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Nonanalytic Landau functionals shaping the finite-size scaling of fluctuations and response functions in and out of equilibrium

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Pith's one-line read Nonanalytic terms in Landau functionals set the finite-size scaling of fluctuations and response functions at the critical point.

desk verdict A clean, honest derivation of finite-size scaling exponents for nonanalytic Landau functionals, with the caveat that the nonequilibrium example leans on an untested power-law spectral density assumption. read the letter →

arxiv 2502.06226 v2 pith:43FNIPN4 submitted 2025-02-10 cond-mat.stat-mech

Krzysztof Ptaszynski , Massimiliano Esposito This is my paper · ORCID
classification cond-mat.stat-mech
keywords Landautheorynonanalyticfunctionalfinite-sizescalinglargedeviationmolecularzipperCurie-Weissmodelphasetransitionmagneticsusceptibility
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

Landau theory usually assumes the free-energy functional is an analytic power series in the order parameter. This paper considers systems where the Landau functional (or, out of equilibrium, the quasipotential) contains a nonanalytic term—an odd or noninteger power of the absolute value of the order parameter—and shows that at a continuous transition this term controls how fluctuations and responses grow with system size. In the generalized molecular zipper, the variance of closed links scales as $\langle\Delta n^2\rangle\sim N^{2\alpha/(\alpha+1)}$; in the nonequilibrium Curie-Weiss model, the magnetization second moment scales as $\langle M^2\rangle\sim N^{2(1+\nu)/(2+\nu)}$ and the susceptibility obeys the same law. Because the scaling exponents are set by the nonanalytic exponent, finite-size measurements could reveal the presence and form of such terms, which otherwise show up only in the thermodynamic limit.

What carries the argument

The load-bearing object is the nonanalytic term in the Landau functional or quasipotential. In the zipper, the free energy density is $F(q)=(k_B T\ln g-\epsilon)q+\epsilon q^{\alpha+1}/(\alpha+1)-k_B T\ln g$, and at $T_c$ the linear term vanishes, leaving $\propto q^{\alpha+1}$; in the Curie-Weiss case the quasipotential expands as $V(m)=\frac12(1-T_c/T_1)m^2+B|m|^{2+\nu}+O(m^4)$. The argument works by approximating moments as Laplace-type integrals $\int m^k e^{-N V(m)}\,dm$, which at the critical point are dominated by the nonanalytic term; gamma-function prefactors then produce power laws in $N$ with exponents controlled by $\alpha$ or $\nu$.

What would settle it

For the nonequilibrium Curie-Weiss model with $\nu=1$, compute $\langle M^2\rangle$ at $T_1=T_c$ from the exact stationary probabilities (38) for $N$ up to $10^6$; Eq. (47) predicts a log-log slope of $4/3$, whereas the quartic-dominated scaling would give $3/4$. A measured slope of $3/4$ would falsify the claim that the nonanalytic term controls the finite-size scaling.

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Extended reading notes

Core claim

The paper's central claim is that when the Landau functional, or its nonequilibrium counterpart the quasipotential, contains a nonanalytic term—an odd or noninteger power of the absolute value of the order parameter—that term, rather than the usual analytic terms, fixes the finite-size scaling of fluctuations and response functions at a continuous transition. In the generalized molecular zipper the variance of closed links obeys $\langle\Delta n^2\rangle\sim N^{2\alpha/(\alpha+1)}$, and in the nonequilibrium Curie-Weiss model the magnetization second moment obeys $\langle M^2\rangle\sim N^{2(1+\nu)/(2+\nu)}$, with the magnetic susceptibility asymptotically equal to $\langle M^2\rangle/(k_B T_c)$. The same nonanalytic exponent also sets the order of the transition and the asymptotic Binder cumulant.

Load-bearing premise

The predicted exponents rest on the two baths coupling to the spins through exact power-law functions of frequency at low frequencies, with different exponents; if a real bath's low-frequency behavior departs from that, the nonanalytic term $B|m|^{2+\nu}$ can change or vanish, and with it the finite-size scaling.

Editorial extensions

If this is right

  • For the molecular zipper, measuring the variance of closed links at $T=T_c$ as a function of $N$ gives a log-log slope $2\alpha/(\alpha+1)$, from which the nonanalytic exponent $\alpha$—and hence the order of the transition—can be read off.
  • For the nonequilibrium Curie-Weiss model, the magnetization second moment and the magnetic susceptibility share the scaling exponent $2(1+\nu)/(2+\nu)$, so a finite-size measurement of either quantity probes the nonanalytic bath-induced term $B|m|^{2+\nu}$.
  • The Binder cumulant at criticality approaches a value that depends only on $\nu$, not on $N$ or on the amplitude $B$, giving a parameter-free finite-size fingerprint of the nonanalytic term.
  • The equilibrium-like relation $\chi\approx \langle M^2\rangle/(k_B T_c)$ holds asymptotically even out of equilibrium, so the susceptibility inherits the nonanalytic scaling of the magnetization fluctuations.

