pith. sign in

arxiv: 2605.16503 · v1 · pith:XZT5XZMVnew · submitted 2026-05-15 · ❄️ cond-mat.stat-mech · cond-mat.soft

From bulk to interface dynamics, in and out of equilibrium

Lila Sarfati , Julien Tailleur , Fr\'ed\'eric van Wijland This is my paper

Pith reviewed 2026-05-19 21:18 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft
keywords interface dynamicsfluctuating hydrodynamicsactive matterphase separationnon-equilibrium systemsdynamical actionstochastic field theories
0
0 comments X

The pith

A dynamical-action approach derives linear interface relaxation and fluctuations from bulk hydrodynamics for both equilibrium and non-equilibrium systems.

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

This paper develops a method to extract the dynamics of weakly deformed interfaces directly from the fluctuating hydrodynamics of the phase-separating fields. By choosing an appropriate interface definition and employing the dynamical-action formalism for path probabilities, it obtains the linear relaxation and fluctuations around the interface. The approach works for equilibrium models and extends to their active, non-equilibrium counterparts, providing a controlled alternative to common approximations. Readers interested in phase separation and pattern formation would value a unified derivation that avoids uncontrolled steps in driven systems.

Core claim

We study the dynamics of weakly deformed interfaces separating two stable phases, starting from the fluctuating hydrodynamics of the phase-separating fields. Using a well-chosen definition for the interface and the dynamical-action formalism to represent path probabilities, we derive the linear relaxation of the interface and the fluctuations around it for a large class of models. Our method applies to equilibrium dynamics, where it recovers and complements existing results, but also extends to their non-equilibrium counterparts. We explain how non-linear terms can be systematically computed and illustrate their derivations in the case of (active) model A.

What carries the argument

The dynamical-action formalism for representing path probabilities, which encodes the likelihood of field trajectories and permits projection of the bulk stochastic equations onto interface coordinates.

If this is right

  • The linear relaxation rate and fluctuation spectrum of the interface height follow directly from the bulk hydrodynamic parameters.
  • Nonlinear corrections to the interface equation can be computed order by order from successive expansions of the same action.
  • The derivation recovers known equilibrium interface results as a special case while remaining valid for active systems.
  • A common ansatz for obtaining interface equations is valid at equilibrium but produces uncontrolled errors in active field theories.

Where Pith is reading between the lines

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

  • The projection method could be applied to derive interface equations in other driven systems with multiple conserved quantities.
  • Numerical simulations of active phase separation could test the predicted linear fluctuation spectrum against the analytic expressions.
  • The framework might generalize to curved interfaces or to dynamics beyond the linear regime by retaining higher-order terms.

Load-bearing premise

The interface remains only weakly deformed and the two phases stay stable and distinct.

What would settle it

Simulate the full field equations for a small sinusoidal perturbation of the interface in an active phase-separating system such as active model A and check whether the measured relaxation rate matches the rate obtained from the bulk parameters via the projection.

Figures

Figures reproduced from arXiv: 2605.16503 by Fr\'ed\'eric van Wijland, Julien Tailleur, Lila Sarfati.

Figure 1
Figure 1. Figure 1: Fluctuations of the interface between two sta [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Left) Relaxation time of a one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relaxation time of a one-dimensional inter [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relaxation of a one-dimensional interface in a [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
read the original abstract

We study the dynamics of weakly deformed interfaces separating two stable phases, starting from the fluctuating hydrodynamics of the phase-separating fields. Using a well-chosen definition for the interface and the dynamical-action formalism to represent path probabilities, we derive the linear relaxation of the interface and the fluctuations around it for a large class of models. Our method applies to equilibrium dynamics, where it recovers and complements existing results, but also extends to their non-equilibrium counterparts. We explain how non-linear terms can be systematically computed and illustrate their derivations in the case of (active) model A. We highlight the danger of a popular ansatz used to derive interface dynamics, which was rigorously established in equilibrium but is uncontrolled for active field theories.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript derives the linear relaxation of weakly deformed interfaces and the fluctuations around them, starting from the fluctuating hydrodynamics of phase-separating fields. Using a specific interface definition and the dynamical-action formalism for path probabilities, the approach recovers and complements known equilibrium results while extending to non-equilibrium dynamics; it also shows how non-linear terms can be computed systematically and illustrates this for active Model A, while flagging the uncontrolled nature of a common ansatz in active field theories.

