Suppression of hyperuniformity in hydrodynamic scalar active field theories

Axel Voigt; Nadia Bihari Padhan

REVIEW 3 major objections 5 minor 41 references

Suppression of hyperuniformity in hydrodynamic scalar active field theories

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Contractile activity destroys hyperuniformity in wet phase separation

desk verdict Solid numerical study: contractile stress in active model H suppresses hyperuniformity with a plausible forward-flux mechanism, but the asymptotic k→0 claim is not fully established. read the letter →

arxiv 2411.17409 v1 pith:VY3AONB6 submitted 2024-11-26 cond-mat.soft physics.flu-dyn

Nadia Bihari Padhan , Axel Voigt This is my paper · ORCID

classification cond-mat.softphysics.flu-dyn

keywords hyperuniformityactivemodelHcontractilestressphaseseparationcoarseningarrestspectralenergytransferturbulencehydrodynamicinteractions

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

This paper asks whether hydrodynamic interactions preserve the universal hyperuniform patterns seen in phase-separating scalar field theories. Using direct numerical simulations of model H and active model H in two dimensions, it shows that passive hydrodynamic coupling keeps the order-parameter spectral density $\psi(k)\sim k^4$ at small wavenumbers, just as in Cahn-Hilliard dynamics. In the active contractile case ($\zeta<0$), where coarsening arrests into a statistically steady state, the spectrum flattens to $\psi(k)\sim$ constant, meaning large-scale fluctuations are no longer suppressed and the system is non-hyperuniform. The result matters because it identifies a mechanism—advective spectral transfer driven by the active stress—that can erase a form of hidden order, with implications for how we recognize hyperuniformity in active fluids.

What carries the argument

The machinery is the shell-averaged spectral budget for the order parameter, $\partial_t S(k,t)=I+D+T^\phi+T^{\rm adv}$, together with the flux $\Pi(k)=-\int_0^k T^{\rm adv}(k')dk'$. The advection term $T^{\rm adv}$ redistributes structural energy across scales without adding or removing it; its sign determines the cascade direction. In hyperuniform states the flux is negative (inverse cascade) and vanishes steeply at small $k$, leaving the $\psi(k)\sim k^4$ scaling untouched; in the contractile steady state the flux is positive (forward cascade) and sustains fluctuations at all scales, flattening the spectrum.

What would settle it

A direct test would be to run the contractile active model H in a box several times larger with much longer averaging, and to check whether $\psi(k)$ develops a rising power law $\psi(k)\sim k^\alpha$ with $\alpha>0$ as $k\to 0$ instead of remaining flat; if a positive exponent reappears, the suppression of hyperuniformity is a finite-size or finite-time artifact.

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Extended reading notes

Core claim

The central claim is that adding hydrodynamic advection to a scalar active field theory can qualitatively change its large-scale order. In passive model H, and in active model H with extensile stress ($\zeta>0$), the spectral density obeys $\psi(k)\sim k^4$ as $k\to 0$, so the system is hyperuniform even though coarsening is much faster than in model B. With contractile stress ($\zeta<0$), coarsening arrests into a non-equilibrium steady state and the spectrum becomes flat, $\psi(k)\sim$ constant, so the system is non-hyperuniform. The paper attributes this to a scale-by-scale balance: the advection term transfers structural energy from small to large wavenumbers (a forward cascade) in the contractile case, whereas it transfers energy from large to small wavenumbers (an inverse cascade) in the hyperuniform cases. The transition is robust across symmetric and asymmetric quenches and across box sizes up to $8\pi\times 8\pi$.

Load-bearing premise

The claim rests on the assumption that the flat spectra measured in finite simulation boxes, up to $8\pi\times 8\pi$, and in time-averaged steady states represent what would happen in an infinite system at asymptotically small wavenumbers.

