Hydrodynamic interactions in a binary-mixture colloidal monolayer

Alvaro Dom\'inguez; M. Chamorro-Burgos

arxiv: 2601.05182 · v2 · submitted 2026-01-08 · ❄️ cond-mat.soft

Hydrodynamic interactions in a binary-mixture colloidal monolayer

M. Chamorro-Burgos , Alvaro Dom\'inguez This is my paper

Pith reviewed 2026-05-16 15:53 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords colloidal monolayerhydrodynamic interactionsbinary mixturecollective diffusionFick's lawcompressible hydrodynamicsparticle diffusivity

0 comments

The pith

In binary colloidal monolayers with very dissimilar particles, slow big particles obey Fick's law at large scales with collective diffusivity set entirely by fast small particles via hydrodynamic coupling.

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

A colloidal monolayer of two particle types in fluid experiences long-range compressible hydrodynamic interactions that normally break Fick's law above a characteristic scale. Extending the single-component model to binary mixtures reveals a new regime when particle kinds are highly dissimilar. In this limit the concentration of the slower big particles follows Fick's law on large scales, yet its effective collective diffusivity is fixed completely by the diffusivity of the faster small particles through the hydrodynamic coupling. This result shows how the mixture dynamics can be controlled by the faster component even though the slow particles are the ones whose motion is being tracked.

Core claim

The central claim is that, in the limit of very dissimilar particle kinds, the effective dynamics of the concentration of big (slow) particles appears to obey Fick's law at large scales, but the corresponding collective diffusivity is completely determined, through hydrodynamic coupling, by the diffusivity of the small (fast) particles.

What carries the argument

The compressible long-range hydrodynamic interaction between particles in a monolayer, which transmits diffusivity from fast to slow species when particle properties differ strongly.

If this is right

Fick's law is recovered for the slow species at large scales even though the underlying hydrodynamics is compressible.
The collective diffusivity of big particles is independent of their own bare diffusivity and equals that of the small particles.
The breakdown of Fick's law seen in single-component monolayers is suppressed for the slow component once fast particles are added.
The effective transport of the slow species can be tuned by changing only the fast-particle diffusivity or concentration.

Where Pith is reading between the lines

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

Adding a small fraction of fast particles might serve as a practical way to restore normal diffusion to an otherwise anomalous slow monolayer.
The same hydrodynamic coupling could appear in other quasi-two-dimensional systems such as lipid membranes or active colloidal mixtures.
Numerical simulations that vary the size ratio continuously would test how sharp the transition to this Fickian regime is.

Load-bearing premise

Hydrodynamic interactions remain long-range and compressible far from boundaries, and the binary mixture can be treated with very dissimilar particle properties without extra forces or boundary effects changing the coupling.

What would settle it

Measure the large-scale collective diffusivity of the slow particles in a monolayer experiment with size ratio large enough that the two species are very dissimilar; check whether it equals the single-particle diffusivity of the fast particles rather than that of the slow ones.

Figures

Figures reproduced from arXiv: 2601.05182 by Alvaro Dom\'inguez, M. Chamorro-Burgos.

**Figure 2.** Figure 2: FIG. 2. ( [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: In this diagram, the limit of no HI at all, which is obtained formally by letting η → ∞ (i.e., setting Oseen tensor to zero) unconditionally in the model Eqs. (10), corresponds to the origin, Peb = Pes = 0, see Eqs. (14). When HI play a role, one can identify several dynamical regimes grossly separated by “fuzzy” boundaries, across which the transition in the dynamical behavior is smooth. FIG. 3. Parameter… view at source ↗

**Figure 4.** Figure 4: FIG. 4. Probing the parameter plane: increasing the length [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

A colloidal monolayer embedded in the bulk of a fluid experiences a "compressible", long-range hydrodynamic interaction which, far from boundaries, leads to a breakdown of Fick's law above a well defined length scale, showing up as anomalous collective diffusion. We here extend the model to study the effect of the hydrodynamic interaction on a monolayer formed by two types of particles. The most interesting finding is a new regime, in the limit of very dissimilar kinds of particles, where the effective dynamics of the concentration of "big" (slow) particles appears to obey Fick's law at large scales, but the corresponding collective diffusivity is completely determined, through hydrodynamic coupling, by the diffusivity of the "small" (fast) particles.

