Generalized Dynamics for Second Kind of Soft-Matter Quasicrystals

Tian-You Fan

REVIEW 4 major objections 4 minor 2 references

Generalized Dynamics for Second Kind of Soft-Matter Quasicrystals

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Ten equations govern 7-, 9-, and 14-fold soft-matter quasicrystals.

desk verdict The three equation systems for 7-, 9-, and 14-fold second-kind soft-matter quasicrystals are stated as new results, but the derivation that would make them checkable is not in the paper; as submitted, this is a claim without proof. read the letter →

arxiv 1908.06430 v2 pith:YUDV2FDA submitted 2019-08-18 cond-mat.soft

Tian-You Fan This is my paper · ORCID

classification cond-mat.soft

keywords soft-matterquasicrystalssecondkindtwo-dimensionalsix-dimensionalembeddingspacegeneralizeddynamicsphononfieldfirstphasonequationofstate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives complete generalized-dynamics equation systems for the second kind of two-dimensional quasicrystals in soft matter: the 7-, 9-, and 14-fold symmetry classes. Each system is ten coupled nonlinear partial differential equations in ten fields: phonon displacement $\mathbf{u}$, first phason $\mathbf{v}$, second phason $\mathbf{w}$, fluid velocity $\mathbf{V}$, mass density $\rho$, and pressure $p$. The equations are a continuity equation, generalized Navier-Stokes momentum equations, phonon equations of motion from symmetry breaking, first- and second-phason dissipation equations, and an equation of state. If correct, they are closed hydrodynamic descriptions for materials whose quasiperiodic order needs a six-dimensional embedding space, and they predict wave propagation for $\mathbf{u}$ and $\mathbf{V}$ while both phason fields diffuse.

What carries the argument

The central mechanism is the six-dimensional embedding space, concretely used in the Landau-Anderson density expansion with phase angle $\Phi_n = \mathbf{G}^n_\parallel\cdot\mathbf{u} + \mathbf{G}^n_{\perp 1}\cdot\mathbf{v} + \mathbf{G}^n_{\perp 2}\cdot\mathbf{w}$. This identity turns the three field variables into the order parameters of the theory. The remaining machinery is a Hamiltonian strain energy that separates phonon, first-phason, second-phason, and pairwise-coupling terms, followed by a Poisson-bracket/Langevin procedure for reversible and dissipative fluxes and an equation of state to close the system.

What would settle it

Find, by experiment or simulation, one 7-, 9-, or 14-fold soft-matter quasicrystal and measure its low-frequency, long-wavelength response. The theory predicts exactly one propagating phonon-like branch plus two purely diffusive phason branches, with a symmetry-dependent coupling pattern; observing ballistic propagation of either phason field, or a different number of phason modes, would rule the systems out. A cheaper check is to compute the dispersion relations from equations (5), (7), and (9) and compare them with molecular-dynamics or colloidal-model simulations built from the six-dimensional embedding potential.

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Extended reading notes

Core claim

The central claim is that the low-energy order of second-kind two-dimensional soft-matter quasicrystals is carried by three fields, not two: phonon $\mathbf{u}$, first phason $\mathbf{v}$, and second phason $\mathbf{w}$, because the diffraction of these symmetries requires a six-dimensional embedding space $E_6 = E_{2\parallel}\oplus E_{2\perp 1}\oplus E_{2\perp 2}$. The paper combines group-representation elastic constants for each symmetry with a Poisson-bracket/generalized-Langevin Hamiltonian and a soft-matter equation of state to obtain equation systems (5), (7), and (9) for the 7-, 14-, and 9-fold cases. It states that each system is mathematically consistent and solvable, that $\mathbf{u}$ and $\mathbf{V}$ propagate as waves while $\mathbf{v}$ and $\mathbf{w}$ diffuse, and that the 9-fold system is close to the earlier 18-fold system but with small differences in coupling.

