Pith. sign in

REVIEW 2 major objections 3 minor 43 references

Exact height distribution in one-dimensional Edwards-Wilkinson interface with diffusing diffusivity

T0 review · 2 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For a one-dimensional Edwards–Wilkinson interface whose diffusivity is the square of a Brownian motion, the paper obtains the exact single-point height distribution, whose scaled form is symmetric and decays as a non-Gaussian exponential.

desk verdict The exact z=2 height distribution is a genuine new result and the derivation is clean; the advertised z>1 generalization is asserted rather than proven. read the letter →

arxiv 2502.01153 v2 pith:M2EQW6S2 submitted 2025-02-03 cond-mat.stat-mech

David S. Dean , Satya N. Majumdar , Sanjib Sabhapandit This is my paper · ORCID
classification cond-mat.stat-mech MSC 60J6560H1082C3182C41
keywords Edwards-WilkinsonmodeldiffusingdiffusivityBrownianfunctionalFeynman-KacformulaheightdistributionexponentialtailRousechaindynamicalexponent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An interface described by the Edwards–Wilkinson equation normally has Gaussian height fluctuations. When the diffusivity is itself stochastic—taken here as $D(t)=B^2(t)$ with $B(t)$ a Brownian motion—the paper shows the height at a fixed point still obeys an exact scaling law, but the scaled distribution $G(H)$ is no longer Gaussian. The typical height grows like $t^{3/4}$, and $G(H)$ is symmetric: it tends to a nonzero constant quadratically near $H=0$ and decays like $e^{-a|H|}/\sqrt{|H|}$ for large $H$, with $a=0.6592248$. The same exponential-tail form is argued to hold for a family of linear interface models with dynamical exponent $z>1$. The result is presented as the first exact tagged-monomer height distribution in an interacting chain driven by time-dependent stochastic diffusivity, connecting diffusing-diffusivity physics to interface growth.

What carries the argument

The load-bearing object is the Brownian functional $\mathcal{V}=\int_0^1 B^2(u)(1-u)^{-1/2}\,du$, whose distribution $Q(\mathcal{V})$ controls the height distribution through Gaussian conditioning. The paper computes the Laplace transform of $Q(\mathcal{V})$ by a backward Feynman–Kac equation adapted to the time-dependent weight $(1-u)^{-1/2}$. The solution uses the ansatz $\phi_p(x,w)=f(w)e^{-g(w)x^2/2}$, which reduces the PDE to a Riccati equation for $g$; a Hopf–Cole transformation $s''(w)-4p\,s(w)/\sqrt{w}=0$ turns it into a linear equation solved by modified Bessel functions $I_{\pm 2/3}$. The explicit Laplace transform (44) then feeds the Fourier representation of $G(H)$ in Eq. (46), from which the small- and large-$H$ behaviors are read off.

What would settle it

Numerically integrate the exact integral (46) at large $H$ and check whether $\log G(H)+a|H|+\tfrac12\log|H|$ approaches a constant with $a=0.6592248$; equivalently, compute $F(iq)=(iq)^{2/3}I_{-2/3}(8iq/(3\sqrt{2}))$ near $q=a$ and verify the branch-point approximation $F(iq)\approx b(q^2+a^2)$ with $b=0.7592287$. If a different singularity is closer to the real axis, or the branch structure differs, the exponential tail fails.

Watch

Extended reading notes

Core claim

The central result is an exact expression for the single-point height distribution of the one-dimensional Edwards–Wilkinson equation with $D(t)=B^2(t)$ and flat initial condition. Writing the variance for a fixed diffusivity history as $V(t)=t^{3/2}\,\mathcal{V}/\sqrt{2\pi\Gamma}$, where $\mathcal{V}=\int_0^1 B^2(u)(1-u)^{-1/2}\,du$, the paper computes the Laplace transform of the distribution of $\mathcal{V}$ by an adapted backward Feynman–Kac method. This yields the scaling form $p(h,t)=(2\pi\Gamma)^{1/4}\,t^{-3/4}\,G((2\pi\Gamma)^{1/4}h/t^{3/4})$, with $G(H)$ given exactly by the Fourier integral $G(H)=\frac{3^{1/3}}{\sqrt{2}\,\Gamma(1/3)}\int_{-\infty}^{\infty}\frac{dk}{2\pi}\frac{e^{ikH}}{\sqrt{k^{2/3}I_{-2/3}(8k/(3\sqrt{2}))}}$. From that integral the paper extracts $G(H)\simeq d_0+(d_2/24)H^2$ near $H=0$ with $d_0=0.387$, and $G(H)\simeq \text{const}\cdot e^{-a|H|}/\sqrt{|H|}$ as $|H|\to\infty$, with $a=0.6592248$. It further asserts the same exponential-tail structure for general linear interface models $\partial_t h=-\Gamma(-\partial_x^2)^{z/2}h+\sqrt{2D(t)}\,\eta$ for any $z>1$, with a $z$-dependent decay exponent.

