This manuscript numerically studies passive tracer transport in two-dimensional chaotic point-vortex flows and compares it to canonical Lévy-walk dynamics (Secs. 2–3). By varying the number of vortices $N$ in a Hamiltonian point-vortex model and the Lévy-walk flight-time exponent $\beta$, the authors span regimes from near-normal diffusion to strong superdiffusion. They analyze ensemble MSD scaling, VACF, residence-time statistics based on a low-speed threshold, displacement PDFs (including Lévy-stable fits), and an ergodicity-breaking (EB) parameter derived from TAMSD variability (Eqs. (3)–(6), Secs. 3.2–3.7). The main kinematic message—that higher-$N$ vortex flows show Lévy-like signatures (superdiffusive MSD, heavy-tailed PDFs, power-law residence times) similar to Lévy walks with $1<\beta<2$—is plausible and potentially interesting. The central conceptual claim is that matching kinematic diagnostics does not imply equivalence in ergodicity: EB decreases with $N$ in the vortex system but increases with stronger Lévy-walk superdiffusion (Sec. 3.7). However, the quantitative robustness of the exponent estimates, tails, and especially the EB trends is currently difficult to assess due to limited sampling, incomplete model/numerics specification (domain, boundary handling, singularity treatment, integrator), and insufficiently rigorous uncertainty quantification and fitting methodology. Addressing these points would substantially strengthen the paper’s main conclusion about the limits of Lévy-walk coarse-graining for deterministic chaotic advection (Sec. 4).

• • For the standard Lévy walk model with constant speed $v_0$ and flight times $\tau$ drawn from a heavy‑tailed distribution $P(\tau) \sim \tau^{-\beta-1}$ for $\tau \geq \tau_{\min}$, the transport regime is determined by $\beta$ such that for $1 < \beta < 2$ the process is superdiffusive with theoretical mean‑squared‑displacement exponent $\alpha = 3 - \beta$, whereas for $\beta > 2$ the mean flight time is finite and the process becomes normally diffusive with $\alpha = 1$.
• _Reference(s):_ 2

• • According to continuous‑time random walk theory, a power‑law residence‑time distribution $P(\tau_{\rm res}) \sim \tau_{\rm res}^{-\gamma}$ with $1 < \gamma < 2$ implies a divergent mean residence time and leads to superdiffusive transport with anomalous diffusion exponent related by $\alpha = 2 - (\gamma - 1)$, whereas $\gamma > 2$ corresponds to a finite mean trapping time and near‑normal diffusion.
• _Reference(s):_ 4

• • For Lévy walks, stronger superdiffusion (smaller tail exponent $\beta$) is theoretically associated with stronger ergodicity breaking, such that the ergodicity‑breaking parameter EB increases as $\beta$ decreases from the normal‑diffusive regime ($\beta > 2$) to strongly superdiffusive cases ($1 < \beta < 2$), because the dynamics become dominated by rare, extremely long ballistic flights that enhance trajectory‑to‑trajectory variability.
• _Reference(s):_ 6

This section audits symbolic/analytic mathematical consistency (algebra, derivations, dimensional/unit checks, definition consistency).

Maths relevance: light

The paper contains a small set of standard transport-statistics definitions (MSD, VACF, TAMSD, EB) and a few scaling-law claims relating power-law exponents to anomalous diffusion. The main analytic concern is an asserted scaling relation linking a residence-time tail exponent to the MSD exponent without specifying the underlying theoretical assumptions needed to make that link verifiable.

### Checked items

• ✔ Point-vortex tracer advection equation (Eq. (1), Sec. 2.1.1, p.3)

• Claim: Tracer velocity is the superposition of point-vortex induced velocities: $\dot{\mathbf{r}} = \sum \Gamma_k/(2\pi) \hat{z} \times (\mathbf{r} - \mathbf{r}_k)/|\mathbf{r} - \mathbf{r}_k|^2$.
• Checks: algebra/structure, dimensional/units consistency, notation consistency
• Verdict: PASS; confidence: high; impact: moderate
• Assumptions/inputs: Circulation $\Gamma_k$ has the usual physical dimension of area/time if $\mathbf{r}$ is a length., The 2D cross product with $\hat{z}$ is interpreted as a $90^\circ$ rotation in the plane.
• Notes: Vector form and scaling with $1/|\mathbf{r}-\mathbf{r}_k|$ are consistent; dimensional check passes if $\Gamma_k$ carries appropriate units. The paper later mixes physical units with $\Gamma_k$ drawn from $N(0,1)$ without stating nondimensionalization (flagged separately).

