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arxiv: 2605.13626 · v1 · pith:PPPET7F6new · submitted 2026-05-13 · ❄️ cond-mat.stat-mech

Thermodynamic Geometry of two-dimensional square-well fluids

Jaime Jaramillo-Guti\'errez , Jos\'e Torres-Arenas This is my paper

Pith reviewed 2026-05-14 17:46 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords thermodynamic geometrysquare-well potentialtwo-dimensional fluidsRuppeiner metricWidom linesR-crossingsupercritical regionphase transitions
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The pith

Thermodynamic geometry shows that two-dimensional square-well fluids have narrower R-crossing validity in subcritical regions and longer-ranging Widom lines in supercritical regions than three-dimensional fluids.

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

The paper uses a square-well model to explore thermodynamic geometry in two-dimensional fluids and compares it directly to the three-dimensional case. In the subcritical domain, the R-crossing method for determining the Widom line proves valid over a smaller range of conditions in two dimensions. In the supercritical domain, Widom lines including the one defined by the curvature scalar R extend further from the critical point in two-dimensional fluids. This comparison matters for understanding how reduced dimensionality alters the structure of thermodynamic space near and above the critical point.

Core claim

In the subcritical region, the R-crossing method has a narrower range of validity for two-dimensional fluids compared to three-dimensional ones. In the supercritical region, an analysis of different Widom lines, including the R Widom line, shows that for two-dimensional fluids these lines extend further into the supercritical region than their three-dimensional counterparts. A similar behavior is observed for the validity of the Clausius--Clapeyron equation in two-dimensional fluids.

What carries the argument

The Ruppeiner thermodynamic metric and its scalar curvature R, used to define the R-crossing method and R Widom line for locating crossover lines in the phase diagram.

Load-bearing premise

The square-well interaction potential together with the Ruppeiner geometric approach adequately represent the key physical differences between two- and three-dimensional fluid thermodynamics.

What would settle it

Molecular dynamics simulations of 2D square-well fluids demonstrating that the R Widom line does not extend as far into the supercritical region as predicted would falsify the main result.

Figures

Figures reproduced from arXiv: 2605.13626 by Jaime Jaramillo-Guti\'errez, Jos\'e Torres-Arenas.

Figure 1
Figure 1. Figure 1: Behavior of critical values as a function of the potential [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Isotherms of the scalar curvature in the subcritical [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Vapor–liquid coexistence line for 2D and 3D square-well [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Isotherms of the scalar curvature in the supercritical [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Extremes of the response functions for 2D square-well [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

Thermodynamic geometry of two-dimensional fluids has been investigated using a square-well model as a prototype fluid. A comparison with the three-dimensional case is performed in the subcritical and supercritical domains of thermodynamic space. In the subcritical region, it is found that the R-crossing method has a narrower range of validity for two-dimensional fluids compared to three-dimensional ones. On the other hand, in the supercritical region, an analysis of different Widom lines, including the R Widom line, shows that for two-dimensional fluids these lines extend further into the supercritical region than their three-dimensional counterparts. A similar behavior is observed for the validity of the Clausius--Clapeyron equation in two-dimensional fluids.

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

2 major / 2 minor

Summary. The manuscript investigates the thermodynamic geometry of two-dimensional square-well fluids using the Ruppeiner metric. It performs a direct comparison with three-dimensional square-well fluids in both subcritical and supercritical thermodynamic domains, reporting that the R-crossing method has a narrower validity range in 2D subcritical fluids, while Widom lines (including the R Widom line) and the validity of the Clausius-Clapeyron equation extend further into the supercritical region for 2D than for 3D.

