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arxiv: 2605.22520 · v1 · pith:J3SWGV2Vnew · submitted 2026-05-21 · ❄️ cond-mat.stat-mech

The cell fluid model with Curie-Weiss interactions: special cases and analytical results

O. A. Dobush , M. P. Kozlovskii , I. V. Pylyuk , R. V. Romanik , M. A. Shpot This is my paper

Pith reviewed 2026-05-22 03:54 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords cell fluid modelCurie-Weiss interactionsvan der Waals lattice gasequation of statecritical pointbinodal curvespinodal curvedeformed exponential
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The pith

In the strong-repulsion limit the cell fluid model reduces exactly to the van der Waals lattice gas with a Curie-Weiss order parameter.

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

The paper examines analytically tractable limits of a cell fluid model whose particles interact through competing global attraction of strength J1 and local repulsion of strength J2. In the regime J2 much larger than J1 the authors obtain closed-form expressions for the critical parameters, the full equation of state, and the binodal and spinodal curves. These expressions coincide with those of the van der Waals lattice gas, and the order parameter obeys the standard mean-field Curie-Weiss equation. The same limiting procedure also yields a Landau expansion whose form and symmetry match the classical mean-field lattice gas. The work further extends the validity of the underlying asymptotic expansion of the deformed exponential to the marginal line J1 equal to J2, thereby legitimizing the ideal-gas limit.

Core claim

For J2 ≫ J1 explicit expressions are derived for the critical point parameters, the equation of state, and the binodal and spinodal curves. The equation of state is found to be in full agreement with that of the van der Waals lattice gas, and the order parameter satisfies the standard Curie-Weiss equation. In a neighborhood of the critical point a Landau expansion has the same form and symmetry as that of the classical lattice gas within the mean-field approximation. The asymptotic expansion of the deformed exponential function is extended to the marginal thermodynamic-stability case J1 = J2, rendering the ideal-gas limit formally legitimate.

What carries the argument

The asymptotic expansion of the deformed exponential function that governs the cell model's grand potential and thereby produces the explicit reduction to van der Waals thermodynamics when J2 greatly exceeds J1.

If this is right

  • The equation of state coincides exactly with the van der Waals lattice-gas equation when J2 ≫ J1.
  • The order parameter satisfies the standard Curie-Weiss mean-field equation.
  • Binodal and spinodal curves are available in closed form.
  • The Landau expansion near the critical point has identical form and symmetry to the mean-field lattice gas.

Where Pith is reading between the lines

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

  • The closed-form expressions supply exact benchmarks that numerical simulations of competing-interaction fluids can be tested against.
  • The extension to J1 = J2 indicates that the model's thermodynamic description remains internally consistent even when strict repulsion dominance is relaxed.
  • Analogous reductions to van der Waals behavior may appear in other microscopic models that combine long-range attraction with short-range repulsion.

Load-bearing premise

The asymptotic expansion of the deformed exponential function remains valid when extended to the marginal case of equal coupling constants J1 equal to J2.

What would settle it

Compute the equation of state numerically for a sequence of increasing J2/J1 ratios and test whether the results converge to the closed-form van der Waals expression; systematic deviation at large ratios would falsify the claimed reduction.

Figures

Figures reproduced from arXiv: 2605.22520 by I. V. Pylyuk, M. A. Shpot, M. P. Kozlovskii, O. A. Dobush, R. V. Romanik.

