arxiv: 2605.03404 · v1 · submitted 2026-05-05 · ❄️ cond-mat.soft · cond-mat.stat-mech

Recognition: unknown

Sparkling bubbles in chiral active fluids

Alessandro Petrini, Lorenzo Caprini, Rapha\"el Maire, Umberto Marini Bettolo Marconi

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:35 UTC · model grok-4.3

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

keywords chiral active fluidsodd interactionsrotating bubblessparkling behaviorhydrodynamic theoryactive matterinhomogeneous phases

0 comments

The pith

Chirality in inertial active fluids produces rotating bubbles that break up and reform spontaneously in a sparkling pattern.

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

This paper investigates inertial chiral active fluids made of repulsive particles that exchange angular momentum through odd transverse forces. Chirality drives formation of an inhomogeneous phase containing rotating bubbles, with this structure preferred at an optimal packing fraction. The bubbles prove dynamically unstable, repeatedly fragmenting and reforming while the system remains in steady state, yielding a sparkling appearance similar to supersaturated liquids. A coarse-grained hydrodynamic theory obtained from the underlying particle model accounts for both the bubble creation and the instability, exposing the nonlinear character of the collective dynamics and indicating that experiments with granular spinners or spinning colloids could confirm the effect.

Core claim

Chirality induces an inhomogeneous phase consisting of rotating bubbles whose formation is favored at an optimal packing fraction. In this regime bubbles may be dynamically unstable, breaking up and reforming in the steady state and thereby showing spontaneous sparkling-like behavior reminiscent of supersaturated liquids. Bubbles and sparkling bubbles are predicted by a coarse-grained hydrodynamic theory derived from the microscopic particle model with odd interactions, revealing the intrinsic non-linearity of these collective phenomena.

What carries the argument

Coarse-grained hydrodynamic theory derived from the microscopic model of repulsive particles with odd transverse forces, which produces the inhomogeneous phase of rotating bubbles and their dynamic instability.

If this is right

Rotating bubbles appear in an inhomogeneous phase induced by chirality.
Bubble formation is favored at an optimal packing fraction.
Bubbles display dynamic instability and spontaneously break up and reform in steady state.
The sparkling behavior arises from the intrinsic nonlinearity captured by the hydrodynamic theory.
The phenomena are expected to be verifiable in granular spinners or spinning colloids.

Where Pith is reading between the lines

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

Similar sparkling dynamics may appear in other active systems that include transverse or odd forces.
Bubble lifetime and reformation rate could be tuned by varying particle density or the strength of the odd interactions.
The instability mechanism might connect to other non-equilibrium phase behaviors such as activity-driven clustering.
Quantitative tests could measure how the frequency of sparkling events scales with packing fraction according to the theory.

Load-bearing premise

The coarse-grained hydrodynamic theory derived from the microscopic particle model with odd interactions accurately captures bubble formation and instability without extra microscopic details or fitting parameters that would change the qualitative behavior.

What would settle it

Direct observation in a granular spinner or spinning colloid experiment of whether rotating bubbles at optimal packing fraction repeatedly break apart and reform rather than remaining stable over long times.

Figures

Figures reproduced from arXiv: 2605.03404 by Alessandro Petrini, Lorenzo Caprini, Rapha\"el Maire, Umberto Marini Bettolo Marconi.

**Figure 1.** Figure 1: Sparkling bubbles induced by odd interactions (SBIO). (a)–(c) Snapshot configurations of the homogeneous, bubble view at source ↗

**Figure 2.** Figure 2: BIO and SBIO phases. (a) Average vorticity view at source ↗

**Figure 3.** Figure 3: Vorticity induced by sparkling behavior. (a) Vorticity distribution view at source ↗

read the original abstract

We study an inertial chiral active fluid, formed by repulsive particles that transfer angular momentum through odd interactions, i.e. transverse forces. Chirality induces an inhomogeneous phase, consisting of rotating bubbles, whose formation is favored at an optimal packing fraction. In this regime, we discover that bubbles may be dynamically unstable, breaking up and reforming in the steady state, thereby showing a spontaneous sparkling-like behavior reminiscent of supersaturated liquids. Bubbles and sparkling bubbles are predicted by a coarse-grained hydrodynamic theory, revealing the intrinsic non-linearity of these collective phenomena, and call for experimental verifications in granular spinners or spinning colloids.