Significance. If the central derivations are rigorous, the work supplies a systematic route from bulk fluctuating hydrodynamics to interface equations that applies across equilibrium and active non-equilibrium models. The explicit treatment of non-linear corrections and the caution regarding the popular ansatz constitute clear strengths that could influence studies of phase separation in active matter.

major comments (2)
  1. [§4] §4 (non-equilibrium projection): the derivation of linear interface relaxation relies on projecting the dynamical action onto interface coordinates under the assumption of weak deformation and stable phases; however, the manuscript does not provide explicit bounds showing that higher-order non-linear terms (illustrated for active Model A) remain negligible or controlled when the same projection is applied out of equilibrium, leaving open the possibility of uncontrolled corrections analogous to those in the criticized ansatz.
  2. [§5.2] §5.2 (active Model A illustration): the computation of non-linear terms is presented as systematic, yet the section does not quantify the truncation error or demonstrate that the retained linear relaxation and fluctuation spectra are insensitive to the neglected contributions under the stated weak-deformation condition.
minor comments (2)
  1. [Notation] The notation distinguishing the interface height function from the underlying bulk order-parameter field is introduced gradually; a consolidated table of symbols at the beginning of the methods section would improve readability.
  2. [Introduction] A few references to prior interface derivations in active systems (e.g., works on active Model B) appear only in passing; expanding the discussion in the introduction would better situate the new results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the strengths of our approach in deriving interface dynamics from bulk fluctuating hydrodynamics. We address the major comments point by point below, and have made revisions to the manuscript to clarify the points raised.

read point-by-point responses

  1. Referee: §4 (non-equilibrium projection): the derivation of linear interface relaxation relies on projecting the dynamical action onto interface coordinates under the assumption of weak deformation and stable phases; however, the manuscript does not provide explicit bounds showing that higher-order non-linear terms (illustrated for active Model A) remain negligible or controlled when the same projection is applied out of equilibrium, leaving open the possibility of uncontrolled corrections analogous to those in the criticized ansatz.

    Authors: We agree that providing more explicit discussion on the control of higher-order terms would improve the manuscript. Our derivation uses the dynamical action formalism, where the projection is exact for the linear response under the weak deformation assumption, as the interface coordinate is treated as the collective variable. The non-linear terms arise as perturbative corrections in the amplitude of the deformation. This structure is preserved out of equilibrium because the action is defined similarly. We have revised the manuscript to include a brief discussion in §4 on why the corrections remain controlled by the small parameter (deformation amplitude), similar to standard hydrodynamic derivations, and note that full bounds would require a more mathematical treatment beyond the scope of this work but the perturbative expansion is systematic. revision: partial

  2. Referee: §5.2 (active Model A illustration): the computation of non-linear terms is presented as systematic, yet the section does not quantify the truncation error or demonstrate that the retained linear relaxation and fluctuation spectra are insensitive to the neglected contributions under the stated weak-deformation condition.

    Authors: We thank the referee for this observation. In §5.2, we illustrate the systematic computation by deriving the first non-linear correction explicitly for active Model A. The linear relaxation and fluctuation spectra are obtained from the leading terms in the expansion and are independent of the higher-order contributions at this order. To address the concern, we have added a sentence quantifying that the truncation error is of order O(δh²) where δh is the interface deformation, which is small by assumption, ensuring the leading spectra are unaffected. This demonstrates the insensitivity at the linear level. revision: yes

