure the distribution of entry velocities $V^*$ along a long orbit and check whether integrating the binomial model in Eq. (5) over that distribution reproduces $\gamma = -4$ without direct simulation.
The paper does not report convergence diagnostics; an ensemble-based check would clarify whether the single-orbit histogram of $10^{11}$ collisions is a stationary sampling of the chaotic sea.

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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

Short version: this is a plausible numerical observation that successive-collision burst sizes in the low-energy chaotic sea of the Fermi-Ulam model decay as a power law with exponent near -4 and collapse under H -> H/epsilon. If it holds up, it's a useful addition to the known scaling picture for that model. But right now it's one long orbit, a fitted exponent, and an ad hoc rescaling; the stochastic model in the paper does not produce a power law, and there's no convergence or ergodicity check. I'd send it to a referee, but I'd expect them to ask for real statistical support.

What's actually new: Leonel and Oliveira are extending their own established scaling framework. The specific claim about the burst-size distribution and the H/epsilon collapse for multiple collisions is not in refs [14,15,16]. The numerical data in Figs 3-4 do show a consistent decay over a few decades for the three epsilon values shown, and the exponent is quoted with a digit in parentheses, so it's not a made-up number. The paper is honest that this is a numerical study.

The soft spots are real and they are the load-bearing ones. (1) The distribution comes from a single orbit of 10^11 collisions. No convergence test, no variation of initial conditions, no error bars on the histogram. The rare large bursts that determine the tail are exactly what is most sensitive to under-sampling, and the FUM phase space has islands and sticky regions. The claim that this is the stationary distribution of the chaotic sea is an assumption. (2) The binomial model in Eqs (5)-(6) doesn't support the power law. p and q as defined don't sum to 1 unless V*=0, and even a proper binomial gives Gaussian/exponential tails, not a power law. So the theoretical justification is not there. (3) The collapse H -> H/epsilon is demonstrated for three epsilon values, but there is no stated fitting range or measure of collapse quality.

These are addressable. A few different orbits, a convergence plot, and a clearer statement about what the stochastic model is and isn't doing would go a long way. As it stands, the central claim is conditional, but it's a condition that a referee can reasonably ask to be met. I'd think of this as a useful data point for rare-event statistics in Hamiltonian chaos, not a framework shift.

Recommendation: engage with it. Send to peer review. The question is whether the numerical evidence for the exponent and collapse survives closer scrutiny; that's a fair question for the referee process. The paper is short, so the requested additions won't be excessive.

Referee Report

4 major / 4 minor

Summary. The paper studies the Fermi–Ulam (bouncing ball) model and focuses on rare 'successive collisions' in the low-energy chaotic regime, where the particle hits the moving wall several times in rapid succession. The authors simulate a single very long orbit (10^11 collisions), record the numbers of successive impacts, and report that the probability distributions for the number of right/left moves decay as power laws with exponent γ ≈ -4 and that the distributions for different control parameters ε collapse under the rescaling H → H/ε. They also propose a binomial model, Eqs. (5)–(6), for the probability of a given sequence of right/left moves during a burst.

Significance. If the scaling invariance and the universal γ = -4 power-law tail were rigorously established, the result would be a valuable characterization of rare, extreme events in a paradigmatic Hamiltonian system and would connect to broader questions of extreme-value statistics in chaotic transport. The paper makes an explicit empirical claim (power-law exponent) that is falsifiable, and the collapse in Figs. 3–4 is visually plausible for the three ε values shown. However, the manuscript does not provide a valid theoretical derivation of the exponent, and the numerical evidence lacks convergence tests and error bars. The central claim is therefore currently a fit-level observation rather than a supported scaling law.

major comments (4)

