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arxiv: 1907.06819 · v1 · pith:KTHWV34Hnew · submitted 2019-07-16 · ❄️ cond-mat.stat-mech

Brownian motion with alternately fluctuating diffusivity: Stretched-exponential and power-law relaxation

Tomoshige Miyaguchi , Takashi Uneyama , Takuma Akimoto This is my paper

Pith reviewed 2026-05-24 20:59 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Brownian motionfluctuating diffusivityalternating renewal processstretched-exponential relaxationpower-law relaxationsojourn time distributionrelaxation function
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The pith

Brownian motion with alternating diffusivity exhibits stretched-exponential relaxation at short times and power-law decay at long times.

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

The authors study a model of Brownian motion in which the diffusivity switches between fast and slow states, with the times spent in each state drawn from exponential or power-law distributions. They use the theory of alternating renewal processes to derive the behavior of a relaxation function given by the time integral of the diffusivity. This function is shown to correspond to the position correlation function in a harmonic potential or the self-intermediate scattering function in free space. The theory predicts that short-time relaxation is either exponential or stretched-exponential, depending on the power-law index of the sojourn times, while long-time relaxation follows a power law with an exponential cutoff. The forms also depend on whether the initial state is an equilibrium or non-equilibrium ensemble.

Core claim

For Brownian motion with diffusivity alternating between fast and slow states with sojourn-time distributions that are exponential or power-law, the relaxation function shows exponential or stretched-exponential decay at short times according to the power law index, and power-law decay with exponential cutoff at long times, with dependence on the initial ensemble.

What carries the argument

The relaxation function expressed as the integral of diffusivity over time, analyzed using alternating renewal process theory.

If this is right

  • If the sojourn-time distribution is a power law with index between 0 and 1, the short-time relaxation is stretched exponential rather than simple exponential.
  • The long-time behavior is a power law decay with exponential cutoff regardless of the sojourn distribution details.
  • Equilibrium and non-equilibrium initial ensembles lead to different prefactors in the relaxation but similar functional forms.
  • The predicted relaxation applies both to particles in harmonic traps via position correlations and to free particles via scattering functions.

Where Pith is reading between the lines

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

  • The model may describe relaxation in systems like molecular motors or particles in heterogeneous media that switch mobility states.
  • Single-molecule experiments measuring mean squared displacement over wide time ranges could test the crossover from stretched-exponential to power-law regimes.
  • Generalizing to multiple diffusivity states could produce more complex relaxation spectra with multiple stretched exponents.

Load-bearing premise

The relaxation function is expressed as an integral of the diffusivity over time and can be directly related to a position correlation function or self-intermediate scattering function.

What would settle it

A numerical simulation with power-law sojourn times of index greater than one that instead shows stretched-exponential short-time relaxation, rather than simple exponential decay, would falsify the predicted dependence on the power-law index.

Figures

Figures reproduced from arXiv: 1907.06819 by Takashi Uneyama, Takuma Akimoto, Tomoshige Miyaguchi.

Figure 1
Figure 1. Figure 1: FIG. 1. (Color online) Schematic illustration of the switch [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Relaxation function Φ( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Relaxation function Φ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) Relaxation function Φ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online)Schematic illustration of the forwar [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

We investigate Brownian motion with diffusivity alternately fluctuating between fast and slow states. We assume that sojourn-time distributions of these two states are given by exponential or power-law distributions. We develop a theory of alternating renewal processes to study a relaxation function which is expressed with an integral of the diffusivity over time. This relaxation function can be related to a position correlation function if the particle is in a harmonic potential, and to the self-intermediate scattering function if the potential force is absent. It is theoretically shown that, at short times, the exponential relaxation or the stretched-exponential relaxation are observed depending on the power law index of the sojourn-time distributions. In contrast, at long times, a power law decay with an exponential cutoff is observed. The dependencies on the initial ensembles (i.e., equilibrium or non-equilibrium initial ensembles) are also elucidated. These theoretical results are consistent with numerical simulations.

