arxiv: 2605.15158 · v1 · submitted 2026-05-14 · ❄️ cond-mat.soft

Recognition: 2 theorem links

· Lean Theorem

Duality Between Chemical Potential Dynamics and Reaction-Diffusion Systems

Daniel Zhou , Erwin Frey

Authors on Pith no claims yet

Pith reviewed 2026-05-15 03:14 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords dualitychemical potentialreaction-diffusionpattern formationphase separationmass conservationattracting manifoldCahn-Hilliard

0 comments

The pith

Chemical-potential theories embed as slow manifolds in mass-conserving reaction-diffusion systems

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

This paper constructs an exact duality showing that phase-field models driven by chemical-potential gradients with conserved order parameters are recovered as the slow dynamics on an attracting manifold of a mass-conserving reaction-diffusion system. Conversely, any mass-conserving reaction-diffusion system whose nullcline is attractive recovers a chemical-potential representation exactly in the fast-interconversion limit, with the constitutive relation fixed by that nullcline. The mapping resolves non-invertibility of the chemical potential in phase-separating regimes by lifting the description to an extended two-field system with conserved total density. Gradient terms in the original model translate into an intrinsic reaction-diffusion length set by the auxiliary field. The same construction supplies an explicit dictionary that equates the Maxwell equal-area rule with reactive turnover balance and produces closed-form traveling-wave velocities for nonreciprocal extensions.

Core claim

McRD is the broader class: every chemical-potential theory with conserved order parameters embeds as the slow dynamics on an attracting manifold of an McRD system; conversely, every McRD with attractive nullcline admits an exact chemical-potential representation in the fast-interconversion limit, with the constitutive relation set by the nullcline. The construction resolves the generic non-invertibility of the chemical-potential as a function of density in phase-separating regimes by embedding it as an attracting manifold in an extended two-field description with conserved total density.

What carries the argument

The attracting manifold of the extended two-field McRD system, on which the chemical-potential dynamics emerge as slow motion with the nullcline fixing the density-chemical-potential relation.

If this is right

The Maxwell equal-area construction for phase coexistence is exactly equivalent to the reactive turnover-balance condition.
Gradient stiffness maps to an intrinsic reaction-diffusion length set by the auxiliary field, yielding a diagonal-diffusion normal form whose interface profile matches the original Cahn-Hilliard model.
The duality extends to weakly nonconservative dynamics, unifying reaction-arrested coarsening and mesa splitting.
It supplies a closed-form velocity law for traveling waves in nonreciprocal Cahn-Hilliard dynamics that matches simulations.

Where Pith is reading between the lines

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

Numerical schemes could switch between the two representations to exploit conservation in one regime and local kinetics in another.
The same manifold construction may apply to other active-matter models where slow effective dynamics arise from fast interconversion.
Experimental measurements of interface velocities in multicomponent biological systems could test whether the predicted turnover-balance condition holds.

Load-bearing premise

An attracting manifold must exist for the slow dynamics and the nullcline must be attractive in the fast-interconversion limit.

What would settle it

A concrete chemical-potential model whose interface profiles or traveling-wave speeds cannot be reproduced by any McRD system on a slow manifold, or an McRD simulation whose recovered chemical potential deviates from the original constitutive relation.

Figures

Figures reproduced from arXiv: 2605.15158 by Daniel Zhou, Erwin Frey.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Characteristic coarsening length Λ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Convergence of the dual two-field dynamics to Cahn–Hilliard with reactive turnover in the stiff limit. Snapshots of the [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: shows a single set of symbols. To quantify the agreement, we define the normalized L 2 deviation e = R Ω dx (c + m − ϕCH) 2 R Ω dx (ϕCH) 2 , (79) where Ω denotes the spatial domain. For D˜m = 10, one obtains e ∼ 10−12 across all probed parameter points, confirming that the two formulations yield pointwise identical profiles to within numerical precision. Regime structure in the phase diagram.— [PITH_FULL_… view at source ↗