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 reports rotating bubbles in inertial chiral active fluids that form at optimal density and then sparkle via spontaneous break-up and reformation, captured by a coarse-grained hydro theory from the odd-interaction particle model.

read the letter

The central new observation is that chirality plus inertia produces an inhomogeneous phase of rotating bubbles whose dynamic instability leads to a steady-state sparkling behavior, not previously reported in this class of systems. The work links a microscopic model of repulsive particles with transverse odd forces to a hydrodynamic description that reproduces both bubble formation and the instability, highlighting the role of nonlinearity in the collective dynamics. That connection is the main substantive advance over earlier clustering or odd-viscosity studies in active matter. The call for experiments in granular spinners or spinning colloids is straightforward and useful. The hydro prediction is presented as emerging directly from the particle model without obvious fitting parameters, which strengthens the claim if the closure steps are shown explicitly. The stress-test concern about whether the coarse-graining preserves the relevant nonlinearities is reasonable to raise, but the paper appears to address it by deriving the instability criterion from the same odd-stress terms that drive the bubbles in the first place; no obvious post-hoc adjustment is visible. Minor soft spot is that the abstract and main claims rest on the fidelity of that derivation, so readers will want the explicit equations and parameter ranges in the supplement or main text to verify robustness. This is squarely for people working on chiral or odd active fluids and non-equilibrium hydrodynamics. A reader already following odd interactions or inertial active matter will pick up the new instability mechanism and the suggested experimental route. It is coherent on its own terms and deserves peer review; the idea is sharp enough and the proposed link between scales is worth referee scrutiny even if the derivation needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript studies an inertial chiral active fluid of repulsive particles interacting via odd transverse forces. Chirality induces an inhomogeneous phase of rotating bubbles favored at an optimal packing fraction; in this regime the bubbles are reported to be dynamically unstable, repeatedly breaking up and reforming to produce a spontaneous 'sparkling' steady state reminiscent of supersaturated liquids. Both the bubble formation and the instability are claimed to be predicted by a coarse-grained hydrodynamic theory derived from the underlying particle model, underscoring the intrinsic nonlinearity of the collective phenomena.

Significance. If the hydrodynamic derivation faithfully reproduces the instability without uncontrolled approximations or fitting, the result would be significant for active-matter hydrodynamics. It supplies an analytic route to a nonlinear instability in an odd-stress system and links microscopic chirality to macroscopic sparkling dynamics, with clear experimental implications for granular spinners and spinning colloids. The existence of a closed hydrodynamic description itself constitutes a strength, as it enables falsifiable predictions beyond direct simulation.

major comments (2)

[§3] §3 (Hydrodynamic derivation): The central claim that the coarse-grained theory predicts both bubble formation and the subsequent dynamic instability rests on the closure and truncation steps that map the microscopic odd-interaction model onto the hydrodynamic equations. Without the explicit form of the odd-stress tensor, the density-gradient expansion, and the retained nonlinear terms, it is impossible to verify that the instability criterion is not an artifact of the approximation. This mapping is load-bearing for the assertion that the sparkling behavior is intrinsic.
[§4] §4 (Simulation-hydro comparison): The manuscript asserts that the hydrodynamic instability reproduces the break-up/reformation cycles seen in particle simulations, yet no quantitative comparison (e.g., growth rates, selected wavelengths, or parameter values entering the instability threshold) is provided. Direct side-by-side plots or tables of the dispersion relation versus simulation observables are required to confirm that the coarse-graining preserves the relevant nonlinearities.

minor comments (2)

The abstract and introduction use 'sparkling-like' without a quantitative metric (e.g., bubble lifetime distribution or reformation frequency); a brief operational definition would improve clarity.
Figure captions should explicitly state the packing fraction, inertia parameter, and odd-interaction strength used in each panel to allow readers to reproduce the reported optimal-packing regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the constructive suggestions. We address the major comments point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (Hydrodynamic derivation): The central claim that the coarse-grained theory predicts both bubble formation and the subsequent dynamic instability rests on the closure and truncation steps that map the microscopic odd-interaction model onto the hydrodynamic equations. Without the explicit form of the odd-stress tensor, the density-gradient expansion, and the retained nonlinear terms, it is impossible to verify that the instability criterion is not an artifact of the approximation. This mapping is load-bearing for the assertion that the sparkling behavior is intrinsic.

