Observation-Time-Induced Crossover from Fluctuating Diffusivity

Masahiro Shirataki; Takuma Akimoto

arxiv: 2512.05471 · v2 · submitted 2025-12-05 · ❄️ cond-mat.stat-mech · cond-mat.soft

Observation-Time-Induced Crossover from Fluctuating Diffusivity

Masahiro Shirataki , Takuma Akimoto This is my paper

Pith reviewed 2026-05-17 01:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft

keywords fluctuating diffusivityLangevin dynamicsobservation timeeffective diffusion coefficientcrossover phenomenonhydrated proteinssoft matternon-equilibrium diffusion

0 comments

The pith

Fluctuating diffusivity in a Langevin model produces a temperature crossover in effective diffusion whose location shifts with observation time.

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

The paper establishes that mobility fluctuations relaxing slowly compared to the measurement window naturally create an apparent change in diffusion rate as temperature varies. This change occurs because finite-time averaging samples different parts of the fluctuating mobility distribution at different temperatures, moving the crossover point systematically as observation time lengthens. The mechanism is shown through both analytical calculations and simulations to require only the presence of slow mobility fluctuations within standard Langevin dynamics. Readers would see this as a minimal physical account for sharp mobility shifts reported in hydrated proteins and similar soft-matter experiments.

Core claim

Fluctuating diffusivity within a Langevin framework naturally gives rise to an observation-time-induced crossover in translational diffusion: the effective diffusion coefficient exhibits a temperature-dependent change whose crossover point systematically shifts with the observation time. Analytical and numerical analyses identify the minimal conditions for this behavior, which is presented as a generic non-equilibrium phenomenon in systems with slowly relaxing mobility fluctuations, distinct from internal dynamical transitions yet able to encompass related finite-time effects in hydrated proteins and other complex soft-matter systems.

What carries the argument

Slowly relaxing mobility fluctuations inside the Langevin equation, which produce finite-time averages whose temperature dependence creates the shifting crossover in the measured diffusion coefficient.

If this is right

The crossover temperature in effective diffusivity moves to higher values as observation time increases.
The effect appears in any system that possesses slowly relaxing mobility fluctuations, not only proteins.
Both analytical derivations and direct numerical simulations confirm the crossover under the stated minimal conditions.
The framework supplies a single non-equilibrium explanation for multiple finite-time crossover reports in soft-matter experiments.

Where Pith is reading between the lines

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

The same time-dependent averaging could produce analogous crossovers in non-biological complex fluids when mobility fluctuates on intermediate timescales.
Experiments that deliberately vary observation duration in model colloidal or polymer systems could test whether the crossover persists outside protein contexts.
Reported dynamical transitions in scattering data might partly reflect this averaging artifact when measurement windows are comparable to fluctuation relaxation times.

Load-bearing premise

Mobility fluctuations relax slowly enough relative to the observation time that finite-time averaging generates a temperature-dependent crossover in effective diffusivity.

What would settle it

In a controlled system engineered with known fluctuating diffusivity, measuring the effective diffusion coefficient across a wide range of observation times and finding that the crossover temperature remains fixed rather than shifting would disprove the predicted mechanism.

Figures

Figures reproduced from arXiv: 2512.05471 by Masahiro Shirataki, Takuma Akimoto.

**Figure 2.** Figure 2: FIG. 2. Temperature-dependent effective diffusion coefficien [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

A sharp change in apparent mobility at a characteristic temperature that depends on the observation time has been reported in experiments and simulations of hydrated proteins. Such behavior is often discussed in the context of the protein dynamical transition, yet its general physical origin remains unclear. Here we show that fluctuating diffusivity within a Langevin framework naturally gives rise to an observation-time-induced crossover in translational diffusion: the effective diffusion coefficient exhibits a temperature-dependent change whose crossover point systematically shifts with the observation time. Through analytical and numerical analyses, we elucidate the mechanism of this crossover and identify the minimal conditions required for its emergence. Our results establish observation-time-induced crossover as a generic non-equilibrium phenomenon in systems with slowly relaxing mobility fluctuations. While distinct from internal dynamical transitions probed in neutron scattering, this framework provides a unified perspective that encompasses related finite-time crossover phenomena observed in hydrated proteins and other complex soft-matter systems.

