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REVIEW 2 major objections 4 minor 67 references

Observation time dependent mean first passage time of diffusion and sub-diffusion processes

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A finite observation window turns the mean first passage time into a model-specific fingerprint of subdiffusion, capable of distinguishing processes that share the same mean square displacement.

desk verdict Solid, useful paper: finite-observation-time MFPT cleanly separates subdiffusion models that share the same MSD, and the central asymptotic claim survives scrutiny, though the FBM simulation needs reporting details. read the letter →

arxiv 1908.02952 v1 pith:6BJZL5DL submitted 2019-08-08 cond-mat.stat-mech

Ji-Hyun Kim , Hunki Lee , Sanggeun Song , Hye Ran Koh , Jaeyoung Sung This is my paper · ORCID
classification cond-mat.stat-mech MSC 60G2260J6082C3135R11
keywords meanfirstpassagetimeobservationdependencesubdiffusionfractionaldiffusionequationscaledBrownianmotionWilemski-Fixmanapproximationsurvivalprobability
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 finite observation window changes the mean first passage time into a model-specific quantity, and that this dependence is a sharper experimental fingerprint of subdiffusion than the mean square displacement. For a particle diffusing in $[-L,L]$, the windowed mean first passage time $\langle t\rangle_T$ is defined by averaging only first-passage events that occur before $T$. In the short-window limit every model examined gives the same linear law $\langle t\rangle_T\simeq T$, but for long windows the fractional diffusion equation keeps growing as $T^{1-\alpha}$ and diverges relative to the others, whereas scaled Brownian motion, fractional Brownian motion, and the Wilemski-Fixman approximation saturate at finite values that scale as $L^{2/\alpha}$. Because the four subdiffusion models share the same mean square displacement, the observation-time dependence can separate them where standard MSD analysis cannot. The paper therefore proposes the $T$-dependent MFPT as a practical observable for identifying the mechanism of subdiffusion in experiments.

What carries the argument

The load-bearing object is the identity $\langle t\rangle_T = \frac{\int_0^T dt\, S(t) - T S(T)}{1-S(T)}$, which converts any measured or computed survival probability into the finite-window first passage time, plus the eigenfunction representations of $S(t)$ for each model: Mittag-Leffler series for the fractional diffusion equation, exponential series in a rescaled time for scaled Brownian motion, Davies-Harte simulated trajectories for fractional Brownian motion, and the Wilemski-Fixman convolution (Eqs. 26-29) for the stationary Markov approximation. The large-$T$ separation of models is carried by how $S(t)$ decays: algebraically for FDE, stretched-exponentially for SBM and WF, exponentially for FBM.

What would settle it

Simulate a continuous-time random walk with waiting-time exponent $\alpha$ on $[-L,L]$, or numerically invert the full Laplace-domain survival probability of the FDE, and compare the resulting $\langle t\rangle_T$ with the leading term of Eq. (17) across a wide range of $\tilde T=T/(L^2/D_\alpha)^{1/\alpha}$. If the ratio $\langle t\rangle_T/(L^2 T^{1-\alpha}/D_\alpha)$ does not approach the predicted prefactor as $\tilde T\to\infty$, or if the higher-order corrections grow rather than decay, the termwise-expansion step is invalid and the FDE branch of the claim needs revision.

Watch

Extended reading notes

Core claim

The central claim is that the $T$-dependent MFPT of a particle starting at the center of an interval $[-L,L]$ has a universal small-$T$ form and a transport-specific large-$T$ form. Small $T$ acts as a low-pass filter: only first-passage times shorter than $T$ contribute, so the conditional average of those times approaches $T$ for every model. At large $T$, the way the survival probability decays takes over: for the FDE, $S(t)\sim t^{-\alpha}$ gives $\langle t\rangle_T\simeq \text{const}\times L^2 T^{1-\alpha}/D_\alpha$ with no finite infinite-time limit, while for SBM, FBM, and the WF approximation the survival probability decays faster and $\langle t\rangle_T$ converges to a finite limit that scales as $L^{2/\alpha}$. The paper's main results are the asymptotic expansions, Eqs. (17) and (20), governing the FDE branch in the large- and small-$\tilde T$ regimes, plus the corresponding saturated limits for the other models.

Load-bearing premise

The load-bearing premise is that the long-time asymptotic expansion of the generalized Mittag-Leffler function (the special function that appears in the fractional diffusion equation's eigenfunction expansion) can be substituted term by term into the infinite eigenfunction series and integrated, with no stated convergence or uniformity condition for that interchange.

Editorial extensions

If this is right

  • A finite observation window converts the FDE's nonexistent mean first passage time into a measurable quantity that grows like $T^{1-\alpha}$ at long $T$.
  • At fixed $\alpha$, $\langle t\rangle_T/T$ depends only on $\tilde T=T/(L^2/D_\alpha)^{1/\alpha}$, so time-dependence and length-dependence measurements contain the same information.
  • At fixed $T$ in the small-$L$ limit, the FDE's MFPT scales as $L^2$ while SBM, FBM, and the WF approximation scale as $L^{2/\alpha}$, giving a system-size test that does not require long observation.
  • In the small-$T$ regime all models collapse to $\langle t\rangle_T\simeq T$, so only windows long compared with the diffusion time scale can discriminate mechanisms.
  • The WF approximation tracks FBM behavior only for $\alpha$ near 1; at smaller $\alpha$ its Markovian assumption misrepresents the first-passage statistics.