Editorial extensions

If this is right

If the claim holds, hyperuniformity is not a universal late-time property of all phase-separating scalar fields; active hydrodynamic stress can destroy it.
The spectral flux analysis offers a diagnostic: the sign and small-$k$ scaling of $\Pi(k)$ distinguish hyperuniform from non-hyperuniform coarsening, so one can predict the fate of a system from the advective transfer rather than waiting for long-time spectra.
Because the contractile case shows a forward cascade and anti-hyperuniform vorticity, the same mechanism may link active turbulence to a loss of order-parameter hyperuniformity in other wet active systems.
The robustness of the result to asymmetric quenches and box size suggests that the transition will be observed in experiments on contractile active emulsions or cytoskeletal networks, not only in simulations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could test the cascade mechanism by forcing the flow externally to reverse the direction of $T^{\rm adv}$ and checking whether hyperuniformity is recovered in the contractile regime.
This finding suggests that hyperuniformity may be a fragile property in fluids with momentum conservation, and that the hyperuniformity of a field can be tuned by controlling the sign of the active stress.
A natural extension would be to three dimensions, where the inverse-cascade phenomenology differs and the arrest may be less complete, so the suppression might be weaker.
The flat spectrum at steady state resembles equipartition of the order-parameter modes; a statistical-mechanics derivation of the flat spectrum from the balance $T^{\rm adv}=-(I+D+T^\phi)$ could turn the numerical observation into a solvable closure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The new result here is that active model H with contractile active stress (ζ<0) loses hyperuniformity in the steady state, while passive model H and extensile active model H keep it. The paper also derives a spectral budget for the order parameter and shows that the advective flux runs forward (small k to large k) in the contractile case and inverse in the hyperuniform cases. That is a concrete, testable mechanism, and I don't think anyone has shown it for hydrodynamic scalar active field theories before.

The numerics are careful as far as they go: the spectral budget follows from the equations, the runs include asymmetric quench and larger box checks, and the flux analysis is a diagnostic rather than a fitted result. The coarsening exponents for model B (1/3) and model H (1) match known values. Good.

Soft spots, in order of softness. The central claim that ψ(k) → constant as k→0 is not fully established. With a 2π box the lowest shell is k∈[1,2], which is not the asymptotic regime; the 8π box gets k_min=0.25 but uses the same unit-width shells, so the first shell still overlaps the domain-scale peak. The 2D spectra in Fig. 2(b) and 6(d) mitigate this—they show a flat, ring-free amplitude down to the accessible k, not just a shell-average artifact—but the paper can only claim a plateau over the accessible range, not the true k→0 limit. The text says 'for the limit k→0' as if that were resolved. That should be softened.

Also, the power-law exponents (α≈4, flux exponents 6 and 4) are quoted without error bars; some spectra are single-time snapshots; and no code or data are provided. These are standard referee requests, not fatal flaws.

The citation pattern is fine; the self-citations to the authors' prior work on active-scalar turbulence are relevant and not padding.

Bottom line: this is a real within-field result, worth refereeing seriously. I'd ask for error bars, a more careful discussion of the accessible-wavenumber limitation, and possibly narrower spectral shells in the large-box run. The mechanism is plausible and the paper should be published in some form.

Referee Report

3 major / 5 minor

Summary. The paper reports direct numerical simulations of passive and active model H in two dimensions, in which a scalar order parameter is coupled to a Navier-Stokes fluid through passive or active stresses. It claims that passive model H (ζ=σ) and extensile active model H (ζ>σ) remain hyperuniform during coarsening, with spectral density ψ(k)∼k^4, whereas contractile active stress (ζ<0) arrests coarsening and drives the order-parameter spectral density to ψ(k)∼const in the steady state, i.e., a non-hyperuniform state. A scale-by-scale budget for the order-parameter spectrum is derived, and the advective transfer term is interpreted as an inverse cascade (negative flux) in the hyperuniform cases and a forward cascade (positive flux) in the contractile case. The paper also reports checks with asymmetric initial compositions and larger simulation boxes.

Significance. If the central claim is correct, the paper is an important counterpoint to recent results on universal hyperuniformity in dry scalar active field theories: it shows that momentum-conserving hydrodynamics combined with contractile activity can destroy the k→0 suppression of concentration fluctuations. The spectral-budget derivation is explicit and does not fit parameters to the target hyperuniformity metric, and the flux diagnostics provide a physically interpretable mechanism. The finite-size and asymmetric-quench checks are valuable. The main risk is that the conclusion is currently supported by a low-wavenumber estimator whose first shell overlaps the domain-scale peak; this must be addressed before the suppression claim is fully established.

major comments (3)