Circularity Check

0 steps flagged

Derivation from bulk fluctuating hydrodynamics is self-contained with no circular reductions

full rationale

The paper starts from the fluctuating hydrodynamics of phase-separating fields and uses the dynamical-action formalism to represent path probabilities, projecting onto a well-chosen interface definition to obtain linear relaxation and fluctuations. This holds for equilibrium (recovering prior results) and extends to non-equilibrium cases, with systematic inclusion of non-linear terms shown explicitly for active Model A. The central claims rest on the bulk equations plus controlled projection under stated assumptions of weak deformation and phase stability; no load-bearing step reduces by construction to a fit, self-citation, or imported ansatz, and the paper itself flags uncontrolled approximations in alternative approaches.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the fluctuating hydrodynamics of phase-separating fields and the dynamical-action formalism; these are treated as established background rather than derived inside the paper.

axioms (2)
  • domain assumption Fluctuating hydrodynamics provides a valid description of the phase-separating fields both in and out of equilibrium.
    Invoked at the start of the derivation to represent the bulk dynamics from which the interface equations are obtained.
  • domain assumption The dynamical-action formalism correctly encodes path probabilities for the chosen class of models.
    Used to convert the stochastic bulk dynamics into an action whose saddle or expansion yields the interface relaxation and fluctuations.

pith-pipeline@v0.9.0 · 5654 in / 1135 out tokens · 72246 ms · 2026-05-19T21:18:10.331252+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

126 extracted references · 126 canonical work pages · 5 internal anchors

  1. [1]

    (1) with a nonequilibrium drivew[ϕ] = −λ(∇ϕ)2, leading to: ∂tϕ=− δF δϕ −λ(∇ϕ) 2 + √ 2T η(r, z, t),(146) whereF[ϕ] remains given by Eq

    Deriving the Edwards-Wilkinson equation The dynamics of active model A (AMA) is de- fined by Eq. (1) with a nonequilibrium drivew[ϕ] = −λ(∇ϕ)2, leading to: ∂tϕ=− δF δϕ −λ(∇ϕ) 2 + √ 2T η(r, z, t),(146) whereF[ϕ] remains given by Eq. (2). Note that we keep the notationTfor the amplitude of the noise, even thoughTis not, thermodynamically speaking, a tempera...

    work page

  2. [2]

    (149), withu 0 =ℓ 0

    Deriving a KPZ equation The full action for active model A is given in Eq. (149), withu 0 =ℓ 0. We shall now follow the same steps as for passive model A and implement a perturbation expansion in powers of √ T. To pro- ceed, we change variables from ¯hto ¯h′ to eliminate time derivatives from the non-Gaussian part of the action: ¯h= ¯h′ + ¯χ⊥, ∂zχ⊥ 1 +⟨∂ ...

    work page

  3. [3]

    Motivations for introducing active model B+ Let us now turn to the simplest nonequilibrium generalization of model B. In active matter, the motility-induced phase separation [100–106] under- gone by pairwise repulsive self-propelled particles, presents features strikingly similar to equilibrium phase separation, in spite of being an intrinsically nonequil...

    work page

  4. [4]

    (170) The lack of steady-state current enforces the stronger constraint jc =−∂ z −∂2 z mc +f ′(mc) + 2λ−ζ 2 (∂zmc)2 = 0

    Dynamical action for interface dynamics The mean-field steady-state profile of the active model B+ satisfies ∂2 z −∂2 z mc +f ′(mc) + 2λ−ζ 2 (∂zmc)2 = 0. (170) The lack of steady-state current enforces the stronger constraint jc =−∂ z −∂2 z mc +f ′(mc) + 2λ−ζ 2 (∂zmc)2 = 0. (171) The corresponding dynamical action truncated to quadratic order reads S0 = 1...

    work page

  5. [5]

    This breaking of time-reversal symmetry is expected to generate a KPZ nonlin- earity

    KPZ nonlinearity in model A and symmetry of the potential In section V A 3, we have shown that, in the pres- ence of an asymmetric potential with degenerate minima, thermal fluctuations generate a nonvanish- ing drift velocity. This breaking of time-reversal symmetry is expected to generate a KPZ nonlin- earity. Let us motivate this by looking at one of t...

    work page

  6. [6]

