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arxiv: 2604.13852 · v1 · submitted 2026-04-15 · ❄️ cond-mat.stat-mech · math-ph· math.MP

First Passage Times for Variable-Order Time-Fractional Diffusion

Wancheng Li , Daniel S. Han This is my paper

Pith reviewed 2026-05-10 11:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords variable-order fractional diffusionfirst passage timessurvival probabilityasymptotic analysisanomalous diffusionheterogeneous transport
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The pith

Survival probability in variable-order time-fractional diffusion decays as t to the power of minus the global minimum of α(x), corrected by a power of ln t.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives the long-time asymptotic form of the first passage time distribution for diffusion where the fractional order α depends on position. For smooth α(x) on a finite interval with absorbing and reflecting boundaries, the survival probability behaves as Ψ(t) ∼ C t^{-α_*} / (ln t)^ν at large t, where α_* is the smallest value of α(x) and the power ν is set by the location and local shape of that minimum. This reduces to the familiar pure power-law decay when α is constant, and supplies a testable signature for identifying spatially varying anomalous transport from experimental first-passage data. The result is confirmed by exact Laplace-domain solutions and Monte Carlo trajectories for both linear and nonlinear α(x) profiles.

Core claim

For any sufficiently smooth position-dependent fractional order α(x) on a bounded domain equipped with absorbing and reflecting boundaries, the survival probability of the variable-order time-fractional diffusion satisfies the asymptotic Ψ(t) ∼ C t^{-α_*} / (ln t)^ν, where α_* equals the global minimum of α(x) and the exponent ν is fixed by the position and functional form of α near that minimum.

What carries the argument

Asymptotic inversion of the Laplace-space survival probability, in which the global minimum of α(x) supplies the dominant singularity and the local expansion of α around that minimum produces the logarithmic correction to the power-law decay.

If this is right

  • When α(x) is constant the logarithmic power vanishes and the decay collapses to the standard t^{-α} form.
  • The value of ν distinguishes whether the minimum occurs in the interior or at a boundary and encodes the leading-order curvature or flatness of α there.
  • Observed first-passage statistics can therefore be inverted to extract the spatial profile of α(x) in heterogeneous media.
  • The same asymptotic structure holds for both linear and nonlinear α(x) profiles, as verified by direct simulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the minimum of α(x) is degenerate or flat over an interval, the power of ln t will change, offering a classification of different heterogeneity types.
  • Analogous dominance by an extremum of a position-dependent exponent is likely to appear in other variable-order or heterogeneous anomalous-transport equations.
  • Controlled experiments that impose a known α(x) variation and track long-time survival could directly test the predicted dependence on the shape of the minimum.

Load-bearing premise

α(x) is smooth enough on a finite domain with mixed absorbing and reflecting boundaries that the global minimum alone determines the leading late-time asymptotics.

What would settle it

A numerical or experimental survival curve for a known smooth α(x) whose measured decay exponent differs from -α_* or lacks the predicted logarithmic factor.

Figures

Figures reproduced from arXiv: 2604.13852 by Daniel S. Han, Wancheng Li.

Figure 1
Figure 1. Figure 1: FIG. 1. Survival probability Ψ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the excellent correspondence between the asymptotic prediction (40) and Monte Carlo simu￾lations of the underlying random walk governed by (1) when the fractional exponent varies as (39). The PDF obtained via Monte Carlo in the lower inset of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $\alpha(x)$ varies with position. For any sufficiently smooth $\alpha(x)$ on a finite domain with absorbing and reflecting boundaries, we show that the survival probability decays as $\Psi(t)\sim C\,t^{-\alpha_*}/(\ln t)^{\nu}$, where $\alpha_*$ is the minimum value of the fractional exponent and $\nu$ is determined by the location and shape of the minimum. For a constant fractional exponent $\nu=0$ and this provides a theoretical prediction that can identify spatially heterogeneous anomalous transport in experiments. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $\alpha(x)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript derives an asymptotic expression for the survival probability in first-passage problems governed by a time-fractional diffusion equation with spatially varying order α(x). Specifically, for sufficiently smooth α(x) on a finite interval with mixed absorbing/reflecting boundaries, Ψ(t) ∼ C t^{-α_*} (ln t)^{-ν} as t → ∞, where α_* = min α(x) and the exponent ν is fixed by the local behavior of α near its minimum. The derivation is checked against closed-form Laplace-space solutions for linear and quadratic α(x) and against direct Monte Carlo realizations of the underlying stochastic process.

Significance. The result is significant for the field of anomalous diffusion because it identifies the global minimum of the local exponent as the sole determinant of the ultimate power-law decay, with a sub-leading logarithmic factor that encodes the geometry of that minimum. This furnishes a falsifiable signature that could be used to infer spatial heterogeneity from measured first-passage histograms, and it recovers the classical constant-order result when ν = 0. The provision of both exact Laplace inversions and stochastic simulations constitutes a strong internal consistency check.

minor comments (3)
  1. [Abstract] The abstract refers to 'nonlinear profiles' while the body uses 'quadratic'; a uniform terminology would avoid minor confusion.
  2. [Section 5] The precise discretization scheme and time-stepping rule employed for the Monte Carlo trajectories are not stated; adding one sentence or a short appendix would improve reproducibility.
  3. [Figures 2 and 3] Figure captions for the comparison plots should explicitly list the numerical value of α_* used in each panel.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The summary accurately captures the main result on the asymptotic decay of the survival probability determined by the global minimum of α(x). No specific major comments were provided in the report, so we have no points to address individually. We will incorporate minor improvements to clarity and presentation in the revised version.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The manuscript derives the long-time asymptotic form of the survival probability directly from the variable-order time-fractional diffusion PDE with position-dependent α(x), using local expansions around the global minimum of α(x) together with Laplace-space solutions for linear/quadratic profiles and Monte Carlo validation. No load-bearing step reduces to a self-definition, a fitted parameter renamed as a prediction, or a self-citation chain; the claimed tail follows from the stated PDE, smoothness, and boundary conditions without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard variable-order time-fractional diffusion equation together with smoothness of α(x) and standard absorbing/reflecting boundary conditions; no free parameters, new entities, or ad-hoc postulates are introduced in the abstract.

axioms (2)
  • domain assumption The underlying process obeys a space-dependent variable-order time-fractional diffusion equation.
    This is the governing model from which the asymptotic is derived.
  • domain assumption α(x) is sufficiently smooth on a finite interval with absorbing and reflecting boundaries.
    Required for the minimum of α(x) to dominate the long-time survival probability.

pith-pipeline@v0.9.0 · 5431 in / 1434 out tokens · 42560 ms · 2026-05-10T11:54:41.641588+00:00 · methodology

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Reference graph

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