Load-bearing premise

The exponential-tail claim for large $|H|$ rests on the nearest singularities of the integrand in Eq. (46) being square-root branch points at $k=\pm ia$ with $F(k)\approx b(k^2+a^2)$; the paper verifies this numerically but does not prove it, and the analogous tail for general $z$ is asserted without derivation.

Editorial extensions

If this is right

  • At late times, the tagged-monomer position distribution in a Rouse chain with $D(t)=B^2(t)$ is the same non-Gaussian scaling law, so diffusing-diffusivity anomalies that are known for a single particle persist in an interacting polymer.
  • The typical height scales as $t^{3/4}$, intermediate between the $t^{1/2}$ of ordinary Edwards–Wilkinson growth and the linear-in-$t$ scaling of a free particle with $D(t)=B^2(t)$.
  • The scaled distribution has a nonzero density at $H=0$ with a quadratic correction, and an exponential rather than Gaussian tail, so large-height events are far more likely than in the constant-diffusivity Edwards–Wilkinson model.
  • Every linear interface model with dynamical exponent $z>1$, including $z=4$ Mullins–Herring, inherits the exponential tail, with a $z$-dependent decay exponent.
  • The exact Fourier representation (46) gives a closed-form route to all moments of the scaled height through the coefficients $d_n$ in Eq. (49).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An extension the paper leaves implicit: in a finite Rouse chain of length $L$, the variance functional is dominated by the slowest relaxation mode, so the exponential tail should survive only for $t\ll L^z$ and cross over to Gaussian behavior at longer times; this can be tested by Brownian dynamics simulation.
  • The same adapted Feynman–Kac machinery applies to other positive self-similar diffusivity processes such as $|B(t)|^\alpha$; one would expect a family of exact scaling functions whose tail exponents depend on $\alpha$, interpolating between the results here and the single-particle case.
  • Because the exact $G(H)$ decays exponentially rather than quadratically in the exponent, the large-deviation rate for atypical heights is linear in $|H|$; this is a concrete prediction that could be checked in experiments or simulations of stochastic-diffusivity interface growth.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper studies the one-dimensional Edwards-Wilkinson (EW) interface in the presence of a stochastic diffusivity D(t)=B^2(t), where B(t) is a one-dimensional Brownian motion. By adapting the backward Feynman-Kac formalism to treat the explicit time-dependent kernel in the variance functional, the authors obtain the exact Laplace transform of the distribution of the scaled variance, Eq. (44), and hence an exact integral representation for the scaled height distribution G(H), Eq. (46). They derive the large-|H| exponential tail G(H) ~ const * e^{-a|H|}/sqrt(|H|) with a=0.6592248, and a quadratic approach to G(0) at small H. The paper also sketches a generalization to a family of linear interface models with dynamical exponent z>1, claiming an exponential tail for all z>1. The z=2 result is exact and internally consistent; the main weakness is the unsupported general-z claim and the numerically verified rather than rigorously proven branch-point structure used for the tail.

Significance. If the results hold, the z=2 exact solution is a significant contribution to the diffusing-diffusivity literature, providing a rare exact tagged-monomer/height distribution for an interacting system driven by a noise with time-dependent stochastic diffusivity. The explicit integral representation (46), the exact Laplace transform (44), and the scaling form (50) are analytically checkable and likely to be useful. The paper's derivation of the Laplace transform is self-contained and the connection to Brownian functionals is clean. However, the advertised robustness of exponential tails for all z>1 is not substantiated, and the asymptotic analysis at z=2 rests on a numerically checked but unproved singularity structure. These gaps do not undermine the exact z=2 result but do limit the paper's broader claims.