Load-bearing premise

The whole derivation rests on the prior assertion that 7-, 9-, and 14-fold quasicrystal order is correctly captured by a six-dimensional embedding space with one phonon and two phason fields, and on the group-representation elastic constants built on that space; if that assertion or those constants are wrong, equations (5), (7), and (9) do not describe any real material.

Editorial extensions

If this is right

The 7-, 14-, and 9-fold soft-matter quasicrystals, if synthesized, would have complete governing equations ready for initial- and boundary-value problems.
The phonon field and fluid velocity behave as propagating waves, while the first and second phason fields diffuse, giving distinct relaxation signatures in scattering or rheology.
The coupling patterns differ by symmetry: the 7- and 14-fold cases couple phonons to second phasons and the two phasons, while the 9-fold case decouples phonons from phasons but couples the two phason fields.
Without the equation of state the systems are not closed, so any thermodynamic relation linking pressure, density, and temperature completes the dynamics.
The 9-fold system is close to the previously reported 18-fold system, so the construction unifies the dynamics of all second-kind soft-matter quasicrystals within one framework.

Reading between the lines

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

If the six-dimensional embedding description is right, equations (5), (7), and (9) imply testable dispersion relations: one propagating phonon branch and two overdamped phason branches per wavevector, with coupling-induced avoided crossings where phonon and second phason interact.
The same six-dimensional embedding construction could be applied to solid versions of 7-, 9-, and 14-fold quasicrystals by omitting the fluid velocity and equation-of-state sector, an extension the paper does not spell out.
Because 7-, 9-, and 14-fold soft-matter quasicrystals have not yet been observed, the equations are predictive: they specify that colloidal or micellar experiments should measure two phason diffusion coefficients, $\Gamma_v$ and $\Gamma_w$, once such phases are made.
A useful consistency check would be to send the second-phason field and its couplings to zero; the systems should then reduce to the first-kind generalized hydrodynamics, testing the internal consistency of the Hamiltonian formulation.

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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

Two things to know. First, this is a legitimate extension within Fan's own program: it writes down explicit generalized dynamics equation systems (5), (7), (9) for 7-, 9-, and 14-fold second-kind soft-matter quasicrystals, which are not in the earlier 18-fold paper. Second, the paper's central claim—that these equations are derived—is not supported by the text. The appendix gives a generic Poisson-bracket form (A5) and then states that after 'further simplifications' and adding the equation of state one obtains (5), (7), (9). The actual reduction, including how the elastic energy functional derivatives become the gradients and Laplacians, how the scalar viscosity enters, and how the dissipation coefficients Γ_u, Γ_v, Γ_w are assigned, is missing. Since the signs and placements of the coupling constants R and G are exactly what distinguish the three symmetry cases, this is not a harmless omission.

What is genuinely new: the explicit systems for the three symmetries, with different coupling patterns. The constitutive laws are borrowed from Hu et al.'s group representation analysis, and the paper makes a reasonable distinction: 7/14 include phonon-second-phason coupling, 9 does not, while all three have first-second phason coupling. If those constitutive laws are right, the structural skeleton of the equations is plausible. The paper also identifies ten field variables and ten equations, and the wave/diffusion assignment for u, V versus v, w is standard.

Soft spots, in order of severity. (1) Missing derivation, as above; this is load-bearing. (2) The typesetting is badly garbled—many equations contain placeholder glyphs, so even the stated final systems cannot be trusted as written. (3) No validation: no known-limit check, no numerical solution, no comparison to the 18-fold case in detail. The closing note that unpublished solutions 'examine the equations' is an appeal to work not in the manuscript and should not count. (4) The paper leans heavily on Ref [1], a Chinese-language paper, for the equation of state and the 18-fold system; that is a citation-pattern choice, not a flaw by itself, but it makes the present paper hard to check without that source.

