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Deriving phase field crystal theory from dynamical density functional theory: consequences of the approximations

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Pith's one-line read Standard phase field crystal theory cannot be derived from density functional dynamics without adding spurious states.

desk verdict This is the first honest derivation audit of PFC from DDFT that follows the approximations through to their phase-diagram consequences, and the two main artifacts it identifies are real. read the letter →

arxiv 1908.02537 v1 pith:T3YLVNBP submitted 2019-08-07 cond-mat.soft cond-mat.stat-mechnlin.PS

Andrew J. Archer , Daniel J. Ratliff , Alastair M. Rucklidge , Priya Subramanian This is my paper · ORCID
classification cond-mat.softcond-mat.stat-mechnlin.PS MSC 82C2682D1535Q82
keywords phasefieldcrystaldynamicaldensityfunctionaltheoryGEM-4fluidspinodalstripegradientexpansionlogarithmicidealgastermone-modeapproximation
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 follows every approximation in the standard derivation of phase field crystal (PFC) theory from dynamical density functional theory (DDFT), and shows where each step changes the physics. Its central conclusion is that the PFC model is not a faithful reduction of the accurate DDFT: dropping the $\nabla\cdot[n\nabla L n]$ term and Taylor-expanding the ideal-gas logarithm introduces a second spinodal, so the liquid freezes and then melts again, and makes stripe and down-hexagon phases the equilibrium structures. Replacing the nonlocal operator $L$ by a gradient expansion instead creates a region of the phase diagram where no solution exists above a certain average density. These artefacts are demonstrated for the two-dimensional GEM-4 fluid, where the starting DDFT is known to be accurate.

What carries the argument

The load-bearing object is the linear operator $L$, defined by $L n(x)=-n(x)+\rho_0\int c^{(2)}(x,x_2)n(x_2)\,dx_2$, whose Fourier eigenvalues $\sigma(k)$ control the linear stability of the liquid; in DDFT it is a nonlocal convolution with the pair direct correlation function. The paper tracks the fate of the term $\nabla\cdot[n\nabla L n]$ in the dynamics, which is linked to the density-dependent mobility and the logarithmic ideal-gas term. The PFC derivation replaces $L$ by the local gradient expansion $L_{\mathrm{grad}}n = r n - \gamma(1+\nabla^2)^2 n$ and replaces $\log(1+n)$ by its truncated Taylor polynomial; the number of real roots of the polynomial versus the one root of the logarithm is the mechanism behind the spurious second spinodal. The paper also constructs a one-mode ansatz for $\varphi=\log(\rho/\rho_0)$, using the decay of the convolution's Fourier coefficients at $|k|\ge 2$ to show why a few modes in $\varphi$ capture sharply peaked density profiles.

What would settle it

Simulate or experimentally measure two-dimensional GEM-4 at temperatures $k_BT/\epsilon > 0.1$ across the density range where PFC predicts stripe, down-hexagon, and second-liquid states; if any of those appear as equilibrium phases, the claim that they are artefacts is wrong.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the standard PFC approximations are not harmless when applied to a microscopically accurate DDFT. The term $\nabla\cdot[n\nabla L n]$, usually discarded as higher order, contributes at the same order as retained terms and helps stabilise the crystal relative to stripes. Dropping it forces a constant mobility and a polynomial replacement for $\log(1+n)$; the polynomial has two roots where the logarithm has one, which is exactly the origin of the second spinodal and the erroneous stripe and down-hexagon equilibria. Replacing the convolution $L$ by the gradient operator $L_{\mathrm{grad}}$ makes the density minimum tend to zero at finite chemical potential, with $\log(1+n)\to -\infty$ balanced by $\gamma n_{xxxx}\to +\infty$, so branches of solutions terminate. The paper therefore claims that PFC models are successful as phenomenological models but cannot be derived as accurate microscopic approximations for the density profile.

Load-bearing premise

The random-phase approximation for the free energy is an accurate description of the GEM-4 fluid in the regime studied, so the extra phases predicted by PFC are artefacts rather than states the baseline theory already misses.