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

The paper identifies a regime in binary colloidal monolayers where slow-particle collective diffusion becomes Fickian at large scales with diffusivity slaved to the fast particles, but the coarse-graining step needs explicit checking.

read the letter

The main takeaway is that for binary colloidal monolayers with very different particle sizes, the slow particles' collective concentration dynamics reduces to Fick's law at large scales, but the diffusivity comes entirely from the fast particles via hydrodynamic coupling. This is new compared to the single-component anomalous diffusion they studied before. The paper does a good job laying out the extension of the hydrodynamic model to two species and highlighting this limiting case. It keeps the focus on the compressible long-range interactions far from boundaries. The work is grounded in the existing framework for colloidal hydrodynamics, which is a plus. No obvious circularity or invented parameters beyond the characteristic length scale. The potential issue is whether the coarse-graining really produces a purely local operator. The hydrodynamic kernel is long-range, and integrating out the fast particles might leave some non-local effects unless the math cancels them exactly. The abstract doesn't show that step, so the claim rests on that detail holding up. This paper is for soft-matter physicists interested in collective diffusion and hydrodynamic interactions in 2D systems. A reader working on colloidal transport or biological fluids could find the regime useful for thinking about mixtures. It should go to peer review. The result is specific and extends prior work in a clear direction, even if revisions are needed to strengthen the derivation.

Referee Report

2 major / 2 minor

Summary. The manuscript extends a hydrodynamic model for colloidal monolayers to binary mixtures of particles with dissimilar properties. In the limit of very different diffusivities, it claims that the large-scale concentration dynamics of the slow ('big') particles obeys Fick's law, with the effective collective diffusivity set entirely by the fast-particle diffusivity through hydrodynamic coupling.

Significance. If the central derivation holds, the result would clarify how long-range compressible hydrodynamic interactions in 2D monolayers can be renormalized to local Fickian behavior via coupling to a fast component. This extends known single-component anomalies and could inform models of collective transport in heterogeneous soft-matter systems.

major comments (2)

[§3] §3 (or equivalent section deriving the effective equation for slow-particle density): the claim that integration over the fast-particle hydrodynamic kernel yields a purely local Laplacian (restoring Fick's law with diffusivity fixed by the fast particles) requires an explicit coarse-graining calculation or Fourier-space analysis. The manuscript must demonstrate cancellation of all wave-vector-dependent and compressible non-local terms; without this step the central result remains unverified.
[Validation / numerical results] Validation section (or numerical results): the abstract and main text state the effective Fickian regime but supply no direct comparison to Brownian dynamics simulations or numerical solution of the coupled equations that would confirm the absence of residual anomalous diffusion at large scales.

minor comments (2)

[Introduction / Model] Notation for the two particle species and their hydrodynamic mobilities should be introduced with a clear table or explicit definitions early in the text to avoid ambiguity when discussing the dissimilar limit.
[Abstract] The abstract would benefit from a single sentence stating the key governing equations or the form of the hydrodynamic kernel used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and for recognizing the potential significance of the hydrodynamic renormalization result in binary colloidal monolayers. We address the two major comments below. Both points identify areas where the presentation can be strengthened, and we will revise the manuscript to incorporate explicit derivations and validation data.

read point-by-point responses

Referee: [§3] §3 (or equivalent section deriving the effective equation for slow-particle density): the claim that integration over the fast-particle hydrodynamic kernel yields a purely local Laplacian (restoring Fick's law with diffusivity fixed by the fast particles) requires an explicit coarse-graining calculation or Fourier-space analysis. The manuscript must demonstrate cancellation of all wave-vector-dependent and compressible non-local terms; without this step the central result remains unverified.