    (2) with the potential of Eq

    Curvature correction in model A Diehl, Kroll and Wagner [66] have shown at the level of the statics that the interfacial free energy for the Ginzburg-Landauϕ 4 field theory (the free energy of Eq. (2) with the potential of Eq. (38)) is given, in the limit of low temperature, by F[h] =σ Z r p 1 + (∇rh)2 .(B4) It has been argued [29] that the proper dynami-...

    work page

  7. [7]

    The only possible combination is − 2 T 2 Z r,t Z r′,t′ (∇rh)2(r, t)¯h(r′, t′)∇rh(r′, t′)· Z z,z ′ ∂2 z mc(z) ∂zmc(z′) σ × ¯χ⊥(r, z, t)∇r∂zχ⊥(r′, z′, t′) ′ c,0

    Let us turn to the second cumulant: such a contribution would involve two bulk fluctuation fields. The only possible combination is − 2 T 2 Z r,t Z r′,t′ (∇rh)2(r, t)¯h(r′, t′)∇rh(r′, t′)· Z z,z ′ ∂2 z mc(z) ∂zmc(z′) σ × ¯χ⊥(r, z, t)∇r∂zχ⊥(r′, z′, t′) ′ c,0 . (B9) This corresponds to the Feynman diagram h h ¯h h where in the notations of Sec. VI A 2, non-...

    work page

  8. [8]

    Detailed derivation of the first order Let us start from Eq. (92). Because the first eigen- value of Ω 0 is zero, the first term dominates in the long wavelength limitq→0. However, to control this expansion, we need to account for the depen- dence of L−1 q onq, and properly estimate the orders inqof the remaining terms. To make the argu- ment fully explic...

    work page

  9. [9]

    We find it useful to work in a system of finite size 2L

    It is immediately visible that these eigenfunctions do not vanish at infinity and therefore do not have a well-defined integral for a strictly infinite system. We find it useful to work in a system of finite size 2L. Then, taking theL→ ∞limit whenever possible (in tanh terms), the normalization reads N2 k = 4L(2k2 + 1)(k2 + 2)−12 r 2 τ (k2 + 1).(C4) With ...

    work page

  10. [10]

    All relevant scalar products have been estimated asL→ ∞: 2q e01,L −1 q e01 =O(1) (C13) 2q ˜ek,L −1 q e01 =O 1√ L 1 |k| (C14) 2q ˜ek,L −1 q ˜el =O 1 L 1 |kl| (C15) 2q ˜e0,L −1 q e01 =O( √ L) (C16) 2q ˜e0,L −1 q ˜ek =O(1) 1 |k| (C17) 2q ˜e0,L −1 q ˜e0 =O(L) (C18) wheree 01 stands for eithere 0 ore 1,kandlare both nonzero, and the scalings ink, lindicated on...

    work page

  11. [11]

    We can however provide the reader with some intuition on the behavior of the dispersion relation to the next order

    Intuition for next order correction With the expansion above, we obtain the leading order inqof the dispersion relation: iω= 2σq3 A(0) (C23) For higher orders inq, the eigenvectors other than ∂zmc start playing a role. We can however provide the reader with some intuition on the behavior of the dispersion relation to the next order. We be- gin by identify...

    work page

  12. [12]

    A. J. Bray, Theory of phase-ordering ki- netics, Advances in Physics51, 481 (2002), https://doi.org/10.1080/00018730110117433

    work page doi:10.1080/00018730110117433 2002

  13. [13]

    Halpin-Healy and Y.-C

    T. Halpin-Healy and Y.-C. Zhang, Kinetic rough- ening phenomena, stochastic growth, directed polymers and all that. aspects of multidisciplinary statistical mechanics, Physics reports254, 215 (1995)

    work page 1995

  14. [14]

    Perlin and W

    M. Perlin and W. W. Schultz, Capillary effects on surface waves, Annual review of fluid mechanics 32, 241 (2000)

    work page 2000

  15. [15]

    T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, Boundary integral methods for multicomponent fluids and multiphase materials, Journal of Com- putational Physics169, 302 (2001)

    work page 2001

  16. [16]