Editorial extensions

If this is right

  • PFC phase diagrams for crystallising soft matter contain a second liquid spinodal, a stripe phase, and down-hexagons that are not present in the accurate DDFT, so quantitative predictions of crystal thermodynamics away from the coexistence region should not be trusted.
  • The gradient-expanded DDFT (DDFT-5) has a finite limit of validity: above a certain average density there is no smooth equilibrium profile, because the density touches zero and no solution exists.
  • Agreement between DDFT and PFC is quantitative only for small-amplitude states close to the spinodal; beyond that, agreement is at best qualitative.
  • A one-mode approximation for $\log \rho(x)$ reproduces the full DDFT-3 stripe and hexagon branches almost exactly, even when the density varies by orders of magnitude.
  • Extending the gradient expansion to higher order (EOF) improves the liquid compressibility but does not remove the no-solution singularity or the second spinodal; it only delays them.

Reading between the lines

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

  • If the one-mode accuracy for $\log\rho$ carries to other potentials, a practical route to faithful PFC-type theories would be to write the free energy in terms of $\varphi=\log(\rho/\rho_0)$ and discard high-wavenumber convolution coefficients, rather than expanding in $n$.
  • The same mechanism, a polynomial replacing a logarithm with one root, implies that binary PFC models derived by the same route inherit analogous spurious re-entrant melting and lamellar phases; the paper notes that the $\nabla\cdot[n\nabla L n]$ term is also dropped in binary derivations.
  • The second-spinodal artefact should appear generically for any even-order truncation of the logarithm, since the relevant equation keeps $n=0$ and $n=1$ as roots; this could be checked numerically with higher-order truncations of a model free energy.
  • For systems where the pair direct correlation function is not well described by the random-phase approximation, the quantitative size of the artefacts may differ, but the structural argument about roots and the gradient expansion should persist.
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Editorial analysis

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

Referee Report

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Summary. The paper systematically derives phase field crystal (PFC) theory from dynamical density functional theory (DDFT), identifying and naming each approximation in the chain: truncation of the functional Taylor expansion of the excess free energy, the Ramakrishnan–Yussouff/RPA treatment, replacement of the nonlocal convolution operator L by a gradient expansion, and finally the suite of changes that converts a DDFT into a PFC model (dropping the ∇·[n∇Ln] term, assuming constant mobility, and replacing the ideal-gas logarithm by a truncated polynomial). The authors construct a hierarchy of models DDFT-0 through DDFT-5 and PFC-β through PFC-ϵ, giving the free energy, chemical potential, and dynamics for each. They illustrate the consequences for the two-dimensional GEM-4 fluid, for which RPA-based DDFT is known to be accurate. Two central artifacts are documented: (i) expanding the logarithm and dropping ∇·[n∇Ln] introduces a second spinodal, producing erroneous stripe and down-hexagon phases and a re-entrant uniform liquid; (ii) making the gradient expansion while retaining the logarithm leads the density to approach zero at isolated points, with the logarithmic singularity balanced by a divergent fourth derivative, after which no real solutions exist.

Significance. This is a significant and carefully executed negative result for the PFC literature. The claim that standard polynomial-in-density PFC models cannot be derived as quantitatively faithful reductions of DDFT is supported by two arguments that are largely independent of the specific pair potential: the second spinodal follows from the algebraic structure of the truncated logarithm (Sec. IV.A), and the no-solution region follows from a local balance between a logarithmic singularity and a fourth derivative (Sec. IV.B and Fig. 8). The numerical work is reproducible: the continuation method is fully described in Appendix B, the singularity is verified at the grid level in Fig. 8(b), and a sample Matlab code is provided as supplementary material. The authors are appropriately careful in scoping their claims to the considered approximation sequence and to the GEM-4 model, and they explicitly flag the reliance on literature support for RPA accuracy in Sec. III. The constructive one-mode log-density result in Sec. V is an important positive step, as it indicates a principled route toward accurate PFC-type descriptions.

minor comments (5)
  1. [Abstract, Sec. VI] The sentence 'we find it impossible to derive the PFC model as an accurate approximation to DDFT' is stronger than what is demonstrated, since the demonstration concerns the standard polynomial-in-density PFC models and the GEM-4 system; consider adding the qualifier 'for the density field n(x) within the approximation sequence considered here' in the abstract to prevent overgeneralization.
  2. [Sec. IV.A] The phrase 'which involves the second derivative of Eq. (51) with respect to n' is imprecise: the spinodal condition (75) involves the derivative of the chemical potential, which after cancellation of the quadratic term is the second derivative of the log expansion minus a constant; rephrasing would improve clarity.
  3. [Eq. (76)] The notation Lgrad-8 in Eq. (76) and in the surrounding text is slightly awkward because of the hyphen and the use of the same subscript 'grad'; consider a cleaner notation such as L_8 or L_grad^(8) for readability.
  4. [Sec. III, Fig. 5] In the caption of Fig. 5, the circled region of quantitative agreement between DDFT-3 and PFC-γ is mentioned but not visually obvious in the small panels; adding a zoomed inset or a more prominent marker would help the reader locate this claim.
  5. [Sec. III, Eq. (70)] The observation that equilibria with the same µ do not necessarily have the same mean density ¯n is important for interpreting the phase diagrams in Fig. 7; a brief additional remark on how the phase diagrams are constructed from the grand-potential comparison would be helpful for readers unfamiliar with this convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PFC artifact analysis is derived from the stated approximations and checked against an externally supported DDFT baseline.