Who this is for: specialists already inside the author's quasicrystal-hydrodynamics program. A general soft-matter reader gets little, because the equations are hypothetical and unverified. As submitted, I would not send it to peer review. The right route is revision with the full derivation written out, clean equations, and at least one sanity check (e.g., reducing to the 18-fold case or a simple numerical solution). Then it could be a useful contribution.

Referee Report

4 major / 4 minor

Summary. This paper claims to derive the equations of generalized dynamics for soft-matter quasicrystals with 7-, 14-, and 9-fold symmetry, which are classified as second-kind two-dimensional quasicrystals. The author introduces a six-dimensional embedding space with phonon, first phason, and second phason fields, cites group-representation constitutive laws, and states the final governing systems as Eqs. (5), (7), and (9). The derivation is said to be based on a Poisson bracket method with the Hamiltonian given in the appendix, but the actual reduction is not shown; the paper also discusses possible solutions in unpublished follow-up work.

Significance. If the four-field generalized dynamics equations were correct and fully derived, they would extend the hydrodynamics of soft-matter quasicrystals to the second kind and provide a starting point for studying 7-, 9-, and 14-fold quasicrystals, which have not yet been observed. The paper, however, does not supply a checkable derivation: the transition from the Hamiltonian and constitutive laws to the final PDE systems is asserted rather than demonstrated, and the constitutive laws themselves are cited from earlier group-representation work. The value of the manuscript is therefore conditional, and as submitted it is better viewed as a presentation of candidate equations than a derivation.

major comments (4)

[Appendix A5 and Eqs. (5), (7), (9)] The central claim is not supported by a derivation. The appendix gives the Hamiltonians (A1)-(A4) and the generic Poisson-bracket form (A5), then states that 'After further simplifications, from Equation (A5) and including the equation of state, we can obtain the generalized dynamics Equations (5), (7) and (9).' The functional derivatives appearing in (A5) are never evaluated, so the reader cannot verify how the elastic constants L, M, T, K1, K2, R, G, the viscosity eta, and the dissipation coefficients Gamma_u, Gamma_v, Gamma_w enter with the correct signs and placements. Since the differences among (5), (7), and (9) consist precisely of the coupling terms, the paper's main assertion that these equations were derived remains unverified.
[Sec. 3 Eq. (4), Sec. 4 Eq. (6), Sec. 5 Eq. (8)] The constitutive laws are introduced by citing group representation theory from Ref. [3], but they are not derived in this manuscript and no explicit irreducible-representation analysis is given. In particular, the sign patterns of the phonon-second phason coupling R and the first-second phason coupling G appear to differ among (4), (6), and (8), yet the manuscript does not show how these differences follow from the symmetry groups of 7-, 14-, and 9-fold quasicrystals. If the six-dimensional embedding space or the elastic-constant structure is applied incorrectly, the dynamics equations would not govern the intended materials. The manuscript needs to provide or reproduce the symmetry derivation for each allowed constant.
[Eqs. (5), (7), (9) as printed] The central equation systems are partially illegible in the submitted text. For example, phrases such as 'wuw wVM u L M Ryx x x y y', 'uwww u u v vVV K w R Gtxy...', and 'vvv w w wVV T v G Gtxy...' do not form well-defined partial differential equations. This prevents a referee from checking closure, signs, or consistency of the systems. Each subequation should be typeset cleanly and labeled (a) through (j) as referenced in the text, so that the claimed ten-field, ten-equation structure can be verified.
[Sec. 6 and final appended note] The closing note states that after publication the author and his group obtained solutions 'which examine the equations' and 'the examination shows the equations are correct and effective', referring to work not contained in the manuscript. This is an appeal to unavailable follow-up publications and cannot substitute for the missing derivation or for any in-paper verification. It should be removed or replaced with actual evidence, such as a well-posedness statement or a reproducible numerical test.

minor comments (4)