[3.1, Eq. (13), Fig. 6(f)] The unit-width shell average in Eq. (13) means the lowest plotted wavenumber is not in the asymptotic regime. In the 2π box, k_min=1, so the first shell averages 1≤|k′|<2; in the 8π box, k_min=0.25, but with the same shell width the first shell averages 0.25≤|k′|<1.25. Since the steady-state spectral peak for the contractile case is near k*≈1, the apparent plateau ψ(k)≈const and the value H≈1 may be contamination from the peak rather than a true k→0 behavior. Please recompute the low-k spectral density with shells narrow compared with k* and with k_min well below k*, and show box-size convergence of the asymptotic value before claiming suppression of hyperuniformity.
[3.2, Eq. (15)] The hyperuniformity metric H(t)=ψ(1,t)/ψ(k_peak,t) inherits the same shell-averaging problem. A minimum H≈10^-5 and a final H≈1 are based on the first shell value ψ(1,t), which includes the peak; these quantitative statements are therefore not robust. The metric should be evaluated at a k_min far below the peak, or replaced by an extrapolated k→0 value, using narrow spectral shells.
[3.1 and 3.3, Figs. 1, 2, and 3] The scaling exponents (α≈4 for hyperuniform cases, flux exponents ≈6 and ≈4, and growth exponents ≈1 and ≈1/3) are quoted without error bars, fit ranges, or the number of decades used, and Fig. 3(a) is a single snapshot at t=8 while Fig. 3(c) is time-averaged. Since the inverse-versus-forward cascade distinction is a central interpretive claim, please provide quantitative fit details and time-averaged spectra for both the passive and active cases.

minor comments (5)

[3.2] The sentence 'we compute H at each time instance and show the evolution H(t) in Fig.2(b)' should refer to Fig.2(d), since the H(t) panel is labeled (d) in the figure.
[1] The first sentence of the Introduction contains the typo 'Hyperuniformy'; it should read 'Hyperuniformity'.
[3.3] The phrase 'as per the law of equipartition of energy' is not justified by a constant ψ(k); a constant spectral density is not, by itself, equipartition.
[3.3, Eq. (21)] The integral ∫₀^∞ T_adv(k)dk=0 should specify that T_adv(k) is a shell-integrated transfer rate and that the integration is over the scalar wavenumber axis, since k is used both as a vector magnitude and as a shell index.
[General] There are several typographical errors that should be corrected: 'looses' in Section 3.3, 'visulization' in the Fig. 6 caption, and 'represtative' in Section 3.5.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: spectral-density and flux conclusions are direct measurements from DNS of the stated model, not fitted inputs or self-citation-chain results.

full rationale

The central claims are quantitative measurements from the model defined in Eqs. (1)-(5): ψ(k)=S(k)/k with S(k) in Eq. (13) is the shell-averaged order-parameter spectrum computed from the pseudospectral solution of Eqs. (8)-(12), and the exponents α≃4 (or ψ≃const) are read directly from these spectra. The spectral budget Eq. (16) is derived from the model equations, and the flux Π(k) in Eq. (21) is a diagnostic computed from the simulated advection term T_adv; it is not a free parameter fitted to reproduce ψ(k) or H(t). No parameter is fitted to the target hyperuniformity metric, and H(t) in Eq. (15) is simply a ratio of measured spectral values. Self-citations to [11] and [21] supply model context, numerical methods, and previously observed active-turbulence/hyperuniform states; they are not load-bearing for the new transition claim, which is supported by the independent DNS and spectral-budget analysis in Figs. 1-3 and 5-6. A skeptical concern about finite-size/shell-width contamination of the lowest shell (noted in Section 3.5 for 8π boxes) is a possible validity limitation of the k→0 asymptotics, not a circularity: it does not make the conclusion equivalent to its inputs by construction.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard active model H equations and on numerical methods rather than on new postulates. The simulation parameters are chosen by hand or from prior literature, not fitted to the target hyperuniformity result, and no new particles, forces, or conserved quantities are introduced.

free parameters (7)

Activity coefficient zeta = 0, 1, 1.5, -0.1, -1, -2
Controls passive (zeta equals sigma), extensile (zeta greater than sigma), and contractile (zeta less than zero) active stress; the sign of zeta is the key control parameter for the reported transition.
Mobility M = 5e-4
Sets the diffusive time scale; chosen following reference [32], not fitted to the hyperuniformity result.
Kinematic viscosity nu = 1.5
Sets the Reynolds number; fixed for all runs.
Interface width parameter epsilon = 0.03
Sets the diffuse interface width; fixed for all runs.
Initial noise amplitude = 0.1
Uniform noise in the interval [-0.1, 0.1] for the initial order-parameter field.
Box size and resolution = 2 pi with 1024^2 points, plus 4 pi and 8 pi tests
Finite-size checks confirm the scaling behavior but limit the low-wavenumber resolution.
Order-parameter offset phi_0 = 0, 0.1, 0.3, 0.5
Varies the quench asymmetry; the reported non-hyperuniform behavior is independent of phi_0.