[Section 3, Eq. (7)] The cumulative distribution H(n1) is not defined consistently: the right-hand side sums PN(n1) over n1 from 1 to N, but both n1 and N appear on the left and right without a clear relation, and the left-hand side depends on n1 while the sum is over n1. The paper never integrates Eq. (5) to obtain the claimed power-law tail. Even if Eq. (5) were valid, it cannot produce a power law with exponent -4: for p=q<1/2 the binomial tail is exponential, and for p=q=1/2 the central decay is ~N^{-1/2}. Thus the stochastic model provides no theoretical support for the central claim.
[Section 3, Eq. (6)] The probabilities p and q as defined satisfy p+q = (ε - V*)/ε, which is not 1 unless V*=0, so Eq. (5) is not a normalized probability distribution. This is not a harmless normalization issue: the paper uses p and q to interpret the velocity histogram and to frame the subsequent binomial calculation. A corrected probabilistic model must either define p and q as conditional probabilities that sum to one or include the explicit V* dependence in the normalization. As written, the theoretical framework is internally inconsistent.
[Section 3, 'we can start an initial condition and follow a very long orbit'] The entire numerical claim rests on a single orbit of 10^11 collisions with no convergence test, no variation of the initial condition, and no estimate of the statistical error of the histograms. The tail exponent is dominated by rare large bursts, which are precisely the quantities most sensitive to sticky islands and to under-sampling in the mixed phase space. The quoted uncertainties (e.g., γ = -3.98(3)) are only the fit errors for one histogram, not a statement about sampling or ergodicity. The authors should demonstrate that the exponent and the collapse are stable with respect to (i) different initial conditions, (ii) lengthening the orbit, and (iii) splitting the orbit into independent blocks.
[Section 3, Figs. 3 and 4] The power-law fit is presented without specifying the fitting range or the number of points used, and the collapse H(n)/ε is judged only visually. A quantitative test (e.g., a measure of residual collapse or a two-parameter scaling ansatz allowing for a cutoff) is needed to support the claim that the distributions are 'scaling invariant'. The paper also does not test whether the exponent depends on a lower cutoff in n; if the power law is only asymptotic, the stated universality would need to be qualified.

minor comments (4)

[Throughout] There are several typos: 'experince' in Section 3, 'Phycsicis' in Ref. [21], and in the Conclusion the exponent appears as 'γ−4' rather than 'γ = -4'.
[Section 2, Eqs. (1)–(4)] The mapping is presented without a derivation; the authors refer to Refs. [14,18]. While this is acceptable, a short definition of the phase variables and the collision conditions would make the paper more self-contained.
[Section 3, Fig. 2 and text] The symbol H is used both for the histogram of velocities (Fig. 2) and for the cumulative distribution in Eq. (7); please use distinct notation to avoid confusion.
[Abstract and Conclusion] The abstract states the distribution is 'scaling invariant' but the paper only demonstrates this for the specific distributions shown in Figs. 3–4, not for the full distribution over all possible n; the claim should be scoped accordingly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the claimed power law and scaling collapse are empirical numerical characterizations, not derived from fitted inputs or self-citations.

full rationale

The central claim—that the distribution of successive collisions decays as a power law with exponent γ ≈ −4 and collapses under H → H/ε—is presented as a numerical observation. The paper states: “A power law fitting gives an exponent similar to the two distributions” and “The numerical distribution for the successive collisions is shown in Fig. 3,” with the collapse obtained by “a straightforward transformation of H(n) → H(n)/ε.” This is a data-fitting and rescaling procedure, not a derivation from an input assumption whose conclusion is already contained in the premise. The binomial model in Eqs. (5)–(6) is introduced as a qualitative analogy (“remarkably similar to a binomial distribution, as foreseen by Eq. (5)”), but it is never used to derive the exponent γ = −4, and the paper does not claim that the binomial model predicts the power-law tail. Therefore the exponent and collapse are not equivalent to the model by construction. The self-citations [14, 15, 16] are used for background (mapping construction, known scaling invariance of average velocity, and convergence of Lyapunov exponents) and are not load-bearing for the new distribution result. Concerns about single-orbit sampling, lack of convergence tests, and the internal inconsistency p + q ≠ 1 in Eq. (6) are validity or correctness issues, not circularity. Accordingly, the paper does not exhibit a circular derivation chain.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on: (i) the standard bouncing-ball map, (ii) an untested ergodicity/stationarity assumption for a single long orbit, (iii) an ad hoc binomial probability model that is not used to derive the power law, and (iv) prior scaling results from the same authors. The free parameters are the fitted exponent and the ad hoc scaling exponent for ε; the fitting range is undisclosed.