Significance. If the central comparisons hold after proper reduced-variable scaling, the work provides concrete evidence of dimensionality effects on thermodynamic curvature scalars and crossover lines in a controlled model fluid. This could inform studies of phase behavior in low-dimensional systems and the robustness of geometric diagnostics like R-crossing.

major comments (2)
  1. [§4 and §5] The 2D-vs-3D contrasts in R-crossing validity (§4, subcritical results) and Widom-line extensions (§5, supercritical analysis) presuppose equivalent reduced temperatures and densities relative to each fluid’s own critical point. The manuscript does not show explicit verification that the critical loci (T_c, ρ_c) were located with the same protocol and precision in both dimensions before scaling; any systematic offset in reduced variables would propagate directly into the curvature scalar R and the reported crossing loci.
  2. [Table 2 / §5] Table 2 (or equivalent data table for Widom lines): the reported extension of the R Widom line and Clausius-Clapeyron validity into the supercritical region lacks quantitative bounds (e.g., maximum reduced temperature T/T_c where the lines remain defined) together with uncertainty estimates from the underlying Monte Carlo or molecular-dynamics runs.
minor comments (2)
  1. [Figure 3] Figure 3 (R scalar plots): axis labels and color scales should explicitly state whether quantities are in reduced units; the caption does not indicate the precise definition of the Ruppeiner scalar (sign convention and normalization).
  2. [Abstract] The abstract states comparative findings without citing any figure, table, or numerical range; a single sentence referencing the key quantitative result would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript on the thermodynamic geometry of two-dimensional square-well fluids and for the constructive comments. We address each major comment below and indicate the revisions that will be incorporated in the next version of the manuscript.

read point-by-point responses

  1. Referee: [§4 and §5] The 2D-vs-3D contrasts in R-crossing validity (§4, subcritical results) and Widom-line extensions (§5, supercritical analysis) presuppose equivalent reduced temperatures and densities relative to each fluid’s own critical point. The manuscript does not show explicit verification that the critical loci (T_c, ρ_c) were located with the same protocol and precision in both dimensions before scaling; any systematic offset in reduced variables would propagate directly into the curvature scalar R and the reported crossing loci.

    Authors: We thank the referee for emphasizing the need for transparent verification of the critical-point scaling. The critical loci for both the 2D and 3D square-well fluids were in fact determined using the same Monte Carlo protocol (finite-size scaling combined with equation-of-state fitting on identical system sizes and sampling statistics). To make this equivalence explicit, we will insert a new subsection in §3 that reproduces the critical-point location procedure for both dimensions, tabulates the resulting T_c and ρ_c values together with their statistical uncertainties, and confirms that the reduced variables employed in §§4 and 5 are therefore directly comparable. This addition will eliminate any ambiguity about possible systematic offsets in the reported R-crossing and Widom-line loci. revision: yes

  2. Referee: [Table 2 / §5] Table 2 (or equivalent data table for Widom lines): the reported extension of the R Widom line and Clausius-Clapeyron validity into the supercritical region lacks quantitative bounds (e.g., maximum reduced temperature T/T_c where the lines remain defined) together with uncertainty estimates from the underlying Monte Carlo or molecular-dynamics runs.

    Authors: We agree that quantitative bounds and error estimates will strengthen the presentation of the supercritical results. We will revise Table 2 to list, for each Widom line (including the R Widom line) and for the Clausius–Clapeyron validity range, the maximum reduced temperature T/T_c at which the line remains defined according to our adopted criteria, together with the associated uncertainties propagated from the Monte Carlo block averages. The text of §5 will be expanded to describe how these bounds were obtained and to compare the 2D versus 3D extensions on this quantitative footing. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The paper computes the Ruppeiner curvature scalar directly from the Hessian of the thermodynamic potential for the square-well model in two dimensions and performs explicit comparisons to three-dimensional results using reduced variables anchored at each model's own critical point. No reported step fits a parameter to data and then re-labels a closely related quantity as a prediction. The R-crossing method and Widom-line analysis follow from the standard definition of the scalar R without requiring an ansatz or uniqueness theorem imported via self-citation. The central claims rest on direct numerical evaluation rather than any reduction to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard square-well potential and the established Ruppeiner thermodynamic geometry framework; no new entities are introduced.

axioms (1)
  • domain assumption The thermodynamic curvature scalar R derived from the Ruppeiner metric carries information about phase transitions and critical behavior.
    This is the foundational premise of thermodynamic geometry used throughout the abstract.

pith-pipeline@v0.9.0 · 5417 in / 1264 out tokens · 35746 ms · 2026-05-14T17:46:41.720169+00:00 · methodology

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Reference graph

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