    J. Casademunt, Viscous fingering as a paradigm of interfacial pattern formation: Recent results and new challenges, Chaos: An Interdisciplinary Jour- nal of Nonlinear Science14, 809 (2004)

    work page 2004

  17. [17]

    Lindner and C

    A. Lindner and C. Wagner, Viscoelastic surface instabilities, Comptes Rendus Physique10, 712 (2009)

    work page 2009

  18. [18]

    Gallaire and P.-T

    F. Gallaire and P.-T. Brun, Fluid dynamic insta- bilities: theory and application to pattern forming in complex media, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences375, 20160155 (2017)

    work page 2017

  19. [19]

    Hallatschek, P

    O. Hallatschek, P. Hersen, S. Ramanathan, and D. R. Nelson, Genetic drift at expanding frontiers promotes gene segregation, Proceedings of the Na- tional Academy of Sciences104, 19926 (2007)

    work page 2007

  20. [20]

    Douezan, K

    S. Douezan, K. Guevorkian, R. Naouar, S. Dufour, D. Cuvelier, and F. Brochard-Wyart, Spreading dynamics and wetting transition of cellular aggre- gates, Proceedings of the National Academy of Sci- ences108, 7315 (2011)

    work page 2011

  21. [21]

    Joanny and S

    J.-F. Joanny and S. Ramaswamy, A drop of active matter, Journal of fluid mechanics705, 46 (2012). 30

    work page 2012

  22. [22]

    Sep´ ulveda, L

    N. Sep´ ulveda, L. Petitjean, O. Cochet, E. Grasland-Mongrain, P. Silberzan, and V. Hakim, Collective cell motion in an ep- ithelial sheet can be quantitatively described by a stochastic interacting particle model, PLoS computational biology9, e1002944 (2013)

    work page 2013

  23. [23]

    Maitra, P

    A. Maitra, P. Srivastava, M. Rao, and S. Ra- maswamy, Activating membranes, Physical review letters112, 258101 (2014)

    work page 2014

  24. [24]

    Nikola, A

    N. Nikola, A. P. Solon, Y. Kafri, M. Kardar, J. Tailleur, and R. Voituriez, Active particles with soft and curved walls: Equation of state, ratch- ets, and instabilities, Physical review letters117, 098001 (2016)

    work page 2016

  25. [25]

    Tjhung, C

    E. Tjhung, C. Nardini, and M. E. Cates, Clus- ter phases and bubbly phase separation in active fluids: Reversal of the ostwald process, Physical Review X8, 031080 (2018)

    work page 2018

  26. [26]

    V. Soni, E. S. Bililign, S. Magkiriadou, S. Sacanna, D. Bartolo, M. J. Shelley, and W. T. Irvine, The odd free surface flows of a colloidal chiral fluid, Nature physics15, 1188 (2019)

    work page 2019

  27. [27]

    P´ erez-Gonz´ alez, R

    C. P´ erez-Gonz´ alez, R. Alert, C. Blanch-Mercader, M. G´ omez-Gonz´ alez, T. Kolodziej, E. Bazellieres, J. Casademunt, and X. Trepat, Active wetting of epithelial tissues, Nature physics15, 79 (2019)

    work page 2019

  28. [28]

    Cagnetta, M

    F. Cagnetta, M. R. Evans, and D. Marenduzzo, Statistical mechanics of a single active slider on a fluctuating interface, Physical Review E99, 042124 (2019)

    work page 2019

  29. [29]

    Agudo-Canalejo, S

    J. Agudo-Canalejo, S. W. Schultz, H. Chino, S. M. Migliano, C. Saito, I. Koyama-Honda, H. Sten- mark, A. Brech, A. I. May, N. Mizushima,et al., Wetting regulates autophagy of phase-separated compartments and the cytosol, Nature591, 142 (2021)

    work page 2021

  30. [30]

    Fausti, E

    G. Fausti, E. Tjhung, M. E. Cates, and C. Nar- dini, Capillary interfacial tension in active phase separation, Phys. Rev. Lett.127, 068001 (2021)

    work page 2021

  31. [31]

