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arxiv: 2502.20705 · v5 · pith:MDC3GK3Snew · submitted 2025-02-28 · ❄️ cond-mat.stat-mech · physics.bio-ph

First passage time properties of diffusion with a broad class of stochastic diffusion coefficients

Go Uchida , Hiromi Miyoshi , Hitoshi Washizu This is my paper

Pith reviewed 2026-05-23 02:25 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-ph
keywords first passage timestochastic diffusion coefficientergodic processesabsorbing boundarydiffusion-limited reactionsheterogeneous environmentsLevy-Smirnov distributionone-dimensional diffusion
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The pith

A stochastic diffusion coefficient produces more early arrivals at an absorbing boundary than diffusion at the matching ensemble average.

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

The paper establishes that particles undergoing diffusion with a positive stochastic diffusion coefficient in a one-dimensional semi-infinite domain reach an absorbing boundary with probability one. The stochastic coefficient yields higher efficiency for early arrivals than a constant coefficient set to the ensemble average of the stochastic one. Efficiency improves further when the supremum of the stochastic coefficient is larger, even at fixed ensemble average. For ergodic coefficients the mean first passage time diverges while the distribution converges to the Levy-Smirnov form because the time average converges to the ensemble average at a rate fixed by the fluctuation timescale. These findings bear on transport efficiency in heterogeneous media or fluctuating particles, such as those involved in diffusion-limited reactions.

Core claim

For diffusion in a one-dimensional semi-infinite domain with an absorbing boundary, particles will eventually reach the absorbing boundary with probability one. A stochastic diffusion coefficient provides higher transport efficiency in an early arrival of particles at the boundary than would be expected under diffusion whose diffusion coefficient is the ensemble average of the stochastic one. A stochastic diffusion coefficient with a larger supremum exhibits more efficient transport even if ensemble averages are the same. For ergodic diffusion coefficients the mean first passage time diverges, the enhancement of early-arrival efficiency diminishes over long times, and the first passage time

What carries the argument

The stochastic diffusion coefficient, whose time average converges to its ensemble average at a speed set by the fluctuation timescale.

If this is right

  • Particles reach the absorbing boundary with probability one for any positive stochastic diffusion coefficient.
  • Early-arrival efficiency exceeds that of the ensemble-average coefficient and increases with the coefficient supremum.
  • For ergodic coefficients the mean first passage time diverges.
  • The first passage time distribution converges to the Levy-Smirnov distribution in the long-time limit.
  • The early-arrival enhancement diminishes as time increases because of convergence between time and ensemble averages.

Where Pith is reading between the lines

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

  • In three-dimensional geometries outside a spherical absorber the same qualitative distinction between short-time efficiency and long-time divergence may hold, though the precise distribution will differ.
  • Experiments that control the fluctuation timescale of the diffusion coefficient could directly test how that timescale sets the crossover between enhanced early transport and the long-time regime.
  • Models of diffusion-limited reactions that assume constant coefficients will underestimate short-time encounter rates in fluctuating environments.
  • Efficient long-distance transport to distant targets may require non-ergodic or non-Markovian fluctuations in the diffusion coefficient.

Load-bearing premise

The time-averaged diffusion coefficient converges to the ensemble average at a speed determined by the fluctuation timescale.

What would settle it

Measuring the distribution of first passage times for particles driven by a known ergodic stochastic diffusion coefficient at progressively longer observation windows and checking whether the distribution approaches the Levy-Smirnov form.

Figures

Figures reproduced from arXiv: 2502.20705 by Go Uchida, Hiromi Miyoshi, Hitoshi Washizu.