major comments (2)
  1. [Section 5, Eq. (65)] The claim that G_z(H) has an exponential tail for every z>1 is asserted without derivation. After Eq. (65), the paper states that 'it is not difficult to show' the tail, but no singularity analysis, decay exponent, or numerical check for any z≠2 is provided. Since the abstract advertises the exponential tail as robust across the whole family z>1, this is a load-bearing claim. Please provide the actual singularity analysis for the integrand in Eq. (65) for general z, or explicitly restrict the claim to z=2. If the general proof is not available, the abstract and Section 5 should be revised so that the unproved part is not presented as a main result.
  2. [Section 4, Eq. (47)] The large-H tail is derived from the assertion that F(k)=k^{2/3}I_{-2/3}(8k/(3\sqrt{2})) has its nearest singularities on the imaginary axis at k=±ia, with F(k)≈b(k^2+a^2) near k=ia, and is verified numerically in Fig. 2 but not proved. Because the exponential decay rate a and the prefactor depend on this singular structure, the tail result is not fully rigorous. Please provide a proof (e.g., using the fact that the zeros of I_{-ν} are purely imaginary, so the integrand has square-root branch points) or cite a reference where this property is established. Without such a justification, the tail in (47) should be described as a strongly supported conjecture or verified numerically.
minor comments (3)
  1. [Eq. (51)] The small-H line in Eq. (51) contains a sign and coefficient error: from Eq. (48), the H^2 coefficient is -d1/2! ≈ -0.4095, not +d2/24 ≈ +0.4368. This contradicts Fig. 4 and should be corrected.
  2. [Fig. 3 and Fig. 4] The vertical axis in Figs. 3 and 4 is labeled P(H), but the plotted function is G(H) as defined in Eq. (46). Please relabel to avoid notation inconsistency.
  3. [Eq. (55)] The symbol Γ is used both for the coupling constant in Eq. (14) and for the Gamma function in the definition of A_z. The sentence after Eq. (55) clarifies this, but the formula would be less prone to misreading if the coupling constant were denoted by a different symbol (e.g., \gamma).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the exact z=2 result is derived from the PDE and Bessel-function constants rather than from fits or self-citations; the Section 5 z>1 tail is asserted without proof and Eq. (51) has a coefficient typo, but neither is circular.

full rationale

The derivation chain is self-contained. For a fixed history of D(t), the variance is computed directly from the Gaussian correlator in Eqs. (18)-(20); Brownian self-similarity then reduces the variance to the functional V = ∫ B^2(u)/√(1-u) du in Eq. (22), with no use of the target distribution. Q(V) is obtained by solving the backward Feynman-Kac PDE (32) with terminal condition ψ_p(x,1)=1. The Gaussian ansatz (34) is a standard exact solution method; the constants are fixed by initial conditions and the Hopf-Cole linearization, so the Laplace transform (44) is solved, not assumed. The large-H tail uses the exact representation (46) and the nearest imaginary-axis zero of the explicitly given function F(k)=k^{2/3} I_{-2/3}(8k/(3√2)); the constants a=0.6592248 and b=0.7592287 come from this fixed Bessel function (and are numerically checked in Fig. 2), not from fitting G(H), so the asymptotic prediction is not forced by a fitted input. The self-citations [31,32,34] are method/background prior work and are not used to forbid alternatives or to import the target result. Flagged but not circular: Section 5 asserts for all z>1 that G_z(H) has an exponential tail ('It is not difficult to show...') without supplying the singularity analysis or a proof for any z≠2; this is an omitted proof, not a circular reduction. Also, Eq. (51)'s small-H line 'd0 + d2/24 H^2' is inconsistent with Eq. (48), whose coefficient is -d1/2 ≈ -0.4095, but this is a sign/coefficient typo in a formula, not a circular step. No fitted parameter is renamed as a prediction, and no self-citation chain carries the central claim.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameter is fitted to the target distribution. The constants a and b in the asymptotic tail are determined by zeros and limits of the Bessel function I_{-2/3}, not by fitting G(H), and the coupling Gamma is an input scale. The stochastic diffusivity D(t)=B^2(t) is taken from prior single-particle literature, and no new particles, forces, dimensions, or conserved quantities are introduced.

