Pith. sign in

REVIEW 2 major objections 4 minor 160 references

Capillary wave formation in conserved active emulsions

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read In a conserved active emulsion, repulsive chemotaxis causes a stationary or oscillatory interfacial instability; the oscillatory instability creates persistent capillary waves with theoretically predicted and numerically verified wavelength and velocity.

desk verdict A genuinely new sharp-interface dispersion framework for capillary waves in a conserved-plus-nonconserved active emulsion pair; the numerics support the instability classification, but the quantitative predictions at small l need firmer grounding before they are fully trusted. read the letter →

arxiv 2505.20028 v1 pith:BNF2TP7J submitted 2025-05-26 cond-mat.soft

Florian Ra{ss}hofer , Simon Bauer , Alexander Ziepke , Ivan Maryshev , Erwin Frey This is my paper · ORCID
classification cond-mat.soft
keywords capillarywavesactivealonginstabilityinterfaceboundariescondensates
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

The paper studies a two-ingredient model of an active liquid. One ingredient, call it droplet material, can separate into dense and dilute phases, just like oil and water. The second ingredient is a chemical signal that the droplet material also produces. In addition, droplet material moves away from places where the signal is strong, a repulsive chemotaxis. Because the signal diffuses slowly, there is a delay: material avoids the chemical, but the chemical is made by the material. This predator-prey-like loop can make a flat interface between dense and dilute phases unstable.

The key new phenomenon is a capillary wave, a traveling ripple along the interface. Unlike ordinary thermal capillary waves, these ripples do not relax away; they propagate steadily and can carry material along the interface. The authors linearize the model around a stationary interface and derive how fast each ripple mode grows, which mode is fastest, and how fast it travels. Finite-element simulations confirm the predicted phase diagram, mode numbers, and wave speeds to about 12 percent mean deviation.

The paper also shows that, near onset, waves can saturate into conveyor belts or rotating gears, and that a gear placed between two walls can crawl if the walls are different. The long-term behavior is more complex: monochromatic waves eventually mix into several modes. The analysis assumes the chemical varies slowly compared with the interface thickness, and some speed comparisons were made only for parameter sets that actually produced stable waves, so the strongest quantitative claims concern the initial linear instability.

Extended reading notes

Core claim

The paper's load-bearing claim is that repulsive chemotactic coupling between a conserved phase-separating field and a non-conserved chemical field produces two interfacial instabilities, one stationary (Mullins-Sekerka-like) and one oscillatory, and that the oscillatory instability yields persistent capillary waves. The quantitative version is the self-consistency condition Eq. (56), whose fastest-growing mode k_max predicts the capillary wavelength and whose Im(sigma_k)/k predicts the phase velocity (Eq. (60), Sec. V C). If the paper is correct, active emulsions with slow chemical signaling exhibit self-sustained traveling interfacial waves with linearly predictable onset, wavelength, and speed.

Load-bearing premise

The sharp-interface boundary conditions assume the chemical field is essentially constant across the interface (l >> 1), so equilibrium Gibbs-Thomson relations, Eqs. (53)-(54) and App. B, can be imposed on a non-equilibrium interface. The dispersion relation and all derived onset, wavelength, and velocity predictions inherit this assumption. The authors apply the theory even for l as small as 1.75, where they note accuracy degrades; if the local equilibrium boundary conditions fail at intermediate l, the quantitative predictions would shift.

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 / 4 minor

Summary. The paper studies a two-field model in which a conserved, phase-separating density field φ is coupled to a non-conserved chemical field ψ through repulsive chemotaxis, Eqs. (22). The authors identify two types of interfacial instabilities of flat and circular domains: a stationary, Mullins-Sekerka-like instability and an oscillatory instability that can produce persistent capillary waves. The central theoretical object is the sharp-interface self-consistency relation Eq. (56), from which they extract onset conditions, the fastest-growing wavenumber k_max, and the phase velocity v_k = |Im(σ_k)|/k, Eq. (60). These predictions are compared with finite-element simulations of the full model, with good agreement for phase boundaries in Fig. 8 and mode numbers in Fig. 9 for small n, and with mean 12% deviation for wave velocities in Fig. 10. The paper also characterizes the long-time secondary instability of capillary waves and demonstrates boundary-induced crawling of rotating gears.

Significance. If the central claim holds, the paper provides a general and quantitative framework for interfacial dynamics in conserved active emulsions: two distinct instabilities, one stationary and one oscillatory, with linear predictions for onset, wavelength, and phase velocity that are validated in selected regimes. The derivation is internally consistent, contains no fitted parameters in the eigenvalue problem, and reproduces the equilibrium |k|^3 capillary decay in the Pe=0 limit, Eq. (57). A clear strength is the reproducibility infrastructure: COMSOL simulation files, Mathematica notebooks, and Julia evaluation scripts are deposited on Zenodo (Ref. [105]). The significance is, however, conditional on the sharp-interface expansion being controlled in the regime where capillary waves are actually characterized, which is the main open risk assessed in the major comments.

major comments (2)
  1. [Sec. IV A 1, App. B, Sec. V A, Figs. 8--11] The sharp-interface boundary conditions, Eqs. (53)--(54), are derived under the assumption l >> 1, i.e., the chemical field is essentially constant across the interface and chemotactic fluxes are negligible on the interface width scale; this is stated explicitly in App. B and Sec. III A. Yet the oscillatory instability and all quantitative wave predictions are characterized at l = 1.75 (Figs. 4, 5, 10, 11) and for velocities in the range l ∈ [1, 3] (Fig. 10). At these values the diffusion length of the chemical field is comparable to the interface width, so the local-equilibrium Gibbs-Thomson condition is not under control. The paper acknowledges that accuracy degrades for small l (Sec. V A, Sec. V B), but it still uses Eq. (56) to make quantitative statements about k_max and v_kmax in this regime. This is load-bearing for the central claim that Eq. (56) predicts onset, wavelength, and velocity of capillary waves. I recommend a direct check: perform a linear stability analysis of the full diffuse-interface equations (22) around the numerical stationary profiles for the same parameters and compare the resulting σ_k with Eq. (56); alternatively, demonstrate that the predictions are robust at larger l (e.g., l ≥ 5) and restrict the quantitative claims to that regime.
  2. [Sec. V C, Fig. 10, App. K] The validation of the capillary wave velocity is based on 15 parameter combinations that were selected post hoc: the authors state that M was chosen by requiring a stable wave pattern to emerge, and that Pe and l were adjusted if no stable pattern was found (Sec. V C, App. K). Thus the comparison in Fig. 10(b) tests Eq. (60) only on favorable cases, not as an out-of-sample prediction. Moreover, the measured velocities are obtained from finite-amplitude, transiently stable wave patterns, whereas the theoretical prediction is linear and infinitesimal. The 12% mean deviation is encouraging, but the evidence does not establish that v_kmax predicts the velocity generically in the oscillatory regime. A cleaner test would be to fix all parameters a priori from the linear stability diagram, or to measure the phase velocity of small-amplitude perturbations before nonlinear saturation sets in.
minor comments (4)
  1. [Sec. V C / Fig. 14] The angular velocities for the three rightmost data points in Fig. 14(b) are stated in the text to have been obtained by extrapolation from previous points; this should also be stated in the figure caption, and the reported value 2πv/ω ≈ 2.33 should be flagged as relying on an extrapolated ω.
  2. [Sec. VIII D, Eq. (62)] The symbol M is used both for the dimensionless mobility in Eq. (22) and as a placeholder coupling in the general model class Eq. (62); renaming one of them would avoid confusion.
  3. [Sec. V B, Fig. 9] The statement that the theory 'slightly underestimates the wavenumber' at n ≥ 5 is plausible, but the comparison in Fig. 9 uses mode numbers inferred from nonlinear growth of white-noise perturbations; a direct visualization of the full dispersion relation against the simulation spectra would make the comparison more transparent.
  4. [Sec. II] The phrase 'Mullins-Sekerka-like' is used for the stationary instability, but the physical origin here is long-range chemical repulsion rather than the diffusion field around a moving solidification front; a sentence qualifying the analogy would help readers.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Eq. (56) is an in-paper eigenvalue problem with no fitted constants; wavelength and wave-speed predictions are validated against independent finite-element simulations of the full model. Self-citations are motivational, not load-bearing.