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗

read the original abstract

Pattern formation in soft, active, and biological matter is described by two ostensibly distinct continuum frameworks: phase-field theories driven by chemical-potential gradients, and mass-conserving reaction-diffusion (McRD) dynamics governed by local interconversion kinetics. Here we establish a constructive, equation-level duality valid in the nonlinear, far-from-equilibrium regime. McRD is the broader class: every chemical-potential theory with conserved order parameters embeds as the slow dynamics on an attracting manifold of an McRD system; conversely, every McRD with attractive nullcline admits an exact chemical-potential representation in the fast-interconversion limit, with the constitutive relation set by the nullcline. The construction resolves the generic non-invertibility of the chemical-potential as a function of density in phase-separating regimes by embedding it as an attracting manifold in an extended two-field description with conserved total density. Gradient stiffness maps faithfully onto an intrinsic reaction-diffusion length set by the auxiliary field, yielding a diagonal-diffusion normal form whose interface profile matches the original Cahn-Hilliard model by construction. The duality yields an explicit dictionary for phase coexistence: the Maxwell equal-area construction is exactly equivalent to the reactive turnover-balance condition. It extends to weakly nonconservative dynamics, unifying reaction-arrested coarsening and mesa splitting, and to multicomponent theories with broken Maxwell symmetry. As a concrete payoff, the dual sharp-interface picture yields a closed-form velocity law for traveling waves in nonreciprocal Cahn-Hilliard dynamics, in quantitative agreement with simulations.

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 builds a workable equation-level duality between chemical-potential phase fields and McRD systems, with the main caveat that manifold attractiveness needs explicit checks in the spinodal.

read the letter

The core result is a direct mapping that embeds any conserved chemical-potential model as the slow dynamics on an attracting manifold of a mass-conserving reaction-diffusion system, and recovers the chemical-potential form from McRD when the nullcline is attractive in the fast-interconversion limit. They fix the usual non-invertibility problem by lifting to a two-field description whose fast variable relaxes to the nullcline, and they show that gradient stiffness becomes an intrinsic reaction length. This gives an exact dictionary: the Maxwell equal-area rule matches the reactive turnover balance, and they extract a closed-form traveling-wave velocity for nonreciprocal Cahn-Hilliard that agrees with simulations. The construction also covers weakly nonconservative extensions and multicomponent cases with broken symmetry. That is the concrete payoff, and it is new at the level of explicit nonlinear mappings rather than just formal analogies. The paper does a clean job showing how the two frameworks can trade analytical and numerical tools once the dictionary is in place. The soft spot is the attractiveness assumption. The stress-test note is right to flag that non-monotonic chemical potentials inside the spinodal could produce a Jacobian without strictly negative transverse eigenvalues at finite rates, which would break hyperbolicity or conservation in the reduction. The abstract states the nullcline is attractive but does not spell out the spectral condition here, so that step needs verification in the full derivations. If it holds, the duality is solid; if not, the embedding is only approximate in parts of the phase diagram. This is for modelers in active matter and biological pattern formation who already work with one framework and want to borrow results from the other. A reader looking for transferable techniques or a new way to compute interface speeds will find it useful. I would send it to peer review; the construction is explicit enough that referees can check the manifold stability directly.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes a constructive, equation-level duality between chemical-potential gradient theories (e.g., Cahn-Hilliard models with conserved order parameters) and mass-conserving reaction-diffusion (McRD) systems. McRD is positioned as the broader class: every chemical-potential theory embeds as the slow dynamics on an attracting manifold of an McRD system, while every McRD system with an attractive nullcline recovers an exact chemical-potential representation in the fast-interconversion limit, with the constitutive relation fixed by the nullcline. The construction resolves non-invertibility via a two-field lift, maps gradient stiffness to an intrinsic reaction-diffusion length, equates the Maxwell equal-area rule to reactive turnover balance, and extends to weakly nonconservative dynamics and multicomponent cases, yielding a closed-form traveling-wave velocity law that matches simulations.

Significance. If the duality holds, it unifies two central frameworks for pattern formation in soft, active, and biological matter, enabling direct transfer of analytical tools such as sharp-interface reductions and coexistence criteria between conservative and reactive descriptions. The explicit, parameter-free dictionary and the quantitative agreement of the derived velocity law with simulations are notable strengths that could facilitate new predictions for interface dynamics and coarsening arrest.

major comments (1)

[Abstract and slow-manifold construction] Abstract and slow-manifold construction: the central duality requires that the nullcline manifold remain attracting with strictly negative transverse eigenvalues even for non-monotonic constitutive relations inside the spinodal. The provided text states the result for attractive nullclines but does not supply an explicit spectral condition on the fast-reaction Jacobian or a verification that hyperbolicity and conservation are preserved at finite interconversion rates; without this, the exact embedding may fail for the non-monotonic cases that are physically central to phase separation.

minor comments (2)

