pith. sign in

arxiv: 2604.02855 · v1 · submitted 2026-04-03 · ❄️ cond-mat.stat-mech

Structure Functions and Intermittency for Coarsening Systems

Pradeep Kumar Yadav , Mahendra K. Verma , Sanjay Puri This is my paper

Pith reviewed 2026-05-13 18:26 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords coarseningstructure functionsintermittencyTDGL equationCahn-Hilliard equationanomalous scalingdomain growthsharp interfaces
0
0 comments X

The pith

Structure functions scale with exponent 1 in TDGL and CH coarsening due to sharp interfaces.

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

The paper takes structure functions and intermittency concepts from turbulence and applies them to domain coarsening modeled by the time-dependent Ginzburg-Landau and Cahn-Hilliard equations. It shows that the presence of sharp interfaces forces the structure functions S_q to scale as r to the power of ζ_q, with the specific value ζ_q equal to 1. This produces anomalous scaling rather than ordinary diffusive behavior. A reader would care because the result links two separate areas of statistical physics and supplies a concrete diagnostic for how domains grow and roughen over time.

Core claim

As a consequence of sharp interfaces, the structure functions scale as S_q ~ r^{ζ_q}, where r is the distance between two points. For the TDGL and CH models, ζ_q = 1, indicating anomalous scaling.

What carries the argument

The structure function S_q(r) whose scaling exponent ζ_q is fixed at 1 by the sharp interfaces that separate growing domains.

If this is right

  • Both the TDGL and CH models produce the same anomalous exponent ζ_q = 1.
  • Intermittency in coarsening systems can be quantified directly from the structure functions.
  • Energy-transfer ideas previously applied to these models gain a complementary diagnostic through the structure-function scaling.
  • The exponent serves as a signature that distinguishes sharp-interface coarsening from smoother interface dynamics.

Where Pith is reading between the lines

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

  • The same scaling might be measurable in laboratory phase-separation experiments by tracking pair correlations at varying distances.
  • Other coarsening models with different conservation laws could be checked to see whether ζ_q remains 1 or changes.
  • The approach could be extended to characterize how interface roughness evolves during the coarsening process.

Load-bearing premise

That sharp interfaces alone set the scaling exponent to exactly 1 without other dynamical features of the models changing the result.

What would settle it

A direct numerical simulation of the TDGL or CH equation that measures ζ_q different from 1 at late times would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.02855 by Mahendra K. Verma, Pradeep Kumar Yadav, Sanjay Puri.

Figure 1
Figure 1. Figure 1: Plot of the order parameter ψ(x, t) vs. x, depicting a typical order parameter profile of the TDGL equation. The locations of kinks and anti-kinks are denoted by σn. −− −− + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + R(t) −− −− −− −− −− −− −− −− −− −− −− −− −−−− −− −− + + + + + + + + + + + + + + + + + + + + + + + −− −− −− −− −− ξb L −− −− −− −− −− −− −− l c −− −− −− −− −− −− −− −− −− −− −− −… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of domain of size R(t) at time t, with an interface of width ξ, in a system of linear size L. The + region, where ψ > 0, is surrounded by the − region, where ψ < 0. The circumference of the + domain is denoted by lc. (b) Nonlinear energy transfers We focus on the unbiased case with ψ(x, t = 0) = 0+small fluctuations. Equation (2.3) in Fourier space is ∂ψˆ(k, t) ∂t = k 2α  ψˆ(k, t) − k 2ψˆ(k, t)… view at source ↗
Figure 3
Figure 3. Figure 3: Plots showing the evolution of ψ(x, t): (a) For 1D TDGL equation, ψ(x, t) vs. x at t = 0, 20, 500. (b)-(c) For 2D TDGL equation, density plots of ψ(x, t) at t = 5 and 50. Points with ψ > 0 are marked yellow, and points with ψ < 0 are unmarked. (d) For 1D CH equation, ψ(x, t) vs. x at t = 10, 20, 107 . (e)-(f) For 2D CH equation, density plots of ψ(x, t) at t = 100 and 1500. (b) Structure functions for the … view at source ↗
Figure 4
Figure 4. Figure 4: 1D TDGL equation: (a) Profile ψ(x, t = 500) vs. x. (b) For the profile in (a), structure function Sq(r, t) vs. r for q = 2 to 6. The inset shows ζq vs. q for r < ξ. (c) For the data in (b), normalized structure function LSq(r, t)/(2qN0r) ≃ 1 with N0 = 2. The symbols used are the same as those in (b). (d) Hardened ψ(x, t = 500) vs. x. (e)-(f) Sq(r, t) and normalized Sq(r, t) for the hardened profile in (d).… view at source ↗
Figure 5
Figure 5. Figure 5: 2D TDGL equation: (a) Density plot of the hardened field [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 1D and 2D CH equations: (a) For 1D CH equation, [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: 1D and 2D CH equations: Structure function [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 1D CH equation: Sq(r, t) vs. r for the unhardened profile in [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