assumptions (5)

domain assumption Incompressible, equal-density, matched-viscosity Navier-Stokes equations describe the fluid.
Section 2 states these simplifications; the central result for wet active matter depends on this hydrodynamic description.
domain assumption The active stress tensor in Eq. (5) with zeta not equal to sigma defines active model H.
Taken from references [17,21,24]; this is the model whose hyperuniformity is studied.
standard math Fourier pseudospectral discretization with 1/2-dealiasing and ETDRK2 integration converges to the PDE solution.
Section 2, numerical method; the validity of the direct numerical simulations rests on this.
standard math Shell-averaged spectra and the flux Pi(k) defined in Eq. (21) faithfully represent the direction of scale-to-scale transfer.
Section 3.3; the mechanism argument uses the sign of Pi(k) to distinguish inverse and forward cascades.
domain assumption The time-averaged steady state for zeta less than zero is statistically stationary and adequately sampled by 700 datasets.
Section 3.3; without stationarity, the flat low-wavenumber spectrum could be a transient.

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Cite this review

Pith. "Pith review of Suppression of hyperuniformity in hydrodynamic scalar active field theories." pith.science (2026). https://pith.science/paper/VY3AONB6

@misc{pith2026241117409,
  author       = {Pith},
  title        = {Pith review of: Suppression of hyperuniformity in hydrodynamic scalar active field theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VY3AONB6}},
  note         = {Machine review of arXiv:2411.17409}
}

read the original abstract

The coarsening dynamics at late times in phase-separating systems lead to universally hyperuniform patterns. This is well known for scalar field theories, such as the Cahn-Hilliard equation, but has also been shown for dry scalar active field theories. We demonstrate the role of hydrodynamic interactions in influencing hyperuniformity in a wet active system described by active model H. Our direct numerical simulations reveal that, while (passive) model H shows hyperuniformity in the coarsening regime, the interplay of activity and hydrodynamic interactions suppresses hyperuniformity in active model H, especially when the activity generates contractile stress in the fluid.

Figures

Figures reproduced from arXiv: 2411.17409 by the authors.

**Figure 1.** (a) Pseudocolor plots of ϕ for ζ = 0, σ and 1.5 taken during the coarsening process, where various scalings have been observed. (b) The corresponding pseudocolor plots of the quantity |ϕˆ(k, t)| 2 shown in the (kx, ky) plane; the colorbars are in logarithmic scale. (c) Pseudocolor plots of ϕ for ζ = −0.1, −1, −2 taken in the statistical steady states after the coarsening process is arrested; (d) the pseudocolor plot… view at source ↗

**Figure 2.** ζ = −1: (a) Pseudocolor plots of ϕ at different representative times. (b) The pseudocolor plots of the quantity |ϕˆ(k, t)| 2 shown in the (kx, ky) plane, with the colorbar in logarithmic scale. (c) The 1D spectral densities ψ(k, t) at the corresponding representative times. (d) Logarithmic plot of the hyperuniformity metric H(t). negative values for the contractile stress [17]. For ζ = −0.1, −1 and −2, coarsening in… view at source ↗

**Figure 3.** (a) Shell-averaged spectral forms of different contributions in Eq.16 for ζ = σ at t = 8, during the regime where L(t) ∼ t 1 . (b) Plot versus k of the flux associated with the advection term. The horizontal axis is in the logarithmic scale. The inset shows the absolute value of the flux in the logarithmic scale. The k → 0 region is fitted to a power-law with exponent ≃ 6. (c) Spectral forms of different contributio… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: (a) The plot of kinetic time series for ζ = σ, 0.1, −1, −2. (b) Log-log plots of the kinetic energy spectrum for various values of ζ. The dotted black line represents a fit with ∼ k 5/3 . 3.4. Turbulence in hyperuniform and non-hyperuniform systems The flow is entirely…