[Duality construction] The mapping of gradient stiffness to the auxiliary-field reaction-diffusion length is stated to be faithful, but the explicit normal-form transformation and its preservation of the interface profile should be shown in a dedicated equation block for clarity.
[Multicomponent extension] The extension to multicomponent theories with broken Maxwell symmetry is mentioned; a brief statement of the modified coexistence condition would strengthen the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive comment on the slow-manifold construction. We address the point below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

Referee: [Abstract and slow-manifold construction] Abstract and slow-manifold construction: the central duality requires that the nullcline manifold remain attracting with strictly negative transverse eigenvalues even for non-monotonic constitutive relations inside the spinodal. The provided text states the result for attractive nullclines but does not supply an explicit spectral condition on the fast-reaction Jacobian or a verification that hyperbolicity and conservation are preserved at finite interconversion rates; without this, the exact embedding may fail for the non-monotonic cases that are physically central to phase separation.

Authors: We agree that an explicit spectral condition on the fast-reaction Jacobian would strengthen the presentation of the duality. In the revised manuscript we will add a dedicated paragraph (and a brief note in the abstract) stating that the nullcline must remain attracting, which requires the fast-reaction Jacobian to possess strictly negative eigenvalues in all directions transverse to the nullcline manifold. This spectral gap is the precise mathematical content of our attractiveness assumption and guarantees that the slow manifold is asymptotically stable. For the non-monotonic constitutive relations inside the spinodal that are central to phase separation, the construction continues to hold whenever this condition is satisfied; we have verified it explicitly via linearization around the nullcline in the phase-separating examples. Hyperbolicity of the slow dynamics and exact conservation of total density are preserved at finite interconversion rates by the two-field lift, which we will confirm in a short appendix by showing that the lifted system remains hyperbolic in the normal coordinates and that the total-density equation decouples exactly from the transverse dynamics. revision: yes

Circularity Check

0 steps flagged

Explicit constructive duality from governing equations; no reduction to inputs by definition or self-citation

full rationale

The paper derives the claimed duality by direct embedding of chemical-potential theories as slow dynamics on an attracting manifold of an explicitly constructed McRD system, and conversely recovers the chemical-potential representation from the fast-interconversion limit with the constitutive relation set by the nullcline. Equivalences such as the Maxwell construction equaling the reactive turnover-balance condition follow from algebraic comparison of the resulting equations rather than from any fitted parameter or self-referential definition. No load-bearing step invokes a prior self-citation whose content is itself unverified, nor renames a known result, nor calls a fit a prediction. The attractiveness of the nullcline is stated as an assumption required for the reduction to hold, but this is an external condition on the equations rather than a circularity in the derivation chain itself. The construction is therefore self-contained against the governing PDEs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard dynamical-systems assumptions about the existence and attractiveness of slow manifolds and nullclines; no free parameters are introduced and no new entities are postulated.

axioms (2)

domain assumption Existence of an attracting manifold on which the slow dynamics of the chemical-potential theory is recovered
Invoked to embed every chemical-potential theory as the slow dynamics of an McRD system.
domain assumption Attractive nullcline in the fast-interconversion limit
Required for the converse exact chemical-potential representation with constitutive relation set by the nullcline.

pith-pipeline@v0.9.0 · 5567 in / 1452 out tokens · 79903 ms · 2026-05-15T03:14:42.711089+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

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

every McRD with attractive nullcline admits an exact chemical-potential representation in the fast-interconversion limit, with the constitutive relation set by the nullcline
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

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

Maxwell equal-area construction is exactly equivalent to the reactive turnover-balance condition

What do these tags mean?

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

Reference graph

Works this paper leans on

80 extracted references · 80 canonical work pages

[1]

The source–sink term is expanded in the same way as before, s(ϕ) =s α +s ′ α δϕ+O(δϕ 2),(C14) ForD m = 0 (so thatD c =M), the linearized dynamics read ∂tδc=M∇ 2δc−A c δc−A m δm+s α +s ′ αδϕ,(C15a) ∂tδm=A c δc+A m δm.(C15b) Assuming fast relaxation of the auxiliary mode, we im- pose quasi-stationarity ofmon the droplet scale, 0 =A c δc+A m δm=⇒δm=− Ac Am δ...

work page
[2]

II D) Equations solved.For Figs

Numerical comparison: Cahn–Hilliard vs.κ-free dual McRD dynamics (Sec. II D) Equations solved.For Figs. 3, 4, and 5, we solve two systems in parallel: (i) the nondimensionalized Cahn–Hilliard equation [Eq. (3)], ∂tϕ=∇ 2 ϕ3 −ϕ−˜κ∇ 2ϕ ; (E1) (ii) theκ-free dual McRD system [Eq. (20)], with pa- rameters chosen to satisfy the matching conditions of Eq. (29). ...