full rationale

The paper's load-bearing claims—onset of instability, capillary wavelength Λ = 2π/k_max, and phase velocity v_kmax—are not reductions of inputs. The dispersion relation follows from an in-paper eigenvalue problem: linearized bulk dynamics, Eqs. (49); matching and Gibbs-Thomson boundary conditions, Eqs. (53)–(54), derived within App. B; and the mass-conservation self-consistency condition, Eq. (56). No parameter is fitted to simulation outputs; σ_k, k_max (Eq. 58), and v_k = |Im(σ_k)|/k (Eq. 60) are computed from the model parameters (M, P e, l, Δϕ) alone. Validation is genuinely independent: the full nonlinear PDEs, Eq. (22), are simulated with COMSOL, the simulated fastest mode is extracted from the interface power spectrum via an independent exponential fit (Eq. 59, App. J), and the 12% mean velocity deviation (Fig. 10) is a test outcome, not a constructed agreement. The mildest circularity-adjacent items are self-citations: the model form 'follow[s] the derivation given in Refs. [15, 16]' (Sec. I), and related group work is cited for context ([60,61,65,66,94,102,110]). These are motivational, not load-bearing; neither the oscillatory-instability mechanism nor the dispersion relation is imported from them. The manuscript's explicit caveats—the quasi-static total-density reduction is described as 'an uncontrolled approximation' (Sec. I B); the sharp-interface boundary conditions are justified only for l >> 1 (App. B, Sec. III A); the theory 'tends to slightly overestimate the regime of the oscillatory instability' where l ≈ 1 (Sec. V A); velocities are compared 'in a parameter regime where the sharp interface approximation loses accuracy' (Sec. V C); and nonlinear stabilization of finite-amplitude waves is 'beyond the scope of this manuscript' (Sec. VI)—flag a genuine validity risk at the l ≈ 1.75 capillary-wave operating point, but they do not make any equation equivalent to its input by construction. No fitted parameter is renamed as a prediction, and no result is imported through a self-citation chain. Score 1 reflects the presence of non-load-bearing self-citations in model motivation and context; the central derivation is self-contained.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

There are no fitted-to-data constants; instead the dimensionless control parameters (Pe, l, M) are scanned by hand, and the central predictions rest on four modeling assumptions: equal substrate/product diffusivities with fast chemical relaxation, linear reaction kinetics, constant mobility with K(rho)=K rho, and the sharp-interface local-equilibrium closure. No new entities are posited.

free parameters (4)
  • Péclet number Pe = Scanned over about 0.1 to 1; wave regime near Pe ~ 0.4
    Dimensionless chemotactic strength; no experimental fit, chosen by hand to explore regimes.
  • Diffusion length l = Scanned over about 1 to 10; wave regime near l ~ 1.75 to 3
    Dimensionless chemical screening length; central control parameter for onset, wavelength, and velocity.
  • Relative mobility M = Scanned over about 10 to 27; wave regime M >= O(10)
    Ratio of density diffusivity to chemical diffusivity; slow signal propagation is required for oscillatory instability.
  • Density contrast Delta phi and critical density phi_c = Delta phi = 0.6, phi_c = 0.7 in simulations
    Chosen by hand for the representative parameter sets; sets surface tension and coexistence densities.
assumptions (5)
  • domain assumption Substrate and product diffusivities are equal, Ds = Dp, allowing elimination of the substrate via s = n0 - p (Eq. 13).
    Section I B; the authors explicitly call the resulting quasi-steady-state approximation uncontrolled.
  • domain assumption The reaction kinetics are linear, R = beta rho - lambda p (Eq. 17b).
    Chosen as a minimal mass-action model for enzyme-catalyzed conversion; the authors present it as a linearized limit of more general kinetics.
  • domain assumption Constant mobility M and chemotactic sensitivity K(rho) = K rho (Eq. 22a).
    Simplifies the model to a generalized Keller-Segel-Cahn-Hilliard form; the authors acknowledge more general functional forms are possible.
  • domain assumption Sharp interface limit l >> 1 with local equilibrium and equilibrium Gibbs-Thomson boundary conditions at the interface (Eqs. 53-54, App. B).
    Load-bearing for the dispersion relation; the authors apply it even where l ~ 1.75 and note accuracy degrades.
  • domain assumption The free energy is symmetric with phi_c = 0.7 for the main simulations (Eq. 24).
    A symmetric Ginzburg-Landau double-well potential is assumed for tractability; the authors note generalizations are possible.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Capillary wave formation in conserved active emulsions." pith.science (2026). https://pith.science/paper/BNF2TP7J

@misc{pith2026250520028,
  author       = {Pith},
  title        = {Pith review of: Capillary wave formation in conserved active emulsions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BNF2TP7J}},
  note         = {Machine review of arXiv:2505.20028}
}
read the original abstract

The dynamics of phase-separated interfaces shape the behavior of both passive and active condensates. While surface tension in equilibrium systems minimizes interface length, non-equilibrium fluxes can destabilize flat or constantly curved interfaces, giving rise to complex interface morphologies. Starting from a minimal model that couples a conserved, phase-separating species to a self-generated chemical field, we identify the conditions under which interfacial instabilities may emerge. Specifically, we show that non-reciprocal chemotactic interactions induce two distinct types of instabilities: a stationary (non-oscillatory) instability that promotes interface deformations, and an oscillatory instability that can give rise to persistent capillary waves propagating along the boundaries of phase-separated domains. To characterize these phenomena, we develop a perturbative framework that predicts the onset, wavelength, and velocity of capillary waves, and quantitatively validate these predictions through numerical simulations. Beyond the linear regime, our simulations reveal that capillary waves undergo a secondary instability, leading to either stationary or dynamically evolving superpositions of different wave modes. Finally, we investigate whether capillary waves can facilitate directed mass transport, either along phase boundaries (conveyor belts) or through self-sustained liquid gears crawling along a solid wall. Taken together, our results establish a general framework for interfacial dynamics in active phase-separating systems and suggest new strategies for controlling mass transport in soft matter and biological condensates.

Figures

Figures reproduced from arXiv: 2505.20028 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: (b) and Video S6 [103]. Although this setup can, in principle, produce crawling motion under certain con￾ditions, the gravity-like interaction often destabilizes the system, causing the gear to collide with the bottom wall. Therefore, to enhance the gear’s stability, …
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p038_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p042_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p043_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p051_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p052_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21 [PITH_FULL_IMAGE:figures/full_fig_p053_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22 [PITH_FULL_IMAGE:figures/full_fig_p054_22.png]
Figure 24
Figure 24. Figure 24: As for the other parameter sets (cf [PITH_FULL_IMAGE:figures/full_fig_p055_24.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

160 extracted references · 56 canonical work pages

  1. [105]

    AB ,\ title title COMSOL Multiphysics® v

    author author C. AB ,\ title title COMSOL Multiphysics® v. 6.1 ,\ www.comsol.com journal journal Stockholm \ ( year 2023 ) NoStop

    2023

  2. [1]

    Kim , author W

    author author D. Kim , author W. Kim , author E. Park , author N. Mattern ,\ and\ author J. Eckert ,\ title title Phase separation in metallic glasses ,\ https://doi.org/10.1016/j.pmatsci.2013.04.002 journal journal Progress in Materials Science \ volume 58 ,\ pages 1103 ( year 2013 ) NoStop

    work page doi:10.1016/j.pmatsci.2013.04.002 2013

  3. [2]

    author author J. T. \ Lee , author C. Jo ,\ and\ author M. D. \ Volder ,\ title title Bicontinuous phase separation of lithium-ion battery electrodes for ultrahigh areal loading ,\ https://doi.org/10.1073/pnas.2007250117 journal journal Proceedings of the National Academy of Sciences \ volume 117 ,\ pages 21155 ( year 2020 ) NoStop

    work page doi:10.1073/pnas.2007250117

  4. [3]

    Benabdelhalim \ and\ author D

    author author H. Benabdelhalim \ and\ author D. Brutin ,\ title title Phase separation and spreading dynamics of French vinaigrette ,\ https://doi.org/10.1063/5.0075453 journal journal Physics of Fluids \ volume 34 ,\ pages 012120 ( year 2022 ) NoStop

    work page doi:10.1063/5.0075453

  5. [4]

    Bartolucci , author D

    author author G. Bartolucci , author D. M. \ Busiello , author M. Ciarchi , author A. Corticelli , author I. D. \ Terlizzi , author F. Olmeda , author D. Revignas ,\ and\ author V. M. \ Schimmenti ,\ title title Phase behavior of Cacio and Pepe sauce ,\ journal journal arXiv \ https://doi.org/10.48550/arxiv.2501.00536 10.48550/arxiv.2501.00536 ( year 2024...

    work page doi:10.48550/arxiv.2501.00536

  6. [5]

    author author A. A. \ Hyman , author C. A. \ Weber ,\ and\ author F. Jülicher ,\ title title Liquid-Liquid Phase Separation in Biology ,\ https://doi.org/10.1146/annurev-cellbio-100913-013325 journal journal Annual Review of Cell and Developmental Biology \ volume 30 ,\ pages 39 ( year 2014 ) NoStop

    doi:10.1146/annurev-cellbio-100913-013325

  7. [6]

    author author S. F. \ Banani , author H. O. \ Lee , author A. A. \ Hyman ,\ and\ author M. K. \ Rosen ,\ title title Biomolecular condensates: organizers of cellular biochemistry ,\ https://doi.org/10.1038/nrm.2017.7 journal journal Nature Reviews Molecular Cell Biology \ volume 18 ,\ pages 285 ( year 2017 ) NoStop

    doi:10.1038/nrm.2017.7 2017

  8. [7]