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

1 major / 0 minor

Summary. The manuscript studies Brownian motion with diffusivity alternating between fast and slow states whose sojourn times follow exponential or power-law distributions. Using alternating renewal process theory, the authors analyze a relaxation function defined via the time integral of the fluctuating diffusivity. This function is asserted to relate to the position correlation function (harmonic trap) and the self-intermediate scattering function (free particle). They derive that short-time relaxation is exponential or stretched-exponential depending on the power-law index of the sojourn distributions, while long-time behavior is a power-law decay with exponential cutoff; initial-ensemble dependence is also obtained. Analytic predictions are stated to agree with numerical simulations.

Significance. If the derivations are exact, the work supplies closed-form short- and long-time asymptotics for relaxation under alternating diffusivity, together with explicit dependence on the sojourn-time index and initial conditions. This is a concrete advance for renewal-based models of fluctuating diffusivity. The combination of analytic renewal theory with direct simulation checks is a positive feature.

major comments (1)
  1. Abstract: the relaxation function is defined via an integral of diffusivity and asserted to relate directly to the free-particle SISF. The exact SISF is the path average of exp(−(k²/2)∫₀ᵗ D(s)ds), which is the Laplace transform of the distribution of the integral, not in general a simple function of its average. If the alternating-renewal analysis computes only moments or the distribution of the integral and then substitutes into an exponential of the average, the claimed short-time stretched-exponential and long-time power-law forms do not automatically carry over to the SISF (or to the harmonic-trap correlation function). This assumption is load-bearing for all physical interpretations and for the dependence on the sojourn-time index; the manuscript must state the precise definition of the relaxation function and demonstrate whether the mapping is exact or approximate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract: the relaxation function is defined via an integral of diffusivity and asserted to relate directly to the free-particle SISF. The exact SISF is the path average of exp(−(k²/2)∫₀ᵗ D(s)ds), which is the Laplace transform of the distribution of the integral, not in general a simple function of its average. If the alternating-renewal analysis computes only moments or the distribution of the integral and then substitutes into an exponential of the average, the claimed short-time stretched-exponential and long-time power-law forms do not automatically carry over to the SISF (or to the harmonic-trap correlation function). This assumption is load-bearing for all physical interpretations and for the dependence on the sojourn-time index; the manuscript must state the precise definition of the relaxation function and demonstrate whether the mapping is exact or approximate.

    Authors: We agree that the exact SISF requires the full path average of the exponential, i.e., the Laplace transform of the distribution of the integrated diffusivity. Our manuscript defines the relaxation function via the time integral of D(t) and derives its statistical properties (primarily the average) using alternating renewal theory; the abstract states only that this function 'can be related to' the SISF and the harmonic-trap correlation. The reported short- and long-time asymptotics therefore apply directly to the relaxation function as defined. In the revised manuscript we will (i) give an explicit mathematical definition of the relaxation function, (ii) state clearly that the connection to the SISF is through the mean integrated diffusivity (corresponding to a first-cumulant approximation), and (iii) note the conditions under which the derived functional forms carry over to the full SISF. This clarification addresses the load-bearing assumption without altering the technical results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation via alternating renewal theory is self-contained

full rationale

The paper defines a relaxation function as the time integral of the fluctuating diffusivity and applies alternating renewal process analysis to obtain its short-time (exponential or stretched-exponential) and long-time (power-law with exponential cutoff) asymptotics as functions of the sojourn-time power-law index and initial ensemble. These steps operate directly on the defined integral without any reduction to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The stated relation of the integral to the SISF or position correlation function is presented as an external mapping rather than an internal re-expression, and the results are cross-checked against independent numerical simulations, confirming the derivation chain stands on its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on the domain assumption that diffusivity takes only two discrete values whose sojourn times follow exponential or power-law distributions; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Sojourn-time distributions of the fast and slow states are exponential or power-law
    Stated as the modeling assumption that enables the alternating renewal analysis.
  • domain assumption Relaxation function equals the time integral of instantaneous diffusivity
    Invoked to connect the switching process to correlation and scattering functions.

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Reference graph

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    Equilibrium ensemble For the equilibrium ensemble, the process starts at t = −∞ , and thus the system is in equilibrium at t = 0. The diffusivity D(0) is therefore at + or − state with equilibrium fractions peq + and peq − . These fractions peq ± are simply given by peq ± = µ± µ+ + µ− , (17) where µ± are mean sojourn times for ρ± (τ ). From this expression...

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