full rationale

The paper's central claim is negative: the standard PFC approximations to DDFT introduce spurious phases and a no-solution region. This claim is not an input to the derivation. The two headline artifacts are obtained by direct algebra from the model equations: the second PFC spinodal follows because the linearized PFC equation (75) with sigma(k)=0 has roots n_liq=0 and n_liq=1 for any even truncation of log(1+n), independently of the pair potential (Sec. IV.A); the DDFT-5 no-solution region follows from balancing log(1+n) against a fourth-derivative term in Eq. (77), with the balance explicitly verified at grid points in Fig. 8 (Sec. IV.B). The comparison between DDFT-3 and the PFC models involves no parameter fitting to the target phase diagram: the GEM-4 parameters are fixed by the linear dispersion relation sigma(1)=0 and the curvature at k=1 (Appendix A, Table II), and phase stability is computed from the grand potential (70), not imposed. The one-mode log-density approximation in Sec. V is derived by projecting Eq. (79) onto k=0 and k=1 modes and is then tested against the full numerical solution; it is not a fit to that solution. The only soft spot is the accuracy of the RPA/DDFT-3 baseline, which the paper explicitly scopes in Sec. III: 'A full assessment of the validity of the RY/RPA approximation... is beyond the scope of the present study' and 'there are examples in the literature where this approximation is reliable and others where it works badly.' For GEM-4 the RPA accuracy is supported by independent references [42,44]; the presence of a self-citation [45] in this list is not load-bearing because the central artifact arguments (the second spinodal and the singularity) do not depend on RPA accuracy. No equation in the paper reduces by construction to its own input, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The derivations rely on standard DDFT and DFT machinery plus the RPA ansatz for GEM-4. The only hand-set constants are the PFC gradient-expansion coefficient γ (fitted to the DDFT dispersion relation) and the spinodal reference parameters; no new physical entities are postulated. The central negative result is not obtained by fitting PFC parameters to DDFT outcomes.

free parameters (4)
  • γ (gradient expansion coefficient) = 4.3692 (2D GEM-4)
    Fitted to the second derivative d²σ/dk² at k=1 of the GEM-4 dispersion relation, defining Lgrad = -γ(1+∇²)². This local fit is the standard PFC choice; the no-solution result is demonstrated for this representative fit and is deferred, not eliminated, by higher-order fits (EOF).
  • R (GEM-4 range) = 5.0962 (2D)
    Chosen, together with ρ0βϵ, so that σ(1)=0 and dσ/dk(1)=0, placing the reference liquid at the linear stability threshold. This fixes the length scale of the model system, a setup choice rather than a parameter fitted to the target conclusion.
  • ρ0βϵ (GEM-4 coupling) = 0.2455 (2D)
    Same spinodal condition as R; sets the reference density and inverse temperature product. All subsequent phase diagrams are mapped from this reference.
  • EB (EOF coefficient) = 14.383 (2D)
    Coefficient in the eighth-order fitting operator Lgrad-8 of Ref. [9], used in the discussion to show that higher-order gradient terms defer but do not remove the singularity. Not central to the main quantitative claims.
assumptions (5)
  • domain assumption DDFT-0, Eq. (5), with mobility M(ρ)=Dρ, correctly describes the slow dynamics of the Brownian fluid.
    Starting point of the derivation; the paper does not test this equation and accepts the standard DDFT framework.
  • domain assumption The excess free energy Fex can be represented by a functional Taylor expansion truncated at O(c(4)).
    Used to obtain DDFT-1 (Eqs. 14, 25). This truncation is standard in PFC derivations, but its validity for GEM-4 at the studied state points is assumed.
  • domain assumption The RPA/RY approximation (c(3)=c(4)=0) is accurate for GEM-4 in the regime studied (kBT/ϵ > 0.1).
    Necessary so that DDFT-3 is a reliable benchmark; the paper cites Refs. [42,44,45] but does not re-establish accuracy. This is the main load-bearing external input.
  • standard math Equilibrium phases are identified by the minimum of the specific grand potential Ω/A at fixed chemical potential.
    Used to construct the phase diagrams in Fig. 7; standard in density functional theory.
  • domain assumption The periodic solution branches of interest persist under parameter continuation and can be resolved with the stated grid sizes.
    The numerical method assumes smooth branches; the paper checks residuals at grid points for the singular cases, supporting this assumption for the presented results.