    Adkins, I

    R. Adkins, I. Kolvin, Z. You, S. Witthaus, M. C. Marchetti, and Z. Dogic, Dynamics of active liquid interfaces, Science377, 768 (2022)

    work page 2022

  32. [32]

    Besse, G

    M. Besse, G. Fausti, M. E. Cates, B. Delamotte, and C. Nardini, Interface roughening in nonequi- librium phase-separated systems, Phys. Rev. Lett. 130, 187102 (2023)

    work page 2023

  33. [33]

    M. E. Pallar` es, I. Pi-Jaum` a, I. C. Fortunato, V. Grazu, M. G´ omez-Gonz´ alez, P. Roca-Cusachs, J. M. de la Fuente, R. Alert, R. Sunyer, J. Casade- munt,et al., Stiffness-dependent active wetting enables optimal collective cell durotaxis, Nature Physics19, 279 (2023)

    work page 2023

  34. [34]

    Langford and A

    L. Langford and A. K. Omar, Theory of capil- lary tension and interfacial dynamics of motility- induced phases, Physical Review E110, 054604 (2024)

    work page 2024

  35. [35]

    Fins Carreira, A

    A. Fins Carreira, A. Wysocki, C. Ybert, M. Leoc- mach, H. Rieger, and C. Cottin-Bizonne, How to steer active colloids up a vertical wall, Nature com- munications15, 1710 (2024)

    work page 2024

  36. [36]

    Y. Zhao, R. Zakine, A. Daerr, Y. Kafri, J. Tailleur, and F. van Wijland, Active young-dupr´ e equation: How self-organized currents stabilize partial wet- ting, arXiv preprint arXiv:2405.20651 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  37. [37]

    Caballero, A

    F. Caballero, A. Maitra, and C. Nardini, Interface dynamics of wet active systems, Physical Review Letters134, 087105 (2025)

    work page 2025

  38. [38]

    B. C. Sessa, F. Cao, R. A. Pelcovits, T. R. Powers, and G. Duclos, Interfacial instability of confined three-dimensional active droplets, Physical Review Research8, 013120 (2026)

    work page 2026

  39. [39]

    Laprade, L

    J. Laprade, L. B. Frechette, C. Amey, A. T. Cusi, A. Baskaran, W. B. Rogers, and G. Duclos, The coarsening of biomimetic condensates in an active fluid is non-self-similar, Nature physics , 1 (2026)

    work page 2026

  40. [40]

    Bausch, V

    R. Bausch, V. Dohm, H. K. Janssen, and R. K. P. Zia, Critical dynamics of an interface in 1 +ϵdi- mensions, Phys. Rev. Lett.47, 1837 (1981)

    work page 1981

  41. [41]

    Kawasaki and T

    K. Kawasaki and T. Ohta, Kinetic Drumhead Model of Interface. I, Progress of Theoretical Physics67, 147 (1982)

    work page 1982

  42. [42]

    Kawasaki and T

    K. Kawasaki and T. Ohta, Kinetic Drumhead Models of Interface. II, Progress of Theoretical Physics68, 129 (1982)

    work page 1982

  43. [43]

    Kawasaki, Variational approach to dynamics of interfaces and quantized vortex lines, Physica A: Statistical Mechanics and its Applications119, 17 (1983)

    K. Kawasaki, Variational approach to dynamics of interfaces and quantized vortex lines, Physica A: Statistical Mechanics and its Applications119, 17 (1983)

    work page 1983

  44. [44]

    Kawasaki and T

    K. Kawasaki and T. Ohta, Kinetics of fluctua- tions for systems undergoing phase transitions- interfacial approach, Physica A: Statistical Me- chanics and its Applications118, 175 (1983)

    work page 1983

  45. [45]

    Ohta, Scaling law in the ordering process of quenched thermodynamically unstable systems, Annals of Physics158, 31 (1984)

    T. Ohta, Scaling law in the ordering process of quenched thermodynamically unstable systems, Annals of Physics158, 31 (1984)

    work page 1984

  46. [46]