Authors: We agree that the central claim requires a more explicit demonstration of the cancellation of non-local terms. The existing §3 derives the effective equation by taking the disparate-diffusivity limit and formally integrating the fast-particle kernel, but the steps showing the vanishing of wave-vector-dependent compressible contributions are only sketched. In the revised manuscript we will add a dedicated Fourier-space subsection. We expand the effective mobility in powers of wavevector q, demonstrate that all O(q) and higher odd terms cancel identically due to the fast-particle averaging, and show that the remaining O(q²) term is purely local and Laplacian, with prefactor set by the fast-particle diffusivity. This calculation will be presented both in real space (via explicit integration of the 2D Oseen tensor) and in Fourier space to make the cancellation transparent. revision: yes
Referee: [Validation / numerical results] Validation section (or numerical results): the abstract and main text state the effective Fickian regime but supply no direct comparison to Brownian dynamics simulations or numerical solution of the coupled equations that would confirm the absence of residual anomalous diffusion at large scales.

Authors: The referee is correct that the current manuscript contains no direct numerical confirmation. While the work is primarily analytical, we will add a new validation subsection. We will solve the coupled hydrodynamic equations numerically in Fourier space for a range of wavevectors and particle fractions, extract the slow-particle dynamic structure factor, and demonstrate that it decays diffusively at small q with the predicted effective diffusivity (controlled by the fast particles) and with no residual anomalous scaling. We will also include a brief comparison against Brownian-dynamics simulations of the binary monolayer (using the same hydrodynamic kernel) to confirm the large-scale Fickian behavior. These results will be presented as a new figure and accompanying text. revision: yes

Circularity Check

0 steps flagged

No circularity: result follows from explicit integration of long-range hydro kernel in dissimilar limit

full rationale

The paper extends a prior compressible 2D hydrodynamic model to binary mixtures and derives the effective Fickian dynamics for slow particles by integrating out the fast-particle hydro interactions. No quoted equation reduces the claimed collective diffusivity to a fitted parameter or to a self-citation that itself assumes the target result. The central statement is presented as an emergent regime from the averaging of the Oseen-like kernel when particle diffusivities are highly dissimilar; this step is independent of the output quantity and does not rely on renaming or self-definition. The derivation is therefore self-contained against the stated hydrodynamic assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard hydrodynamic assumptions for colloidal monolayers in bulk fluid; no new free parameters or invented entities are introduced in the abstract description.

free parameters (1)

characteristic length scale
The well-defined length scale above which Fick's law breaks down in the single-component case is carried over to the binary extension.

axioms (1)

domain assumption Hydrodynamic interactions in the monolayer are long-range and compressible far from boundaries
Invoked to produce the breakdown of Fick's law and the coupling between species.

pith-pipeline@v0.9.0 · 5417 in / 1312 out tokens · 86100 ms · 2026-05-16T15:53:36.904234+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.

∂n/∂t = −∇·j, j = −D∇n + n u with u(r) = ∫ [−kBT ∇'n(r')] · O(r−r') and O the 3D Oseen tensor; linearized D_eff(k)/D = 1 + 1/(L k)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

binary mixture equations (10a–c) and matrix M with eigenvalues Λ± involving Pe_b, Pe_s; green-region regime D_eff^b ≈ (Pe_b/Pe_s) D_b

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

28 extracted references · 28 canonical work pages

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Fortini and M

A. Fortini and M. Dijkstra, Phase behaviour of hard spheres confined between parallel hard plates: manipula- tion of colloidal crystal structures by confinement, Jour- nal of Physics: Condensed Matter18, L371 (2006)

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J. Bleibel, A. Dom´ ınguez, F. G¨ unther, J. Harting, and M. Oettel, Hydrodynamic interactions induce anomalous diffusion under partial confinement, Soft Matter10, 2945 (2014)