work page
[3]

IV A) Equations solved.In Figs

Dual reaction–diffusion model of Cahn–Hilliard with reactive turnover (Sec. IV A) Equations solved.In Figs. 6 and 7 we compare two systems side by side: (i) the Cahn–Hilliard with reactive turnover equation [Eq. (65)] with a linear source, ∂tϕ=M∇ 2 −r(ϕ−ϕ eq) + (ϕ−ϕ eq)3 −κ∇ 2ϕ +k 2 −k 1ϕ; (E2) (ii) the dual two-field system [Eq. (68)], with parame- ters ...

work page
[4]

V C) Equations solved.We integrate the original nonrecip- rocal two-component conserved dynamics [Eqs

Comparison to numerical simulations: nonreciprocal traveling waves (Sec. V C) Equations solved.We integrate the original nonrecip- rocal two-component conserved dynamics [Eqs. (98)] di- rectly; the dual McRD representation [Eqs. (100)] is used only for the analytical sharp-interface construction and not for numerical integration. We did verify its matchin...

work page
[5]

P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys.49, 435 (1977)

work page 1977
[6]

Bray, Advances in Physics43, 357–459 (1994)

A. Bray, Advances in Physics43, 357–459 (1994)

work page 1994
[7]

J. W. Cahn and J. E. Hilliard, The Journal of Chemical Physics28, 258 (1958)

work page 1958
[8]

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

work page 2013
[9]

M. J. Bowick, N. Fakhri, M. C. Marchetti, and S. Ra- maswamy, Phys. Rev. X12, 010501 (2022)

work page 2022
[10]

M. E. Cates and C. Nardini, Rep. Prog. Phys.88, 056601 (2025)

work page 2025
[11]

Gompper, R

G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nar- dini, F. Peruani, H. L¨ owen, R. Golestanian, U. B. Kaupp, L. Alvarez, T. Kiørboe, E. Lauga, W. C. K. Poon, A. DeSimone, S. Mui˜ nos-Landin, A. Fischer, N. A. S¨ oker, F. Cichos, R. Kapral, P. Gaspard, M. Ripoll, F. Sagues, A. Doostmohammadi, J. M. Yeomans, I. S. Aranson, C. Bechinger, H. Stark, C. K....

work page 2020
[12]

M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.65, 851 (1993)

work page 1993
[13]

K. S. Korolev, M. Avlund, O. Hallatschek, and D. R. Nelson, Rev. Mod. Phys.82, 1691 (2010)

work page 2010
[14]

Frey, Physica A389, 4265 (2010)

E. Frey, Physica A389, 4265 (2010)

work page 2010
[15]

Edelstein-Keshet, W

L. Edelstein-Keshet, W. R. Holmes, M. Zajac, and M. Dutot, Philosophical Transactions of the Royal So- ciety B: Biological Sciences368, 20130003 (2013)

work page 2013
[16]

A. B. Goryachev and M. Leda, Molecular Biology of the Cell28, 370 (2017)

work page 2017
[17]

Frey and H

E. Frey and H. Weyer, Annual Review of Biophysics 55, 10.1146/annurev-biophys-030822-031638 (2026), re- view in Advance first posted online February 10, 2026; in press

work page doi:10.1146/annurev-biophys-030822-031638 2026
[18]

Ishihara, M

S. Ishihara, M. Otsuji, and A. Mochizuki, Phys. Rev. E 75, 015203 (2007)

work page 2007
[19]

Y. Mori, A. Jilkine, and L. Edelstein-Keshet, Biophysical Journal94, 3684 (2008)

work page 2008
[20]

Otsuji, S

M. Otsuji, S. Ishihara, C. Co, K. Kaibuchi, A. Mochizuki, and S. Kuroda, PLoS Computational Biology3, e108 (2007)

work page 2007
[21]

A. B. Goryachev and A. V. Pokhilko, FEBS Letters582, 1437 (2008)

work page 2008
[22]

Halatek and E

J. Halatek and E. Frey, Nature Physics14, 507 (2018)

work page 2018
[23]

Brauns, J

F. Brauns, J. Halatek, and E. Frey, Physical Review X 10, 041036 (2020)

work page 2020
[24]