free parameters (3)

Power-law exponent γ = -4 (fit gives -3.98(3) for n1, -3.99(5) for n2)
The exponent is obtained by fitting the numerical histograms in Fig. 3; it is not derived from the model. The paper rounds the fitted values to -4.
Scaling exponent for ε in the collapse H(n) → H(n)/ε = -1 (chosen by hand)
The collapse is achieved by rescaling the histograms by 1/ε; this exponent is an ansatz, not derived, and is validated only by visual overlap of curves for three ε values.
Power-law fitting range = not stated
The range of n over which the power law is fitted is not specified, which is a free choice that can affect the fitted exponent.

assumptions (4)

standard math The two-dimensional map (1) with elastic collisions and sinusoidal wall motion accurately models the bouncing ball dynamics in the low-energy regime.
This is the standard Fermi-Ulam model mapping, taken from Refs. [14,18].
domain assumption The chaotic sea is ergodic enough that a single orbit of 10^11 collisions yields a converged, stationary histogram of successive-collision events.
Section 3 states 'we can start an initial condition and follow a very long orbit', with no convergence test, multiple orbits, or error bars.
ad hoc to paper The probability of a right/left move after entering the collision zone is p = q = (ε - V*)/(2ε) and successive collisions are independent, giving the binomial distribution Eq. (5).
This is an estimate introduced in Section 3; it is not derived from the map and is used only to justify the shape of Fig. 2(a), not the power-law tail.
domain assumption The first invariant spanning curve and the scaling properties of the chaotic sea from Refs. [15,16] remain valid for the low-energy domain explored here.
The paper relies on these prior results to frame the scaling invariance claim, citing Refs. [15,16].

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Cite this review

Pith. "Pith review of Rare events for low energy domain in bouncing ball model." pith.science (2026). https://pith.science/paper/2BLVUZB4

@misc{pith2026241116945,
  author       = {Pith},
  title        = {Pith review of: Rare events for low energy domain in bouncing ball model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2BLVUZB4}},
  note         = {Machine review of arXiv:2411.16945}
}

read the original abstract

The probability distribution for multiple collisions observed in the chaotic low energy domain in the bouncing ball model is shown to be scaling invariant concerning the control parameters. The model considers the dynamics of a bouncing ball particle colliding elastically with two rigid walls. One is fixed, and the other one moves periodically in time. The dynamics is described by a two-dimensional mapping for the variables velocity of the particle and phase of the moving wall. For a specific combination of velocity and phase, the particle may experience a type of rare collision named successive collisions. We show that a power law describes the probability distribution of the multiple impacts and is scaling invariant to the control parameter.

Figures

Figures reproduced from arXiv: 2411.16945 by the authors.

**Figure 3.** Plot of the probability distribution for the successive reflection for (a) [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Plot of the normalized probability distribution for the successive re [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

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Bessi, A., 2017. On the statistical properties of viral misinformation in online social media. Physica A: Statistical Mechanics and Its Applica- tions 469, 459–470

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work page 1993

[4] [4]

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Ditlevsen, P.D., Johnsen, S.J., 2010. Tipping points: early warning and wishful thinking. Geophysical Research Letters 37

work page 2010

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Eckmann, J.P., Ruelle, D., 1985. Ergodic theory of chaos and strange attractors. Reviews of modern physics 57, 617

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work page 2022