Equation Systems of Generalized Hydrodynamics for Soft-Matter Quasicrystals

Tian-You Fan

REVIEW 5 major objections 3 minor 3 references

Equation Systems of Generalized Hydrodynamics for Soft-Matter Quasicrystals

T0 review · 5 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The dynamics of soft-matter quasicrystals are governed by four closed PDE systems, one per symmetry class, built from phonons, phasons, and a new fluid phonon.

desk verdict A hydrodynamics extension for soft-matter quasicrystals, but the governing equations are asserted, not derived, and the same content already appeared in Chinese in 2016. read the letter →

arxiv 1908.06425 v2 pith:QVB5EDFS submitted 2019-08-18 cond-mat.soft

Tian-You Fan This is my paper · ORCID

classification cond-mat.soft

keywords soft-matterquasicrystalsgeneralizedhydrodynamicsfluidphononphasonequationofstatequasicrystalsymmetryPoissonbracketmethod

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 argues that soft-matter quasicrystals—materials that flow like fluids yet keep quasiperiodic atomic order—are described by four closed systems of partial differential equations, one for each symmetry class: 12-fold, 18-fold, 5/10-fold, and 8-fold. To close the systems, the author introduces a new elementary excitation, the fluid phonon velocity field, alongside the usual phonon and phason fields, and appends a polynomial equation of state linking pressure to mass density. The paper presents equations (6), (8), (9), and (10) as the final governing systems for plane fields, containing mass conservation, generalized Navier–Stokes momentum equations, phonon wave equations, phason dissipation equations, and the equation of state. A sympathetic reader would care because these systems give the partial-differential-equation basis for solving deformation and flow problems in the observed 12- and 18-fold soft-matter quasicrystals and in the predicted 5/10-fold and 8-fold ones.

What carries the argument

The named new object is the fluid phonon $V_i$, a velocity field that carries fluid motion in the soft-matter quasicrystal on the same footing as phonon displacements $u_i$ and phason displacements $w_i$. The argument runs through the Poisson-bracket Hamiltonian method: a Hamiltonian with phonon, phason, coupling, fluid kinetic, and density-variation terms yields the differential-variational equations (A5), and dropping higher-order variational derivatives of the Hamiltonian with respect to $u$ and $w$ is the step that turns (A5) into the displayed PDE systems. Two further pieces do essential work: replacing the solid stress–strain constitutive relation with a Newtonian-fluid viscous law, and appending a polynomial equation of state $p=f(\rho)$, obtained by modifying a columnar-liquid-crystal equation of state, which closes the system.

What would settle it

Linearize the 12-fold system (6) around a uniform state and derive its dispersion relations; if the number or shape of propagating modes does not match measured phonon and phason spectra in a known colloidal 12-fold quasicrystal—for instance, a missing fluid-phonon branch or a phason diffusion law with the wrong exponent—the system is falsified. A more direct check is to redo the omitted reduction from equation (A5) to equation (6) and see whether any dropped variational-derivative term survives at the order kept in the displayed equations.

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Extended reading notes

Core claim

The paper's central claim is that the plane-field dynamics of soft-matter quasicrystals are governed by explicit nonlinear PDE systems, one for each symmetry class, and that each system is mathematically closed because the number of unknown fields equals the number of equations. For 12-fold symmetry the unknowns are the mass density, pressure, the two phonon displacement components, the two phason displacement components, and the two fluid-velocity components; equations (6a)–(6h) give mass conservation, momentum balance, phonon wave equations, phason diffusion equations, and the equation of state. The 18-fold system (8) adds a second phason field, giving ten fields and ten equations, while equations (9) and (10) cover the 5/10-fold and 8-fold classes with phonon–phason coupling included. The derivation begins with Poisson-bracket Hamiltonian equations in the appendix and, after omitting higher-order variational-derivative terms, produces the printed systems; the paper states explicitly that the intermediate algebra is not included.