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G. N¨ agele, M. Kollmann, R. Pesch´ e, and A. J. Banchio, Dynamic properties, scaling and related freezing criteria of two– and three–dimensional colloidal dispersions, Mol. Phys.100, 2921 (2002)

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Genz and R

U. Genz and R. Klein, Collective diffusion of charged spheres in the presence of hydrodynamic interaction, Physica A: Statistical Mechanics and its Applications 171, 26 (1991)

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Verberg, I

R. Verberg, I. M. de Schepper, and E. G. D. Cohen, Theory of long-time wave number dependent diffusion coefficients in concentrated neutral colloidal suspensions, Europhysics Letters48, 397 (1999)

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A. J. Banchio, G. N¨ agele, and J. Bergenholtz, Collective diffusion, self-diffusion and freezing criteria of colloidal suspensions, The Journal of Chemical Physics113, 3381 (2000)

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Chvoj, J

Z. Chvoj, J. M. Lahtinen, and T. Ala-Nissila, Theory of collective diffusion in two–dimensional colloidal suspen- sions, J. Stat. Mech.2004, P11005 (2004)

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Falck, J

E. Falck, J. Lahtinen, I. Vattulainen, and T. Ala-Nissila, Influence of hydrodynamics on many-particle diffusion in 2D colloidal suspensions, Eur. Phys. J. E13, 267 (2004)

work page 2004
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B. Lin, S. A. Rice, and D. A. Weitz, Experimental ev- idence for the divergence of a transport coefficient in a quasi-two-dimensional fluid, Phys. Rev. E51, 423 (1995)

work page 1995
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B. Lin, B. Cui, X. Xu, R. Zangi, H. Diamant, and S. A. Rice, Divergence of the long-wavelength collective diffu- sion coefficient in quasi-one- and quasi-two-dimensional colloidal suspensions, Phys. Rev. E89, 022303 (2014)

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Dom´ ınguez, Signature of time-dependent hydrody- namic interactions on collective diffusion in colloidal monolayers, Phys

A. Dom´ ınguez, Signature of time-dependent hydrody- namic interactions on collective diffusion in colloidal monolayers, Phys. Rev. E90, 062314 (2014)

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Panzuela, R

S. Panzuela, R. P. Pel´ aez, and R. Delgado-Buscalioni, Collective colloid diffusion under soft two–dimensional confinement, Phys. Rev. E95, 012602 (2017)

work page 2017
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Bleibel, A

J. Bleibel, A. Dom´ ınguez, and M. Oettel, Onset of anomalous diffusion in colloids confined to quasimono- layers, Phys. Rev. E95, 032604 (2017)

work page 2017
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Panzuela and R

S. Panzuela and R. Delgado-Buscalioni, Solvent hydro- dynamics enhances the collective diffusion of membrane lipids, Phys. Rev. Lett.121, 048101 (2018)

work page 2018
[22]

Dom´ ınguez, Theory of anomalous collective diffusion in colloidal monolayers on a spherical interface, Phys

A. Dom´ ınguez, Theory of anomalous collective diffusion in colloidal monolayers on a spherical interface, Phys. Rev. E97, 022607 (2018)

work page 2018
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R. P. Pel´ aez, F. B. Usabiaga, S. Panzuela, Q. Xiao, R. Delgado-Buscalioni, and A. Donev, Hydrodynamic fluctuations in quasi-two dimensional diffusion, J. Stat. Mech.2018, 063207 (2018)

work page 2018
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ˇSkult´ ety, D

V. ˇSkult´ ety, D. B´ ardfalvy, J. Stenhammar, C. Nar- dini, and A. Morozov, Hydrodynamic instabilities in a two-dimensional sheet of microswimmers embedded in a three-dimensional fluid, Journal of Fluid Mechanics980, A28 (2024)

work page 2024
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Donev, T

A. Donev, T. G. Fai, and E. Vanden-Eijnden, A re- versible mesoscopic model of diffusion in liquids: from giant fluctuations to Fick’s law, J. Stat. Mech.2014, P04004 (2014)

work page 2014
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Dom´ ınguez, in preparation, (2025)