Authors: We agree that additional explicit details on the derivation steps will strengthen the presentation. In the revised manuscript we will expand the relevant section to display the explicit form of the odd-stress tensor, the order of the density-gradient expansion employed in the coarse-graining, and the complete list of nonlinear terms retained after truncation. These additions will make the mapping from the microscopic model transparent and allow direct verification that the instability criterion originates from the retained nonlinearities rather than from uncontrolled approximations. revision: yes
Referee: [§4] §4 (Simulation-hydro comparison): The manuscript asserts that the hydrodynamic instability reproduces the break-up/reformation cycles seen in particle simulations, yet no quantitative comparison (e.g., growth rates, selected wavelengths, or parameter values entering the instability threshold) is provided. Direct side-by-side plots or tables of the dispersion relation versus simulation observables are required to confirm that the coarse-graining preserves the relevant nonlinearities.

Authors: We accept that quantitative side-by-side validation is necessary. In the revised manuscript we will add figures that overlay the dispersion relation obtained from linear stability analysis of the hydrodynamic equations with growth rates extracted from the particle simulations. We will also include a table that reports selected wavelengths, instability thresholds, and the corresponding parameter values (packing fraction, interaction strength) for both the hydrodynamic prediction and the simulations, thereby confirming that the coarse-graining retains the essential nonlinearities responsible for the sparkling dynamics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; hydro derivation presented as independent from micro model

full rationale

The paper derives a coarse-grained hydrodynamic theory from the microscopic inertial chiral fluid model with odd transverse forces, then uses the resulting equations to predict both bubble formation at optimal packing and their dynamic instability leading to sparkling behavior. The abstract explicitly frames these as predictions revealing intrinsic nonlinearity, with no quoted steps showing self-definition (e.g., instability criterion defined in terms of itself), fitted parameters from data renamed as predictions, or load-bearing self-citations whose prior results reduce to the current claim. The mapping from micro to macro is asserted to preserve relevant nonlinearities without evidence of uncontrolled approximations that would force the outcome by construction. This is the most common honest finding for such coarse-graining papers when the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The model presupposes repulsive particles with odd transverse forces and inertial dynamics; the hydrodynamic closure is an additional modeling step whose assumptions are not stated.

pith-pipeline@v0.9.0 · 5403 in / 1048 out tokens · 37660 ms · 2026-05-07T13:35:11.466168+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

87 extracted references · 5 canonical work pages

[1]

Marchetti, J

M. Marchetti, J. Joanny, S. Ramaswamy, T. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013)

2013
[2]

Elgeti, R

J. Elgeti, R. G. Winkler, and G. Gompper, Rep. Prog. Phys.78, 056601 (2015)

2015
[3]

Bechinger, R

C. Bechinger, R. Di Leonardo, H. L¨ owen, C. Reichhardt, G. Volpe, and G. Volpe, Rev. Mod. Phys.88, 045006 (2016)

2016
[4]

Fodor, R

´E. Fodor, R. L. Jack, and M. E. Cates, Annual Review of Condensed Matter Physics13, 215 (2022)

2022
[5]

M. E. Cates and J. Tailleur, Annu. Rev. Condens. Matter Phys.6, 219 (2015)

2015
[6]

Fily and M

Y. Fily and M. C. Marchetti, Phys. Rev. Lett.108, 235702 (2012)

2012
[7]

Digregorio, D

P. Digregorio, D. Levis, A. Suma, L. F. Cugliandolo, G. Gonnella, and I. Pagonabarraga, Phys. Rev. Lett. 121, 098003 (2018)

2018
[8]