• ✔ Lévy-walk flight-time heavy-tail definition (Eq. (2), Sec. 2.1.2, p.3)

• Claim: Flight times satisfy $P(\tau) \sim \tau^{-\beta-1}$ for $\tau \geq \tau_{\min}$.
• Checks: definition consistency, normalization/constraints
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: $P(\tau)$ is a probability density; '$\sim$' denotes asymptotic tail behavior.
• Notes: Asymptotic form is consistent as a tail statement. Normalization requires $\beta > 0$, which is satisfied by the $\beta$ values used in the paper.

• ✔ Stated Lévy-walk MSD scaling exponent relation (Sec. 2.1.2, p.3 (text following Eq. (2)); reiterated Sec. 3.2, p.7)

• Claim: For $1 < \beta < 2$, the Lévy walk yields superdiffusion with $\alpha = 3 - \beta$; for $\beta > 2$, $\alpha = 1$.
• Checks: internal consistency, limiting/sanity cases
• Verdict: PASS; confidence: low; impact: moderate
• Assumptions/inputs: Constant-speed flights with direction resets; $\alpha$ denotes MSD exponent via $\langle \Delta r^2 \rangle \propto t^\alpha$.
• Notes: No derivation is provided in the paper, so only internal sanity checks are possible: $\beta\to 2$ gives $\alpha\to 1$ and $\beta\to1$ gives $\alpha\to2$ (ballistic), which are consistent with the regime descriptions.

• ✔ Ensemble-averaged MSD definition (Eq. (3), Sec. 2.3.1, p.4)

• Claim: $\langle \Delta r^2(\Delta t) \rangle = \langle |\mathbf{r}(t+\Delta t) - \mathbf{r}(t)|^2 \rangle$ averaged over start times and trajectories.
• Checks: definition consistency
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: Time lag $\Delta t$ is within observation window.
• Notes: Definition is standard and internally consistent with later use of $\alpha$ as the log-log slope.

• ✔ Normalized velocity autocorrelation definition (Eq. (4), Sec. 2.3.2, p.4)

• Claim: $C_v(\Delta t) = \langle \mathbf{v}(t)\cdot\mathbf{v}(t+\Delta t) \rangle / \langle |\mathbf{v}(t)|^2 \rangle$.
• Checks: definition consistency, normalization/constraints
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: Averaging procedure over $t$/trajectories is consistent in numerator and denominator.
• Notes: Produces $C_v(0)=1$ if the same averaging is used; consistent with a normalized VACF.

• ✔ Residence-time tail model and mean-divergence criterion (Sec. 2.3.3 (definition), p.4; Sec. 3.4 (tail claim), p.9)

• Claim: Residence-time tails follow $P(\tau_{\rm res}) \sim \tau_{\rm res}^{-\gamma}$; for $1 < \gamma < 2$ the mean residence time diverges.
• Checks: normalization/constraints, limiting/sanity cases
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $P(\tau_{\rm res})$ is a PDF with tail exponent $\gamma$., Tail behavior dominates large-$\tau$ integrals.
• Notes: For a tail $P(\tau) \sim \tau^{-\gamma}$, the mean behaves like $\int \tau\cdot\tau^{-\gamma} d\tau = \int \tau^{1-\gamma}d\tau$, which diverges for $\gamma \leq 2$. Thus the stated divergence for $1<\gamma<2$ is correct.

• ⚠ Claimed $\alpha$–$\gamma$ relation connecting trapping to MSD scaling (Sec. 3.4, p.9 (text: 'This result can be connected... via the relation $\alpha = 2 - (\gamma - 1)$ for $1 < \gamma < 2$.'))