assumptions (4)
  • domain assumption Brownian motion B(t) and the interface noise eta(x,t) are statistically independent Gaussian processes, with B(0)=0.
    Used throughout Section 2 to average over histories of D(t) independently of the linear response of the height.
  • domain assumption The continuum EW equation (14) correctly captures the late-time height and position statistics of the discrete Rouse chain for t much larger than 1/kappa.
    This is the bridge from the Rouse model to the exactly solved continuum model; the hydrodynamic limit is standard but not proven in the paper.
  • domain assumption The system is infinite with flat initial condition h(x,0)=0, and the k-integral converges without an ultraviolet cutoff for z>1.
    This is used to derive Eq. (20) and to restrict the general-z analysis to z>1 in Section 5; finite-size and lattice effects are excluded.
  • standard math The backward Feynman-Kac equation (32) has a unique solution of the Gaussian ansatz form (34) for the relevant p>=0.
    The PDE derivation in Section 3 assumes uniqueness and interchange of expectations; this is a standard but unproved background fact.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Exact height distribution in one-dimensional Edwards-Wilkinson interface with diffusing diffusivity." pith.science (2026). https://pith.science/paper/M2EQW6S2

@misc{pith2026250201153,
  author       = {Pith},
  title        = {Pith review of: Exact height distribution in one-dimensional Edwards-Wilkinson interface with diffusing diffusivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2EQW6S2}},
  note         = {Machine review of arXiv:2502.01153}
}
abstract

We study the height distribution of a one-dimensional Edwards-Wilkinson interface in the presence of a stochastic diffusivity $D(t)=B^2(t)$, where $B(t)$ represents a one-dimensional Brownian motion at time $t$. The height distribution at a fixed point is space is computed analytically. The typical height $h(x,t)$ at a given point in space is found to scale as $t^{3/4}$ and the distribution $G(H)$ of the scaled height $H=h/t^{3/4}$ is symmetric but with a nontrivial shape: while it approaches a nonzero constant quadratically as $H\to 0$, it has a non-Gaussian tail that decays exponentially for large $H$. We show that this exponential tail is rather robust and holds for a whole family of linear interface models parametrized by a dynamical exponent $z>1$, with $z=2$ corresponding to the Edwards-Wilkinson model.

Figures

Figures reproduced from arXiv: 2502.01153 by the authors.

Figure 1
Figure 1. A schematic trajectory of a Brownian motion B(u ′ ) propagating from time u ′ = u to time u ′ = 1, starting at B(u) = x. In the first infinitesimal time step du, the Brownian motion moves from x to x + dx. Inverting this Laplace transform ψp(0, 0) with respect to p, one then obtains Q(V ). To derive the partial differential equation for ψp(x, u), defined in (26), we proceed as follows. Here, it is convenient to look… view at source ↗
Figure 2
Figure 2. The red solid line plots the function F(iq) as a function of q whereas the blue dashed line plots the approximation b(−q 2 + a 2 ). Both the solid and the dashed lines are in excellent agreement near q = ±a, with a = 0.6592248 . . . . expression for G(H) in (46) is exact for all H. Unfortunately, it is hard to perform the integral to obtain an explicit expression for G(H). However, once can easily extract the asympt… view at source ↗
Figure 3
Figure 3. The red points are obtained from the exact numerical integration of (46). The blue dashed and magenta dot-dashed lines represent the two forms given the last line of (47) respectively. where dn = 3 1/3 p 2Γ(1/3) Z ∞ −∞ dk 2π k 2n r k 2/3 I−2/3  8k 3 √ 2  . (49) The numerical value of the first few coefficients are given by d0 = 0.387 . . . , d1 = 0.819 . . . , d2 = 10.483 . . . , and d3 = 342.279 . . . [PITH_FUL… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The points are obtained from exact numerical integration of (46), while the solid line plots (48) keeping upto O(H6 ) terms, with the coefficients dn obtained numerically. 5. Generalization to other linear interface models In this section, we generalize the computation…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 40 canonical work pages

  1. [1]

    Rouse P E 1953 A theory of the linear viscoelastic properties of dilute solutions of coiling polymers J. Chem. Phys.211272

    work page 1953

  2. [2]

    De Gennes P 1976 Dynamics of entangled polymer solutions. I. the Rouse modelMacromolecules 9587

    work page 1976

  3. [3]

    Edwards S F and Wilkinson D 1982 The surface statistics of a granular aggregateProc. R. Soc. London A38117

    work page 1982

  4. [4]