Drescher, K

K. Drescher, K. C. Leptos, I. Tuval, T. Ishikawa, T. J. Pedley, and R. E. Goldstein, Phys. Rev. Lett.102, 168101 (2009)

2009
[9]

A. J. Mathijssen, N. Figueroa-Morales, G. Junot, ´E. Cl´ ement, A. Lindner, and A. Z¨ ottl, Nat. Commun. 10, 3434 (2019)

2019
[10]

Pellicciotta, O

N. Pellicciotta, O. S. Bagal, M. C. Cannarsa, S. Bianchi, and R. Di Leonardo, Phys. Rev. Lett.135, 138302 (2025)

2025
[11]

Buttinoni, J

I. Buttinoni, J. Bialk´ e, F. K¨ ummel, H. L¨ owen, C. Bechinger, and T. Speck, Phys. Rev. Lett.110, 238301 (2013)

2013
[12]

Bricard, J.-B

A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Nature503, 95 (2013)

2013
[13]

Ginot, I

F. Ginot, I. Theurkauff, D. Levis, C. Ybert, L. Bocquet, L. Berthier, and C. Cottin-Bizonne, Phys. Rev. X5, 011004 (2015)

2015
[14]

Ketzetzi, L

S. Ketzetzi, L. Caprini, V. Willems, L. Alvarez, H. L¨ owen, and L. Isa, ACS nano19, 29430 (2025)

2025
[15]

I. S. Aranson, D. Volfson, and L. S. Tsimring, Phys. Rev. E75, 051301 (2007)

2007
[16]

Kudrolli, Phys

A. Kudrolli, Phys. Rev. Lett.104, 088001 (2010)

2010
[17]

Kumar, H

N. Kumar, H. Soni, S. Ramaswamy, and A. Sood, Nat. Commun.5, 4688 (2014)

2014
[18]

Koumakis, A

N. Koumakis, A. Gnoli, C. Maggi, A. Puglisi, and R. Di Leonardo, New J. Phys.18, 113046 (2016)

2016
[19]

Baconnier, D

P. Baconnier, D. Shohat, C. H. L´ opez, C. Coulais, V. D´ emery, G. D¨ uring, and O. Dauchot, Nat. Phys.18, 1234 (2022)

2022
[20]

M. ´A. L´ opez-Casta˜ no, A. M. Seco, A. M. Seco, ´A. Rodr´ ıguez-Rivas, and F. Vega Reyes, Phys. Fluids 35, 033321 (2023)

2023
[21]

A. P. Antonov, M. Musacchio, H. L¨ owen, and L. Caprini, Nat. Commun.16, 7235 (2025)

2025
[22]

Cavagna and I

A. Cavagna and I. Giardina, Annu. Rev. Condens. Matter Phys.5, 183 (2014)

2014
[23]

L¨ owen, Eur

H. L¨ owen, Eur. Phys. J.225, 2319 (2016)

2016
[24]

Liebchen and D

B. Liebchen and D. Levis, Europhys. Lett.139, 67001 (2022)

2022
[25]

Mecke, J

J. Mecke, J. O. Nketsiah, R. Li, and Y. Gao, National Science Open3, 20230086 (2024)

2024
[26]

Scholz, M

C. Scholz, M. Engel, and T. P¨ oschel, Nat. Commun.9, 931 (2018)

2018
[27]

Vega Reyes, M

F. Vega Reyes, M. A. L´ opez-Casta˜ no, and´A. Rodr´ ıguez- Rivas, Commun. Phys.5, 256 (2022)

2022
[28]

Siebers, A

F. Siebers, A. Jayaram, P. Bl¨ umler, and T. Speck, Sci. Adv.9, eadf5443 (2023)

2023
[29]

Farhadi, S

S. Farhadi, S. Machaca, J. Aird, B. O. T. Maldonado, S. Davis, P. E. Arratia, and D. J. Durian, Soft Matter 14, 5588 (2018)

2018
[30]

Caprini, I

L. Caprini, I. Abdoli, U. Marini Bettolo Marconi, and H. L¨ owen, Newton1, 100253 (2025)

2025
[31]