**Figure 5.** Figure 5: ζ = −1: (a) Pseudocolor plots of ϕ for different value of ϕ0 at the same represtative times in the steady state. (b) The corresponding pseudocolor plots of | ˆ ϕ(k, t)| 2 shown in the (kx, ky) plane, with the colorbar in logarithmic scale. (c) The time evolution of the…

**Figure 6.** Figure 6: (a) Pseudocolor plot of ϕ for ζ = σ at time t = 6. The size of the simulation domain is (8π, 8π) with 40962 collocation points. A zoom-in plot is shown for the area bounded by the black box. (b) The corresponding pseudocolor plot of |ϕˆ(k, t)| 2 shown in the (kx, ky) p…

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Works this paper leans on

41 extracted references · 41 canonical work pages

[1]

Hyperuniform states of matter

Salvatore Torquato. Hyperuniform states of matter. Phys. Reports, 745:1–95, 2018

work page 2018
[2]

Hyperuniformity and its generalizations

Salvatore Torquato. Hyperuniformity and its generalizations. Physical Review E , 94(2):022122, 2016

work page 2016
[3]

Non-equilibrium dynamic hyperuniform states

Yusheng Lei and Ran Ni. Non-equilibrium dynamic hyperuniform states. Journal of Physics: Condensed Matter, 2024

work page 2024
[4]

Universal hyperuniformity in active field theories

Yuanjian Zheng, Michael A Klatt, and Hartmut L¨ owen. Universal hyperuniformity in active field theories. Physical Review Research, 6(3):L032056, 2024

work page 2024
[5]

Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility

Filippo De Luca, Xiao Ma, Cesare Nardini, and Michael E Cates. Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility. arXiv preprint arXiv:2405.00508, 2024

work page Pith review arXiv 2024
[6]

Scalar φ 4 field theory for active-particle phase separation

Raphael Wittkowski, Adriano Tiribocchi, Joakim Stenhammar, Rosalind J Allen, Davide Marenduzzo, and Michael E Cates. Scalar φ 4 field theory for active-particle phase separation. Nature communications, 5(1):4351, 2014

work page 2014
[7]

Capillary interfacial tension in active phase separation

Giordano Fausti, Elsen Tjhung, ME Cates, and Cesare Nardini. Capillary interfacial tension in active phase separation. Physical review letters , 127(6):068001, 2021

work page 2021
[8]

Random scalar fields and hyperuniformity

Zheng Ma and Salvatore Torquato. Random scalar fields and hyperuniformity. Journal of Applied Physics, 121(24), 2017

work page 2017

Show all 41 references

[9]

H. Tomita. Preservation of isotropy at the mesoscopic stage of phase separation processes. Prog. Theor. Phys., 85:47–56, 1991

1991
[10]

The kinetics of precipitation from supersaturated solid solutions

Ilya M Lifshitz and Vitaly V Slyozov. The kinetics of precipitation from supersaturated solid solutions. Journal of physics and chemistry of solids , 19(1-2):35–50, 1961. Suppression of hyperuniformity in hydrodynamic scalar active field theories 14

1961
[11]

Nonequilibrium hyperuniform states in active turbulence

Rainer Backofen, Abdelrahman YA Altawil, Marco Salvalaglio, and Axel Voigt. Nonequilibrium hyperuniform states in active turbulence. Proceedings of the National Academy of Sciences , 121(24):e2320719121, 2024

2024
[12]

Hyperuniform active chiral fluids with tunable internal structure

Bo Zhang and Alexey Snezhko. Hyperuniform active chiral fluids with tunable internal structure. Physical Review Letters, 128(21):218002, 2022

2022
[13]

Theory of dynamic critical phenomena

Pierre C Hohenberg and Bertrand I Halperin. Theory of dynamic critical phenomena. Reviews of Modern Physics , 49(3):435, 1977

1977
[14]

Diffuse-interface methods in fluid mechanics

Daniel M Anderson, Geoffrey B McFadden, and Adam A Wheeler. Diffuse-interface methods in fluid mechanics. Annual review of fluid mechanics , 30(1):139–165, 1998

1998
[15]

Conservative multigrid methods for cahn– hilliard fluids

Junseok Kim, Kyungkeun Kang, and John Lowengrub. Conservative multigrid methods for cahn– hilliard fluids. Journal of Computational Physics , 193(2):511–543, 2004