Reading between the lines

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

  • The paper does not pursue it, but because $T$ acts as a low-pass filter, the derivative $d\langle t\rangle_T/dT$ could in principle recover the underlying first-passage time distribution, turning the finite-window experiment into a spectroscopy of first-passage times.
  • The exchange between $T$ and $L$ established in Appendix D suggests a fixed-window experiment with varying compartment size can mimic a time scan, which may be easier to realize in microfluidic or porous-media assays.
  • For the FDE branch, the predicted local slope $d\log\langle t\rangle_T/d\log T\to 1-\alpha$ gives a finite-time estimator of the anomalous exponent; a two-window measurement would not require waiting for the divergent infinite-time limit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. This paper studies the mean first passage time conditioned on the observation window [0,T]. For a particle in an interval with absorbing boundaries, the authors use the exact relation (5) between the T-dependent MFPT and the survival probability, and evaluate it for normal diffusion, the fractional diffusion equation (FDE), scaled Brownian motion (SBM), fractional Brownian motion (FBM), and the Wilemski-Fixman approximation to SBM/FBM. The main claims are (i) in the small-T limit the T-dependent MFPT is proportional to T for all models, and (ii) in the large-T limit the FDE model shows a growing MFPT ~ T^{1-α}, whereas SBM, FBM and the WF approximation saturate to finite values whose travel-length scaling is L^{2/α}, in contrast to the L^2 scaling of normal diffusion and the FDE. The paper also provides asymptotic expansions for the FDE branch, Eqs. (17) and (20), and discusses the relationship between T- and L-dependence through the scaling variable T̃.

Significance. The central proposal—that the observation-time dependence of the MFPT can discriminate subdiffusion models that share the same MSD—is interesting and potentially useful experimentally. The mathematical core is largely analytic and parameter-free: Eq. (5) is exact, the FDE and SBM survival probabilities follow from standard eigenfunction expansions, and no fitted constants appear. The FBM simulation is benchmarked against the known MSD and the semi-infinite first-passage tail. If the requested simulation details are supplied, the paper would constitute a solid contribution to the first-passage literature.

major comments (2)
  1. [Section 2.4, Figures 1-3] The FBM results, which are the only simulation-based part of the core comparison, are not reproducible from the information given. The text states only that the Davies-Harte algorithm was used and that the MSD and the semi-infinite first-passage tail are reproduced; it does not report the number of realizations, the time step or integration scheme, the rule for detecting boundary crossings (required because FBM has continuous trajectories), the maximum simulation time used to construct S(t), or error estimates for the plotted curves. Please provide these details, or make the simulation data available, before the FBM branch of the central claim can be assessed.
  2. [Section 2.2, Eq. (17)] The derivation of the large-T expansion is presented as 'one can show' after substituting Eq. (16) into Eq. (15), with no statement of the conditions under which the termwise operations and the subsequent summation over eigenmodes are justified. Although the result is correct and can be justified by the uniform large-argument behavior of E_{α,β} when λ_k z ≥ λ_0 z, the manuscript should include this argument (at least in an appendix) and should also typeset Eq. (17) so that all Gamma-function arguments, powers of z and the coefficients R_n are unambiguous; as printed, several terms cannot be read with confidence.
minor comments (4)
  1. [Appendix B] The text 'Substituting equation (B.3) into equation (26)' should refer to the formula for the T-dependent MFPT, Eq. (5) or Eq. (32), not the WF integral equation (26).
  2. [Section 3, reference [63]] Reference [63] is listed as 'to be published'; for the interpretive claim about search efficiency in Section 3, please update the reference to the published version or cite a published source.
  3. [Figure 3(d)] The numerical value φ=5.67 for FBM with α=0.3 is reported without an uncertainty or a fit range; please state how φ was extracted from the simulation data.
  4. [General] The small-T universal result ⟨t_T⟩≈T is stated for all models, but the text does not explain why the conditional first-passage distribution concentrates near T as T→0; a brief remark on this point would help readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central results are derived from model definitions, exact survival-probability identities, and standard asymptotics; the only self-citation is interpretive and non-load-bearing.