Figure 1
Figure 1. Figure 1: Phase diagrams of the cell fluid model with Curie-Weiss-type interaction potential ( [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The pP ˚ , n¯q phase diagram of the cell fluid model with equal attraction and repulsion parameters, J1 “ J2, showing pressure isotherms (solid lines) at ten values of the reduced temperature T listed in the legend. The first two coexistence lines (binodals) related to the three lowest-density phases are indicated by dotted curves. Their apex points correspond to the critical points labelled by k “ 1 and k… view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the function Epµc; yq from (59) (thin solid curves) and Epyq at µ0 “ µc from (28) (dots) at seven values of the parameter p listed in the legend. For p ă pc “ 4, there is a simple single maximum. At p “ pc, the maximum becomes flat, corresponding to the critical point, with its height Ecpycq “ ln 2´1{2 from (58). For p ą pc, the central extremum smoothly becomes a minimum and two new symmetric max… view at source ↗
Figure 4
Figure 4. Figure 4: Plots of the function Eˆpµc; sq from (71) at the same values of p as in [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The function Epyq from (49) at p “ 4.75 and f “ 10, for seven values of µ growing from 4.468 to 4.53. Gray curves show the full family, while selected ones are highlighted for µ “ µ1 » 4.478 and µ “ µ2 » 4.522 (black curves with inflection points marked by blue circles), and µ “ µc “ 4.5 (dark-green curve, with its two equal-height maxima connected by the green dashed horizontal line). The thick blue curve… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of analytical results for the CFM in the strong-repulsion limit from [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
read the original abstract

Inspired by previous extensive numerical studies of a cell fluid model with Curie-Weiss interactions, we concentrate on some analytically tractable special cases in its description. The key ingredient of the model is a competition between global attraction and local repulsion interactions between particles with coupling constants $J_1$ and $J_2$, respectively. We provide analytical results in several limiting cases, including the ideal-gas limit $J_1=J_2=0$ and the strong-repulsion limit $J_2\gg J_1$. For $J_2\gg J_1$, a detailed analytical study is presented. We derive explicit expressions for the critical point parameters, the equation of state, and the binodal and spinodal curves in closed form. The equation of state is found to be in full agreement with that of the van der Waals lattice gas, and the order parameter satisfies the standard Curie-Weiss equation. In a neighborhood of the critical point, a Landau expansion is shown to have the same form and symmetry as that of the classical lattice gas within the mean-field approximation. Moreover, based on the explicit knowledge of a few leading terms in the asymptotic expansion of the deformed exponential function governing the physics of the cell model, we extend its validity range to include the marginal case of thermodynamic stability, $J_1=J_2$. In particular, this extension makes a consideration of the ideal-gas limit $J_1=J_2=0$ formally legitimate. For the generic marginal case $J_1=J_2\ne0$ systematically avoided in previous works, we present numerical data and phase diagrams that augment their findings for $J_2>J_1$.

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

1 major / 1 minor

Summary. The manuscript analyzes special cases of a cell fluid model with competing global attraction (J1) and local repulsion (J2) interactions. For the limit J2 ≫ J1 it derives explicit closed-form expressions for the critical-point parameters, the equation of state (reported to agree fully with the van der Waals lattice gas), the binodal and spinodal curves, and shows that the order parameter obeys the standard Curie-Weiss equation. A Landau expansion near the critical point is shown to match the classical lattice-gas mean-field form. The authors further extend the leading terms of the asymptotic expansion of the deformed exponential to the marginal-stability line J1 = J2, thereby including the ideal-gas limit J1 = J2 = 0, and supply numerical phase diagrams for the generic marginal case J1 = J2 ≠ 0.

Significance. If the derivations are controlled, the work supplies analytical benchmarks that confirm and extend previous numerical studies of the model. The explicit agreement with the van der Waals and Curie-Weiss mean-field results, together with the closed-form binodal/spinodal expressions, offers concrete reference points for the competition between attraction and repulsion. The extension to the marginal line J1 = J2, if justified, would also make the ideal-gas limit formally accessible within the same framework.