[After Eq. (5)] The notation L=C12 and M=(C11-C12)/2 is confusing because standard isotropic elasticity usually writes C11=L+2M; the paper should state the convention explicitly.
[Sec. 2 and Sec. 5] The 18-fold equations are omitted and referred to Ref. [1], which is in Chinese; this makes the claimed closeness between Eq. (9) and the 18-fold system impossible for most readers to check.
[Secs. 3-5] The repeated statement that an equation system is 'mathematically consistent and solvable' is unsupported; no function space, boundary conditions, or well-posedness theorem is provided.
[Eq. (2)] The Landau-Anderson expansion in Eq. (2) introduces quantities such as rho_G without defining their normalization or dependence on the quasiperiodic density, and Eq. (3) uses phase variables without explaining their geometric relation to the three perpendicular directions.

Circularity Check

2 steps flagged · score 4.0 of 10

The displayed PDE systems are not derived in the paper: the decisive reduction is delegated to the author's own Ref [1], making the central claim self-citation-load-bearing rather than an independent derivation.

self citation load bearing [Appendix, paragraph after Eq. (A5); echoed in Section 7 Conclusion]
"The terms of variational form can be reduced to a differential form based on Ref [1]. After further simplifications, from Equation (A5) and including the equation of state, we can obtain the generalized dynamics Equations (5), (7) and (9) for the plane field of individual two-dimensional quasicrystalline systems of second kind in soft matter."

The paper's central claim is that Equations (5), (7) and (9) were derived. The only derivation actually written out stops at the generic Poisson-bracket form (A5); the conversion of that variational form into the specific Laplacian, gradient, dissipative PDE systems is not exhibited. Instead the text says the reduction is 'based on Ref [1]', and Section 7 states 'the Ref [1] is a basis' for the whole theoretical system. Ref [1] is a prior paper by the same author.
other [Unnumbered paragraph after References]
"After the publication of this paper the author and his group have obtained some solutions of some initial and boundary value problems of equations (5), (7) and (9), which examine the equations, the examination shows the equations are correct and effective, please refer to the new publications of the author and his co-workers in the field."

This is not part of the equation derivation, but it is a self-referential validation claim: the equations are declared correct and effective on the basis of solutions in unspecified future publications by the same author and co-workers. Since no such publication, data set, or independent benchmark is provided, this assertion is circular as evidence and cannot independently confirm the central claim.

full rationale

No fitted parameter is called a prediction, and no external experimental quantity is being used as both input and output, so the most common statistical circularity is absent. The constitutive laws for 7-, 14-, and 9-fold quasicrystals are imported from Hu et al. [3], which is an external, independent input, and the Poisson-bracket Hamiltonian in (A1)-(A4) is stated with the second-kind phason couplings explicitly listed. However, the actual reduction from the generic variational equations (A5) to the displayed PDE systems (5), (7), and (9) is not carried out in the paper; the appendix says the reduction is 'based on Ref [1]', and the conclusion identifies Ref [1] and the author's books [6,7] as the basis of the theoretical system. Ref [1] is a same-author prior publication, so the load-bearing step of the derivation is a self-citation rather than a shown mathematical consequence. Because the equations are not fitted to data and there is no definitional identity between input and output, this is not a 6-10 circularity; but the central derivation is not self-contained and cannot be checked from the manuscript alone, so a moderate score of 4 is appropriate.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced beyond the phonon field u and first/second phason fields v, w, which come from Hu et al [3] and Ref [1].

free parameters (3)

A, B
Material constants in the momentum equations and equation of state (Eqs. 5b,c,j). No derivation or values are given.
Dissipation coefficients (Γ_u, Γ_v, Γ_w)
Phenomenological coefficients in the phonon and phason equations (defined after Eq. 5). No values or estimates.
Elastic constants (L, M, T, K1, K2, R, G)
Phonon and phason elastic constants and coupling constants from group representation theory of Hu et al [3]; treated as inputs, not fitted in this paper.