Load-bearing premise

The paper's key assumption is that the unshown algebra connecting the general Hamiltonian equations in the appendix to the printed PDE systems is correct, and that the terms it deletes as higher order are truly negligible; if either fails, the displayed governing equations do not follow.

Editorial extensions

If this is right

For the observed 12-fold and 18-fold soft-matter quasicrystals, equations (6) and (8) become the working PDE models, so solving them with initial and boundary conditions should describe mass redistribution, deformation, and flow.
The systems predict a dynamical split: phonon and fluid-phonon variables propagate as waves, while phason variables diffuse, a distinction that can be checked by measuring mode spectra.
Without the equation of state the field equations are not closed, so thermodynamics enters as an essential component of quasicrystal hydrodynamics rather than an optional add-on.
For the not-yet-observed 5/10-fold and 8-fold quasicrystals, equations (9) and (10) make concrete predictions about how phonon–phason coupling enters the dynamics, testable if such materials are synthesized.

Reading between the lines

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

If the truncation step from (A5) to (6) is verified, the same Poisson-bracket route could generate governing systems for other quasiperiodic soft materials, such as liquid-crystalline or polymeric quasicrystals with different symmetries.
The modified equation of state may be the most portable piece: it offers a singularity-free pressure–density relation for any complex fluid whose characteristic particle size and initial density are known.
Linearizing equations (6)–(10) around a uniform state would yield dispersion relations; comparing them with scattering or rheology data would provide a direct experimental test that the paper itself does not carry out.
The explicit phonon–phason coupling terms in (9) and (10) suggest that in 5/10-fold and 8-fold soft-matter quasicrystals, flow can directly reorganize quasiperiodic order, an effect that may appear as flow-induced phason rearrangement.

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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

Dear colleague,

Punchline: this is a research announcement, not a derivation. The paper extends Lubensky's solid-quasicrystal hydrodynamics by adding a fluid phonon field and an equation of state, and writes down PDE systems for 12-, 18-, 5/10- and 8-fold soft-matter quasicrystals. That is a reasonable physical goal, and the 12- and 18-fold equations are, as far as I know, the first closed hydrodynamic descriptions offered for those observed phases. The idea of coupling phonon wave propagation, phason diffusion, and fluid velocity through the Poisson bracket is sensible.

What it does well: the physical motivation is clear, the list of elementary excitations is appropriate, and the distinction between solid and fluid constitutive laws is made explicitly. The author also correctly notes that an equation of state is needed for closure, and that this goes beyond the Poisson bracket derivation.

The soft spots are large. Section 2 states 'the derivation details in mathematics are not included.' The Appendix ends at a variational form (A5) with a remark that the terms 'can be reduced to a differential form' and that 'after further simplifications' equations (6)-(10) follow. That reduction is the entire content of the paper, and it is not shown. As the stress-test note says, the variational derivatives of the Hamiltonian must produce exactly the phonon, phason, coupling, viscous, and pressure terms with correct coefficients; one wrong sign changes the system. The omission is load-bearing, not cosmetic.

The equation of state, Eq (2), is justified by saying that with l=8-9 nm 'the theoretical prediction is the best,' but no fit, data, or comparison is shown. That is an ad hoc parameterization. The final note also discloses that the same content was published in Chinese in 2016, so this arXiv posting does not present a new result in the literature, and the reference to 'new publications of the author' for verification is not a citable check. The citation pattern is self-heavy but that alone is not a flaw; the flaw is that the self-citations are used as the authority for the derivation steps that are missing here.

The equation rendering is badly corrupted, which makes independent verification harder, but even with clean typesetting the derivation gap would remain.

Bottom line: if the equations are right, they are useful for people working on soft-matter quasicrystals. But this paper gives no way to check them. It is a conference abstract expanded to arXiv, not a peer-reviewable manuscript. Recommendation: desk reject, and invite the author to submit a full paper with the Poisson-bracket algebra carried out and the equation of state tested against data.