    Siteur , author Q.-X

    author author K. Siteur , author Q.-X. \ Liu , author V. Rottschäfer , author T. v. d. \ Heide , author M. Rietkerk , author A. Doelman , author C. Boström ,\ and\ author J. v. d. \ Koppel ,\ title title Phase-separation physics underlies new theory for the resilience of patchy ecosystems ,\ https://doi.org/10.1073/pnas.2202683120 journal journal Proceedi...

    work page doi:10.1073/pnas.2202683120

Show all 160 references
  1. [8]

    author author C. A. \ Weber , author D. Zwicker , author F. Jülicher ,\ and\ author C. F. \ Lee ,\ title title Physics of active emulsions ,\ https://doi.org/10.1088/1361-6633/ab052b journal journal Reports on Progress in Physics \ volume 82 ,\ pages 064601 ( year 2019 ) ,\ ht...

    arXiv

  2. [9]

    author author W. Ostwald ,\ title title Studien über die Bildung und Umwandlung fester Körper ,\ https://doi.org/10.1515/zpch-1897-2233 journal journal Zeitschrift für Physikalische Chemie \ volume 22U ,\ pages 289 ( year 1897 ) NoStop

    doi

  3. [10]

    author author C. Wagner ,\ title title Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald‐Reifung) ,\ https://doi.org/10.1002/bbpc.19610650704 journal journal Zeitschrift für Elektrochemie, Berichte der Bunsengesellschaft für physikalische Chemie \ volume 65 ,\ pag...

    doi

  4. [11]

    Lifshitz \ and\ author V

    author author I. Lifshitz \ and\ author V. Slyozov ,\ title title The kinetics of precipitation from supersaturated solid solutions ,\ https://doi.org/10.1016/0022-3697(61)90054-3 journal journal Journal of Physics and Chemistry of Solids \ volume 19 ,\ pages 35 ( year 1961 ) NoStop

    doi

  5. [12]

    author author S. C. \ Glotzer \ and\ author A. Coniglio ,\ title title Self-consistent solution of phase separation with competing interactions ,\ https://doi.org/10.1103/physreve.50.4241 journal journal Physical Review E \ volume 50 ,\ pages 4241 ( year 1994 ) NoStop

    doi

  6. [13]

    author author D. Zwicker ,\ title title The intertwined physics of active chemical reactions and phase separation ,\ https://doi.org/10.1016/j.cocis.2022.101606 journal journal Current Opinion in Colloid & Interface Science \ volume 61 ,\ pages 101606 ( year 2022 ) ,\ https://...

    2022

  7. [14]

    Halatek , author F

    author author J. Halatek , author F. Brauns ,\ and\ author E. Frey ,\ title title Self-organization principles of intracellular pattern formation ,\ https://doi.org/10.1098/rstb.2017.0107 journal journal Philosophical Transactions of the Royal Society B: Biological Sciences \ ...

    2017

  8. [15]

    Demarchi , author A

    author author L. Demarchi , author A. Goychuk , author I. Maryshev ,\ and\ author E. Frey ,\ title title Enzyme-Enriched Condensates Show Self-Propulsion, Positioning, and Coexistence ,\ https://doi.org/10.1103/physrevlett.130.128401 journal journal Physical Review Letters \ v...

    arXiv

  9. [16]

    Goychuk , author L

    author author A. Goychuk , author L. Demarchi , author I. Maryshev ,\ and\ author E. Frey ,\ title title Self-consistent sharp interface theory of active condensate dynamics ,\ https://doi.org/10.1103/physrevresearch.6.033082 journal journal Physical Review Research \ volume 6...

    arXiv

  10. [17]

    author author V. S. \ Doan , author I. Alshareedah , author A. Singh , author P. R. \ Banerjee ,\ and\ author S. Shin ,\ title title Diffusiophoresis promotes phase separation and transport of biomolecular condensates ,\ https://doi.org/10.1038/s41467-024-51840-6 journal journ...

    doi

  11. [18]

    Burkart , author M

    author author T. Burkart , author M. C. \ Wigbers , author L. Würthner ,\ and\ author E. Frey ,\ title title Control of protein-based pattern formation via guiding cues ,\ https://doi.org/10.1038/s42254-022-00461-3 journal journal Nature Reviews Physics \ volume 4 ,\ pages 511...

    doi

  12. [19]

    Schumacher , author S

    author author D. Schumacher , author S. Bergeler , author A. Harms , author J. Vonck , author S. Huneke-Vogt , author E. Frey ,\ and\ author L. Søgaard-Andersen ,\ title title The PomXYZ Proteins Self-Organize on the Bacterial Nucleoid to Stimulate Cell Division ,\ https://doi...

    arXiv

  13. [20]

    Bergeler \ and\ author E

    author author S. Bergeler \ and\ author E. Frey ,\ title title Regulation of Pom cluster dynamics in Myxococcus xanthus ,\ https://doi.org/10.1371/journal.pcbi.1006358 journal journal PLoS Computational Biology \ volume 14 ,\ pages e1006358 ( year 2018 ) ,\ https://arxiv.org/a...

    arXiv

  14. [21]

    author author C. P. \ Brangwynne , author C. R. \ Eckmann , author D. S. \ Courson , author A. Rybarska , author C. Hoege , author J. Gharakhani , author F. Jülicher ,\ and\ author A. A. \ Hyman ,\ title title Germline P Granules Are Liquid Droplets That Localize by Controlled...

    doi

  15. [22]

    Bornens ,\ title title The Centrosome in Cells and Organisms ,\ https://doi.org/10.1126/science.1209037 journal journal Science \ volume 335 ,\ pages 422 ( year 2012 ) NoStop

    author author M. Bornens ,\ title title The Centrosome in Cells and Organisms ,\ https://doi.org/10.1126/science.1209037 journal journal Science \ volume 335 ,\ pages 422 ( year 2012 ) NoStop

    2012 doi

  16. [23]

    Zwicker , author M

    author author D. Zwicker , author M. Decker , author S. Jaensch , author A. A. \ Hyman ,\ and\ author F. Jülicher ,\ title title Centrosomes are autocatalytic droplets of pericentriolar material organized by centrioles ,\ https://doi.org/10.1073/pnas.1404855111 journal journal...

    doi

  17. [24]

    Zeng , author Y

    author author M. Zeng , author Y. Shang , author Y. Araki , author T. Guo , author R. L. \ Huganir ,\ and\ author M. Zhang ,\ title title Phase Transition in Postsynaptic Densities Underlies Formation of Synaptic Complexes and Synaptic Plasticity ,\ https://doi.org/10.1016/j.c...

    doi

  18. [25]

    author author I. B. A. \ Smokers , author B. S. \ Visser , author A. D. \ Slootbeek , author W. T. S. \ Huck ,\ and\ author E. Spruijt ,\ title title How Droplets Can Accelerate Reactions-Coacervate Protocells as Catalytic Microcompartments ,\ https://doi.org/10.1021/acs.accou...

    doi

  19. [26]

    author author B. G. \ O'Flynn \ and\ author T. Mittag ,\ title title The role of liquid–liquid phase separation in regulating enzyme activity ,\ https://doi.org/10.1016/j.ceb.2020.12.012 journal journal Current Opinion in Cell Biology \ volume 69 ,\ pages 70 ( year 2021 ) NoStop

    2020 doi

  20. [27]

    Prouteau \ and\ author R

    author author M. Prouteau \ and\ author R. Loewith ,\ title title Regulation of Cellular Metabolism through Phase Separation of Enzymes ,\ https://doi.org/10.3390/biom8040160 journal journal Biomolecules \ volume 8 ,\ pages 160 ( year 2018 ) NoStop

    doi

  21. [28]

    Wunder , author S

    author author T. Wunder , author S. L. H. \ Cheng , author S.-K. \ Lai , author H.-Y. \ Li ,\ and\ author O. Mueller-Cajar ,\ title title The phase separation underlying the pyrenoid-based microalgal Rubisco supercharger ,\ https://doi.org/10.1038/s41467-018-07624-w journal jo...

    doi

  22. [29]

    Wang , author X

    author author H. Wang , author X. Yan , author H. Aigner , author A. Bracher , author N. D. \ Nguyen , author W. Y. \ Hee , author B. M. \ Long , author G. D. \ Price , author F. U. \ Hartl ,\ and\ author M. Hayer-Hartl ,\ title title Rubisco condensate formation by CcmM in -c...

    doi

  23. [30]

    Feric , author N

    author author M. Feric , author N. Vaidya , author T. S. \ Harmon , author D. M. \ Mitrea , author L. Zhu , author T. M. \ Richardson , author R. W. \ Kriwacki , author R. V. \ Pappu ,\ and\ author C. P. \ Brangwynne ,\ title title Coexisting Liquid Phases Underlie Nucleolar S...

    doi

  24. [31]