assumptions (5)

domain assumption Six-dimensional embedding space E6 = E2_parallel ⊕ E2_perp1 ⊕ E2_perp2 for second kind 2D quasicrystals
Section 1, Eq. (1). Taken from Hu et al [3] without derivation.
domain assumption Landau-Anderson symmetry breaking and elementary excitation principle
Section 1 and Appendix. Basis for phonon and phason fields and their dynamics.
domain assumption Constitutive laws for 7-, 9-, 14-fold symmetries from group representation
Equations (4), (6), (8). Stated 'according to the theory of group representation' with citation to Hu et al [3].
ad hoc to paper Equation of state p = (kT/l^3) ρ + A ρ^2 + B ρ^3 (form per Eq. 5j)
Introduced to close the system; A and B are unspecified constants and the form is not derived.
standard math Poisson bracket formalism with Hamiltonian following Lubensky et al.
Appendix (A1)-(A5). Standard method, but the reduction to final equations is not shown.

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Cite this review

Pith. "Pith review of Generalized Dynamics for Second Kind of Soft-Matter Quasicrystals." pith.science (2026). https://pith.science/paper/YUDV2FDA

@misc{pith2026190806430,
  author       = {Pith},
  title        = {Pith review of: Generalized Dynamics for Second Kind of Soft-Matter Quasicrystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUDV2FDA}},
  note         = {Machine review of arXiv:1908.06430}
}

read the original abstract

The equations of generalized dynamics of the first kind of soft-matter quasicrystals are derived and given in the previous report, we now introduce the equations of the dynamics for the second kind of quasicrystals in soft matter.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Fan T Y, Equation Systems of Generalized Hydrodynamics for Soft-Matter Quasicrystals, Appl. Math. Mech.,2016,37(4),331-347 (in Chinese). [2]Fischer S, Exner A, Zielske K, Perlic h J, Deloudi S, Steu er W, Linder P and Foestor S, Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry, Proc Nat Ac Sci, 2011, 108, 1810-1814. [3]Hu C Z, Ding D ...

work page 2016
[6]

[7]Fan T Y , Mathematical Theory of Elas ticity and Relevant Topics of Solid and Soft-Matter Quasicrystals, Beijing, Be ijing Institute Technology Press, 2014, in Chinese

Fan T Y , Mathematical Theory of Elasticity of Quasicrystals and Its Applications, 2010, 1 st Edition, 2016, 2 nd Edition, Beijing, Scie nce Press/Heidelberg, Springer-Verlag. [7]Fan T Y , Mathematical Theory of Elas ticity and Relevant Topics of Solid and Soft-Matter Quasicrystals, Beijing, Be ijing Institute Technology Press, 2014, in Chinese. [8]Lubens...

work page 2010

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Fan T Y, Equation Systems of Generalized Hydrodynamics for Soft-Matter Quasicrystals, Appl. Math. Mech.,2016,37(4),331-347 (in Chinese). [2]Fischer S, Exner A, Zielske K, Perlic h J, Deloudi S, Steu er W, Linder P and Foestor S, Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry, Proc Nat Ac Sci, 2011, 108, 1810-1814. [3]Hu C Z, Ding D ...

work page 2016

[2] [6]

[7]Fan T Y , Mathematical Theory of Elas ticity and Relevant Topics of Solid and Soft-Matter Quasicrystals, Beijing, Be ijing Institute Technology Press, 2014, in Chinese

Fan T Y , Mathematical Theory of Elasticity of Quasicrystals and Its Applications, 2010, 1 st Edition, 2016, 2 nd Edition, Beijing, Scie nce Press/Heidelberg, Springer-Verlag. [7]Fan T Y , Mathematical Theory of Elas ticity and Relevant Topics of Solid and Soft-Matter Quasicrystals, Beijing, Be ijing Institute Technology Press, 2014, in Chinese. [8]Lubens...

work page 2010