In studies of turbulence, there has been extensive use of physical quantities such as {\it energy transfers} and {\it structure functions}. We examine whether these quantities can be useful in understanding problems of domain growth or coarsening, as modeled by the {\it time-dependent Ginzburg-Landau} (TDGL) equation and the {\it Cahn-Hilliard} (CH) equation. This paper has two major themes. First, we review our recent papers on energy transfers in domain growth. Second, we study structure functions and intermittency for coarsening systems. As a consequence of sharp interfaces, the structure functions scale as $S_q \sim r^{\zeta_q}$, where $r$ is the distance between two points. For the TDGL and CH models, $\zeta_q = 1$, indicating {\it anomalous scaling}

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 / 1 minor

Summary. The manuscript reviews energy transfers in domain growth for the TDGL and CH equations and then analyzes structure functions S_q(r) for coarsening systems. It claims that sharp interfaces imply the scaling S_q ~ r^{ζ_q} with ζ_q = 1 for both models, indicating anomalous scaling and intermittency.

Significance. If the central scaling result is rigorously established, the work would usefully import structure-function diagnostics from turbulence into coarsening dynamics, supplying a simple, parameter-free prediction for the intermittency exponents that could be tested across models and dimensions.

major comments (2)
  1. [Abstract] Abstract and the section introducing structure functions: the assertion that ζ_q = 1 follows directly from sharp interfaces is stated without an explicit derivation or scaling argument showing why finite interface width ξ and intra-domain fluctuations remain subdominant throughout the window ξ ≪ r ≪ L(t).
  2. [Structure functions section] The central claim section: no explicit calculation or numerical verification is referenced that demonstrates the interface-crossing probability yields S_q ~ r^1 independently of q; the manuscript must supply this to confirm the exponent is truly q-independent rather than an effective value in a limited range.
minor comments (1)
  1. [Introduction] Define the precise expression for the structure function S_q(r) in terms of the order-parameter field φ at the outset, including any averaging procedure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the points below and have revised the manuscript to include the requested derivations and calculations.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section introducing structure functions: the assertion that ζ_q = 1 follows directly from sharp interfaces is stated without an explicit derivation or scaling argument showing why finite interface width ξ and intra-domain fluctuations remain subdominant throughout the window ξ ≪ r ≪ L(t).

    Authors: We agree that an explicit derivation was missing. In the revised manuscript we have added a scaling argument in the structure functions section. For r ≫ ξ the order parameter takes values ±1 away from interfaces, so |φ(x+r)−φ(x)| equals either 0 or 2. The probability of an interface crossing between the two points scales linearly with r/L(t). Finite-width corrections and intra-domain fluctuations then contribute only subdominant terms that vanish as ξ/r → 0, establishing S_q ∼ r^1 uniformly for all q inside the window ξ ≪ r ≪ L(t). revision: yes

  2. Referee: [Structure functions section] The central claim section: no explicit calculation or numerical verification is referenced that demonstrates the interface-crossing probability yields S_q ~ r^1 independently of q; the manuscript must supply this to confirm the exponent is truly q-independent rather than an effective value in a limited range.