full rationale

The paper's derivation chain is self-contained. The T-dependent MFPT is defined from the first-passage distribution in Eq. (4), then converted exactly into the survival-probability form in Eq. (5). Every model result is obtained by inserting independently defined model propagators or survival probabilities into that identity: Brownian motion in Eq. (9), the fractional diffusion equation in Eq. (15), scaled Brownian motion in Eq. (24), and the Wilemski-Fixman approximation in Eqs. (29a)-(29b). The FDE large-T behavior follows from standard generalized Mittag-Leffler asymptotics, Eq. (16), applied to the eigenfunction series; the leading term is cross-checked against the known result in Ref. [27], and the multidimensional form cites Condamin et al. [35], not the present authors. No fitted parameters appear anywhere in the derivation, and no fitted quantity is renamed as a prediction. Although the small-T linear proportionality is a fairly generic consequence of Eq. (5) once the survival probability has a finite small-time expansion, the paper derives concrete model-dependent prefactors and corrections (Eqs. (11) and (20)) rather than assuming the conclusion. The only self-citation, Ref. [63], supports an interpretive remark about search efficiency and is not load-bearing for any core equation. The FBM simulation is calibrated against the known MSD and the known semi-infinite first-passage tail, so it does not import the paper's conclusions as inputs. The suspected termwise Mittag-Leffler integration would be a correctness or rigor concern, not a circularity, and in any case the leading large-T law is independently supported by the Laplace-domain expression Eq. (19). Overall, no circular step is present; score 0.

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

No parameter is fitted to data; alpha, D_alpha, L, and T are model inputs from the problem setup or from prior literature. The axioms are standard PDE and series methods plus the explicit modeling assumptions for each subdiffusion model. The least standard assumption is the termwise Mittag-Leffler expansion behind Eq (17).

assumptions (5)
  • standard math The eigenfunction expansion solution of the diffusion equation with absorbing boundaries at x=±L is valid and can be Laplace-transformed termwise.
    Section 2.1, Eqs (7)-(9); this is the standard starting point for the Brownian survival probability.
  • domain assumption For the fractional diffusion equation, replacing D1 by D_alpha s^(1-alpha) in the Laplace domain gives the correct survival probability under absorbing boundary conditions.
    Section 2.2, Eq (12). This is the standard fractional diffusion equation solution, but the absorbing-boundary interpretation is a modeling assumption.
  • ad hoc to paper The long-time asymptotic expansion of the generalized Mittag-Leffler function (Eq 16) may be substituted termwise into the infinite series for the survival probability to obtain Eq (17).
    Section 2.2, Eq (17); convergence and uniformity conditions are not stated.
  • domain assumption The scaled Brownian motion Langevin equation (Eq 21) and the fractional Brownian motion autocorrelation (Eq 25) define the models, and the Davies-Harte algorithm generates exact fractional Brownian motion trajectories.
    Sections 2.3-2.4; these assumptions enter the simulation-based results.
  • domain assumption The Wilemski-Fixman approximation, which allows time shifts in Green's functions, is a valid stationary Markovian approximation of scaled Brownian motion and fractional Brownian motion.
    Section 2.5, Eq (27). The authors explicitly note it is approximate for nonstationary or non-Markov processes.

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Pith. "Pith review of Observation time dependent mean first passage time of diffusion and sub-diffusion processes." pith.science (2026). https://pith.science/paper/6BJZL5DL

@misc{pith2026190802952,
  author       = {Pith},
  title        = {Pith review of: Observation time dependent mean first passage time of diffusion and sub-diffusion processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BJZL5DL}},
  note         = {Machine review of arXiv:1908.02952}
}
read the original abstract

The mean first passage time, one of the important characteristics for a stochastic process, is often calculated assuming the observation time is infinite. However, in practice, the observation time, T, is always finite and the mean first passage time (MFPT) is dependent on the length of the observation time. In this work, we investigate the observation time dependence of the MFPT of a particle freely moving in the interval [-L,L] for a simple diffusion model and four different models of subdiffusion, the fractional diffusion equation (FDE), scaled Brown motion (SBM), fractional Brownian motion (FBM), and stationary Markovian approximation model of SBM and FBM. We find that the MFPT is linearly dependent on T in the small T limit for all the models investigated, while the large-T behavior of the MFPT is sensitive to stochastic properties of the transport model in question. We also discuss the relationship between the observation time, T, dependence and the travel-length, L, dependence of the MFPT. Our results suggest the observation time dependency of the MFPT can serve as an experimental measure that is far more sensitive to stochastic properties of transport processes than the mean square displacement.

Figures

Figures reproduced from arXiv: 1908.02952 by the authors.

Figure 1
Figure 1. The observation time dependence of the T-dependent MFPT at various values of . The units of time and length are respectively given by 2 11 () LD    and L. The green, red, yellow, and blue solid lines represent the results for the FDE model, the scaled Brownian motion (SBM), the Wilemski-Fixman (WF) approximation, and the fractional Brownian motion (FBM), respectively. The green dashed line represents the large-T… view at source ↗
Figure 2
Figure 2. The length scale dependence of the T-dependent MFPT at various values of . The units of time and length are respectively given by T and 12 () DT  . The green, red, yellow, and blue solid lines represent the results for the FDE model, the scaled Brownian motion (SBM), the Wilemski-Fixman (WF) approximation, and fractional Brownian motion (FBM), respectively. The green dashed line represents the small-L asymptote f… view at source ↗
Figure 3
Figure 3. Survival probability and normalized displacement distribution. (a) Survival probabilities for four kinds of subdiffusion models at   0.3. The dashed lines are given as an eye guide to show that the long-time behavior of survival probability follows a stretched exponential decay for [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗

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