• Claim: If $1 < \gamma < 2$, then the anomalous diffusion exponent satisfies $\alpha = 2 - (\gamma - 1)$ (i.e., $\alpha = 3 - \gamma$).
• Checks: derivation logic, definition consistency
• Verdict: UNCERTAIN; confidence: medium; impact: moderate
• Assumptions/inputs: An implicit CTRW-like theory is applicable., The measured $\tau_{\rm res}$ is the relevant renewal-time variable in that theory.
• Notes: The paper does not specify the stochastic model mapping from the threshold-defined $\tau_{\rm res}$ (low-speed residence) to a renewal waiting-time or flight-time variable, nor does it provide a derivation of $\alpha$ as a function of $\gamma$. Without these missing steps/assumptions, the formula cannot be verified from the document alone.

• ✔ Time-averaged MSD (TAMSD) definition (Eq. (5), Sec. 2.3.5, p.5)

• Claim: For each trajectory $i$, $\delta_i^2(\Delta t) = (1/(T-\Delta t)) \int_0^{T-\Delta t} |\mathbf{r}_i(t'+\Delta t) - \mathbf{r}_i(t')|^2 dt'$.
• Checks: definition consistency, normalization/constraints
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: $0 < \Delta t < T$.
• Notes: Integral bounds and normalization by $(T-\Delta t)$ are consistent with standard TAMSD definitions.

• ✔ EB parameter definition and equivalence of forms (Eq. (6), Sec. 2.3.5, p.5)

• Claim: $\text{EB}(\Delta t) = \operatorname{Var}[\delta^2(\Delta t)]/(\operatorname{Mean}[\delta^2(\Delta t)])^2 = (\langle (\delta^2)^2 \rangle - \langle \delta^2 \rangle^2)/\langle \delta^2 \rangle^2$.
• Checks: algebra between shown steps, definition consistency
• Verdict: PASS; confidence: high; impact: minor
• Assumptions/inputs: Mean and variance are taken over the ensemble of trajectories (index $i$).
• Notes: Identity $\operatorname{Var}[X]=\langle X^2\rangle-\langle X\rangle^2$ justifies the second form. Typesetting could be clarified with parentheses to avoid ambiguity.

• ✔ Statement $\text{EB}=0$ indicates ergodicity (interpretation of EB) (Sec. 2.3.5, p.5)

• Claim: $\text{EB}=0$ signifies perfect ergodicity; sustained $\text{EB}>0$ indicates heterogeneity across trajectories.
• Checks: definition/interpretation consistency
• Verdict: PASS; confidence: medium; impact: minor
• Assumptions/inputs: Ergodicity is operationalized as equivalence of time-averaged and ensemble-averaged behavior across realizations, measured via TAMSD variability.
• Notes: Given EB is a normalized variance across trajectory-wise TAMSDs, $\text{EB}=0$ indeed means all trajectories share identical TAMSD at that lag, consistent with the stated operational interpretation.

• ⚠ Use of Lévy-stable fits for displacement PDFs vs finite-speed constraint (Sec. 2.3.4, p.5 and Sec. 3.5, p.9)

• Claim: Long-lag displacement PDFs are compared to Gaussian and symmetric Lévy-stable distributions; the $\beta=1.5$ Lévy walk displacement PDF is described by a Lévy-stable distribution with index $\alpha_{\rm stable}\approx 1.5$.
• Checks: constraint/sanity checks, notation consistency
• Verdict: UNCERTAIN; confidence: low; impact: minor
• Assumptions/inputs: Comparison is meant as an empirical fit/approximation over the sampled range.
• Notes: A constant-speed walk over a finite lag $\Delta t$ implies a hard bound $|\Delta x| \leq v_0\Delta t$, while an ideal Lévy-stable law has unbounded support. If the authors mean approximate core/tail-shape fitting on a finite interval, this is fine, but the paper does not clarify this analytic distinction.