Best, [You]

Referee Report

5 major / 3 minor

Summary. The paper proposes closed systems of generalized hydrodynamic equations for soft-matter quasicrystals with 12-, 18-, 5/10-, and 8-fold symmetry. It introduces a 'fluid phonon' field V_i in addition to the usual phonon and phason fields, and it appends an equation of state p(rho) to close the conservation laws. The central claims are that Eqs. (6), (8), (9), and (10) are the final governing equation systems for the plane dynamics of these four symmetry classes, and that these systems are 'consistent with mathematical solvability.' The derivation is said to follow from a Poisson-bracket Hamiltonian formalism, but the manuscript states that the mathematical details are not included, and the Appendix stops at a variational form (A5).

Significance. If the displayed systems are correct, they would fill a genuine gap: no closed hydrodynamic description of soft-matter quasicrystals with fluid phonons and a density-dependent pressure has been established, and the 18-fold case with two phason fields is an interesting extension. The conceptual framework, using the Landau-Lubensky elementary-excitation picture and supplementing it with an equation of state, is physically reasonable, and the author is right that without an equation of state the system is not closed. The paper also correctly distinguishes the fluid constitutive law from the solid one used by Lubensky et al. However, the actual deliverables are the PDE systems themselves, and those are not substantiated: the key reduction is explicitly omitted, the equation of state is justified by an unshown fit, and the claimed solvability is asserted rather than demonstrated. There are no machine-checked derivations, reproducibility artifacts, or numerical validations in the manuscript that would allow an independent check. On the evidence in the manuscript I am unable to vouch for the correctness of any of the four systems.

major comments (5)

[§2 and Appendix (A5)] The central derivation is absent. Section 2 says that 'the derivation details in mathematics are not included,' and the Appendix terminates at the variational system (A5), followed by the statement that the variational terms 'can be reduced to a differential form' and that after further simplifications one obtains (6), (8), (9), and (10). The reduction from (A5) to these PDE systems is not shown: the variational derivatives of the Hamiltonian with respect to u, v, and w must produce exactly the elastic-constant combinations in (6)-(10); the viscous stresses must emerge from the Poisson bracket with the fluid velocity; and the omitted higher-order terms in gradient(delta H/delta u) and gradient(delta H/delta w) must be negligible. Any one of these steps could change signs, coefficients, or couplings, so the displayed equations cannot be verified from the evidence given. The pointer in the Appendix to 'Reference [14] given by Lubensky' is also incorrect, since Ref. [14] in the bibliography is Sommerfeld, not Lubensky (Ref. [19]).
[§1, Eq. (2)] The equation of state is load-bearing, because the paper itself stresses that without it the system is not closed, but its justification is only the sentence 'in our computation, l=8-9 nm, the theoretical prediction is the best.' No data, fitting procedure, comparison, or error estimate is provided, and the modification of Wensink's equation is not derived. The parameter l is therefore effectively free in this manuscript. The constants A and B in the momentum equations are likewise introduced without values or derivation, so the proposed systems contain undetermined parameters that are central to their predictive content.
[§6 and final note] The claim that the equations are 'consistent with mathematical solvability' is asserted without proof or numerical evidence. The final note says that the author's group has obtained many solutions that show the equations are 'correct and effective,' but no citations or data are given. Solving an equation system can test internal consistency, but it cannot replace the missing derivation from the Poisson-bracket Hamiltonian, because it does not establish that the equations follow from the stated model.
[§§4-5, Eqs. (9)-(10)] As typeset here, Eqs. (9) and (10) contain unparsable fragments, for example 'wwwR( A Bxx y y x' and '12( ) R(A B' in Section 5, where parentheses, operators, and signs cannot be resolved. Since (9) and (10) are two of the four central deliverables, the manuscript does not permit a reader to verify them. The systems should be re-typeset or, better, derived explicitly.
[References [12], [20], and [22]] The reader cannot check the new material through the cited literature. Equation (2) is attributed to Ref. [12], an unpublished 2015 conference contribution; the Poisson-bracket treatment is attributed to Ref. [20], a Chinese-language article by the author; and the 18-fold framework is attributed to the author's monograph Ref. [22]. Unless these are generally accessible, the manuscript should include the essential steps itself.