Onsager, Phys

L. Onsager, Phys. Rev.37, 405 (1931)

work page 1931
[25]

Onsager, Phys

L. Onsager, Phys. Rev.38, 2265 (1931)

work page 1931
[26]

H. B. G. Casimir, Rev. Mod. Phys.17, 343 (1945)

work page 1945
[27]

S. R. de Groot and P. Mazur,Non-Equilibrium Thermo- dynamics(North-Holland Publishing Company, Amster- dam, 1962)

work page 1962
[28]

H. B. Callen,Thermodynamics and an Introduction to Thermostatistics, 2nd ed. (John Wiley & Sons, 1985)

work page 1985
[29]

Graham and H

R. Graham and H. Haken, Z. Physik243, 289 (1971)

work page 1971
[30]

Risken, Z

H. Risken, Z. Physik251, 231 (1972)

work page 1972
[31]

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

work page 2015
[32]

Wittkowski, A

R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen, D. Marenduzzo, and M. E. Cates, Nature Com- munications5, 4351 (2014)

work page 2014
[33]

Tjhung, C

E. Tjhung, C. Nardini, and M. E. Cates, Phys. Rev. X 8, 031080 (2018), published 11 June 2018

work page 2018
[34]

S. Saha, J. Agudo-Canalejo, and R. Golestanian, Phys. Rev. X10, 041009 (2020)

work page 2020
[35]

Z. You, A. Baskaran, and M. C. Marchetti, Proc. Natl. Acad. Sci. U.S.A.117, 19767 (2020)

work page 2020
[36]

Tailleur and M

J. Tailleur and M. E. Cates, Physical Review Letters100, 218103 (2008)

work page 2008
[37]

Schl¨ ogl, Z

F. Schl¨ ogl, Z. Physik253, 147 (1972)

work page 1972
[38]

S. M. Allen and J. W. Cahn, Acta Metallurgica27, 1085 (1979)

work page 1979
[39]

Frohoff-H¨ ulsmann and U

T. Frohoff-H¨ ulsmann and U. Thiele, Physical Review Letters131, 107201 (2023)

work page 2023
[40]

Morita and T

Y. Morita and T. Ogawa, Nonlinearity23, 1387 (2010)

work page 2010
[41]

T. A. Roth, H. Weyer, and E. Frey, Interface dynam- ics in mass-conserving reaction–diffusion systems (2026), preprint. arXiv:{arXiv:}

work page 2026
[42]

Weyer, T

H. Weyer, T. A. Roth, and E. Frey, Nature Physics22, 94 (2026), published online 2 Dec 2025

work page 2026
[43]

J. F. Robinson, T. Machon, and T. Speck, Phys. Rev. E 111, 065417 (2025)

work page 2025
[44]

3 and of the phase- space dynamics shown in Fig

See Supplemental Material at [URL will be inserted by publisher] for movies of the spinodal-decomposition and coarsening dynamics shown in Fig. 3 and of the phase- space dynamics shown in Fig. 5

work page
[45]

I. M. Lifshitz and V. V. Slyozov, Journal of Physics and Chemistry of Solids19, 35 (1961)

work page 1961
[46]

Wagner, Zeitschrift f¨ ur Elektrochemie65, 581 (1961)

C. Wagner, Zeitschrift f¨ ur Elektrochemie65, 581 (1961)

work page 1961
[47]

Toffenetti, B

D. Toffenetti, B. Nettuno, H. Weyer, and E. Frey, Escaping Cahn–Hilliard: Active Model B − from three-component reaction diffusion (2026), preprint, arXiv:XXXX.XXXXX [cond-mat.soft]

work page 2026
[48]

Zwicker, Current Opinion in Colloid & Interface Sci- ence61, 101606 (2022)

D. Zwicker, Current Opinion in Colloid & Interface Sci- ence61, 101606 (2022)

work page 2022
[49]

C. A. Weber, D. Zwicker, F. J¨ ulicher, and C. F. Lee, Reports on Progress in Physics82, 064601 (2019)

work page 2019
[50]

S. C. Glotzer, D. Stauffer, and N. Jan, Phys. Rev. Lett. 72, 4109 (1994)

work page 1994
[51]

Puri and H

S. Puri and H. L. Frisch, Journal of Physics A: Mathe- matical and General27, 6027 (1994)

work page 1994
[52]

J. J. Christensen, K. Elder, and H. C. Fogedby, Phys. Rev. E54, R2212 (1996)

work page 1996
[53]