• ⚠ Units/nondimensionalization consistency for vortex vs Lévy-walk datasets (Secs. 2.1.1–2.2, pp.3–4)

• Claim: Vortex simulations and Lévy walks are both discussed with times in seconds, and Lévy walks with $v_0$ in m/s, while $\Gamma_k$ is sampled from a standard normal distribution.
• Checks: dimensional/units consistency, definition consistency
• Verdict: UNCERTAIN; confidence: medium; impact: moderate
• Assumptions/inputs: Either all variables are nondimensional, or $\Gamma_k$ has physical units consistent with Eq. (1).
• Notes: Eq. (1) is dimensionally consistent only if $\Gamma_k$ carries appropriate units and $\mathbf{r}$ is in consistent length units. The paper does not state any nondimensionalization or physical scaling for $\Gamma_k$ and the domain, making the use of 'm/s' and 's' potentially inconsistent.

### Limitations

• Only the mathematics and definitions explicitly present in the provided PDF text were checked; no external theory or derivations were imported.
• Several scaling-law statements (e.g., $\alpha=3-\beta$ and the asserted $\alpha$–$\gamma$ relation) are not derived in the paper; such items can only be assessed for internal consistency and basic sanity limits, otherwise marked UNCERTAIN.
• Figure-based empirical fit claims (e.g., Lévy-stable fit quality) cannot be validated analytically beyond checking basic constraints implied by the model definitions (e.g., finite-speed bounds).

This section audits numerical/empirical consistency: reported metrics, experimental design, baseline comparisons, statistical evidence, leakage risks, and reproducibility.

Twelve numeric/logical consistency checks derived from the text and tables were executed; all 12 passed with no detected discrepancies under the stated tolerances. Several additional quantitative claims remain unverified because they require extracting values from figures or fitting to underlying time-series/PDF data not available in the parsed text.

### Checked items

• ✔ C01_time_steps_total_samples (Page 3, Section 2.2 (Simulation datasets))

• Claim: Each trajectory was integrated for a total duration of $24.75\ {\rm s}$ with a time step of $0.05\ {\rm s}$.
• Checks: duration_step_count_consistency
• Verdict: PASS
• Notes: $T/\Delta t = 24.75/0.05 = 495$ steps exactly; implies 496 stored points if including endpoints.

• ✔ C02_total_tracer_trajectories_count (Page 3, Section 2.2 (Simulation datasets))

• Claim: We simulated the trajectories of 5 passive tracers for each of the four vortex configurations ($N = 5, 10, 20, 40$).
• Checks: count_product_total
• Verdict: PASS
• Notes: Computed total trajectories $= 5\times 4 = 20$.

• ✔ C03_levy_alpha_from_beta_1p2 (Page 3, Section 2.1.2 (Lévy walk model) and Page 4, Section 2.2 ($\beta$ list))

• Claim: For $1 < \beta < 2$, the model produces superdiffusion with theoretical mean squared displacement exponent $\alpha = 3 - \beta$. Dataset includes $\beta = 1.2$.
• Checks: formula_substitution
• Verdict: PASS
• Notes: $\alpha = 3 - 1.2 = 1.8$.

• ✔ C04_levy_alpha_from_beta_1p5 (Page 3, Section 2.1.2 and Page 4, Section 2.2)

• Claim: For $1 < \beta < 2$, $\alpha = 3 - \beta$. Dataset includes $\beta = 1.5$.
• Checks: formula_substitution
• Verdict: PASS
• Notes: $\alpha = 3 - 1.5 = 1.5$.

• ✔ C05_levy_alpha_from_beta_1p8 (Page 3, Section 2.1.2 and Page 4, Section 2.2)

• Claim: For $1 < \beta < 2$, $\alpha = 3 - \beta$. Dataset includes $\beta = 1.8$.
• Checks: formula_substitution
• Verdict: PASS
• Notes: $\alpha = 3 - 1.8 = 1.2$.

• ✔ C06_levy_beta_gt2_implies_alpha1_for_2p5 (Page 3, Section 2.1.2 and Page 4, Section 2.2)

• Claim: For $\beta > 2$, normal diffusion with $\alpha = 1$. Dataset includes $\beta = 2.5$ and text notes normal diffusion ($\alpha = 1.0$).
• Checks: conditional_rule_check
• Verdict: PASS
• Notes: $\beta = 2.5$ satisfies $\beta>2$ and the stated $\alpha=1.0$ matches the rule.