minor comments (3)

[Eq. (6)] Please define the operators and notation in Eq. (6) fully; the symbols gradient, divergence, partial derivatives, and index notation are mixed, and the definition 'gradient = (partial/partial x) i + (partial/partial y) j' appears only in the text after Eq. (6).
[Throughout] Several typographical errors remain: 'calcogenides' should be 'chalcogenides,' 'Notations' in Ref. [17] should be 'Notions,' and 'Clarenden' should be 'Clarendon.'
[Final page] The final page states that the paper was already 'reported by Applied Mathematics and Mechanics, Vol.37, No.4, pp331-344, 2016, in Chinese.' The prior publication should be formally cited, and the relationship of the present submission to that earlier version should be explained.

Circularity Check

1 steps flagged · score 2.0 of 10

No construction-level circularity; the governing PDEs rest on an asserted appendix reduction and on minor self-citations, not on a fit masquerading as prediction.

self citation load bearing [Section 6/Conclusion and Discussion, closing self-assessment after References (arXiv full text p.13)]
"After the publication of this paper the author and his group have obtained many solutions of some initial and boundary value problems of equations (6), (8), (9) and (10), 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 the paper's only offered post-publication check of the central equations, and it reduces to the author's own later work: solving (6), (8), (9) and (10) assumes those equations, so the 'examination' cannot establish that they follow from the Hamiltonian/Poisson-bracket input. The cited 'new publications' are unspecified self-citations, providing no independent falsifiable check.

full rationale

The central derivation chain is Poisson bracket (A5) to plane-field PDEs (6)/(8)/(9)/(10), but the paper explicitly says the algebra is omitted ('the derivation details in mathematics are not included') and the Appendix ends by asserting that variational terms 'can be reduced to a differential form.' That is an omitted derivation, a correctness risk, not a circularity: no displayed equation is shown to be its own input by construction. The Poisson-bracket framework is borrowed from independent sources (Dzyaloshinskii and Volovick [18]; Lubensky et al. [19]), and the phason-mode structure for 18-fold comes from Hu et al. [21], so the central claim is not forced by a self-citation chain. The only genuine circular element is the closing validation by the author's own later solutions, which is not a load-bearing derivation step and therefore warrants only a low score. Because the main derivation is incomplete and the equation of state (2) is a modified Wensink form introduced via the author's own [12], the paper is not fully self-contained; but for circularity specifically the score is 2.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central equations rest on the Poisson bracket route from Lubensky, plus a new fluid phonon field and a phenomenological equation of state with a chosen length scale. Most material constants are inherited from prior continuum theory; the paper itself supplies modeling choices and no fitted data or independent validation.

free parameters (2)

l (characteristic size of soft matter in equation of state Eq (2)) = 8-9 nm (used in the author's computation)
The length scale l enters the modified Wensink equation of state. The text says 'in our computation, l = 8-9 nm, the theoretical prediction is the best', indicating it is chosen to match an unspecified prediction rather than derived.
A and B (mass-density variation constants in momentum equations)
Introduced in equations (6), (8), (9) and (10) as 'material constants due to variation of mass density'. No values or independent measurements are given, and they are not fixed by the derivation.

assumptions (5)

standard math Poisson bracket formalism gives the correct hydrodynamic equations for quasicrystals.
The paper invokes Dzyaloshinskii and Vologodskii [18] and Lubensky et al. [19] without reproducing the derivation in detail.
domain assumption Soft-matter quasicrystals can be described by continuum fields u, w and V with the Landau-Anderson symmetry-breaking principle.
Section 2 states this as the basis of the dynamics, following the pattern for solid quasicrystals.
ad hoc to paper The fluid phonon field V_i is a valid independent elementary excitation in soft-matter quasicrystals.
Introduced by analogy with Landau's superfluid phonon, with no experimental or microscopic justification specific to soft-matter quasicrystals.
ad hoc to paper Equation of state Eq (2) correctly describes the thermodynamics of soft-matter quasicrystals.
The equation is a modification of Wensink's equation and is recommended in the text, but its validity is not demonstrated and the length scale is chosen by an unshown fit.
domain assumption Higher-order terms involving the variational derivatives of the Hamiltonian can be omitted in the plane-field reduction.
Section 2 says the equations are obtained by omitting higher terms, but no estimate of their size or justification for their neglect is given.