Carati and R

D. Carati and R. Lefever, Phys. Rev. E56, 3127 (1997)

work page 1997
[54]

Brauns, H

F. Brauns, H. Weyer, J. Halatek, J. Yoon, and E. Frey, Physical Review Letters126, 104101 (2021)

work page 2021
[55]

Weyer, F

H. Weyer, F. Brauns, and E. Frey, Physical Review E 108, 064202 (2023). 45

work page 2023
[56]

Zwicker, R

D. Zwicker, R. Seyboldt, C. A. Weber, A. A. Hyman, and F. J¨ ulicher, Nature Physics13, 408 (2017)

work page 2017
[57]

B. A. Huberman, The Journal of Chemical Physics65, 2013 (1976)

work page 2013
[58]

S. C. Glotzer, E. A. Di Marzio, and M. Muthukumar, Phys. Rev. Lett.74, 2034 (1995)

work page 2034
[59]

D. S. Cohen and J. D. Murray, Journal of Mathematical Biology12, 237 (1981)

work page 1981
[60]

M. E. Cates, D. Marenduzzo, I. Pagonabarraga, and J. Tailleur, Proceedings of the National Academy of Sci- ences107, 11715–11720 (2010)

work page 2010
[61]

Winter, Y

A. Winter, Y. Liu, A. Ziepke, G. Dadunashvili, and E. Frey, Phys. Rev. E111, 044405 (2025)

work page 2025
[62]

F. C. Thewes, Y. Qiang, O. W. Paulin, and D. Zwicker, Phase separation with non-local interactions (2025), arXiv:2511.05214 [cond-mat.soft]

work page arXiv 2025
[63]

Zwicker, M

D. Zwicker, M. Decker, F. Jaeger, A. A. Hyman, and F. J¨ ulicher, Phys. Rev. E92, 012317 (2015)

work page 2015
[64]

Nardini, ´E

C. Nardini, ´E. Fodor, E. Tjhung, F. van Wijland, J. Tailleur, and M. E. Cates, Phys. Rev. X7, 021007 (2017)

work page 2017
[65]

Brauns and M

F. Brauns and M. C. Marchetti, Phys. Rev. X14, 021014 (2024)

work page 2024
[66]

Frohoff-H¨ ulsmann, J

T. Frohoff-H¨ ulsmann, J. Wrembel, and U. Thiele, Phys. Rev. E103, 042602 (2021)

work page 2021
[67]

Rana and R

N. Rana and R. Golestanian, Phys. Rev. Lett.133, 078301 (2024)

work page 2024
[68]

Fruchart, R

M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Nature592, 363 (2021)

work page 2021
[69]

Raßhofer, S

F. Raßhofer, S. Bauer, A. Ziepke, I. Maryshev, and E. Frey, Phys. Rev. Res.7, 043240 (2025)

work page 2025
[70]

S. C. Glotzer and A. Coniglio, Phys. Rev. E50, 4241 (1994)

work page 1994
[71]

Ohta and K

T. Ohta and K. Kawasaki, Macromolecules19, 2621 (1986)

work page 1986
[72]

Liu and N

F. Liu and N. Goldenfeld, Physical Review A39, 4805 (1989)

work page 1989
[73]

W. W. Mullins and R. F. Sekerka, Journal of Applied Physics34, 323 (1963)

work page 1963
[74]

Gomez, Y

D. Gomez, Y. Mori, and S. Strikwerda, (2025), arXiv:2509.21706 [math-ph]

work page arXiv 2025
[75]

Zhou and E

D. Zhou and E. Frey, Partial duality and hybrid Cahn–Hilliard–reaction–diffusion descriptions of non- reciprocal phase separation (2026), in preparation, arXiv:XXXX.XXXXX [cond-mat.soft]

work page 2026
[76]

A. P. Solon, J. Stenhammar, M. E. Cates, Y. Kafri, and J. Tailleur, New J. Phys.20, 075001 (2018)

work page 2018
[77]

Langford and A

L. Langford and A. K. Omar, The Journal of Chemical Physics163, 10.1063/5.0263060 (2025)

work page doi:10.1063/5.0263060 2025
[78]

L. Li, Z. Sun, and M. Yang, Proc. Natl. Acad. Sci. U. S. A.122, e2505010122 (2025)

work page 2025
[79]

S. Saha, J. Agudo-Canalejo, and R. Golestanian, Physical Review X10, 041009 (2020)

work page 2020
[80]

COMSOL Multiphysics, COMSOL AB (2019)

work page 2019