Ebbens, R

S. Ebbens, R. A. Jones, A. J. Ryan, R. Golestanian, and J. R. Howse, Phys. Rev. E82, 015304 (2010)

2010
[32]

K¨ ummel, B

F. K¨ ummel, B. Ten Hagen, R. Wittkowski, I. Buttinoni, R. Eichhorn, G. Volpe, H. L¨ owen, and C. Bechinger, Phys. Rev. Lett.110, 198302 (2013). 9

2013
[33]

Woolley, Reproduction126, 259 (2003)

D. Woolley, Reproduction126, 259 (2003)

2003
[34]

Lauga, W

E. Lauga, W. R. DiLuzio, G. M. Whitesides, and H. A. Stone, Biophys. J.90, 400 (2006)

2006
[35]

Liao and S

G.-J. Liao and S. H. Klapp, Soft Matter14, 7873 (2018)

2018
[36]

Ma and R

Z. Ma and R. Ni, J. Chem. Phys.156, 021102 (2022)

2022
[37]

Bickmann, S

J. Bickmann, S. Br¨ oker, J. Jeggle, and R. Wittkowski, J. Chem. Phys.156, 194904 (2022)

2022
[38]

Kuroda, T

Y. Kuroda, T. Kawasaki, and K. Miyazaki, Phys. Rev. Res.7, L012048 (2025)

2025
[39]

Liebchen and D

B. Liebchen and D. Levis, Phys. Rev. Lett.119, 058002 (2017)

2017
[40]

Zhang, A

B. Zhang, A. Sokolov, and A. Snezhko, Nat. Commun. 11, 4401 (2020)

2020
[41]

K. L. Kreienkamp and S. H. Klapp, New J. Phys.24, 123009 (2022)

2022
[42]

Liebchen, M

B. Liebchen, M. E. Cates, and D. Marenduzzo, Soft Matter12, 7259 (2016)

2016
[43]

Pisegna and S

G. Pisegna and S. Saha, arXiv preprint arXiv:2509.07630 (2025)

work page arXiv 2025
[44]

Huang, A

Z.-F. Huang, A. M. Menzel, and H. L¨ owen, Phys. Rev. Lett.125, 218002 (2020)

2020
[45]

N. Kruk, J. A. Carrillo, and H. Koeppl, Phys. Rev. E 102, 022604 (2020)

2020
[46]

Liao and S

G.-J. Liao and S. H. Klapp, Soft Matter17, 6833 (2021)

2021
[47]

A. Shee, S. Henkes, and C. Huepe, Soft Matter20, 7865 (2024)

2024
[48]

Ishikawa, T

T. Ishikawa, T. Pedley, K. Drescher, and R. E. Goldstein, J. Fluid Mech.903, A11 (2020)

2020
[49]

M. A. L´ opez-Casta˜ no, A. M. Seco, A. M. Seco, ´A. Rodr´ ıguez-Rivas, and F. Vega Reyes, Phys. Rev. Res. 4, 033230 (2022)

2022
[50]

Caprini, A

L. Caprini, A. Petrini, and U. Marini Bettolo Marconi, J. Stat. Mech.: Theory Exp.2026, 024001 (2026)

2026
[51]

C. B. Caporusso, G. Gonnella, and D. Levis, Phys. Rev. Lett.132, 168201 (2024)

2024
[52]

B. C. Van Zuiden, J. Paulose, W. T. Irvine, D. Bartolo, and V. Vitelli, Proc. Natl. Acad. Sci. U.S.A.113, 12919 (2016)

2016
[53]

Caprini, U

L. Caprini, U. M. B. Marconi, B. Liebchen, and H. L¨ owen, arXiv preprint arXiv:2509.05053 (2025)

work page arXiv 2025
[54]

T. H. Tan, A. Mietke, J. Li, Y. Chen, H. Higinbotham, P. J. Foster, S. Gokhale, J. Dunkel, and N. Fakhri, Na- ture607, 287 (2022)

2022
[55]

Scheibner, A

C. Scheibner, A. Souslov, D. Banerjee, P. Sur´ owka, W. T. Irvine, and V. Vitelli, Nat. Phys.16, 475 (2020)

2020
[56]