    S. Ohta, T. Ohta, and K. Kawasaki, Dynamics of topological defects in critical binary fluids, meta- magnets and 3he-4he mixtures, Physica A: Statis- tical Mechanics and its Applications128, 1 (1984)

    work page 1984

  47. [47]

    Jasnow and R

    D. Jasnow and R. K. Zia, Crossover of interfacial dynamics, Physical Review A36, 2243 (1987)

    work page 1987

  48. [48]

    R. Zia, R. Bausch, H. Janssen, and V. Dohm, Dy- namics of an interface coupled to a diffusive bulk mode, Modern Physics Letters B2, 961 (1988)

    work page 1988

  49. [49]

    Bausch, R

    R. Bausch, R. Kree, A. Lusakowski, and L. Turski, On the dynamics of low-temperature models for phase transitions with conserved order parameter, International Journal of Modern Physics B2, 1537 (1988)

    work page 1988

  50. [50]

    Bausch, V

    R. Bausch, V. Dohm, H. Janssen, and R. Zia, Effects of bulk diffusion on interfacial dynamics, Zeitschrift f¨ ur Physik B Condensed Matter82, 121 (1991)

    work page 1991

  51. [51]

    Shinozaki and Y

    A. Shinozaki and Y. Oono, Dispersion relation around the kink solution of the cahn-hilliard equa- tion, Physical Review E47, 804 (1993)

    work page 1993

  52. [52]

    Shinozaki, Dispersion relation around a kink solution in binary fluids undergoing spinodal de- composition, Physical Review E48, 1984 (1993)

    A. Shinozaki, Dispersion relation around a kink solution in binary fluids undergoing spinodal de- composition, Physical Review E48, 1984 (1993)

    work page 1984

  53. [53]

    Zia, Normal coordinates and curvature terms in an interface hamiltonian, Nuclear Physics B251, 676 (1985)

    R. Zia, Normal coordinates and curvature terms in an interface hamiltonian, Nuclear Physics B251, 676 (1985)

    work page 1985

  54. [54]

    Caballero, E

    N. Caballero, E. Agoritsas, V. Lecomte, and T. Giamarchi, From bulk descriptions to emer- 31 gent interfaces: Connecting the ginzburg-landau and elastic-line models, Physical Review B102, 104204 (2020)

    work page 2020

  55. [55]

    Y. Kuramoto, Instability and turbulence of wavefronts in reaction-diffusion systems, Progress of Theoretical Physics63, 1885 (1980), https://academic.oup.com/ptp/article- pdf/63/6/1885/5221609/63-6-1885.pdf

    work page 1980

  56. [56]

    Kardar, G

    M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett.56, 889 (1986)

    work page 1986

  57. [57]

    Kardar, Nonequilibrium dynamics of interfaces and lines, Physics reports301, 85 (1998)

    M. Kardar, Nonequilibrium dynamics of interfaces and lines, Physics reports301, 85 (1998)

    work page 1998

  58. [58]

    A. J. Bray, A. Cavagna, and R. D. M. Travasso, In- terface fluctuations, burgers equations, and coars- ening under shear, Phys. Rev. E65, 016104 (2001)

    work page 2001

  59. [59]

    Lucassen, H

    M. Lucassen, H. Van Driel, C. M. Smith, and R. Duine, Current-driven and field-driven domain walls at nonzero temperature, Physical Review B—Condensed Matter and Materials Physics79, 224411 (2009)

    work page 2009

  60. [60]

    M. Kardar, Fluctuation-induced forces in and out of equilibrium, inActive Matter and Nonequilib- rium Statistical Physics: Lecture Notes of the Les Houches Summer School: Volume 112, September 2018(Oxford University Press, 2022)

    work page 2018

  61. [61]

    D. S. Dean, P. Gersberg, and P. C. Holdsworth, The effect of driving on model c interfaces, Journal of Statistical Mechanics: Theory and Experiment 2020, 033206 (2020)

    work page 2020

  62. [62]