major comments (1)
  1. [asymptotic extension to marginal stability (J1 = J2)] The central analytical results for J2 ≫ J1 rest on the cell-model partition function expressed via the deformed exponential. To reach the marginal line J1 = J2 (and thereby the ideal-gas limit), the manuscript invokes a truncation to a few leading asymptotic terms. No explicit remainder estimate or proof is supplied showing that omitted higher-order terms remain sub-dominant in the thermodynamic limit as J1 approaches J2 from above. If those terms contribute at the same order to the pressure or chemical potential, the claimed closed-form equation of state, binodal, and spinodal would acquire uncontrolled corrections precisely on the boundary the authors wish to reach. This issue is load-bearing for the legitimacy of the ideal-gas limit and for the numerical augmentation of the marginal case.
minor comments (1)
  1. [abstract] The abstract and introduction would benefit from a clearer separation between the fully analytical results obtained for J2 ≫ J1 and the numerical data presented for the marginal line J1 = J2.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the asymptotic extension. We respond to the major comment below.

read point-by-point responses

  1. Referee: [asymptotic extension to marginal stability (J1 = J2)] The central analytical results for J2 ≫ J1 rest on the cell-model partition function expressed via the deformed exponential. To reach the marginal line J1 = J2 (and thereby the ideal-gas limit), the manuscript invokes a truncation to a few leading asymptotic terms. No explicit remainder estimate or proof is supplied showing that omitted higher-order terms remain sub-dominant in the thermodynamic limit as J1 approaches J2 from above. If those terms contribute at the same order to the pressure or chemical potential, the claimed closed-form equation of state, binodal, and spinodal would acquire uncontrolled corrections precisely on the boundary the authors wish to reach. This issue is load-bearing for the legitimacy of the ideal-gas limit and for the numerical augmentation of the marginal case.

    Authors: We thank the referee for raising this point on the rigor of the truncation. The asymptotic expansion of the deformed exponential follows from its integral representation or recurrence relation, yielding successive corrections that are systematically smaller by powers of the inverse coupling. In the thermodynamic limit the free energy is obtained from the leading saddle-point contribution, so that higher-order terms in the expansion of the deformed exponential enter only as sub-extensive corrections that vanish upon taking the appropriate derivatives for pressure and chemical potential. We nevertheless acknowledge that an explicit uniform remainder bound as J1 approaches J2 from above is not supplied in the present text. In the revised version we will insert a short paragraph that recalls the origin of the expansion, states the expected suppression of the remainder, and adds a direct numerical comparison of the truncated analytic expressions against the full (numerically evaluated) partition function for several values of J1/J2 approaching unity. This will furnish concrete evidence supporting the extension to the marginal line and the ideal-gas limit. revision: partial

Circularity Check

0 steps flagged

No significant circularity in analytical derivations

full rationale

The paper derives closed-form expressions for critical point parameters, equation of state, binodal and spinodal curves in the J2 ≫ J1 limit directly from the cell-model partition function and deformed exponential. These match van der Waals and Curie-Weiss forms via explicit algebra rather than fitting, renaming, or self-referential definition. The extension of the asymptotic expansion to the marginal J1 = J2 case is presented as an approximation using known leading terms to legitimize the ideal-gas limit; this does not force the main results by construction. Prior numerical studies are referenced for inspiration but are not load-bearing for the new analytical outputs. No self-definitional loops, fitted predictions, or uniqueness theorems imported from self-citations reduce the claims to their inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The work rests on the pre-existing cell fluid model with Curie-Weiss interactions and invokes an ad-hoc extension of the deformed exponential's asymptotic series to the marginal stability regime; no new entities are postulated and the only free parameters are the model couplings J1 and J2.

free parameters (1)
  • J1 and J2
    Coupling constants for global attraction and local repulsion; treated as given inputs that define the special cases studied.
axioms (2)
  • domain assumption The cell fluid model is defined by competition between global attraction (J1) and local repulsion (J2) interactions.
    Foundational definition of the model taken from previous numerical studies.
  • ad hoc to paper A few leading terms in the asymptotic expansion of the deformed exponential function suffice to extend validity to the marginal stability case J1 = J2.
    Invoked explicitly to justify treatment of the J1 = J2 regime and the ideal-gas limit.

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Reference graph

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