{"id":"9040a2dd-c1d8-410a-9a3e-3066ff2108be","arxiv_id":"1908.02952","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For finite observation windows, the mean first passage time starts at the window length for every model and then saturates or keeps growing depending on the subdiffusion model.","lead":"This paper calculates how the average first-passage time to hit a boundary changes when the observation window is finite instead of infinite. It shows that this finite-window average can tell apart subdiffusion models that otherwise look identical in their mean squared displacement.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the termwise Mittag-Leffler expansion is justifiable, and the remaining issues are reporting details for the FBM simulation.","rationale":"I read the paper as claiming that the large-T scaling of the finite-observation MFPT separates the fractional diffusion equation from scaled Brownian motion, fractional Brownian motion, and the Wilemski-Fixman approximation, and that the observation-time dependence is more informative than the mean square displacement. The most delicate step is the derivation of equation (17), but the concern about termwise expansion is not load-bearing: because the spectral arguments lambda_k z are bounded below by lambda_0 z, the Mittag-Leffler asymptotic expansion is uniform in k, and the k-sum of the remainder converges absolutely for every fixed truncation order N. The alternative Laplace-domain route via equation (19) provides independent support. Known results, such as the finite MFPT scaling L^{2/alpha} for FBM and the algebraic survival tail for FDE, also back the qualitative comparison. The remaining weakness is that the FBM simulation component is not reproducible from the manuscript: no ensemble size, time discretization, crossing-detection procedure, or error bars are given. That reporting gap reasonably supports a conditional acceptance while the central analytical comparison stands.","tokens_in":24695,"tokens_out":29423,"duration_ms":339118,"concrete_test":"For alpha = 0.5, L = 1, and D_alpha = 1, compute equation (15) directly by summing the first 10^4 eigenmodes, evaluating each generalized Mittag-Leffler function by its power series for x < 30 and by its asymptotic expansion for x > 30, at z = 10, 100, and 1000; compare the resulting <t>_T/T with the first four terms of equation (17). Agreement within 10% would confirm the termwise expansion; disagreement would require revisiting equation (17).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption, that equation (17) rests on an unjustified termwise use of equation (16) inside equation (15), does not survive scrutiny. In equation (15) the argument of each generalized Mittag-Leffler term is -lambda_k z with lambda_k >= lambda_0 = pi^2/4, so every argument is at least lambda_0 z and tends to infinity uniformly in k as z tends to infinity. For 0 < alpha < 1, the remainder after N terms in the large-x expansion of the generalized Mittag-Leffler function is O(x^{-N-1}) uniformly for x >= x_0 > 0. Therefore the summed remainder is bounded by C_N z^{-N-1} sum_k |c_k| lambda_k^{-N-1}, which converges absolutely for N >= 1; letting N grow gives equation (17) as an asymptotic expansion. The Laplace-domain route through equation (19) also bypasses the objection. I therefore find no load-bearing correctness problem in the central FDE-versus-other-models comparison. The genuinely weak spot is reproducibility of the FBM results: the paper gives no number of realizations, no time step, no boundary-crossing interpolation rule, and no error bars, so the FBM curves in Figures 1-3 cannot be independently checked from the text. This is a reporting gap, not a demonstrated failure of the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24986,"tokens_out":23230,"duration_ms":247911,"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":[{"comment":"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.","section":"Section 2.4, Figures 1-3"},{"comment":"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.","section":"Section 2.2, Eq. (17)"}],"minor_comments":[{"comment":"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).","section":"Appendix B"},{"comment":"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.","section":"Section 3, reference [63]"},{"comment":"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.","section":"Figure 3(d)"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically sound in its main derivations; my recommendation of major revision is driven by the reproducibility gap for the FBM simulations and the need for a clearer statement of the main asymptotic formula. The requested changes are local and should not require a full re-derivation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a useful paper. It gives experimentalists a finite-observation-time MFPT probe that distinguishes the fractional diffusion equation from scaled Brownian motion, fractional Brownian motion, and the Wilemski-Fixman approximation even when all four models share the same mean squared displacement. The central claim holds up: in the small-T limit all models give linear-in-T MFPT, while at large T the FDE diverges as T^(1-alpha) and the others saturate to finite values. No fitted constants anywhere.\n\nWhat is actually new: the systematic comparison, the small-T expansion, the higher-order large-T FDE asymptotics in Eq. (17), the finite-MFPT formula for SBM in Eq. (31a), and the finite-domain Wilemski-Fixman treatment. The exact relation in Eq. (5) is a correct integration by parts, and the Brownian and SBM survival probabilities are standard. The one analytical concern that might bother a referee, the termwise use of the Mittag-Leffler expansion inside the eigenfunction sum, does not survive checking: the arguments -lambda_k z go to infinity uniformly in k, so the remainder is O(z^{-N-1}) with an absolutely convergent coefficient sum. That part is fine.