    Authors: We have inserted the explicit calculation: because |Δφ| ∈ {0,2} for sharp interfaces, S_q(r) = 2^q × P(odd number of crossings). For r ≪ L(t) the crossing probability P ∝ r/L(t), so S_q(r) ∼ r^1 with a q-independent exponent. We now reference supporting numerical data from our earlier TDGL and CH studies that confirm the scaling holds across a range of q. Additional plots can be added if required. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling follows directly from sharp-interface assumption

full rationale

The paper derives ζ_q = 1 for structure functions S_q ~ r^{ζ_q} as a direct consequence of sharp interfaces in the TDGL and CH models, where field differences are binary (0 or 2) and interface-crossing probability scales linearly with r in the regime ξ ≪ r ≪ L(t). This is a standard physical argument from the model dynamics and does not reduce to a fitted parameter, self-definition, or load-bearing self-citation. The provided text presents the result as following from the equations' interface properties without invoking prior author work as a uniqueness theorem or smuggling an ansatz. The derivation chain remains self-contained against external benchmarks of interface sharpness.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that sharp interfaces in TDGL and CH systems directly enforce the scaling exponent ζ_q = 1.

axioms (1)
  • domain assumption Sharp interfaces in coarsening systems produce structure function scaling S_q ~ r^1
    Invoked directly in the abstract as the basis for ζ_q = 1

pith-pipeline@v0.9.0 · 5445 in / 1090 out tokens · 24795 ms · 2026-05-13T18:26:42.538730+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

36 extracted references · 36 canonical work pages

  1. [1]

    2004Instabilities, Chaos and Turbulence

    Manneville P. 2004Instabilities, Chaos and Turbulence. London: Imperial College Press

    work page

  2. [2]

    2014 Dissipative Structures and Weak Turbulence

    Manneville P. 2014 Dissipative Structures and Weak Turbulence. San Diego: Academic Press

    work page 2014

  3. [3]

    1995 Turbulence: The Legacy of A

    Frisch U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge: Cambridge University Press

    work page 1995

  4. [4]

    1941a Dissipation of Energy in Locally Isotropic Turbulence.Dokl Acad Nauk SSSR32, 16–18

    Kolmogorov AN. 1941a Dissipation of Energy in Locally Isotropic Turbulence.Dokl Acad Nauk SSSR32, 16–18

    work page

  5. [5]

    1941b The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.Dokl Acad Nauk SSSR30, 301–305

    Kolmogorov AN. 1941b The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.Dokl Acad Nauk SSSR30, 301–305

    work page

  6. [6]

    2008Turbulence in Fluids

    Lesieur M. 2008Turbulence in Fluids. Dordrecht: Springer-Verlag

    work page

  7. [7]

    1978 A simple dynamical model of intermittent fully developed turbulence.J

    Frisch U, Sulem PL, Nelkin M. 1978 A simple dynamical model of intermittent fully developed turbulence.J. Fluid Mech.87, 719–736

    work page 1978

  8. [8]

    1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number.J

    Kolmogorov AN. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number.J. Fluid Mech.13, 82–85

    work page 1962

  9. [9]

    1987 Simple multifractal cascade model for fully developed turbulence

    Meneveau C, Sreenivasan KR. 1987 Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett.59, 1424–1427

    work page 1987

  10. [10]

    1994 Universal scaling laws in fully developed turbulence.Phys

    She ZS, Leveque E. 1994 Universal scaling laws in fully developed turbulence.Phys. Rev. Lett.72, 336–339

    work page 1994

  11. [11]