    Hnisz , author K

    author author D. Hnisz , author K. Shrinivas , author R. A. \ Young , author A. K. \ Chakraborty ,\ and\ author P. A. \ Sharp ,\ title title A Phase Separation Model for Transcriptional Control ,\ https://doi.org/10.1016/j.cell.2017.02.007 journal journal Cell \ volume 169 ,\ ...

    doi

  25. [32]

    author author J. E. \ Henninger , author O. Oksuz , author K. Shrinivas , author I. Sagi , author G. LeRoy , author M. M. \ Zheng , author J. O. \ Andrews , author A. V. \ Zamudio , author C. Lazaris , author N. M. \ Hannett , author T. I. \ Lee , author P. A. \ Sharp , author...

    doi

  26. [33]

    Bracha , author M

    author author D. Bracha , author M. T. \ Walls , author M.-T. \ Wei , author L. Zhu , author M. Kurian , author J. L. \ Avalos , author J. E. \ Toettcher ,\ and\ author C. P. \ Brangwynne ,\ title title Mapping Local and Global Liquid Phase Behavior in Living Cells Using Photo...

    doi

  27. [34]

    author author A. W. \ Fritsch , author A. F. \ Diaz-Delgadillo , author O. Adame-Arana , author C. Hoege , author M. Mittasch , author M. Kreysing , author M. Leaver , author A. A. \ Hyman , author F. Jülicher ,\ and\ author C. A. \ Weber ,\ title title Local thermodynamics go...

    doi

  28. [35]

    Jambon-Puillet , author A

    author author E. Jambon-Puillet , author A. Testa , author C. Lorenz , author R. W. \ Style , author A. A. \ Rebane ,\ and\ author E. R. \ Dufresne ,\ title title Phase-separated droplets swim to their dissolution ,\ https://doi.org/10.1038/s41467-024-47889-y journal journal N...

    doi

  29. [36]

    Zwicker , author A

    author author D. Zwicker , author A. A. \ Hyman ,\ and\ author F. Jülicher ,\ title title Suppression of Ostwald ripening in active emulsions ,\ https://doi.org/10.1103/physreve.92.012317 journal journal Physical Review E \ volume 92 ,\ pages 012317 ( year 2015 ) NoStop

    doi

  30. [37]

    author author J. D. \ Wurtz \ and\ author C. F. \ Lee ,\ title title Chemical-Reaction-Controlled Phase Separated Drops: Formation, Size Selection, and Coarsening ,\ https://doi.org/10.1103/physrevlett.120.078102 journal journal Physical Review Letters \ volume 120 ,\ pages 07...

    arXiv

  31. [38]

    Kirschbaum \ and\ author D

    author author J. Kirschbaum \ and\ author D. Zwicker ,\ title title Controlling biomolecular condensates via chemical reactions ,\ https://doi.org/10.1098/rsif.2021.0255 journal journal Journal of the Royal Society Interface \ volume 18 ,\ pages 20210255 ( year 2021 ) ,\ https...

    2021

  32. [39]

    Donau \ and\ author J

    author author C. Donau \ and\ author J. Boekhoven ,\ title title The chemistry of chemically fueled droplets ,\ https://doi.org/10.1016/j.trechm.2022.11.003 journal journal Trends in Chemistry \ volume 5 ,\ pages 45 ( year 2023 ) NoStop

    2022 doi

  33. [40]

    Bauermann , author G

    author author J. Bauermann , author G. Bartolucci , author J. Boekhoven , author C. A. \ Weber ,\ and\ author F. Jülicher ,\ title title Formation of liquid shells in active droplet systems ,\ https://doi.org/10.1103/physrevresearch.5.043246 journal journal Physical Review Res...

    doi

  34. [41]

    Häfner \ and\ author M

    author author G. Häfner \ and\ author M. M\"uller ,\ title title Reaction-Driven Diffusiophoresis of Liquid Condensates: Potential Mechanisms for Intracellular Organization ,\ https://doi.org/10.1021/acsnano.3c12842 journal journal ACS Nano \ volume 18 ,\ pages 16530 ( year 20...

    doi

  35. [42]

    Zwicker , author R

    author author D. Zwicker , author R. Seyboldt , author C. A. \ Weber , author A. A. \ Hyman ,\ and\ author F. Jülicher ,\ title title Growth and division of active droplets provides a model for protocells ,\ https://doi.org/10.1038/nphys3984 journal journal Nature Physics \ vo...

    arXiv

  36. [43]

    Tailleur \ and\ author M

    author author J. Tailleur \ and\ author M. E. \ Cates ,\ title title Statistical Mechanics of Interacting Run-and-Tumble Bacteria ,\ https://doi.org/10.1103/physrevlett.100.218103 journal journal Physical Review Letters \ volume 100 ,\ pages 218103 ( year 2008 ) ,\ https://arx...

    arXiv

  37. [44]

    Fily \ and\ author M

    author author Y. Fily \ and\ author M. C. \ Marchetti ,\ title title Athermal Phase Separation of Self-Propelled Particles with No Alignment ,\ https://doi.org/10.1103/physrevlett.108.235702 journal journal Physical Review Letters \ volume 108 ,\ pages 235702 ( year 2012 ) ,\ ...

    arXiv

  38. [45]

    author author G. S. \ Redner , author M. F. \ Hagan ,\ and\ author A. Baskaran ,\ title title Structure and Dynamics of a Phase-Separating Active Colloidal Fluid ,\ https://doi.org/10.1103/physrevlett.110.055701 journal journal Physical Review Letters \ volume 110 ,\ pages 055...

    doi

  39. [46]

    author author M. E. \ Cates \ and\ author J. Tailleur ,\ title title Motility-Induced Phase Separation ,\ https://doi.org/10.1146/annurev-conmatphys-031214-014710 journal journal Annual Review of Condensed Matter Physics \ volume 6 ,\ pages 1 ( year 2014 ) ,\ https://arxiv.org...

    arXiv

  40. [47]

    \ Liu , author A

    author author Q.-X. \ Liu , author A. Doelman , author V. Rottschäfer , author M. d. \ Jager , author P. M. J. \ Herman , author M. Rietkerk ,\ and\ author J. v. d. \ Koppel ,\ title title Phase separation explains a new class of self-organized spatial patterns in ecological s...

    doi

  41. [48]

    Tjhung , author C

    author author E. Tjhung , author C. Nardini ,\ and\ author M. E. \ Cates ,\ title title Cluster Phases and Bubbly Phase Separation in Active Fluids: Reversal of the Ostwald Process ,\ https://doi.org/10.1103/physrevx.8.031080 journal journal Physical Review X \ volume 8 ,\ pag...

    arXiv

  42. [49]

    author author S. C. \ Takatori \ and\ author J. F. \ Brady ,\ title title Towards a thermodynamics of active matter ,\ https://doi.org/10.1103/physreve.91.032117 journal journal Physical Review E \ volume 91 ,\ pages 032117 ( year 2015 ) ,\ https://arxiv.org/abs/1411.5776 1411...

    arXiv

  43. [50]

    author author A. P. \ Solon , author J. Stenhammar , author M. E. \ Cates , author Y. Kafri ,\ and\ author J. Tailleur ,\ title title Generalized thermodynamics of phase equilibria in scalar active matter ,\ https://doi.org/10.1103/physreve.97.020602 journal journal Physical R...

    arXiv

  44. [51]

    Singer , author T

    author author G. Singer , author T. Araki ,\ and\ author C. J. \ Weijer ,\ title title Oscillatory cAMP cell-cell signalling persists during multicellular Dictyostelium development ,\ https://doi.org/10.1038/s42003-019-0371-0 journal journal Communications Biology \ volume 2 ,...

    doi

  45. [52]

    Feng \ and\ author M

    author author M. Feng \ and\ author M. K. \ Gilson ,\ title title Enhanced Diffusion and Chemotaxis of Enzymes ,\ https://doi.org/10.1146/annurev-biophys-121219-081535 journal journal Annual Review of Biophysics \ volume 49 ,\ pages 1 ( year 2020 ) NoStop

    doi

  46. [53]

    Jin , author C

    author author C. Jin , author C. Krüger ,\ and\ author C. C. \ Maass ,\ title title Chemotaxis and autochemotaxis of self-propelling droplet swimmers ,\ https://doi.org/10.1073/pnas.1619783114 journal journal Proceedings of the National Academy of Sciences \ volume 114 ,\ page...

    arXiv

  47. [54]

    Liebchen \ and\ author H

    author author B. Liebchen \ and\ author H. Löwen ,\ title title Synthetic Chemotaxis and Collective Behavior in Active Matter ,\ https://doi.org/10.1021/acs.accounts.8b00215 journal journal Accounts of Chemical Research \ volume 51 ,\ pages 2982 ( year 2018 ) NoStop

    doi

  48. [55]

    author author E. F. \ Keller \ and\ author L. A. \ Segel ,\ title title Initiation of slime mold aggregation viewed as an instability ,\ https://doi.org/10.1016/0022-5193(70)90092-5 journal journal Journal of Theoretical Biology \ volume 26 ,\ pages 399 ( year 1970 ) NoStop

    doi

  49. [56]

    author author E. F. \ Keller \ and\ author L. A. \ Segel ,\ title title Model for chemotaxis ,\ https://doi.org/10.1016/0022-5193(71)90050-6 journal journal Journal of Theoretical Biology \ volume 30 ,\ pages 225 ( year 1971 ) NoStop