invented entities (1)

fluid phonon field V_i
purpose: Represents fluid velocity in soft-matter quasicrystals and closes the hydrodynamics alongside phonon and phason fields.
The paper introduces it as a new elementary excitation for soft-matter quasicrystals by analogy with Landau's superfluid phonon, but provides no measurement, dispersion relation, or falsifiable prediction that would independently validate it.

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Cite this review

Pith. "Pith review of Equation Systems of Generalized Hydrodynamics for Soft-Matter Quasicrystals." pith.science (2026). https://pith.science/paper/QVB5EDFS

@misc{pith2026190806425,
  author       = {Pith},
  title        = {Pith review of: Equation Systems of Generalized Hydrodynamics for Soft-Matter Quasicrystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVB5EDFS}},
  note         = {Machine review of arXiv:1908.06425}
}

read the original abstract

The equation systems of generalized hydrodynamics, or generalized dynamics for simplicity, for soft-matter quasicrystals were established. Considering the fluidity of the matter with high order we introduced a new elementary excitation---fluid phonon to the quasicrystals, and the equation of state is necessary and introduced too. By considering other two elementary excitations---phonons and phasons, a theory---generalized hydrodynamics for soft-matter quasicrystals is set up, in which the governing equations of the dynamics are reported in this letter.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

3 extracted references · 3 canonical work pages

[2]

Takano et al., A mesoscopic Archimedean tiling having a new complexity in an ABC star polymer, J Polym Sci Pol Phys, 2005, 43(18), 2427-2432

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work page 2005
[4]

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[18]

Ann Phys (NY), 1980, 125(1), 67-97

Dzyaloshinskii I E, V olovick G E, Poisson brackets in condensed matter physics. Ann Phys (NY), 1980, 125(1), 67-97. [19]Lubensky T C, Ramaswamy S and T oner J, Hydrodynamics of icosahedral quasicrystals, Phys Rev B, 1985, 32, 7444-7452. [20]Fan T Y , Poisson bracket method and it s applications to qu asicrystals, liquid crystals and a kind of soft matter...

work page 1980

Pith tools

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

[1] [2]

Takano et al., A mesoscopic Archimedean tiling having a new complexity in an ABC star polymer, J Polym Sci Pol Phys, 2005, 43(18), 2427-2432

A. Takano et al., A mesoscopic Archimedean tiling having a new complexity in an ABC star polymer, J Polym Sci Pol Phys, 2005, 43(18), 2427-2432. [3]Hayashida K, Dotera T, Takano A, a nd Matsushita Y , Polymeric quasicrystal: mesoscopic quasicrystalline tiling in A B C Star Polymers, Phys. Rev.Lett., 2007,98 195502. 12

work page 2005

[2] [4]

Talapin V D, Shevechenko E V , Bodnarchuk M I, Ye X C, Chen J and Murray C B, Quasicrystalline order in self-assemble d binary nanoparticle superlattices, Nature, 2009, 461, 964-9671. [5]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 A...

work page 2009

[3] [18]

Ann Phys (NY), 1980, 125(1), 67-97

Dzyaloshinskii I E, V olovick G E, Poisson brackets in condensed matter physics. Ann Phys (NY), 1980, 125(1), 67-97. [19]Lubensky T C, Ramaswamy S and T oner J, Hydrodynamics of icosahedral quasicrystals, Phys Rev B, 1985, 32, 7444-7452. [20]Fan T Y , Poisson bracket method and it s applications to qu asicrystals, liquid crystals and a kind of soft matter...

work page 1980