2004
[16]

Uniqueness and regularity for the navier– stokes–cahn–hilliard system

Andrea Giorgini, Alain Miranville, and Roger Temam. Uniqueness and regularity for the navier– stokes–cahn–hilliard system. SIAM Journal on Mathematical Analysis , 51(3):2535–2574, 2019

2019
[17]

Active model h: scalar active matter in a momentum-conserving fluid

Adriano Tiribocchi, Raphael Wittkowski, Davide Marenduzzo, and Michael E Cates. Active model h: scalar active matter in a momentum-conserving fluid. Physical review letters, 115(18):188302, 2015

2015
[18]

Late stages of spinodal decomposition in binary mixtures

Eric D Siggia. Late stages of spinodal decomposition in binary mixtures. Physical review A , 20(2):595, 1979

1979
[19]

Effect of inertia on droplet growth in a fluid

Hiroshi Furukawa. Effect of inertia on droplet growth in a fluid. Physical Review A , 31(2):1103, 1985

1985
[20]

3d spinodal decomposition in the inertial regime

VM Kendon, JC Desplat, P Bladon, and ME Cates. 3d spinodal decomposition in the inertial regime. Physical Review Letters, 83(3):576, 1999

1999
[21]

Novel turbulence and coarsening arrest in active-scalar fluids

Nadia Bihari Padhan, Kolluru Venkata Kiran, and Rahul Pandit. Novel turbulence and coarsening arrest in active-scalar fluids. Soft Matter , 20(17):3620–3627, 2024

2024
[22]

Theories of binary fluid mixtures: from phase-separation kinetics to active emulsions

Michael E Cates and Elsen Tjhung. Theories of binary fluid mixtures: from phase-separation kinetics to active emulsions. Journal of Fluid Mechanics , 836:P1, 2018

2018
[23]

Benchmark computations of diffuse interface models for two-dimensional bubble dynamics

Sebastian Aland and Axel Voigt. Benchmark computations of diffuse interface models for two-dimensional bubble dynamics. International Journal for Numerical Methods in Fluids , 69(3):747–761, 2012

2012
[24]

Transition to turbulence in driven active matter

Aritra Das, JK Bhattacharjee, and TR Kirkpatrick. Transition to turbulence in driven active matter. Physical Review E , 101(2):023103, 2020

2020
[25]

Fully discrete finite element approximations of the navier–stokes–cahn-hilliard diffuse interface model for two-phase fluid flows

Xiaobing Feng. Fully discrete finite element approximations of the navier–stokes–cahn-hilliard diffuse interface model for two-phase fluid flows. SIAM Journal on Numerical Analysis , 44(3):1049–1072, 2006

2006
[26]

Two-dimensional turbulence in symmetric binary- fluid mixtures: Coarsening arrest by the inverse cascade

Prasad Perlekar, Nairita Pal, and Rahul Pandit. Two-dimensional turbulence in symmetric binary- fluid mixtures: Coarsening arrest by the inverse cascade. Scientific Reports, 7(1):44589, 2017

2017
[27]

Activity-induced droplet propulsion and multifractality

Nadia Bihari Padhan and Rahul Pandit. Activity-induced droplet propulsion and multifractality. Physical Review Research, 5(3):L032013, 2023

2023
[28]

Two-dimensional turbulence

Guido Boffetta and Robert E Ecke. Two-dimensional turbulence. Annual review of fluid mechanics, 44(1):427–451, 2012

2012
[29]

Spectral methods in fluid dynamics

Claudio Canuto, M Yousuff Hussaini, Alfio Quarteroni, A Thomas Jr, et al. Spectral methods in fluid dynamics . Springer Science Business Media, 2012

2012
[30]

Numerical methods for the simulation of turbulence

Steven A Orszag. Numerical methods for the simulation of turbulence. The Physics of Fluids , 12(12):II–250, 1969

1969
[31]

Exponential time differencing for stiff systems

Steven M Cox and Paul C Matthews. Exponential time differencing for stiff systems. Journal of Computational Physics , 176(2):430–455, 2002

2002
[32]

Magaletti, F

F. Magaletti, F. Picano, M. Chinappi, L. Marino, and C. M. Casciola. The sharp-interface limit of the Cahn-Hilliard/Navier-Stokes model for binary fluids. J. Fluid Mech. , 714:95–126, 2013