    2009Kinetics of Phase Transitions

    Puri S, Wadhawan V, editors. 2009Kinetics of Phase Transitions. Boca Raton, FL: CRC Press

    work page

  12. [12]

    1994 Theory of phase-ordering kinetics.Adv

    Bray AJ. 1994 Theory of phase-ordering kinetics.Adv. Phys.43, 357–459

    work page 1994

  13. [13]

    1977 Theory of dynamic critical phenomena.Rev

    Hohenberg PC, Halperin BI. 1977 Theory of dynamic critical phenomena.Rev. Mod. Phys. 49, 435–479

    work page 1977

  14. [14]

    2023 Nonlinear energy dissipation and transfers in coarsening systems.Phys

    Verma MK, Agrawal R, Yadav PK, Puri S. 2023 Nonlinear energy dissipation and transfers in coarsening systems.Phys. Rev. E107, 034207. (10.1103/physreve.107.034207)

    work page doi:10.1103/physreve.107.034207 2023

  15. [15]

    2024 Spectral energy transfers in domain growth problems

    Yadav PK, Verma MK, Puri S. 2024 Spectral energy transfers in domain growth problems. Phys. Rev. E110, 044130. (10.1103/physreve.110.044130)

    work page doi:10.1103/physreve.110.044130 2024

  16. [16]

    1982Small-Angle X-Ray Scattering

    Porod G, Glatter O, Kratky O, editors. 1982Small-Angle X-Ray Scattering. New York: Academic Press

    work page

  17. [17]

    1988 Large wave number features of form factors for phase transition kinetics

    Oono Y, Puri S. 1988 Large wave number features of form factors for phase transition kinetics. Mod. Phys. Lett. B02, 861–867. (10.1142/S0217984988000606)

    work page doi:10.1142/s0217984988000606 1988

  18. [18]

    1995 Scaling and intermittency in Burgers turbulence

    Bouchaud JP, Mézard M, Parisi G. 1995 Scaling and intermittency in Burgers turbulence. Phys. Rev. E52, 3656–3674. (10.1103/PhysRevE.52.3656)

    work page doi:10.1103/physreve.52.3656 1995

  19. [19]

    2000 Intermittency exponents and energy spectrum of the Burgers and KPZ equations with correlated noise.Phys

    Verma MK. 2000 Intermittency exponents and energy spectrum of the Burgers and KPZ equations with correlated noise.Phys. A: Stat. Mech. Appl.277, 359–388

    work page 2000

  20. [20]

    Porod G. p. 35. In , p. 35. Academic Press, New York, 1982

    work page 1982

  21. [21]

    2022 Variable energy flux in turbulence.J

    Verma MK. 2022 Variable energy flux in turbulence.J. Phys. A: Math. Theor.55, 013002. (10.1088/1751-8121/ac354e)

    work page doi:10.1088/1751-8121/ac354e 2022

  22. [22]

    2019Energy transfers in Fluid Flows: Multiscale and Spectral Perspectives

    Verma MK. 2019Energy transfers in Fluid Flows: Multiscale and Spectral Perspectives. Cambridge: Cambridge University Press

    work page

  23. [23]

    2018Physics of Buoyant Flows: From Instabilities to Turbulence

    Verma MK. 2018Physics of Buoyant Flows: From Instabilities to Turbulence. Singapore: World Scientific

    work page

  24. [24]

    2025 Characterization of local energy transfer in large-scale intermittent stratified turbulent flows via coarse-graining.Phys

    Foldes R, Marino R, Cerri SS, Camporeale E. 2025 Characterization of local energy transfer in large-scale intermittent stratified turbulent flows via coarse-graining.Phys. Rev. Fluids 10, 043803. (10.1103/PhysRevFluids.10.043803) 16royalsocietypublishing.org/journal/rsta Phil. Trans. R. Soc. A 0000000

    work page doi:10.1103/physrevfluids.10.043803 2025

  25. [25]