    Romano, R

    J. Romano, R. Golestanian, and B. Mahault, Dy- namics of phase-separated interfaces in inhomoge- neous and driven mixtures, Soft Matter21, 9245 (2025)

    work page 2025

  63. [63]

    Langford and A

    L. Langford and A. K. Omar, Phase separation, capillarity, and odd-surface flows in chiral ac- tive matter, Physical Review Letters134, 068301 (2025)

    work page 2025

  64. [64]

    Maire, L

    R. Maire, L. Galliano, A. Plati, and L. Berthier, Hyperuniform interfaces in nonequilibrium phase coexistence, Physical Review Letters135, 227102 (2025)

    work page 2025

  65. [65]

    Z. Sun, L. Li, F. Ye, and M. Yang, Interfacial stability in tensionless phase-separated quorum- sensing systems, arXiv preprint arXiv:2507.15030 (2025)

    work page arXiv 2025

  66. [66]

    Burekovi´ c, F

    S. Burekovi´ c, F. De Luca, M. E. Cates, and C. Nardini, Active cahn–hilliard theory for non- equilibrium phase separation: quantitative macro- scopic predictions and a microscopic derivation, arXiv preprint arXiv:2601.16539 (2026)

    work page arXiv 2026

  67. [67]

    P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys.49, 435 (1977)

    work page 1977

  68. [68]

    M. E. Cates, Active field theories (2019), arXiv:1904.01330 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  69. [69]

    O’Byrne, Nonequilibrium currents in stochastic field theories: A geometric insight, Physical Re- view E107, 054105 (2023)

    J. O’Byrne, Nonequilibrium currents in stochastic field theories: A geometric insight, Physical Re- view E107, 054105 (2023)

    work page 2023

  70. [70]

    O’Byrne and M

    J. O’Byrne and M. E. Cates, Geometric theory of (extended) time-reversal symmetries in stochastic processes: I. finite dimension, Journal of Statis- tical Mechanics: Theory and Experiment2024, 113207 (2024)

    work page 2024

  71. [71]

    O’Byrne and M

    J. O’Byrne and M. Cates, Geometric theory of (extended) time-reversal symmetries in stochastic processes: Ii. field theory, Journal of Statistical Mechanics: Theory and Experiment2025, 053204 (2025)

    work page 2025

  72. [72]

    H. T. Davis, Capillary waves and the mean field theory of interfaces, The Journal of Chemical Physics67, 3636 (1977)

    work page 1977

  73. [73]

    L. A. Turski and J. S. Langer, Dynamics of a dif- fuse liquid-vapor interface, Phys. Rev. A22, 2189 (1980)

    work page 1980

  74. [74]

    R. F. Dashen, B. Hasslacher, and A. Neveu, Non- perturbative methods and extended-hadron mod- els in field theory. ii. two-dimensional models and extended hadrons, Physical Review D10, 4130 (1974)

    work page 1974

  75. [75]

    Gervais and B

    J.-L. Gervais and B. Sakita, Extended particles in quantum field theories, Physical Review D11, 2943 (1975)

    work page 1975

  76. [76]

    Gervais, A

    J.-L. Gervais, A. Jevicki, and B. Sakita, Perturba- tion expansion around extended-particle states in quantum field theory, Physical Review D12, 1038 (1975)

    work page 1975

  77. [77]

    H. W. Diehl, D. M. Kroll, and H. Wagner, The interface in a ginsburg-landau-wilson model: Derivation of the drumhead model in the low- temperature limit, Zeitschrift f¨ ur Physik B Con- densed Matter36, 329 (1980)

    work page 1980

  78. [78]

    Langford and A

    L. Langford and A. K. Omar, The mechanics of nucleation and growth and the surface tensions of active matter, The Journal of Chemical Physics 163(2025)

    work page 2025

  79. [79]

    See sections 2.3 and 2.4 of [1] for models A and B, respectively

    work page

  80. [80]

    Ohta, Private notes for L

    T. Ohta, Private notes for L. Sarfati (2025)

    work page 2025

Showing first 80 references.