Reading between the lines

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

  • Extension: the same Laplace-type argument should produce a variance exponent $2\alpha/(\alpha+1)$ in any equilibrium model whose Landau functional has a leading nonanalytic term $c|q|^{1+\alpha}$ at criticality, so the prediction could be tested by exact enumeration in other solvable lattice models.
  • Extension: if the mechanism survives in finite-dimensional systems, the asymptotic ratio $\chi T_c/\langle M^2\rangle\to 1$ could serve as a nonequilibrium fluctuation-response test even where the scaling exponents are renormalized; the paper leaves this open.
  • Extension: the zipper's nonanalyticity arises at a boundary of the order parameter domain ($q\ge 0$) while the Curie-Weiss term arises from broken symmetry, yet both produce the same type of scaling law, suggesting the controlling feature is the shape of the functional near its minimum rather than the symmetry.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies how nonanalytic terms in Landau functionals (or nonequilibrium quasipotentials) affect the finite-size scaling of fluctuations and response functions at continuous phase transitions. Two analytically tractable mean-field-type models are analyzed. For a generalized molecular zipper, the free-energy density contains a term proportional to q^{α+1} (Eq. 11); at T = T_c the linear term vanishes, and Laplace-type asymptotics yield the variance scaling ⟨Δn²⟩ ∼ N^{2α/(α+1)} (Eq. 19), an energy-variance formula (Eq. 26), and exact numerical agreement. For a nonequilibrium Curie–Weiss model coupled to two baths with power-law spectral densities, the quasipotential has a nonanalytic contribution B|m|^{2+ν} (Eqs. 39–40), leading to magnetization moments scaling as N^{k(1+ν)/(2+ν)} (Eq. 47), a ν-dependent Binder cumulant (Eq. 49), and an asymptotic equilibrium-like susceptibility relation χ ≈ ⟨M²⟩/(k_B T_c) (Eq. 56). These predictions are compared with exact solutions of the stationary master equation, with good agreement for the scaling exponents and deviations of order ten percent attributed to higher-order terms in the quasipotential.

Significance. If the results hold, they provide a concrete, parameter-free way to detect nonanalytic Landau terms from finite-size scaling of fluctuations and response functions: the predicted exponents differ from the standard quartic-dominated values, and the Binder-cumulant formula (Eq. 49) is a sharp, falsifiable prediction depending only on ν. The derivations use no fitted parameters; the coefficient B is expressed in model parameters (Eq. 40), and the numerical checks against exact finite-size solutions are convincing. The paper is self-contained, clearly written, and ships Wolfram Mathematica notebooks and data at a public DOI, which strengthens reproducibility. The main limitation—that the out-of-equilibrium example is conditional on the power-law spectral-density assumption of Eq. (34)—is a scope issue rather than an internal inconsistency, because the paper's claim is that nonanalytic terms can shape finite-size scaling, not that they do so for every bath.

minor comments (4)
  1. [II.B, Eq. (19)] The prefactor in Eq. (19), and similarly the prefactors in Eq. (27), contain ln 2, but the model is defined with a general degeneracy g and T_c = ϵ/(k_B ln g). As written, the formulas are valid only for g = 2. Please replace ln 2 by ln g, or state explicitly that all numerical examples in Sec. II take g = 2. The scaling exponents are unaffected, so this is a consistency and reproducibility issue rather than a correctness issue.
  2. [III.F, Eq. (55)] The notation O(|m|^{2+ν}) in Eq. (55) is ambiguous: the leading term −m/(k_B T_c) is an odd function of m, whereas |m|^{2+ν} is even. The correction term must be odd in m to respect the symmetry ∂_h V(m,0) = −∂_h V(−m,0). Please write the correction with an explicit odd prefactor (for example, −C sign(m)|m|^{2+ν} + …) so the symmetry of the expansion is clear.
  3. [III.A and Conclusions] The out-of-equilibrium demonstration is conditional on the power-law spectral densities in Eq. (34). The statement in Sec. III.A that only low-frequency power-law behavior is needed is helpful, but the paper would benefit from one additional sentence clarifying that for baths with a different low-frequency form, e.g., an analytic spectral density with a finite cutoff, the |m|^{2+ν} term may be absent and the magnetization variance would revert to the quartic-dominated N^{3/2} scaling. This is a scope clarification rather than a request for new numerical calculations.
  4. [General] There are a few presentation issues: the phrase "ofjth order" appears before Eq. (43) (missing space), and the captions of Figs. 3 and 4 do not list the values of g and α used in the calculations. Please add the parameter values to the captions or the main text for full reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: finite-size scaling formulas are derived from explicitly stated model Landau functionals and confirmed by exact numerical solutions.