S. Kole, G. P. Alexander, S. Ramaswamy, and A. Maitra, Phys. Rev. Lett.126, 248001 (2021)

2021
[57]

Braverman, C

L. Braverman, C. Scheibner, B. VanSaders, and V. Vitelli, Phys. Rev. Lett.127, 268001 (2021)

2021
[58]

Shankar and L

S. Shankar and L. Mahadevan, Nat. Phys.20, 1501 (2024)

2024
[59]

S. Kole, G. P. Alexander, A. Maitra, and S. Ramaswamy, PNAS Nexus3, pgae398 (2024)

2024
[60]

C.-T. Lee, T. C. Lubensky, and T. Markovich, arXiv preprint arXiv:2508.04468 (2025)

work page arXiv 2025
[61]

Fruchart, C

M. Fruchart, C. Scheibner, and V. Vitelli, Annu. Rev. Condens. Matter Phys.14, 471 (2023)

2023
[62]

Huang, M

Z.-F. Huang, M. t. Vrugt, R. Wittkowski, and H. L¨ owen, Proc. Natl. Acad. Sci. U.S.A.122, e2511350122 (2025)

2025
[63]

Caprini and U

L. Caprini and U. Marini Bettolo Marconi, New J. Phys. 27, 054401 (2025)

2025
[64]

Caprini and U

L. Caprini and U. Marini Bettolo Marconi, J. Chem. Phys.162, 161101 (2025)

2025
[65]

U. M. B. Marconi, A. Petrini, R. Maire, and L. Caprini, arXiv preprint arXiv:2601.19432 (2026)

work page arXiv 2026
[66]

Shen and J

Z. Shen and J. S. Lintuvuori, Physical Review Letters 130, 188202 (2023)

2023
[67]

Digregorio, I

P. Digregorio, I. Pagonabarraga, and F. Vega Reyes, arXiv preprint arXiv:2504.08533 (2025)

work page arXiv 2025
[68]

Guo, J.-j

R.-x. Guo, J.-j. Li, and B.-q. Ai, Phys. Rev. E111, 014105 (2025)

2025
[69]

H. C. Andersen, J. D. Weeks, and D. Chandler, Phys. Rev. A4, 1597 (1971)

1971
[70]

Massana-Cid, D

H. Massana-Cid, D. Levis, R. J. H. Hern´ andez, I. Pag- onabarraga, and P. Tierno, Phys. Rev. Research3, L042021 (2021)

2021
[71]

On the pressure developed in a liquid dur- ing the collapse of a spherical cavity: Philosophical mag- azine series 6, 34, 94–98,

L. Rayleigh, “On the pressure developed in a liquid dur- ing the collapse of a spherical cavity: Philosophical mag- azine series 6, 34, 94–98,” (1917)

1917
[72]

M. S. Plesset, J. Appl. Mech.16, 228 (1949)

1949
[73]

Banerjee, A

D. Banerjee, A. Souslov, A. G. Abanov, and V. Vitelli, Nat. Commun.8, 1573 (2017)

2017
[74]

Markovich and T

T. Markovich and T. C. Lubensky, Phys. Rev. Lett.127, 048001 (2021)

2021
[75]

Reichhardt and C

C. Reichhardt and C. Reichhardt, EPL137, 66004 (2022)

2022
[76]

X. Lou, Q. Yang, Y. Ding, P. Liu, K. Chen, X. Zhou, F. Ye, R. Podgornik, and M. Yang, Proc. Natl. Acad. Sci. U.S.A.119, e2201279119 (2022)

2022
[77]

Hosaka, R

Y. Hosaka, R. Golestanian, and A. Vilfan, Phys. Rev. Lett.131, 178303 (2023)

2023
[78]

Machado Monteiro, A

G. Machado Monteiro, A. G. Abanov, and S. Ganeshan, SciPost Phys.14, 103 (2023)

2023
[79]

Markovich and T

T. Markovich and T. C. Lubensky, Phys. Rev. E112, 035409 (2025)

2025
[80]

A. R. Poggioli and D. T. Limmer, Phys. Rev. Lett.130, 158201 (2023)

2023

Showing first 80 references.