2013
[33]

H. Abels. On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Rat. Mech. Anal. , 194:463–506, 2009

2009
[34]

Suppression of hyperuniformity in hydrodynamic scalar active field theories 15 Spinodal decomposition in homogeneous and isotropic turbulence

Prasad Perlekar, Roberto Benzi, Herman JH Clercx, David R Nelson, and Federico Toschi. Suppression of hyperuniformity in hydrodynamic scalar active field theories 15 Spinodal decomposition in homogeneous and isotropic turbulence. Physical review letters , 112(1):014502, 2014

2014
[35]

Spectral energy transfers in domain growth problems

Pradeep Kumar Yadav, Mahendra Kumar Verma, and Sanjay Puri. Spectral energy transfers in domain growth problems. Physical Review E , 110(4):044130, 2024

2024
[36]

Cascades and spectra of a turbulent spinodal decomposition in two-dimensional symmetric binary liquid mixtures

Xiang Fan, PH Diamond, L Chac´ on, and Hui Li. Cascades and spectra of a turbulent spinodal decomposition in two-dimensional symmetric binary liquid mixtures. Physical Review Fluids , 1(5):054403, 2016

2016
[37]

Theorie der Alterung von Niederschl¨ agen durch Uml¨ osen (Ostwald-Reifung)

Carl Wagner. Theorie der Alterung von Niederschl¨ agen durch Uml¨ osen (Ostwald-Reifung). Zeitschrift f¨ ur Elektrochemie, 65(7-8):581–591, 1961

1961
[38]

Energy transfers in fluid flows: multiscale and spectral perspectives

Mahendra K Verma. Energy transfers in fluid flows: multiscale and spectral perspectives . Cambridge University Press, 2019

2019
[39]

Cascades and transitions in turbulent flows

Alexandros Alexakis and Luca Biferale. Cascades and transitions in turbulent flows. Physics Reports, 767:1–101, 2018

2018
[40]

Cascade or not cascade? energy transfer and elastic effects in active nematics

Livio Nicola Carenza, Luca Biferale, and Giuseppe Gonnella. Cascade or not cascade? energy transfer and elastic effects in active nematics. Europhysics Letters, 132(4):44003, 2020

2020
[41]

Kinetic energy spectra and flux in turbulent phase-separating symmetric binary- fluid mixtures

Prasad Perlekar. Kinetic energy spectra and flux in turbulent phase-separating symmetric binary- fluid mixtures. Journal of Fluid Mechanics , 873:459–474, 2019

2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Hyperuniform states of matter

Salvatore Torquato. Hyperuniform states of matter. Phys. Reports, 745:1–95, 2018

work page 2018

[2] [2]

Hyperuniformity and its generalizations

Salvatore Torquato. Hyperuniformity and its generalizations. Physical Review E , 94(2):022122, 2016

work page 2016

[3] [3]

Non-equilibrium dynamic hyperuniform states

Yusheng Lei and Ran Ni. Non-equilibrium dynamic hyperuniform states. Journal of Physics: Condensed Matter, 2024

work page 2024

[4] [4]

Universal hyperuniformity in active field theories

Yuanjian Zheng, Michael A Klatt, and Hartmut L¨ owen. Universal hyperuniformity in active field theories. Physical Review Research, 6(3):L032056, 2024

work page 2024

[5] [5]

Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility

Filippo De Luca, Xiao Ma, Cesare Nardini, and Michael E Cates. Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility. arXiv preprint arXiv:2405.00508, 2024

work page Pith review arXiv 2024

[6] [6]

Scalar φ 4 field theory for active-particle phase separation

Raphael Wittkowski, Adriano Tiribocchi, Joakim Stenhammar, Rosalind J Allen, Davide Marenduzzo, and Michael E Cates. Scalar φ 4 field theory for active-particle phase separation. Nature communications, 5(1):4351, 2014

work page 2014

[7] [7]

Capillary interfacial tension in active phase separation

Giordano Fausti, Elsen Tjhung, ME Cates, and Cesare Nardini. Capillary interfacial tension in active phase separation. Physical review letters , 127(6):068001, 2021

work page 2021

[8] [8]

Random scalar fields and hyperuniformity

Zheng Ma and Salvatore Torquato. Random scalar fields and hyperuniformity. Journal of Applied Physics, 121(24), 2017

work page 2017