    1971 doi

  50. [57]

    author author K. J. \ Painter \ and\ author T. Hillen ,\ title title Spatio-temporal chaos in a chemotaxis model ,\ https://doi.org/10.1016/j.physd.2010.09.011 journal journal Physica D: Nonlinear Phenomena \ volume 240 ,\ pages 363 ( year 2011 ) NoStop

    2010 doi

  51. [58]

    Hillen \ and\ author K

    author author T. Hillen \ and\ author K. J. \ Painter ,\ title title A user’s guide to PDE models for chemotaxis ,\ https://doi.org/10.1007/s00285-008-0201-3 journal journal Journal of Mathematical Biology \ volume 58 ,\ pages 183 ( year 2008 ) NoStop

    doi

  52. [59]

    author author M. J. \ Tindall , author P. K. \ Maini , author S. L. \ Porter ,\ and\ author J. P. \ Armitage ,\ title title Overview of Mathematical Approaches Used to Model Bacterial Chemotaxis II: Bacterial Populations ,\ https://doi.org/10.1007/s11538-008-9322-5 journal jou...

    doi

  53. [60]

    Weyer , author D

    author author H. Weyer , author D. Muramatsu ,\ and\ author E. Frey ,\ title title Chemotaxis-induced phase separation ,\ journal journal arXiv \ https://doi.org/10.48550/arxiv.2409.20090 10.48550/arxiv.2409.20090 ( year 2024 a ),\ https://arxiv.org/abs/2409.20090 2409.20090 NoStop

    doi

  54. [61]

    Weyer , author D

    author author H. Weyer , author D. Muramatsu ,\ and\ author E. Frey ,\ title title Coarsening dynamics of chemotactic aggregates ,\ journal journal arXiv \ https://doi.org/10.48550/arxiv.2409.20100 10.48550/arxiv.2409.20100 ( year 2024 b ),\ https://arxiv.org/abs/2409.20100 24...

    doi

  55. [62]

    Zhao , author A

    author author H. Zhao , author A. Košmrlj ,\ and\ author S. S. \ Datta ,\ title title Chemotactic Motility-Induced Phase Separation ,\ https://doi.org/10.1103/physrevlett.131.118301 journal journal Physical Review Letters \ volume 131 ,\ pages 118301 ( year 2023 ) NoStop

    doi

  56. [63]

    Schwarz-Linek , author C

    author author J. Schwarz-Linek , author C. Valeriani , author A. Cacciuto , author M. E. \ Cates , author D. Marenduzzo , author A. N. \ Morozov ,\ and\ author W. C. K. \ Poon ,\ title title Phase separation and rotor self-assembly in active particle suspensions ,\ https://doi...

    arXiv

  57. [64]

    Brauns , author J

    author author F. Brauns , author J. Halatek ,\ and\ author E. Frey ,\ title title Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics ,\ https://doi.org/10.1103/physrevx.10.041036 journal journal Physical Review X \ volume 10 ,\ pages 041036 ( year 2020 ) ,\ ht...

    arXiv

  58. [65]

    Brauns , author H

    author author F. Brauns , author H. Weyer , author J. Halatek , author J. Yoon ,\ and\ author E. Frey ,\ title title Wavelength Selection by Interrupted Coarsening in Reaction-Diffusion Systems ,\ https://doi.org/10.1103/physrevlett.126.104101 journal journal Physical Review L...

    arXiv

  59. [66]

    Frey \ and\ author F

    author author E. Frey \ and\ author F. Brauns ,\ title title Self-organisation of Protein Patterns ,\ journal journal arXiv \ https://doi.org/10.48550/arxiv.2012.01797 10.48550/arxiv.2012.01797 ( year 2020 ),\ https://arxiv.org/abs/2012.01797 2012.01797 NoStop

    2012 doi arXiv

  60. [67]

    Würthner , author F

    author author L. Würthner , author F. Brauns , author G. Pawlik , author J. Halatek , author J. Kerssemakers , author C. Dekker ,\ and\ author E. Frey ,\ title title Bridging scales in a multiscale pattern-forming system ,\ https://doi.org/10.1073/pnas.2206888119 journal journ...

    arXiv

  61. [68]

    Ren , author H

    author author Z. Ren , author H. Weyer , author M. Sandler , author L. Würthner , author H. Fu , author C. B. \ Tangtartharakul , author D. Li , author C. Sou , author D. Villarreal , author J. E. \ Kim , author E. Frey ,\ and\ author S. Jun ,\ title title Robust and resource-...

    doi

  62. [69]

    Weyer , author T

    author author H. Weyer , author T. A. \ Roth ,\ and\ author E. Frey ,\ title title Deciphering the Interface Laws of Turing Mixtures and Foams ,\ journal journal arXiv \ https://doi.org/10.48550/arxiv.2409.20070 10.48550/arxiv.2409.20070 ( year 2024 c ),\ https://arxiv.org/abs...

    doi arXiv

  63. [70]

    author author A. J. \ Bray ,\ title title Theory of phase-ordering kinetics ,\ https://doi.org/10.1080/00018730110117433 journal journal Advances in Physics \ volume 51 ,\ pages 481 ( year 2002 ) NoStop

    2002 doi

  64. [71]

    Cho \ and\ author W

    author author Y. Cho \ and\ author W. M. \ Jacobs ,\ title title Tuning Nucleation Kinetics via Nonequilibrium Chemical Reactions ,\ https://doi.org/10.1103/physrevlett.130.128203 journal journal Physical Review Letters \ volume 130 ,\ pages 128203 ( year 2023 ) ,\ https://arx...

    arXiv

  65. [72]

    Fausti , author E

    author author G. Fausti , author E. Tjhung , author M. E. \ Cates ,\ and\ author C. Nardini ,\ title title Capillary Interfacial Tension in Active Phase Separation ,\ https://doi.org/10.1103/physrevlett.127.068001 journal journal Physical Review Letters \ volume 127 ,\ pages 0...

    arXiv

  66. [73]

    author author W. W. \ Mullins \ and\ author R. F. \ Sekerka ,\ title title Morphological Stability of a Particle Growing by Diffusion or Heat Flow ,\ https://doi.org/10.1063/1.1702607 journal journal Journal of Applied Physics \ volume 34 ,\ pages 323 ( year 1963 ) NoStop

    doi

  67. [74]

    Alert , author A

    author author R. Alert , author A. Martínez-Calvo ,\ and\ author S. S. \ Datta ,\ title title Cellular Sensing Governs the Stability of Chemotactic Fronts ,\ https://doi.org/10.1103/physrevlett.128.148101 journal journal Physical Review Letters \ volume 128 ,\ pages 148101 ( y...

    arXiv

  68. [75]

    Brauns \ and\ author M

    author author F. Brauns \ and\ author M. C. \ Marchetti ,\ title title Nonreciprocal Pattern Formation of Conserved Fields ,\ https://doi.org/10.1103/physrevx.14.021014 journal journal Physical Review X \ volume 14 ,\ pages 021014 ( year 2024 ) ,\ https://arxiv.org/abs/2306.08...

    arXiv

  69. [76]

    author author C. d. \ Junco \ and\ author S. Vaikuntanathan ,\ title title Interface height fluctuations and surface tension of driven liquids with time-dependent dynamics ,\ https://doi.org/10.1063/1.5042251 journal journal The Journal of Chemical Physics \ volume 150 ,\ page...

    arXiv

  70. [77]

    Bedeaux \ and\ author J

    author author D. Bedeaux \ and\ author J. D. \ Weeks ,\ title title Correlation functions in the capillary wave model of the liquid–vapor interface ,\ https://doi.org/10.1063/1.448474 journal journal The Journal of Chemical Physics \ volume 82 ,\ pages 972 ( year 1985 ) NoStop

    doi

  71. [78]

    author author A. J. \ Bray , author A. Cavagna ,\ and\ author R. D. M. \ Travasso ,\ title title Interface fluctuations, Burgers equations, and coarsening under shear ,\ https://doi.org/10.1103/physreve.65.016104 journal journal Physical Review E \ volume 65 ,\ pages 016104 ( ...

    arXiv

  72. [79]

    Jain , author J

    author author S. Jain , author J. Wheeler , author R. Walters , author A. Agrawal , author A. Barsic ,\ and\ author R. Parker ,\ title title ATPase-Modulated Stress Granules Contain a Diverse Proteome and Substructure ,\ https://doi.org/10.1016/j.cell.2015.12.038 journal journ...

    doi

  73. [80]

    \ Youn , author B

    author author J.-Y. \ Youn , author B. J. \ Dyakov , author J. Zhang , author J. D. \ Knight , author R. M. \ Vernon , author J. D. \ Forman-Kay ,\ and\ author A.-C. \ Gingras ,\ title title Properties of Stress Granule and P-Body Proteomes ,\ https://doi.org/10.1016/j.molcel....

    doi

  74. [81]

    author author S. F. \ Banani , author A. M. \ Rice , author W. B. \ Peeples , author Y. Lin , author S. Jain , author R. Parker ,\ and\ author M. K. \ Rosen ,\ title title Compositional Control of Phase-Separated Cellular Bodies ,\ https://doi.org/10.1016/j.cell.2016.06.010 jo...

    doi

  75. [82]

    author author J. A. \ Ditlev , author L. B. \ Case ,\ and\ author M. K. \ Rosen ,\ title title Who's In and Who's Out—Compositional Control of Biomolecular Condensates ,\ https://doi.org/10.1016/j.jmb.2018.08.003 journal journal Journal of Molecular Biology \ volume 430 ,\ pag...