full rationale

The paper's central derivations are asymptotic (Laplace/saddle-point) evaluations of fluctuation moments using Landau-type functionals that are constructed explicitly from the models. For the molecular zipper, the free energy density F(q) in Eq. (11) is derived from the model's energy and entropy, and at T=Tc the linear term vanishes, leaving the nonanalytic q^{alpha+1} term; Eq. (19) follows by direct integration and is independently confirmed by exact finite-size calculations in Fig. 3. Nothing is fitted to the variance data. For the nonequilibrium Curie-Weiss model, the quasipotential V(m) is obtained from the exact detailed-balance product expression Eq. (38), which is derived in the paper; the expansion Eq. (39) is attributed to Ref. [3] (Aron and Chamon), an external source, and the paper explicitly states that the authors verified the quadratic and nonanalytic terms against their exact formula. The coefficient B in Eq. (40) is a function of the model parameters (temperatures, coupling strengths, and spectral exponents), not a fit to the magnetization fluctuations. Eq. (47) is then a direct consequence of replacing V(m) by B|m|^{2+nu} at T1=Tc, where the quadratic coefficient vanishes. Exact numerical solutions of the master equation (Figs. 6 and 7) provide independent support for the scaling prediction. The power-law spectral density in Eq. (34) is a transparent modeling assumption inherited from Ref. [3], not a hidden fit to the predicted observables; the paper states explicitly that only low-frequency power-law behavior is needed. Self-citations (Refs. [20], [25], [35], [36]) are used for background, methodology, or alternative derivations, and none is load-bearing for the main claim. No uniqueness theorem or ansatz is smuggled in via self-citation, and no predicted quantity is equivalent to an input by construction. The derivation chain is therefore self-contained and non-circular.

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

No parameters are fitted to data. All coefficients in the scaling formulas are determined by the model parameters (epsilon, g, alpha, J, T1, T2, gamma_i, alpha_i). The central derivations rest on the large deviation form of the stationary distribution, which is exact or model-defined, and on the assumed power-law spectral densities for the Curie-Weiss example.

assumptions (3)
  • domain assumption Bath spectral densities obey a power-law form C_i(beta_i, omega) proportional to |omega|^{alpha_i} over the relevant low-frequency range.
    Defined in Eq. (34), inherited from Ref. [3]. It generates the nonanalytic term B|m|^{2+nu} in Eq. (39), which is the basis of the Curie-Weiss finite-size scaling predictions.
  • domain assumption At the critical point, the leading nonanalytic term in the quasipotential dominates the Laplace integrals that determine the scaling of moments.
    Used in Section III D to replace V(m) by B|m|^{2+nu}. The neglect of quartic and higher-order terms is justified for nu<2 and large N; numerical comparisons show it is accurate for the leading exponent.
  • standard math The stationary distribution of the one-dimensional magnetization chain is given by the zero-current product formula.
    Eq. (38) follows from the birth-death structure of the master equation with reflecting boundaries; this is an exact result for one-dimensional Markov chains, not an approximation.

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Cite this review

Pith. "Pith review of Nonanalytic Landau functionals shaping the finite-size scaling of fluctuations and response functions in and out of equilibrium." pith.science (2026). https://pith.science/paper/43FNIPN4

@misc{pith2026250206226,
  author       = {Pith},
  title        = {Pith review of: Nonanalytic Landau functionals shaping the finite-size scaling of fluctuations and response functions in and out of equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/43FNIPN4}},
  note         = {Machine review of arXiv:2502.06226}
}
read the original abstract

Landau theory relates phase transitions to the minimization of the Landau functional (e.g., free energy functional), which is expressed as a power series of the order parameter. It has been shown that the critical behavior of certain physical systems can be described using Landau functionals that include nonanalytic terms, corresponding to odd or even noninteger powers of the absolute value of the order parameter. In particular, these nonanalytic terms can determine the order of the phase transition and the values of the critical exponents. Here, we show that such terms can also shape the finite-size scaling behavior of fluctuations of observables (e.g., of energy or magnetization) or the response functions (e.g., heat capacity or magnetic susceptibility) at the continuous phase transition point. We demonstrate this on two examples, the equilibrium molecular zipper and the nonequilibrium version of the Curie--Weiss model.

Figures

Figures reproduced from arXiv: 2502.06226 by the authors.

Figure 1
Figure 1. FIG. 1. Scheme of the molecular zipper with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Temperature-dependence of the order parameter [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The energy variance at the critical point [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. The normalized stationary magnetization as a func [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The Binder cumulant at the critical point [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) The second moment of magnetization [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The ratio of the magnetic susceptibility at the critical [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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Forward citations

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