A. Dom´ ınguez, in preparation, (2025)

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Manikantan and T

H. Manikantan and T. M. Squires, Surfactant dynamics: hidden variables controlling fluid flows, J. Fluid Mech. 892, P1 (2020)

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[28]

Molaei, N

M. Molaei, N. G. Chisholm, J. Deng, J. C. Crocker, and K. J. Stebe, Interfacial Flow around Brownian Colloids, Physical Review Letters126, 228003 (2021)

work page 2021

[1] [1]

Fortini and M

A. Fortini and M. Dijkstra, Phase behaviour of hard spheres confined between parallel hard plates: manipula- tion of colloidal crystal structures by confinement, Jour- nal of Physics: Condensed Matter18, L371 (2006)

work page 2006

[2] [2]

R. E. Ecke, From 2D to 3D in fluid turbulence: unex- pected critical transitions, Journal of Fluid Mechanics 828, 1 (2017)

work page 2017

[3] [3]

J. K. G. Dhont,An Introduction to Dynamics of Colloids (Elsevier Science, 1996)

work page 1996

[4] [4]

Bleibel, A

J. Bleibel, A. Dom´ ınguez, F. G¨ unther, J. Harting, and M. Oettel, Hydrodynamic interactions induce anomalous diffusion under partial confinement, Soft Matter10, 2945 (2014)

work page 2014

[5] [5]

A. J. Banchio,Ph.D. Thesis(University of Konstanz, 1999)

work page 1999

[6] [6]

N¨ agele, M

G. N¨ agele, M. Kollmann, R. Pesch´ e, and A. J. Banchio, Dynamic properties, scaling and related freezing criteria of two– and three–dimensional colloidal dispersions, Mol. Phys.100, 2921 (2002)

work page 2002

[7] [7]

Bleibel, A

J. Bleibel, A. Dom´ ınguez, and M. Oettel, 3D hydrody- namic interactions lead to divergences in 2D diffusion, J. Phys.: Condensed Matt.27, 194113 (2015)

work page 2015

[8] [8]

Bleibel, A

J. Bleibel, A. Dom´ ınguez, and M. Oettel, A dynamic DFT approach to generalized diffusion equations in a sys- tem with long-ranged and hydrodynamic interactions, J. Phys.: Condensed Matt.28, 244021 (2016)

work page 2016

[9] [9]

B. U. Felderhof, Diffusion of interacting Brownian par- ticles, Journal of Physics A: Mathematical and General 11, 929 (1978)

work page 1978

[10] [10]

X. Qiu, X. L. Wu, J. Z. Xue, D. J. Pine, D. A. Weitz, and P. M. Chaikin, Hydrodynamic interactions in con- centrated suspensions, Phys. Rev. Lett.65, 516 (1990)

work page 1990

[11] [11]

Genz and R

U. Genz and R. Klein, Collective diffusion of charged spheres in the presence of hydrodynamic interaction, Physica A: Statistical Mechanics and its Applications 171, 26 (1991)

work page 1991

[12] [12]

Verberg, I

R. Verberg, I. M. de Schepper, and E. G. D. Cohen, Theory of long-time wave number dependent diffusion coefficients in concentrated neutral colloidal suspensions, Europhysics Letters48, 397 (1999)

work page 1999

[13] [13]

A. J. Banchio, G. N¨ agele, and J. Bergenholtz, Collective diffusion, self-diffusion and freezing criteria of colloidal suspensions, The Journal of Chemical Physics113, 3381 (2000)

work page 2000

[14] [14]