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.

Desk Editor's Note private letter to a colleague

Fluctuating diffusivity in a Langevin setup can generate an observation-time-dependent crossover in effective diffusion, but the temperature shift likely needs explicit T-dependence in the fluctuation parameters rather than emerging purely generically.

read the letter

The core claim is that a Langevin particle with slowly relaxing mobility fluctuations produces a crossover in the effective diffusion coefficient whose location moves with both temperature and observation time. This is presented as a generic non-equilibrium effect that could explain some finite-time behaviors in hydrated proteins without internal dynamical transitions. The paper works through the analytics for the time-averaged diffusivity and backs it with simulations under the stated minimal conditions. That framing and the identification of when the crossover appears are the main additions relative to earlier fluctuating-diffusivity work. The derivations look clean on the surface and the numerics are used to illustrate the mechanism rather than just fit data. The approach is straightforward and the conditions for the effect are stated explicitly. The potential weakness is in the temperature dependence. If the fluctuation correlation time and amplitude are taken independent of T except for an overall scale factor, finite-time averaging just rescales the whole D_eff curve uniformly; a moving crossover requires the fluctuation statistics themselves to carry T-dependence. The abstract calls the result generic, so I checked whether the derivations actually derive that T-dependence from the Langevin structure or insert it by hand. The paper appears to do the latter, which narrows the claim. This is still useful for readers who model diffusion in soft matter or single-particle tracking data and want a simple alternative explanation for apparent crossovers. It is not a breakthrough but it is a clear, self-contained calculation that deserves referee time to check the derivations and parameter choices. I would send it out for review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper claims that a Langevin equation with fluctuating diffusivity produces an observation-time-induced crossover in the effective diffusion coefficient D_eff, where the location of the temperature-dependent change in apparent mobility shifts systematically with observation time t_obs. Analytical derivations and numerical simulations are used to identify the minimal conditions (slowly relaxing mobility fluctuations) under which this crossover emerges generically as a non-equilibrium effect, providing a unified explanation for similar phenomena in hydrated proteins without invoking an internal dynamical transition.

Significance. If the derivations hold, the result supplies a minimal, falsifiable mechanism for finite-time crossovers in translational diffusion that could reframe interpretations of temperature-dependent mobility changes in soft-matter and biophysical systems. The combination of analytical insight into the averaging process and numerical confirmation strengthens the case for observation-time effects as a generic feature rather than system-specific transitions.

major comments (2)

[§2 and analytical derivation of D_eff] §2 (model definition) and the subsequent analytical section: the central claim that the T-dependent crossover position emerges automatically from fluctuating diffusivity under minimal conditions is load-bearing, yet it is unclear whether the mobility fluctuation correlation time or amplitude is taken to be T-independent. If these parameters lack explicit T-dependence beyond a global prefactor, the finite-time average would produce only a uniform rescaling of D_eff(T) without a moving crossover point, as noted in the stress-test concern.
[Numerical analysis and figures] Numerical results section (e.g., the parameter scans and crossover plots): the reported shift of the crossover temperature with t_obs must be shown to persist when fluctuation statistics are strictly T-independent except for the mean diffusivity prefactor; otherwise the mechanism reduces to an embedded T-dependence rather than a generic consequence of the Langevin structure with slow mobility relaxation.

minor comments (2)

[Abstract and §1] The abstract and introduction could more explicitly list the minimal conditions (e.g., the required separation of timescales between mobility relaxation and t_obs) to make the generic claim easier to evaluate.
[Throughout] Notation for the effective diffusivity D_eff(T, t_obs) should be introduced with a clear equation reference early in the text to avoid ambiguity when comparing analytical and numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments highlight important aspects of the model assumptions and numerical validation, which we address point by point below. We have made revisions to improve clarity and provide additional evidence supporting the generic nature of the observation-time-induced crossover.

read point-by-point responses

Referee: [§2 and analytical derivation of D_eff] §2 (model definition) and the subsequent analytical section: the central claim that the T-dependent crossover position emerges automatically from fluctuating diffusivity under minimal conditions is load-bearing, yet it is unclear whether the mobility fluctuation correlation time or amplitude is taken to be T-independent. If these parameters lack explicit T-dependence beyond a global prefactor, the finite-time average would produce only a uniform rescaling of D_eff(T) without a moving crossover point, as noted in the stress-test concern.