    2018 doi

  76. [83]

    Doi ,\ https://doi.org/10.1093/acprof:oso/9780199652952.001.0001 title Soft Matter Physics \ ( publisher Oxford University Press ,\ year 2013 ) NoStop

    author author M. Doi ,\ https://doi.org/10.1093/acprof:oso/9780199652952.001.0001 title Soft Matter Physics \ ( publisher Oxford University Press ,\ year 2013 ) NoStop

    2013

  77. [84]

    author author J. W. \ Cahn \ and\ author J. E. \ Hilliard ,\ title title Free Energy of a Nonuniform System. I. Interfacial Free Energy ,\ https://doi.org/10.1063/1.1744102 journal journal The Journal of Chemical Physics \ volume 28 ,\ pages 258 ( year 1958 ) NoStop

    doi

  78. [85]

    author author S. d. \ Groot \ and\ author P. Mazur ,\ https://books.google.de/books?id=HFAIv43rlGkC title Non-equilibrium Thermodynamics ,\ Dover Books on Physics\ ( publisher Dover Publications ,\ year 1984 ) NoStop

    1984

  79. [86]

    Onsager ,\ title title Reciprocal Relations in Irreversible Processes

    author author L. Onsager ,\ title title Reciprocal Relations in Irreversible Processes. I ,\ https://doi.org/10.1103/physrev.37.405 journal journal Physical Review \ volume 37 ,\ pages 405 ( year 1930 ) NoStop

    1930 doi

  80. [87]

    Onsager ,\ title title Reciprocal Relations in Irreversible Processes

    author author L. Onsager ,\ title title Reciprocal Relations in Irreversible Processes. II. ,\ https://doi.org/10.1103/physrev.38.2265 journal journal Physical Review \ volume 38 ,\ pages 2265 ( year 1931 ) NoStop

    1931 doi

  81. [88]

    note If we had chosen an asymmetric free energy or a more complex mobility function, the diffusion constants generally differ in the high- and low-density phases. Stop

  82. [89]

    author author P. L. \ Krapivsky , author S. Redner ,\ and\ author E. Ben-Naim ,\ @noop title A Kinetic View of Statistical Physics \ ( publisher Cambridge University Press ,\ year 2010 ) NoStop

    2010

  83. [90]

    Kondepudi \ and\ author I

    author author D. Kondepudi \ and\ author I. Prigogine ,\ title title Modern Thermodynamics ,\ journal journal Wiley \ https://doi.org/10.1002/9781118698723 10.1002/9781118698723 ( year 2014 ) NoStop

    2014 doi

  84. [91]

    Halatek \ and\ author E

    author author J. Halatek \ and\ author E. Frey ,\ title title Rethinking pattern formation in reaction–diffusion systems ,\ https://doi.org/10.1038/s41567-017-0040-5 journal journal Nature Physics \ volume 14 ,\ pages 507 ( year 2018 ) NoStop

    doi

  85. [92]

    Hänggi , author P

    author author P. Hänggi , author P. Talkner ,\ and\ author M. Borkovec ,\ title title Reaction-rate theory: fifty years after Kramers ,\ https://doi.org/10.1103/revmodphys.62.251 journal journal Reviews of Modern Physics \ volume 62 ,\ pages 251 ( year 1990 ) NoStop

    doi

  86. [93]

    Painter \ and\ author T

    author author K. Painter \ and\ author T. Hillen ,\ title title Volume-filling and quorum-sensing in models for chemosensitive movement ,\ https://api.semanticscholar.org/CorpusID:2456794 journal journal Canadian Applied Mathematics Quarterly \ volume 10 ,\ pages 501 ( year 20...

    2002

  87. [94]

    author author L. A. \ Segel ,\ title title A Theoretical Study of Receptor Mechanisms in Bacterial Chemotaxis ,\ https://doi.org/10.1137/0132054 journal journal SIAM Journal on Applied Mathematics \ volume 32 ,\ pages 653 ( year 1977 ) NoStop

    doi

  88. [95]

    author author J. v. d. \ Kolk , author F. Raßhofer , author R. Swiderski , author A. Haldar , author A. Basu ,\ and\ author E. Frey ,\ title title Anomalous Collective Dynamics of Autochemotactic Populations ,\ https://doi.org/10.1103/physrevlett.131.088201 journal journal Phy...

    arXiv

  89. [96]

    Rapp \ and\ author W

    author author L. Rapp \ and\ author W. Zimmermann ,\ title title Universal aspects of collective behavior in chemotactic systems ,\ https://doi.org/10.1103/physreve.100.032609 journal journal Physical Review E \ volume 100 ,\ pages 032609 ( year 2019 ) NoStop

    doi

  90. [97]

    author author A. V. \ Narla , author J. Cremer ,\ and\ author T. Hwa ,\ title title A traveling-wave solution for bacterial chemotaxis with growth ,\ https://doi.org/10.1073/pnas.2105138118 journal journal Proceedings of the National Academy of Sciences \ volume 118 ,\ pages e...

    arXiv

  91. [98]

    Arumugam \ and\ author J

    author author G. Arumugam \ and\ author J. Tyagi ,\ title title Keller-Segel Chemotaxis Models: A Review ,\ https://doi.org/10.1007/s10440-020-00374-2 journal journal Acta Applicandae Mathematicae \ volume 171 ,\ pages 6 ( year 2021 ) NoStop

    doi

  92. [99]

    author author S. K. \ Baek \ and\ author B. J. \ Kim ,\ title title Free energy of a chemotactic model with nonlinear diffusion ,\ https://doi.org/10.1038/s41598-017-09369-w journal journal Scientific Reports \ volume 7 ,\ pages 8909 ( year 2017 ) ,\ https://arxiv.org/abs/1709...

    arXiv

  93. [100]

    Liu \ and\ author N

    author author F. Liu \ and\ author N. Goldenfeld ,\ title title Dynamics of phase separation in block copolymer melts ,\ https://doi.org/10.1103/physreva.39.4805 journal journal Physical Review A \ volume 39 ,\ pages 4805 ( year 1989 ) NoStop

    doi

  94. [101]

    Kumar \ and\ author S

    author author A. Kumar \ and\ author S. A. \ Safran ,\ title title Fluctuations and Shape Dependence of Microphase Separation in Systems with Long-Range Interactions ,\ https://doi.org/10.1103/physrevlett.131.258401 journal journal Physical Review Letters \ volume 131 ,\ pages...

    doi

  95. [102]

    Yu \ and\ author A

    author author Q. Yu \ and\ author A. Košmrlj ,\ title title Pattern formation of lipid domains in bilayer membranes ,\ journal journal Soft Matter \ https://doi.org/10.1039/d5sm00276a 10.1039/d5sm00276a ( year 2025 ) NoStop

    2025 doi

  96. [103]

    Winter , author Y

    author author A. Winter , author Y. Liu , author A. Ziepke , author G. Dadunashvili ,\ and\ author E. Frey ,\ title title Phase separation on deformable membranes: Interplay of mechanical coupling and dynamic surface geometry ,\ https://doi.org/10.1103/physreve.111.044405 jour...

    doi

  97. [104]

    @noop note See Supplemental Material at URL-will-be-inserted-by-publisher for supplementary Videos S1--S7. Stop

  98. [106]

    Raßhofer , author S

    author author F. Raßhofer , author S. Bauer , author A. Ziepke , author I. Maryshev ,\ and\ author E. Frey ,\ title title Capillary wave formation in conserved active emulsions -- Supplementary material ,\ journal journal Zenodo \ https://doi.org/10.5281/zenodo.15495165 10.528...

    doi

  99. [107]

    author author C. B. \ Muratov ,\ title title Theory of domain patterns in systems with long-range interactions of Coulomb type ,\ https://doi.org/10.1103/physreve.66.066108 journal journal Physical Review E \ volume 66 ,\ pages 066108 ( year 2002 ) NoStop

    doi

  100. [108]

    de Gennes , author F

    author author P. de Gennes , author F. Brochard-Wyart ,\ and\ author D. Quere ,\ https://doi.org/10.1007/978-0-387-21656-0 title Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves \ ( publisher Springer New York ,\ year 2003 ) NoStop

    doi

  101. [109]

    author author D. J. \ Eyre ,\ title title Systems of Cahn-Hilliard Equations ,\ https://doi.org/10.1137/0153078 journal journal SIAM Journal on Applied Mathematics \ volume 53 ,\ pages 1686 ( year 1993 ) NoStop

    1993 doi

  102. [110]

    Garcke \ and\ author A

    author author H. Garcke \ and\ author A. Novick-Cohen ,\ title title A singular limit for a system of degenerate Cahn-Hilliard equations ,\ journal journal Advances in Differential Equations \ volume 5 ,\ https://doi.org/10.57262/ade/1356651336 10.57262/ade/1356651336 ( year 2...