\n\nThe real soft spot is the FBM simulation. The paper gives no number of realizations, no time step, no boundary-crossing interpolation rule, and no error bars, so the blue curves in Figures 1–3 cannot be independently checked from the text. That is a reporting gap, not a demonstrated failure of the central claim, but it should be fixed before publication. The WF derivation is heavy and hard to follow—Appendix C is dense—though it reduces correctly to the Brownian limit. There are also a few minor equation-numbering slips in the appendices (e.g., references to \"equation (19)\" or \"equation (26)\" that do not match the main text), which are cosmetic. The single self-citation, Ref. [63], only supports an interpretive comment about search efficiency and is not load-bearing.\n\nWho this is for: anyone working on first-passage statistics in anomalous diffusion, and experimentalists doing single-particle tracking who need a probe more sensitive than the MSD. It deserves a serious referee. I would accept it for review and ask for simulation details; with those details it should be publishable essentially as is.","headline":"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.","tokens_in":25457,"tokens_out":1489,"would_cite":true,"duration_ms":19102,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G22","60J60","82C31","35R11"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["mean first passage time","observation time dependence","subdiffusion","fractional diffusion equation","scaled Brownian motion","fractional Brownian motion","Wilemski-Fixman approximation","survival probability"],"falsifier":"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.","tokens_in":24524,"feed_emoji":"⏱️","tokens_out":10637,"duration_ms":104306,"temperature":0.7,"pith_summary":"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.","feed_headline":"Observation time separates subdiffusion models that MSD cannot","feed_subtitle":"Short windows give one linear law; long windows separate a diverging power law from finite saturation.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the observation-time-dependent first passage time distribution and the T-dependent MFPT that the paper studies.","marker":"[29]"},{"why":"Supplies the standard first-passage formalism and eigenfunction expansion used to derive the diffusion and WF survival probabilities.","marker":"[30]"},{"why":"Establishes that the fractional diffusion equation has no finite MFPT in the infinite-time limit, the fact the finite-window definition regularizes.","marker":"[27]"},{"why":"Supplies the Mittag-Leffler and Wright function expansions used to obtain the FDE large- and small-T asymptotic results.","marker":"[34]"},{"why":"Provides the general long-time first-passage expansion in finite domains underlying the multidimensional form of Eq. (17).","marker":"[35]"},{"why":"Introduces the Wilemski-Fixman convolution approximation for non-Markovian first passage used in Eqs. (26)-(29).","marker":"[46]"},{"why":"Provides the exact Davies-Harte simulation algorithm used to generate fractional Brownian motion survival probabilities.","marker":"[50]"},{"why":"Supplies the near-boundary displacement scaling and persistence-exponent relation used to interpret differences among models.","marker":"[57]"}],"fun_headline_variants":["Universal small-T law, distinct large-T MFPT per model","Observation time makes MFPT beat MSD for subdiffusion","T-window MFPT: a sharper discriminator of subdiffusion","Watch time sets MFPT: key to separating subdiffusion","MFPT's T-dependence: a new probe for subdiffusion models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Universal small-T law, distinct large-T MFPT per model","Observation time makes MFPT beat MSD for subdiffusion","T-window MFPT: a sharper discriminator of subdiffusion","Watch time sets MFPT: key to separating subdiffusion","MFPT's T-dependence: a new probe for subdiffusion models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000407,"raw_usage":{"total_tokens":2131,"prompt_tokens":979,"completion_tokens":1152,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":1061}},"tokens_in":595,"tokens_out":1152,"duration_ms":11950,"temperature":1.0,"reasoning_tokens":1061,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:29:55.203651+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the observation-time-dependent first passage time distribution and the T-dependent MFPT that the paper studies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard first-passage formalism and eigenfunction expansion used to derive the diffusion and WF survival probabilities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that the fractional diffusion equation has no finite MFPT in the infinite-time limit, the fact the finite-window definition regularizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Mittag-Leffler and Wright function expansions used to obtain the FDE large- and small-T asymptotic results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the general long-time first-passage expansion in finite domains underlying the multidimensional form of Eq. (17)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Wilemski-Fixman convolution approximation for non-Markovian first passage used in Eqs. (26)-(29)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the exact Davies-Harte simulation algorithm used to generate fractional Brownian motion survival probabilities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the near-boundary displacement scaling and persistence-exponent relation used to interpret differences among models."}],"review_version":1}