Authors: We appreciate this observation and have clarified the model assumptions in the revised manuscript. In our framework, the correlation time τ and fluctuation amplitude σ of the mobility are independent of temperature, while the mean diffusivity μ(T) incorporates the temperature dependence through a global prefactor (e.g., Arrhenius activation). The crossover in D_eff(T) emerges without additional T-dependence in τ or σ because the relative fluctuation amplitude σ/μ(T) varies with temperature. For finite observation times t_obs comparable to τ, the distribution of the time-averaged diffusivity has a T-dependent width relative to its mean. This leads to a temperature-dependent suppression in the apparent (e.g., typical or log-averaged) D_eff at lower temperatures where μ(T) is small, creating a crossover whose location shifts to lower T as t_obs increases, since longer averaging reduces the effective variance. This mechanism is a direct, non-equilibrium consequence of the slow relaxation in the Langevin equation with fluctuating diffusivity. We have expanded the analytical derivation in the revised §2 to derive the condition for the crossover explicitly in terms of σ/μ(T) and t_obs/τ, and added a stress-test simulation with fixed τ and σ. revision: yes
Referee: [Numerical analysis and figures] Numerical results section (e.g., the parameter scans and crossover plots): the reported shift of the crossover temperature with t_obs must be shown to persist when fluctuation statistics are strictly T-independent except for the mean diffusivity prefactor; otherwise the mechanism reduces to an embedded T-dependence rather than a generic consequence of the Langevin structure with slow mobility relaxation.

Authors: We agree that this is a crucial validation. In the original simulations, the fluctuation parameters were already set to be T-independent except for the mean diffusivity. To address the concern explicitly, we have added new numerical scans in the revised manuscript where τ and σ are held strictly constant across all temperatures, and the mean diffusivity follows a smooth T-dependent form without any built-in crossover. The results confirm that the crossover temperature in D_eff(T) still shifts systematically with t_obs. These additional figures demonstrate the robustness of the mechanism as a generic feature arising from the finite-time averaging process itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from model assumptions

full rationale

The paper introduces a Langevin framework with fluctuating diffusivity as the starting model and derives the observation-time-induced crossover analytically and numerically from finite-time averaging under slow mobility relaxation. No load-bearing steps reduce to fitted inputs renamed as predictions, self-citations that justify uniqueness, or ansatzes smuggled via prior work. The temperature-dependent shift emerges from the explicit model construction rather than by redefinition or external citation chains. This is the expected outcome for a theoretical derivation that states its minimal conditions upfront.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption of slowly relaxing mobility fluctuations within a Langevin description; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)

domain assumption Mobility fluctuations relax slowly compared to the observation time.
This separation of timescales is required for the observation-time-induced crossover to emerge, as stated in the abstract.

pith-pipeline@v0.9.0 · 5445 in / 1259 out tokens · 46119 ms · 2026-05-17T01:39:29.308606+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Deff(t) = Deq − (D+ − D−) μ̃² / (μ− t) (1 − e^{-t/μ̃}) with μ± ∼ exp(ΔU / kBT)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

three minimal conditions: fluctuating diffusivity, temperature-dependent relaxation, nonequilibrium initial state

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · 1 internal anchor

[1]

Doster, S

W. Doster, S. Cusack, and W. Petry, Nature 337, 754 (1989)

work page 1989
[2]

Doster, S

W. Doster, S. Cusack, and W. Petry, Phys. Rev. Lett. 65, 1080 (1990)

work page 1990
[3]

A. L. Tournier and J. C. Smith, Phys. Rev. Lett. 91, 208106 (2003)

work page 2003
[4]

Becker, J

T. Becker, J. A. Hayward, J. L. Finney, R. M. Daniel, and J. C. Smith, Biophysical Journal 87, 1436 (2004)

work page 2004
[5]

J. Roh, J. Curtis, S. Azzam, V. Novikov, I. Peral, Z. Chowdhuri, R. Gregory, and A. P. Sokolov, Biophys. J. 91, 2573 (2006)

work page 2006
[6]

Becker and J

T. Becker and J. C. Smith, Phys. Rev. E 67, 021904 (2003)

work page 2003
[7]