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Pith. "Pith review of Deriving phase field crystal theory from dynamical density functional theory: consequences of the approximations." pith.science (2026). https://pith.science/paper/T3YLVNBP

@misc{pith2026190802537,
  author       = {Pith},
  title        = {Pith review of: Deriving phase field crystal theory from dynamical density functional theory: consequences of the approximations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3YLVNBP}},
  note         = {Machine review of arXiv:1908.02537}
}
abstract

Phase field crystal (PFC) theory, extensively used for modelling the structure of solids, can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Standard derivations neglect a term of form $\nabla\cdot[n\nabla L n]$, where $n$ is the scaled density profile and $L$ is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in $n$. However, the consequences of doing this are the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and other periodic structures are erroneously predicted to be thermodynamic equilibria. A second approximation is to replace $L$ by a gradient expansion. This leads to the possibility of solutions failing to exist above a certain value of the average density. We illustrate these conclusions with a simple model two-dimensional fluid. The consequences of the PFC approximations are that the phase diagram is both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal. Thus, although PFC models are successful as phenomenological models of crystallisation, we find it impossible to derive the PFC model as an accurate approximation to DDFT, without introducing spurious artefacts. However, making a simple one-mode approximation for the logarithm of the density distribution is surprisingly accurate, which gives a tantalising hint that accurate PFC-type theories may instead be derived for the field $\log(\rho(x))$, rather than for the density profile itself.

Figures

Figures reproduced from arXiv: 1908.02537 by the authors.

Figure 1
Figure 1. Illustrative example of the growth rate k 2σ(k) as a function of wavenumber k. Small amplitude modes with k 2σ(k) < 0 decay exponentially in time, while those with k 2σ(k) > 0 grow exponentially. Throughout we scale lengths so that the maximum growth rate occurs at k = 1. The non-local operator L is most conveniently considered in terms of its Fourier transform, or equivalently, in terms of how it acts on modes of t… view at source ↗
Figure 2
Figure 2. The eigenvalue σ(k) of L plotted as a function of wavenumber k for the GEM-4 potential (solid line) for R = 5.0962 and ρ0β = 0.2455, which is at the threshold where the system becomes linearly unstable. This has σ(0) = −18.75. We also display σ(k) from the gradient expansion of L (dashed line), i.e., a Taylor expansion in Fourier space around k = 1, which is the PFC relation for Lgrad, σ(k) = −γ(1−k 2 ) 2 , with γ … view at source ↗
Figure 3
Figure 3. (a) Liquid density (1 + nliq) and (b) specific grand potential Ωliq/A as a function of the scaled chemical potential µ, for DDFT-3 (solid black line), DDFT-5 (dashed black line), PFC-γ (indistinguishable from DDFT-3) and PFC- (dashed magenta line). equilibria might result from initial conditions via the dy￾namics. The solution corresponding to the uniform density liq￾uid state with n(x) = nliq can readily be found.… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Examples of solutions of DDFT-3 (63) with [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Data summarizing the nature of the stripe and the two hexagonal solutions in the four cases: (a–c) DDFT-3, [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Specific Ω relative to the value for the liquid at the same value of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: The phase diagram for the GEM-4 model as predicted by (a) DDFT-3, (b) DDFT-5, (c) PFC- [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: (a) DDFT-5 stripe density profile 1 + n, its logarithm and other terms in (67), for µ = 3.3688, with 1 + n(xmin) ≈ 5 × 10−7 . The resolution was Nx = 2048 grid points in a domain with one wavelength. (b) Detail around the location of the density minimum at xmin, showin…
Figure 9
Figure 9. Figure 9: Full numerical solutions of DDFT-3 plotted together with the one-mode approximation. (a) Stripes in 1D DDFT-3, [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

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