• ✔ C07_beta_to_alpha_span_claim (Page 4, Section 2.2 (Lévy walk dataset description))

• Claim: $\beta = 1.2, 1.5, 1.8, 2.5$ were chosen to span strong superdiffusion ($\alpha = 1.8$) to normal diffusion ($\alpha = 1.0$).
• Checks: derived_range_endpoints
• Verdict: PASS
• Notes: Endpoints consistent with stated theory: $\beta=1.2\to\alpha=1.8$; $\beta=2.5\to\alpha=1.0$.

• ✔ C08_msd_trapping_relation_alpha_from_gamma_N40 (Page 9, Section 3.4 (Lévy-like trapping statistics))

• Claim: Relation given: $\alpha = 2 - (\gamma - 1)$ for $1 < \gamma < 2$. For $N = 40$, $\gamma \approx 1.7$ implies $\alpha \approx 2 - (1.7 - 1) = 1.3$.
• Checks: formula_substitution
• Verdict: PASS
• Notes: Worked arithmetic is consistent: $\alpha = 2 - (1.7-1) = 1.3$.

• ✔ C09_gamma_uncertainty_range_condition_N40 (Page 9, Table 2 and Section 3.4)

• Claim: For $N = 40$, $\gamma = 1.7 \pm 0.2$ and claim emphasizes $\gamma < 2$ (divergent mean residence time).
• Checks: inequality_with_uncertainty
• Verdict: PASS
• Notes: Conservative upper bound $1.7+0.2=1.9$ remains $<2$.

• ✔ C10_gamma_uncertainty_range_condition_N10 (Page 9, Table 2 and Section 3.4)

• Claim: For $N = 10$, $\gamma = 2.8 \pm 0.4$ and statement says $\gamma > 2$ (finite mean trapping time / exponential cutoff).
• Checks: inequality_with_uncertainty
• Verdict: PASS
• Notes: Conservative lower bound $2.8-0.4=2.4$ remains $>2$.

• ✔ C11_table1_monotonic_increase_alpha_with_N (Page 7, Table 1)

• Claim: Table 1 implies $\alpha$ increases monotonically with $N$: $(N,\alpha) = (5,1.05), (10,1.18), (20,1.35), (40,1.52)$.
• Checks: monotonicity_check
• Verdict: PASS
• Notes: Verified strict monotonic increase: $1.05 < 1.18 < 1.35 < 1.52$.

• ✔ C12_table1_R2_range_check (Page 7, Table 1)

• Claim: Table 1 lists $R^2$ values: $0.97, 0.98, 0.97, 0.96$.
• Checks: boundedness_check
• Verdict: PASS
• Notes: All listed $R^2$ values lie within $[0,1]$.

### Limitations

• Only parsed text (and not underlying numeric datasets) is available; many quantitative claims referenced to figures cannot be recomputed without the raw time series or tabulated fit results.
• Figure-based approximate values (e.g., VACF zero-crossings, EB plateaus, slope fits) cannot be verified without extracting curve data, which is out of scope.
• No checks were proposed that require external datasets, internet access, or long simulations; such items are instead listed as unverified.

## Paper Ratings

| Dimension | Score |

|-----------|:-----:|

| Overall | 5/10 █████░░░░░ |

| Soundness | 5/10 █████░░░░░ |

| Novelty | 6/10 ██████░░░░ |

| Significance | 5/10 █████░░░░░ |

| Clarity | 5/10 █████░░░░░ |

| Evidence Quality | 4/10 ████░░░░░░ |

Justification: The work presents a thoughtful comparative study between chaotic point‑vortex transport and Lévy walks, with a conceptually interesting finding that superdiffusion–ergodicity trends diverge across the two systems. However, key methodological gaps—very small tracer ensembles and no multiple vortex realizations, incomplete specification of domain/boundaries and integrator/singularity handling, and limited uncertainty quantification—undermine confidence in the quantitative claims, especially the EB trends and tail fits. The mathematical audit finds core definitions correct but flags the asserted α–γ link and units/nondimensionalization as uncertain; figure issues and lack of rigorous fitting further reduce clarity and support. Overall, the paper is promising but currently borderline due to insufficient evidence and reporting needed to substantiate its central conclusions.