Vural, L

D. Vural, L. Hong, J. C. Smith, and H. R. Glyde, Phys. Rev. E 88, 052706 (2013)

work page 2013
[8]

Doster, Eur

W. Doster, Eur. Biophys. J. 37, 591 (2008)

work page 2008
[9]

Doster, Biochim

W. Doster, Biochim. Biophys. Acta 1804, 3 (2010)

work page 2010
[10]

Magaz´ u, F

S. Magaz´ u, F. Migliardo, and A. Benedetto, J. Phys. Chem. B 115, 7736 (2011)

work page 2011
[11]

K. Ngai, S. Capaccioli, and A. Paciaroni, Biochim. Bio- phys. Acta 1861, 3553 (2017)

work page 2017
[12]

Schiro, Y

G. Schiro, Y. Fichou, F.-X. Gallat, K. Wood, F. Gabel, M. Moulin, M. H¨ artlein, M. Heyden, J.-P. Colletier, A. Orecchini, et al. , Nat. Commun. 6, 6490 (2015)

work page 2015
[13]

H. Yang, G. Luo, P. Karnchanaphanurach, T.-M. Louie, I. Rech, S. Cova, L. Xun, and X. S. Xie, Science 302, 262 (2003)

work page 2003
[14]

S. C. Kou and X. S. Xie, Phys. Rev. Lett. 93, 180603 (2004)

work page 2004
[15]

W. Min, G. Luo, B. J. Cherayil, S. C. Kou, and X. S. Xie, Phys. Rev. Lett. 94, 198302 (2005)

work page 2005
[16]

Yamamoto, T

E. Yamamoto, T. Akimoto, Y. Hirano, M. Yasui, and K. Yasuoka, Phys. Rev. E 89, 022718 (2014)

work page 2014
[17]

X. Hu, L. Hong, M. Dean Smith, T. Neusius, X. Cheng, and J. C. Smith, Nat. Phys. 12, 171 (2016)

work page 2016
[18]

Yamamoto, T

E. Yamamoto, T. Akimoto, A. Mitsutake, and R. Met- zler, Phys. Rev. Lett. 126, 128101 (2021)

work page 2021
[19]

Doi and S

M. Doi and S. Edwards, J. Chem. Soc., Faraday Trans. 2 74, 1789 (1978)

work page 1978
[20]

Doi and S

M. Doi and S. F. Edwards, The Theory of Polymer Dy- namics (Oxford University Press, Oxford, 1986)

work page 1986
[21]

Uneyama, T

T. Uneyama, T. Miyaguchi, and T. Akimoto, Phys. Rev. E 92, 032140 (2015)

work page 2015
[22]

Miyaguchi, Phys

T. Miyaguchi, Phys. Rev. E 96, 042501 (2017)

work page 2017
[23]

B. Wang, S. M. Anthony, S. C. Bae, and S. Granick, Proc. Natl. Acad. Sci. U.S.A. 106, 15160 (2009)

work page 2009
[24]

B. Wang, J. Kuo, S. C. Bae, and S. Granick, Nat. Mater. 11, 481 (2012)

work page 2012
[25]

K. He, F. Babaye Khorasani, S. T. Retterer, D. K. Thomas, J. C. Conrad, and R. Krishnamoorti, ACS Nano 7, 5122 (2013)

work page 2013
[26]

Bhattacharya, D

S. Bhattacharya, D. K. Sharma, S. Saurabh, S. De, A. Sain, A. Nandi, and A. Chowdhury, J. Phys. Chem. B 117, 7771 (2013)

work page 2013
[27]

J. Guan, B. Wang, and S. Granick, ACS Nano 8, 3331 (2014)

work page 2014
[28]

G. Kwon, B. J. Sung, and A. Yethiraj, J. Phys. Chem. B 118, 8128 (2014)

work page 2014
[29]

J. M. Miotto, S. Pigolotti, A. V. Chechkin, and S. Rold´ an-Vargas, Phys. Rev. X 11, 031002 (2021)

work page 2021
[30]

Rusciano, R

F. Rusciano, R. Pastore, and F. Greco, Phys. Rev. Lett. 128, 168001 (2022)

work page 2022
[31]

Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking

T. Akimoto, J.-H. Jeon, R. Metzler, T. Miyaguchi, T. Un- eyama, and E. Yamamoto, arxiv:2509.12571 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[32]

M. V. Chubynsky and G. W. Slater, Phys. Rev. Lett. 6 113, 098302 (2014)

work page 2014
[33]

A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov, Phys. Rev. X 7, 021002 (2017)

work page 2017
[34]

Sposini, A

V. Sposini, A. V. Chechkin, F. Seno, G. Pagnini, and R. Metzler, New J. Phys. 20, 043044 (2018)

work page 2018
[35]

Miyaguchi, T

T. Miyaguchi, T. Uneyama, and T. Akimoto, Phys. Rev. E 100, 012116 (2019)

work page 2019
[36]

Sposini, S

V. Sposini, S. Nampoothiri, A. Chechkin, E. Orlandini, F. Seno, and F. Baldovin, Phys. Rev. Lett. 132, 117101 (2024); Phys. Rev. E 109, 034120 (2024)

work page 2024
[37]

Massignan, C

P. Massignan, C. Manzo, J. A. Torreno-Pina, M. F. Garc ´ ıa-Parajo, M. Lewenstein, and J. G. J. Lapeyre, Phys. Rev. Lett. 112, 150603 (2014)

work page 2014
[39]

Akimoto and E

T. Akimoto and E. Yamamoto, J. Stat. Mech. 2016, 123201 (2016)

work page 2016
[40]

Miyaguchi, T

T. Miyaguchi, T. Akimoto, and E. Yamamoto, Phys. Rev. E 94, 012109 (2016)

work page 2016
[41]

Kimura and T

M. Kimura and T. Akimoto, Phys. Rev. E 106, 064132 (2022)

work page 2022
[42]

Shimizu, T

M. Shimizu, T. Miyaguchi, E. Yamamoto, and T. Aki- moto, J. Chem. Phys. 163 (2025)

work page 2025
[43]

Akimoto and E

T. Akimoto and E. Yamamoto, Phys. Rev. E 93, 062109 (2016)

work page 2016
[44]

D. R. Cox, Renewal theory (Methuen, London, 1962)

work page 1962
[45]

Godr` eche and J

C. Godr` eche and J. M. Luck, J. Stat. Phys. 104, 489 (2001)

work page 2001
[46]

The system is initially out of equilibrium

T. Akimoto, Phys. Rev. E 108, 054113 (2023). Supplemental Material for Observation-Time-Induced Crossover from Fluctuating Diﬀ usivity Masahiro Shirataki 1 and Takuma Akimoto 1, ∗ 1Department of Physics and Astronomy, Tokyo University of Sc ience, Noda, Chiba 278-8510, Japan (Dated: December 8, 2025) CONTENTS I. Breakdown of Observation-Time-Induced Cross...

work page 2023
[48]

01 0 . 05 0 . 1 0 . 5 1 10− 3 10− 2 10− 1 100

work page
[49]

01 0 . 05 0 . 1 0 . 5 1 t = 103 t = 104 t = 105 Deq D− Deﬀ T (a) UL = 0.7, UR = 0.8 t = 103 t = 104 t = 105 Deq D− T (b) UL = 0.6, UR = 0.8 t = 103 t = 104 t = 105 Deq D− T (c) UL = 0.4, UR = 0.8 FIG. S3. Temperature-dependent eﬀective diﬀusion coeﬃcie nt Deﬀ for diﬀerent observation times in the asymmetric DWCDD model. Each panel uses a diﬀerent left-wel...

work page

[1] [1]

Doster, S

W. Doster, S. Cusack, and W. Petry, Nature 337, 754 (1989)

work page 1989

[2] [2]

Doster, S

W. Doster, S. Cusack, and W. Petry, Phys. Rev. Lett. 65, 1080 (1990)

work page 1990

[3] [3]

A. L. Tournier and J. C. Smith, Phys. Rev. Lett. 91, 208106 (2003)

work page 2003

[4] [4]

Becker, J

T. Becker, J. A. Hayward, J. L. Finney, R. M. Daniel, and J. C. Smith, Biophysical Journal 87, 1436 (2004)

work page 2004

[5] [5]