    2024 Evidence of dual energy transfer driven by magnetic reconnection at subion scales

    Foldes R, Cerri SS, Marino R, Camporeale E. 2024 Evidence of dual energy transfer driven by magnetic reconnection at subion scales. Phys. Rev. E110, 055207. (10.1103/PhysRevE.110.055207)

    work page doi:10.1103/physreve.110.055207 2024

  26. [26]

    Physical Review Research , keywords =

    Zhou M, Bhat P, Loureiro NF, Uzdensky DA. 2019 Magnetic island merger as a mechanism for inverse magnetic energy transfer. Phys. Rev. Research1, 012004. (10.1103/PhysRevResearch.1.012004)

    work page doi:10.1103/physrevresearch.1.012004 2019

  27. [27]

    2024 Large-scale self-organization in dry turbulent atmospheres

    Alexakis A, Marino R, Mininni PD, van Kan A, Foldes R, Feraco F. 2024 Large-scale self-organization in dry turbulent atmospheres. Science383, 1005–1009. (10.1126/science.adg8269)

    work page doi:10.1126/science.adg8269 2024

  28. [28]

    1999 Asymptotic Theory for the Probability Density Functions in Burgers Turbulence.Phys

    E W, Vanden Eijnden E. 1999 Asymptotic Theory for the Probability Density Functions in Burgers Turbulence.Phys. Rev. Lett.83, 2572–2575. (10.1103/PhysRevLett.83.2572)

    work page doi:10.1103/physrevlett.83.2572 1999

  29. [29]

    1965 Handbook of Mathematical Functions

    Abramowitz M, Stegun IA. 1965 Handbook of Mathematical Functions. With Formulas, Graphs, and Mathematical Tables. Courier Corporation

    work page 1965

  30. [30]

    1988 Study of phase-separation dynamics by use of cell dynamical systems

    Oono Y, Puri S. 1988 Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling.Phys. Rev. A38, 434–453. (10.1103/PhysRevA.38.434)

    work page doi:10.1103/physreva.38.434 1988

  31. [31]

    1988 Study of phase-separation dynamics by use of cell dynamical systems

    Puri S, Oono Y. 1988 Study of phase-separation dynamics by use of cell dynamical systems. II. Two-dimensional demonstrations. Phys. Rev. A38, 1542–1565. (10.1103/PhysRevA.38.1542)

    work page doi:10.1103/physreva.38.1542 1988

  32. [32]

    2019 A Technique for Removing Large-scale Variations in Regularly and Irregularly Spaced Data.ApJ874, 75

    Cho J. 2019 A Technique for Removing Large-scale Variations in Regularly and Irregularly Spaced Data.ApJ874, 75. (10.3847/1538-4357/ab06f3)

    work page doi:10.3847/1538-4357/ab06f3 2019

  33. [33]

    2011 Kinetics of Phase Transitions: Numerical Techniques and Simulations

    Puri S. 2011 Kinetics of Phase Transitions: Numerical Techniques and Simulations. In Computational Statistical Physics: Lecture Notes, Guwahati SERC School , pp. 123–160. Springer

    work page 2011

  34. [34]

    2015 New class of turbulence in active fluids.PNAS112, 15048–15053

    Bratanov V, Jenko F, Frey E. 2015 New class of turbulence in active fluids.PNAS112, 15048–15053. (10.1073/pnas.1509304112)

    work page doi:10.1073/pnas.1509304112 2015

  35. [35]

    1993 Pattern formation outside of equilibrium.Rev

    Cross MC, Hohenberg PC. 1993 Pattern formation outside of equilibrium.Rev. Mod. Phys. 65, 851–1112

    work page 1993

  36. [36]

    2009 Pattern formation and dynamics in nonequilibrium systems

    Cross M, Greenside H. 2009 Pattern formation and dynamics in nonequilibrium systems. Cambridge: Cambridge University Press

    work page 2009