  103. [111]

    Würthner , author A

    author author L. Würthner , author A. Goychuk ,\ and\ author E. Frey ,\ title title Geometry-induced patterns through mechanochemical coupling ,\ https://doi.org/10.1103/physreve.108.014404 journal journal Physical Review E \ volume 108 ,\ pages 014404 ( year 2023 ) ,\ https:/...

    arXiv

  104. [112]

    author author W. R. \ Inc. ,\ title title Mathematica ,\ https://www.wolfram.com/mathematica journal journal Champaign, Illinois \ ( year 2024 ) NoStop

    2024

  105. [113]

    author author M. C. \ Cross \ and\ author P. C. \ Hohenberg ,\ title title Pattern formation outside of equilibrium ,\ https://doi.org/10.1103/revmodphys.65.851 journal journal Reviews of Modern Physics \ volume 65 ,\ pages 851 ( year 1993 ) NoStop

    doi

  106. [114]

    Frohoff-Hülsmann \ and\ author U

    author author T. Frohoff-Hülsmann \ and\ author U. Thiele ,\ title title Nonreciprocal Cahn-Hilliard Model Emerges as a Universal Amplitude Equation ,\ https://doi.org/10.1103/physrevlett.131.107201 journal journal Physical Review Letters \ volume 131 ,\ pages 107201 ( year 20...

    doi

  107. [115]

    author author W. v. \ Saarloos ,\ title title Front propagation into unstable states ,\ https://doi.org/10.1016/j.physrep.2003.08.001 journal journal Physics Reports \ volume 386 ,\ pages 29 ( year 2003 ) ,\ https://arxiv.org/abs/cond-mat/0308540 cond-mat/0308540 NoStop

    2003 arXiv

  108. [116]

    author author D. J. \ Griffiths ,\ https://www.cambridge.org/highereducation/books/introduction-to-electrodynamics/3AB220820DBB628E5A43D52C4B011ED4\#overview title Introduction to Electrodynamics ,\ edition 4th \ ed.\ ( publisher Cambridge University Press ,\ year 2019 ) NoStop

    2019

  109. [117]

    Carrère , author J

    author author A. Carrère , author J. d’Alessandro , author O. Cochet-Escartin , author J. Hesnard , author N. Ghazi , author C. Rivière , author C. Anjard , author F. Detcheverry ,\ and\ author J.-P. \ Rieu ,\ title title Microphase separation of living cells ,\ https://doi.or...

    doi

  110. [118]

    Jäger \ and\ author S

    author author W. Jäger \ and\ author S. Luckhaus ,\ title title On explosions of solutions to a system of partial differential equations modelling chemotaxis ,\ https://doi.org/10.1090/s0002-9947-1992-1046835-6 journal journal Transactions of the American Mathematical Society ...

    1992 doi

  111. [119]

    Hoyle ,\ https://doi.org/10.1017/cbo9780511616051 title Pattern Formation \ ( publisher Cambridge University Press ,\ address Cambridge ,\ year 2006 ) NoStop

    author author R. Hoyle ,\ https://doi.org/10.1017/cbo9780511616051 title Pattern Formation \ ( publisher Cambridge University Press ,\ address Cambridge ,\ year 2006 ) NoStop

    2006 doi

  112. [120]

    Greve \ and\ author U

    author author D. Greve \ and\ author U. Thiele ,\ title title An amplitude equation for the conserved-Hopf bifurcation—Derivation, analysis, and assessment ,\ https://doi.org/10.1063/5.0222013 journal journal Chaos: An Interdisciplinary Journal of Nonlinear Science \ volume 34...

    doi

  113. [121]

    Ramm , author A

    author author B. Ramm , author A. Goychuk , author A. Khmelinskaia , author P. Blumhardt , author H. Eto , author K. A. \ Ganzinger , author E. Frey ,\ and\ author P. Schwille ,\ title title A diffusiophoretic mechanism for ATP-driven transport without motor proteins ,\ https:...

    doi

  114. [122]

    author author J. B. S. \ Haldane ,\ title title The origin of life ,\ @noop journal journal Rationalist Annual 148 3-10 \ ( year 1929 ) NoStop

    1929

  115. [123]

    author author A. I. \ Oparin ,\ title title The origin of life ,\ @noop journal journal Dover Publications, Inc. \ ( year 1952 ) NoStop

    1952

  116. [124]

    author author A. D. \ Slootbeek , author M. H. I. v. \ Haren , author I. B. A. \ Smokers ,\ and\ author E. Spruijt ,\ title title Growth, replication and division enable evolution of coacervate protocells ,\ https://doi.org/10.1039/d2cc03541c journal journal Chemical Communica...

    doi

  117. [125]

    author author L. Rayleigh ,\ title title On The Instability Of Jets ,\ https://doi.org/10.1112/plms/s1-10.1.4 journal journal Proceedings of the London Mathematical Society \ volume s1-10 ,\ pages 4 ( year 1878 ) NoStop

    doi

  118. [126]

    Bauermann ,\ title Organization of chemical reactions by phase separation ,\ @noop Ph.D

    author author J. Bauermann ,\ title Organization of chemical reactions by phase separation ,\ @noop Ph.D. thesis ( year 2022 ) NoStop

    2022

  119. [127]

    Cremer , author A

    author author J. Cremer , author A. Melbinger ,\ and\ author E. Frey ,\ title title Growth dynamics and the evolution of cooperation in microbial populations ,\ https://doi.org/10.1038/srep00281 journal journal Scientific Reports \ volume 2 ,\ pages 281 ( year 2012 ) ,\ https:...

    arXiv

  120. [128]

    Laurent , author L

    author author G. Laurent , author L. Peliti ,\ and\ author D. Lacoste ,\ title title Survival of Self-Replicating Molecules under Transient Compartmentalization with Natural Selection ,\ https://doi.org/10.3390/life9040078 journal journal Life \ volume 9 ,\ pages 78 ( year 201...

    doi

  121. [129]

    You , author A

    author author Z. You , author A. Baskaran ,\ and\ author M. C. \ Marchetti ,\ title title Nonreciprocity as a generic route to traveling states ,\ https://doi.org/10.1073/pnas.2010318117 journal journal Proceedings of the National Academy of Sciences \ volume 117 ,\ pages 1976...

    arXiv

  122. [130]

    Fruchart , author R

    author author M. Fruchart , author R. Hanai , author P. B. \ Littlewood ,\ and\ author V. Vitelli ,\ title title Non-reciprocal phase transitions ,\ https://doi.org/10.1038/s41586-021-03375-9 journal journal Nature \ volume 592 ,\ pages 363 ( year 2021 ) ,\ https://arxiv.org/a...

    arXiv

  123. [131]

    author author E. Frey ,\ title title Evolutionary game theory: Theoretical concepts and applications to microbial communities ,\ https://doi.org/10.1016/j.physa.2010.02.047 journal journal Physica A: Statistical Mechanics and its Applications \ volume 389 ,\ pages 4265 ( year ...

    2010 doi

  124. [132]

    author author U. C. \ Täuber ,\ title title Population oscillations in spatial stochastic Lotka-Volterra models: a field-theoretic perturbational analysis ,\ https://doi.org/10.1088/1751-8113/45/40/405002 journal journal Journal of Physics A: Mathematical and Theoretical \ vol...

    arXiv

  125. [133]

    author author M. O. \ Lavrentovich \ and\ author D. R. \ Nelson ,\ title title Asymmetric Mutualism in Two- and Three-Dimensional Range Expansions ,\ https://doi.org/10.1103/physrevlett.112.138102 journal journal Physical Review Letters \ volume 112 ,\ pages 138102 ( year 2014...

    arXiv

  126. [134]

    Reichenbach , author M

    author author T. Reichenbach , author M. Mobilia ,\ and\ author E. Frey ,\ title title Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games ,\ https://doi.org/10.1038/nature06095 journal journal Nature \ volume 448 ,\ pages 1046 ( year 2007 ) ,\ https://...

    arXiv

  127. [135]

    Reichenbach , author M

    author author T. Reichenbach , author M. Mobilia ,\ and\ author E. Frey ,\ title title Self-organization of mobile populations in cyclic competition ,\ https://doi.org/10.1016/j.jtbi.2008.05.014 journal journal Journal of Theoretical Biology \ volume 254 ,\ pages 368 ( year 20...