    Barab´ asi A L and Stanley H E 1995Fractal concepts in surface growth(Cambridge University Press)

    work page

  5. [5]

    Phys.46139

    Krug J 1997 Origins of scale invariance in growth processesAdv. Phys.46139

    work page 1997

  6. [6]

    aspects of multidisciplinary statistical mechanicsPhys

    Halpin-Healy T and Zhang Y C 1995 Kinetic roughening phenomena, stochastic growth, directed polymers and all that. aspects of multidisciplinary statistical mechanicsPhys. Rep.254215

    work page 1995

  7. [7]

    Wang B, Anthony S M, Bae S C and Granick S 2009 Anomalous yet BrownianProc. Nat. Acad. Sci.10615160 URLhttps://doi.org/10.1073/pnas.0903554106 Edwards-Wilkinson interface with diffusing diffusivity17

    work page doi:10.1073/pnas.0903554106 2009

  8. [8]

    Wang B, Kuo J, Bae S C and Granick S 2012 When Brownian diffusion is not gaussianNat. Mater. 11481 URLhttps://doi.org/10.1038/nmat3308

    work page doi:10.1038/nmat3308 2012

Show all 43 references
  1. [9]

    Bhattacharya S, Sharma D K, Saurabh S, De S, Sain A, Nandi A and Chowdhury A 2013 Plasticization of poly(vinylpyrrolidone) thin films under ambient humidity: Insight from single- molecule tracer diffusion dynamicsJ. Phys. Chem. B1177771 pMID: 23777572 (Preprint https://doi.org...

    2013 doi

  2. [10]

    Guan J, Wang B and Granick S 2014 Even hard-sphere colloidal suspensions display fickian yet non-gaussian diffusionACS nano83331

    2014

  3. [11]

    He K, Babaye Khorasani F, Retterer S T, Thomas D K, Conrad J C and Krishnamoorti R 2013 Diffusive dynamics of nanoparticles in arrays of nanopostsACS nano75122

    2013

  4. [12]

    Czajka P, Antosiewicz J M and D lugosz M 2019 Effects of hydrodynamic interactions on the near-surface diffusion of spheroidal moleculesACS Omega417016

    2019

  5. [13]

    Alexandre A, Lavaud M, Fares N, Millan E, Louyer Y, Salez T, Amarouchene Y, Gu´ erin T and Dean D S 2023 Non-gaussian diffusion near surfacesPhys. Rev. Lett.130077101 URL https://link.aps.org/doi/10.1103/PhysRevLett.130.077101

    2023 doi

  6. [14]

    Han Y, Alsayed A, Nobili M, Zhang J, Lubensky T C and Yodh A G 2006 Brownian motion of an ellipsoidScience314626

    2006

  7. [15]

    Lett.8530003 URLhttps://doi.org/10.1209/ 0295-5075/85/30003

    Munk T, Hofling F, Frey E and Franosch T 2009 Effective perrin theory for the anisotropic diffusion of a strongly hindered rodEurophys. Lett.8530003 URLhttps://doi.org/10.1209/ 0295-5075/85/30003

    2009

  8. [16]

    Chubynsky M V and Slater G W 2014 Diffusing diffusivity: A model for anomalous, yet Brownian, diffusionPhys. Rev. Lett.113098302 URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.113.098302

    2014

  9. [17]

    Phys.22063056

    Sposini V, Grebenkov D S, Metzler R, Oshanin G and Seno F 2020 Universal spectral features of different classes of random-diffusivity processesNew J. Phys.22063056

    2020

  10. [18]

    Chechkin A V, Seno F, Metzler R and Sokolov I M 2017 Brownian yet non-gaussian diffusion: from superstatistics to subordination of diffusing diffusivitiesPhys. Rev. X7021002

    2017

  11. [19]

    Tyagi N and Cherayil B J 2017 Non-gaussian Brownian diffusion in dynamically disordered thermal environmentsJ. Phys. Chem. B1217204

    2017

  12. [20]

    Lanoisel´ ee Y and Grebenkov D S 2018 A model of non-gaussian diffusion in heterogeneous media J. Phys. A: Math. Theor.51145602

    2018

  13. [21]

    Barkai E and Burov S 2020 Packets of diffusing particles exhibit universal exponential tailsPhys. Rev. Lett.124060603

    2020

  14. [22]

    Pacheco-Pozo A and Sokolov I M 2021 Large deviations in continuous-time random walksPhys. Rev. E103042116