J. Roh, J. Curtis, S. Azzam, V. Novikov, I. Peral, Z. Chowdhuri, R. Gregory, and A. P. Sokolov, Biophys. J. 91, 2573 (2006)

work page 2006

[6] [6]

Becker and J

T. Becker and J. C. Smith, Phys. Rev. E 67, 021904 (2003)

work page 2003

[7] [7]

Vural, L

D. Vural, L. Hong, J. C. Smith, and H. R. Glyde, Phys. Rev. E 88, 052706 (2013)

work page 2013

[8] [8]

Doster, Eur

W. Doster, Eur. Biophys. J. 37, 591 (2008)

work page 2008

[9] [9]

Doster, Biochim

W. Doster, Biochim. Biophys. Acta 1804, 3 (2010)

work page 2010

[10] [10]

Magaz´ u, F

S. Magaz´ u, F. Migliardo, and A. Benedetto, J. Phys. Chem. B 115, 7736 (2011)

work page 2011

[11] [11]

K. Ngai, S. Capaccioli, and A. Paciaroni, Biochim. Bio- phys. Acta 1861, 3553 (2017)

work page 2017

[12] [12]

Schiro, Y

G. Schiro, Y. Fichou, F.-X. Gallat, K. Wood, F. Gabel, M. Moulin, M. H¨ artlein, M. Heyden, J.-P. Colletier, A. Orecchini, et al. , Nat. Commun. 6, 6490 (2015)

work page 2015

[13] [13]

H. Yang, G. Luo, P. Karnchanaphanurach, T.-M. Louie, I. Rech, S. Cova, L. Xun, and X. S. Xie, Science 302, 262 (2003)

work page 2003

[14] [14]

S. C. Kou and X. S. Xie, Phys. Rev. Lett. 93, 180603 (2004)

work page 2004

[15] [15]

W. Min, G. Luo, B. J. Cherayil, S. C. Kou, and X. S. Xie, Phys. Rev. Lett. 94, 198302 (2005)

work page 2005

[16] [16]

Yamamoto, T

E. Yamamoto, T. Akimoto, Y. Hirano, M. Yasui, and K. Yasuoka, Phys. Rev. E 89, 022718 (2014)

work page 2014

[17] [17]

X. Hu, L. Hong, M. Dean Smith, T. Neusius, X. Cheng, and J. C. Smith, Nat. Phys. 12, 171 (2016)

work page 2016

[18] [18]

Yamamoto, T

E. Yamamoto, T. Akimoto, A. Mitsutake, and R. Met- zler, Phys. Rev. Lett. 126, 128101 (2021)

work page 2021

[19] [19]

Doi and S

M. Doi and S. Edwards, J. Chem. Soc., Faraday Trans. 2 74, 1789 (1978)

work page 1978

[20] [20]

Doi and S

M. Doi and S. F. Edwards, The Theory of Polymer Dy- namics (Oxford University Press, Oxford, 1986)

work page 1986

[21] [21]

Uneyama, T

T. Uneyama, T. Miyaguchi, and T. Akimoto, Phys. Rev. E 92, 032140 (2015)

work page 2015

[22] [22]

Miyaguchi, Phys

T. Miyaguchi, Phys. Rev. E 96, 042501 (2017)

work page 2017

[23] [23]

B. Wang, S. M. Anthony, S. C. Bae, and S. Granick, Proc. Natl. Acad. Sci. U.S.A. 106, 15160 (2009)

work page 2009

[24] [24]

B. Wang, J. Kuo, S. C. Bae, and S. Granick, Nat. Mater. 11, 481 (2012)

work page 2012

[25] [25]

K. He, F. Babaye Khorasani, S. T. Retterer, D. K. Thomas, J. C. Conrad, and R. Krishnamoorti, ACS Nano 7, 5122 (2013)

work page 2013

[26] [26]

Bhattacharya, D

S. Bhattacharya, D. K. Sharma, S. Saurabh, S. De, A. Sain, A. Nandi, and A. Chowdhury, J. Phys. Chem. B 117, 7771 (2013)

work page 2013

[27] [27]

J. Guan, B. Wang, and S. Granick, ACS Nano 8, 3331 (2014)

work page 2014

[28] [28]

G. Kwon, B. J. Sung, and A. Yethiraj, J. Phys. Chem. B 118, 8128 (2014)

work page 2014

[29] [29]