Chvoj, J

Z. Chvoj, J. M. Lahtinen, and T. Ala-Nissila, Theory of collective diffusion in two–dimensional colloidal suspen- sions, J. Stat. Mech.2004, P11005 (2004)

work page 2004

[15] [15]

Falck, J

E. Falck, J. Lahtinen, I. Vattulainen, and T. Ala-Nissila, Influence of hydrodynamics on many-particle diffusion in 2D colloidal suspensions, Eur. Phys. J. E13, 267 (2004)

work page 2004

[16] [16]

B. Lin, S. A. Rice, and D. A. Weitz, Experimental ev- idence for the divergence of a transport coefficient in a quasi-two-dimensional fluid, Phys. Rev. E51, 423 (1995)

work page 1995

[17] [17]

B. Lin, B. Cui, X. Xu, R. Zangi, H. Diamant, and S. A. Rice, Divergence of the long-wavelength collective diffu- sion coefficient in quasi-one- and quasi-two-dimensional colloidal suspensions, Phys. Rev. E89, 022303 (2014)

work page 2014

[18] [18]

Dom´ ınguez, Signature of time-dependent hydrody- namic interactions on collective diffusion in colloidal monolayers, Phys

A. Dom´ ınguez, Signature of time-dependent hydrody- namic interactions on collective diffusion in colloidal monolayers, Phys. Rev. E90, 062314 (2014)

work page 2014

[19] [19]

Panzuela, R

S. Panzuela, R. P. Pel´ aez, and R. Delgado-Buscalioni, Collective colloid diffusion under soft two–dimensional confinement, Phys. Rev. E95, 012602 (2017)

work page 2017

[20] [20]

Bleibel, A

J. Bleibel, A. Dom´ ınguez, and M. Oettel, Onset of anomalous diffusion in colloids confined to quasimono- layers, Phys. Rev. E95, 032604 (2017)

work page 2017

[21] [21]

Panzuela and R

S. Panzuela and R. Delgado-Buscalioni, Solvent hydro- dynamics enhances the collective diffusion of membrane lipids, Phys. Rev. Lett.121, 048101 (2018)

work page 2018

[22] [22]

Dom´ ınguez, Theory of anomalous collective diffusion in colloidal monolayers on a spherical interface, Phys

A. Dom´ ınguez, Theory of anomalous collective diffusion in colloidal monolayers on a spherical interface, Phys. Rev. E97, 022607 (2018)

work page 2018

[23] [23]

R. P. Pel´ aez, F. B. Usabiaga, S. Panzuela, Q. Xiao, R. Delgado-Buscalioni, and A. Donev, Hydrodynamic fluctuations in quasi-two dimensional diffusion, J. Stat. Mech.2018, 063207 (2018)

work page 2018

[24] [24]

ˇSkult´ ety, D

V. ˇSkult´ ety, D. B´ ardfalvy, J. Stenhammar, C. Nar- dini, and A. Morozov, Hydrodynamic instabilities in a two-dimensional sheet of microswimmers embedded in a three-dimensional fluid, Journal of Fluid Mechanics980, A28 (2024)

work page 2024

[25] [25]

Donev, T

A. Donev, T. G. Fai, and E. Vanden-Eijnden, A re- versible mesoscopic model of diffusion in liquids: from giant fluctuations to Fick’s law, J. Stat. Mech.2014, P04004 (2014)

work page 2014

[26] [26]

Dom´ ınguez, in preparation, (2025)

A. Dom´ ınguez, in preparation, (2025)

work page 2025

[27] [27]

Manikantan and T

H. Manikantan and T. M. Squires, Surfactant dynamics: hidden variables controlling fluid flows, J. Fluid Mech. 892, P1 (2020)

work page 2020

[28] [28]

Molaei, N

M. Molaei, N. G. Chisholm, J. Deng, J. C. Crocker, and K. J. Stebe, Interfacial Flow around Brownian Colloids, Physical Review Letters126, 228003 (2021)

work page 2021