    2021

  15. [23]

    Hamdi O, Burov S and Barkai E 2024 Laplace’s first law of errors applied to diffusive motionEur. Phys. J. B971

    2024

  16. [24]

    Singh R and Burov S 2024 The emergence of laplace universality in correlated processesarXiv preprint arXiv:2410.23112

    2024 arXiv

  17. [25]

    Gu´ eneau M, Majumdar S N and Schehr G 2025 Large deviations in switching diffusion: from free cumulants to dynamical transitionsarXiv preprint arXiv:2501.13754

    2025 arXiv

  18. [26]

    Hidalgo-Soria M, Barkai E and Burov S 2021 Cusp of non-gaussian density of particles for a diffusing diffusivity modelEntropy23231

    2021

  19. [27]

    Sposini V, Chechkin A and Metzler R 2018 First passage statistics for diffusing diffusivityJ. Phys. A: Math Theor5204LT01

    2018

  20. [28]

    Jain R and Sebastian K 2017 Diffusion in a crowded, rearranging environmentJ. Chem. Phys. 146214102

    2017

  21. [29]

    Jain R and Sebastian K L 2016 Diffusing diffusivity: survival in a crowded rearranging and bounded domainJ. Phys. Chem. B1209215

    2016

  22. [30]

    Phys.161 Edwards-Wilkinson interface with diffusing diffusivity18

    Yin Q, Li Y, Marchesoni F, Nayak S and Ghosh P K 2021 Non-gaussian normal diffusion in low dimensional systemsFront. Phys.161 Edwards-Wilkinson interface with diffusing diffusivity18

    2021

  23. [31]

    Santra I, Basu U and Sabhapandit S 2021 Active Brownian motion with directional reversalsPhys. Rev. E104L012601 URLhttps://link.aps.org/doi/10.1103/PhysRevE.104.L012601

    2021 doi

  24. [32]

    Santra I, Basu U and Sabhapandit S 2022 Effect of stochastic resetting on Brownian motion with stochastic diffusion coefficientJ. Phys. A: Math. Theor.55414002

    2022

  25. [33]

    Sci.892076

    Majumdar S N 2005 Brownian functionals in physics and computer scienceCurr. Sci.892076

    2005

  26. [34]

    Boyer D and Dean D S 2011 On the distribution of estimators of diffusion constants for Brownian motionJ. Phys. A: Math. Theor.44335003 URLhttps://dx.doi.org/10.1088/1751-8113/ 44/33/335003

    2011 doi

  27. [35]

    Phys.62225

    Bray A J, Majumdar S N and Schehr G 2013 Persistence and first-passage properties in nonequilibrium systemsAdv. Phys.62225

    2013

  28. [36]

    Majumdar S N and Bray A J 2001 Spatial persistence of fluctuating interfacesPhys. Rev. Lett.86 3700

    2001

  29. [37]

    Dean D S, Majumdar S N and Schawe H 2021 Position distribution in a generalized run-and-tumble processPhys. Rev. E103012130

    2021

  30. [38]

    Mullins W W 1957 Theory of thermal groovingJ. Appl. Phys.28333

    1957

  31. [39]

    Herring C 1950 Effect of change of scale on sintering phenomenaJ. Appl. Phys.21301

    1950

  32. [40]

    Panja D 2010 Generalized langevin equation formulation for anomalous polymer dynamicsJ. Stat. Mech.2010L02001 URLhttps://dx.doi.org/10.1088/1742-5468/2010/02/L02001

    2010 doi

  33. [41]

    Nampoothiri S, Orlandini E, Seno F and Baldovin F 2021 Polymers critical point originates brownian non-gaussian diffusionPhys. Rev. E104L062501 URLhttps://link.aps.org/doi/ 10.1103/PhysRevE.104.L062501

    2021 doi

  34. [42]

    Nampoothiri S, Orlandini E, Seno F and Baldovin F 2022 Brownian non-gaussian polymer diffusion and queuing theory in the mean-field limitN. J. Phys.24023003 URLhttps: //dx.doi.org/10.1088/1367-2630/ac4924

    2022 doi

  35. [43]

    Kardar M, Parisi G and Zhang Y C 1986 Dynamic scaling of growing interfacesPhys. Rev. Lett. 56889 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.56.889

    1986 doi

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.