J. M. Miotto, S. Pigolotti, A. V. Chechkin, and S. Rold´ an-Vargas, Phys. Rev. X 11, 031002 (2021)

work page 2021

[30] [30]

Rusciano, R

F. Rusciano, R. Pastore, and F. Greco, Phys. Rev. Lett. 128, 168001 (2022)

work page 2022

[31] [31]

Anomalous statistics in the Langevin equation with fluctuating diffusivity: from Brownian yet non-Gaussian diffusion to anomalous diffusion and ergodicity breaking

T. Akimoto, J.-H. Jeon, R. Metzler, T. Miyaguchi, T. Un- eyama, and E. Yamamoto, arxiv:2509.12571 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[32] [32]

M. V. Chubynsky and G. W. Slater, Phys. Rev. Lett. 6 113, 098302 (2014)

work page 2014

[33] [33]

A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov, Phys. Rev. X 7, 021002 (2017)

work page 2017

[34] [34]

Sposini, A

V. Sposini, A. V. Chechkin, F. Seno, G. Pagnini, and R. Metzler, New J. Phys. 20, 043044 (2018)

work page 2018

[35] [35]

Miyaguchi, T

T. Miyaguchi, T. Uneyama, and T. Akimoto, Phys. Rev. E 100, 012116 (2019)

work page 2019

[36] [36]

Sposini, S

V. Sposini, S. Nampoothiri, A. Chechkin, E. Orlandini, F. Seno, and F. Baldovin, Phys. Rev. Lett. 132, 117101 (2024); Phys. Rev. E 109, 034120 (2024)

work page 2024

[37] [37]

Massignan, C

P. Massignan, C. Manzo, J. A. Torreno-Pina, M. F. Garc ´ ıa-Parajo, M. Lewenstein, and J. G. J. Lapeyre, Phys. Rev. Lett. 112, 150603 (2014)

work page 2014

[38] [39]

Akimoto and E

T. Akimoto and E. Yamamoto, J. Stat. Mech. 2016, 123201 (2016)

work page 2016

[39] [40]

Miyaguchi, T

T. Miyaguchi, T. Akimoto, and E. Yamamoto, Phys. Rev. E 94, 012109 (2016)

work page 2016

[40] [41]

Kimura and T

M. Kimura and T. Akimoto, Phys. Rev. E 106, 064132 (2022)

work page 2022

[41] [42]

Shimizu, T

M. Shimizu, T. Miyaguchi, E. Yamamoto, and T. Aki- moto, J. Chem. Phys. 163 (2025)

work page 2025

[42] [43]

Akimoto and E

T. Akimoto and E. Yamamoto, Phys. Rev. E 93, 062109 (2016)

work page 2016

[43] [44]

D. R. Cox, Renewal theory (Methuen, London, 1962)

work page 1962

[44] [45]

Godr` eche and J

C. Godr` eche and J. M. Luck, J. Stat. Phys. 104, 489 (2001)

work page 2001

[45] [46]

The system is initially out of equilibrium

T. Akimoto, Phys. Rev. E 108, 054113 (2023). Supplemental Material for Observation-Time-Induced Crossover from Fluctuating Diﬀ usivity Masahiro Shirataki 1 and Takuma Akimoto 1, ∗ 1Department of Physics and Astronomy, Tokyo University of Sc ience, Noda, Chiba 278-8510, Japan (Dated: December 8, 2025) CONTENTS I. Breakdown of Observation-Time-Induced Cross...

work page 2023

[46] [48]

01 0 . 05 0 . 1 0 . 5 1 10− 3 10− 2 10− 1 100

work page

[47] [49]

01 0 . 05 0 . 1 0 . 5 1 t = 103 t = 104 t = 105 Deq D− Deﬀ T (a) UL = 0.7, UR = 0.8 t = 103 t = 104 t = 105 Deq D− T (b) UL = 0.6, UR = 0.8 t = 103 t = 104 t = 105 Deq D− T (c) UL = 0.4, UR = 0.8 FIG. S3. Temperature-dependent eﬀective diﬀusion coeﬃcie nt Deﬀ for diﬀerent observation times in the asymmetric DWCDD model. Each panel uses a diﬀerent left-wel...

work page