Figure 1
Figure 1. Figure 1: FIG. 1. Time dependence of the variance of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. First passage time distributions (FPTDs) at different values of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The dependence of the excess cumulative probability [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

This study investigates the first passage time (FPT) properties of particles with a broad class of positive stochastic diffusion coefficients (DCs), representing diffusion in heterogeneous environments or of particles with conformational fluctuations. We demonstrate that for diffusion in a one-dimensional semi-infinite domain with an absorbing boundary, particles will eventually reach the absorbing boundary with probability one. We also show that a stochastic DC provides higher transport efficiency in an early arrival of particles at the boundary than would be expected under diffusion whose DC is the ensemble average of the stochastic DC. Furthermore, a stochastic DC with a larger supremum exhibits a more efficient transport even if ensemble averages are the same. For ergodic DCs, we show three more properties: the mean FPT diverges, the enhancement of early-arrival efficiency diminishes over long times, and the FPT distribution converges to a L\'evy-Smirnov distribution in the long-time limit. These properties are shown to arise from the convergence of the time-averaged DC to the ensemble average, with the convergence speed determined by the DC's fluctuation time scale. We finally discuss the similarities and differences of FPT properties between three-dimensional diffusion outside a spherical absorbing boundary and the one-dimensional diffusion. Our results indicate that fluctuations in DCs may need to be non-Markov and/or non-ergodic to allow efficient transport of particles to distant targets. Our results also suggest that fluctuations in a DC play an important role, for example, in diffusion-limited reactions triggered by single molecules in physics, chemistry, or biology.

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 / 1 minor

Summary. The manuscript examines first passage time (FPT) properties of diffusion in a one-dimensional semi-infinite domain with an absorbing boundary, where the diffusion coefficient is a positive stochastic process D(t) independent of position. It claims that particles reach the boundary with probability one; that stochastic D(t) yields higher early-arrival efficiency than diffusion with the constant ensemble-average diffusivity (with larger supremum of D(t) further improving efficiency for fixed average); and that for ergodic D(t) the mean FPT diverges, the early-time advantage vanishes at long times, and the FPT distribution converges to the Lévy-Smirnov form because the time-averaged diffusivity converges to the ensemble average at a rate set by the fluctuation timescale. Similarities and differences with three-dimensional diffusion outside a spherical absorber are discussed, leading to the suggestion that non-Markovian and/or non-ergodic fluctuations may be required for efficient transport to distant targets.

Significance. If the derivations hold, the work clarifies how fluctuations in diffusivity can produce short-time transport advantages that are invisible when only the ensemble average is considered, with direct relevance to diffusion-limited reactions triggered by single molecules. The explicit link between ergodicity, convergence of time averages, and recovery of the standard Lévy-Smirnov tail is a useful technical clarification. The observation that efficient long-range transport may require non-ergodic fluctuations supplies a concrete modeling criterion for heterogeneous media or conformational dynamics.

minor comments (1)
  1. The abstract and introduction would benefit from a brief statement of the precise stochastic representation used for the position process (e.g., the integral form X(t) = ∫ sqrt(2D(s)) dW(s)) to make the subsequent claims immediately traceable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their accurate summary of our results, and their positive recommendation to accept the paper. We are pleased that the referee recognizes the relevance of our findings to diffusion-limited reactions and the technical clarification regarding ergodicity and the Lévy-Smirnov distribution.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The central claims follow from standard properties of the SDE X(t) = ∫ sqrt(2D(s)) dW(s) with positive stochastic D(t): absorption probability 1 because ∫D(s)ds → ∞ a.s.; early-arrival advantage from convexity of short-time survival; long-time Lévy-Smirnov limit and diverging mean FPT from the definition of ergodicity (time-average convergence to ⟨D⟩). No equations reduce to fitted inputs renamed as predictions, no load-bearing self-citations, and no ansatz or uniqueness imported from prior author work. The abstract and skeptic analysis confirm the steps are independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard assumptions in stochastic diffusion modeling; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Diffusion occurs in a one-dimensional semi-infinite domain with an absorbing boundary
    The setup for the FPT analysis.
  • domain assumption The diffusion coefficient is a positive stochastic process
    Core modeling assumption for heterogeneous environments.

pith-pipeline@v0.9.0 · 5812 in / 1379 out tokens · 64204 ms · 2026-05-23T02:25:06.265312+00:00 · methodology

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Reference graph

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