    2008 arXiv

  128. [136]

    Agudo-Canalejo \ and\ author R

    author author J. Agudo-Canalejo \ and\ author R. Golestanian ,\ title title Active Phase Separation in Mixtures of Chemically Interacting Particles ,\ https://doi.org/10.1103/physrevlett.123.018101 journal journal Physical Review Letters \ volume 123 ,\ pages 018101 ( year 201...

    arXiv

  129. [137]

    Saha , author S

    author author S. Saha , author S. Ramaswamy ,\ and\ author R. Golestanian ,\ title title Pairing, waltzing and scattering of chemotactic active colloids ,\ https://doi.org/10.1088/1367-2630/ab20fd journal journal New Journal of Physics \ volume 21 ,\ pages 063006 ( year 2019 )...

  130. [138]

    author author T. H. \ Tan , author A. Mietke , author J. Li , author Y. Chen , author H. Higinbotham , author P. J. \ Foster , author S. Gokhale , author J. Dunkel ,\ and\ author N. Fakhri ,\ title title Odd dynamics of living chiral crystals ,\ https://doi.org/10.1038/s41586-...

    arXiv

  131. [139]

    Brandenbourger , author X

    author author M. Brandenbourger , author X. Locsin , author E. Lerner ,\ and\ author C. Coulais ,\ title title Non-reciprocal robotic metamaterials ,\ https://doi.org/10.1038/s41467-019-12599-3 journal journal Nature Communications \ volume 10 ,\ pages 4608 ( year 2019 ) ,\ ht...

    arXiv

  132. [140]

    Saha , author J

    author author S. Saha , author J. Agudo-Canalejo ,\ and\ author R. Golestanian ,\ title title Scalar Active Mixtures: The Nonreciprocal Cahn-Hilliard Model ,\ https://doi.org/10.1103/physrevx.10.041009 journal journal Physical Review X \ volume 10 ,\ pages 041009 ( year 2020 )...

    arXiv

  133. [141]

    Frohoff-Hülsmann , author J

    author author T. Frohoff-Hülsmann , author J. Wrembel ,\ and\ author U. Thiele ,\ title title Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling ,\ https://doi.org/10.1103/physreve.103.042602 journal journal Ph...

    arXiv

  134. [142]

    Schüler , author S

    author author D. Schüler , author S. Alonso , author A. Torcini ,\ and\ author M. Bär ,\ title title Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations ,\ https://doi.org/10.1063/1.4905017 journal journal Cha...

    doi

  135. [143]

    author author R. FitzHugh ,\ title title Impulses and Physiological States in Theoretical Models of Nerve Membrane ,\ https://doi.org/10.1016/s0006-3495(61)86902-6 journal journal Biophysical Journal \ volume 1 ,\ pages 445 ( year 1961 ) NoStop

    doi

  136. [144]

    Nagumo , author S

    author author J. Nagumo , author S. Arimoto ,\ and\ author S. Yoshizawa ,\ title title An Active Pulse Transmission Line Simulating Nerve Axon ,\ https://doi.org/10.1109/jrproc.1962.288235 journal journal Proceedings of the IRE \ volume 50 ,\ pages 2061 ( year 1962 ) NoStop

    1962

  137. [145]

    author author E. M. \ Izhikevich ,\ https://doi.org/10.7551/mitpress/2526.001.0001 title Dynamical Systems in Neuroscience \ ( publisher MIT press ,\ year 2007 ) NoStop

    2007 doi

  138. [146]

    Zheng \ and\ author J

    author author Q. Zheng \ and\ author J. Shen ,\ title title Pattern formation in the FitzHugh–Nagumo model ,\ https://doi.org/10.1016/j.camwa.2015.06.031 journal journal Computers & Mathematics with Applications \ volume 70 ,\ pages 1082 ( year 2015 ) NoStop

    2015 doi

  139. [147]

    Shoji , author K

    author author H. Shoji , author K. Yamada , author D. Ueyama ,\ and\ author T. Ohta ,\ title title Turing patterns in three dimensions ,\ https://doi.org/10.1103/physreve.75.046212 journal journal Physical Review E \ volume 75 ,\ pages 046212 ( year 2007 ) NoStop

    doi

  140. [148]

    Rocşoreanu , author A

    author author C. Rocşoreanu , author A. Georgescu ,\ and\ author N. Giurgiţeanu ,\ title title The FitzHugh-Nagumo Model, Bifurcation and Dynamics ,\ journal journal Mathematical Modelling: Theory and Applications \ https://doi.org/10.1007/978-94-015-9548-3 10.1007/978-94-015-...

    doi

  141. [149]

    Bär , author A

    author author M. Bär , author A. K. \ Bangia ,\ and\ author I. G. \ Kevrekidis ,\ title title Bifurcation and stability analysis of rotating chemical spirals in circular domains: Boundary-induced meandering and stabilization ,\ https://doi.org/10.1103/physreve.67.056126 journa...

    doi

  142. [150]

    author author A. S. \ Mikhailov \ and\ author K. Showalter ,\ title title Control of waves, patterns and turbulence in chemical systems ,\ https://doi.org/10.1016/j.physrep.2005.11.003 journal journal Physics Reports \ volume 425 ,\ pages 79 ( year 2006 ) NoStop

    2005 doi

  143. [151]

    Ziepke , author S

    author author A. Ziepke , author S. Martens ,\ and\ author H. Engel ,\ title title Wave propagation in spatially modulated tubes ,\ https://doi.org/10.1063/1.4962173 journal journal The Journal of Chemical Physics \ volume 145 ,\ pages 094108 ( year 2016 ) ,\ https://arxiv.org...

    arXiv

  144. [152]

    Gelens , author G

    author author L. Gelens , author G. A. \ Anderson ,\ and\ author J. E. \ Ferrell ,\ title title Spatial trigger waves: positive feedback gets you a long way ,\ https://doi.org/10.1091/mbc.e14-08-1306 journal journal Molecular Biology of the Cell \ volume 25 ,\ pages 3486 ( yea...

    doi

  145. [153]

    author author A. M. \ Turing ,\ title title The chemical basis of morphogenesis ,\ https://doi.org/10.1098/rstb.1952.0012 journal journal Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences \ volume 237 ,\ pages 37 ( year 1952 ) NoStop

    1952

  146. [154]

    author author P. C. \ Hohenberg \ and\ author B. I. \ Halperin ,\ title title Theory of dynamic critical phenomena ,\ https://doi.org/10.1103/revmodphys.49.435 journal journal Reviews of Modern Physics \ volume 49 ,\ pages 435 ( year 1977 ) NoStop

    doi

  147. [155]

    Cebrián-Lacasa , author P

    author author D. Cebrián-Lacasa , author P. Parra-Rivas , author D. Ruiz-Reynés ,\ and\ author L. Gelens ,\ title title Six decades of the FitzHugh–Nagumo model: A guide through its spatio-temporal dynamics and influence across disciplines ,\ https://doi.org/10.1016/j.physrep....

    arXiv

  148. [156]

    Ohta , author M

    author author T. Ohta , author M. Mimura ,\ and\ author R. Kobayashi ,\ title title Higher-dimensional localized patterns in excitable media ,\ https://doi.org/10.1016/0167-2789(89)90230-3 journal journal Physica D: Nonlinear Phenomena \ volume 34 ,\ pages 115 ( year 1989 ) NoStop

    doi

  149. [157]

    author author D. M. \ Petrich \ and\ author R. E. \ Goldstein ,\ title title Nonlocal contour dynamics model for chemical front motion ,\ https://doi.org/10.1103/physrevlett.72.1120 journal journal Physical Review Letters \ volume 72 ,\ pages 1120 ( year 1994 ) ,\ https://arxi...

    arXiv

  150. [158]

    author author R. E. \ Goldstein , author D. J. \ Muraki ,\ and\ author D. M. \ Petrich ,\ title title Interface proliferation and the growth of labyrinths in a reaction-diffusion system ,\ https://doi.org/10.1103/physreve.53.3933 journal journal Physical Review E \ volume 53 ,...

    arXiv

  151. [159]

    author author R. C. \ Tolman ,\ title title The Effect of Droplet Size on Surface Tension ,\ https://doi.org/10.1063/1.1747247 journal journal The Journal of Chemical Physics \ volume 17 ,\ pages 333 ( year 1949 ) NoStop

    1949 doi

  152. [160]

    author author E. M. \ Blokhuis \ and\ author J. Kuipers ,\ title title Thermodynamic expressions for the Tolman length ,\ https://doi.org/10.1063/1.2167642 journal journal The Journal of Chemical Physics \ volume 124 ,\ pages 074701 ( year 2006 ) NoStop

    doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.