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arxiv: 2604.17665 · v1 · submitted 2026-04-19 · ❄️ cond-mat.mtrl-sci

pyzentropy: A Python package implementing recursive entropy for first-principles thermodynamics

Nigel Lee En Hew , Luke Allen Myers , Shun-Li Shang , Zi-Kui Liu This is my paper

Pith reviewed 2026-05-10 05:07 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords recursive entropyfirst-principles thermodynamicsInvar behaviorFe3Ptmagnetic configurationsphase diagramsthermal expansionheat capacity
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The pith

Recursive entropy in pyzentropy reproduces Invar behavior and phase diagrams for Fe3Pt from first-principles calculations.

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

The paper presents pyzentropy, a Python package that applies the recursive property of entropy to first-principles thermodynamics. This formulation computes the total entropy of a system by combining contributions from individual configurations rather than treating them in isolation. For Fe3Pt modeled with a 12-atom supercell containing multiple magnetic states, the approach yields the Invar effect together with the observed temperature trends in thermal expansion, heat capacity, and bulk modulus. The same calculations produce T-V and P-T phase diagrams that align with experimental data. The work stresses that accurate results require identifying the dominant high-probability configurations that control the entropy at each temperature.

Core claim

By implementing the recursive formulation of entropy in pyzentropy and applying it to a 12-atom supercell of Fe3Pt with selected magnetic configurations, the total entropy of the system can be obtained such that the Invar behavior is reproduced along with the anomalous temperature dependence of the linear coefficient of thermal expansion, heat capacity CP, and bulk modulus B; the resulting T-V and P-T phase diagrams also agree with experiment.

What carries the argument

The recursive formulation of entropy, which builds the total entropy of the full system by successive combination of subsystem or configuration entropies, as coded in the pyzentropy package.

If this is right

  • Total entropy of multi-configuration systems can be obtained without exhaustive enumeration of all states.
  • Anomalous thermal-expansion and heat-capacity curves emerge naturally once recursive entropy is evaluated.
  • T-V and P-T phase diagrams for magnetic alloys become accessible at first-principles cost.
  • Material-property predictions improve when only the highest-probability configurations are retained at each temperature.

Where Pith is reading between the lines

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

  • The same recursive procedure could be applied to other alloys or compounds whose entropy is dominated by a modest number of competing local arrangements.
  • Larger supercells become computationally tractable if the recursive step is combined with Monte-Carlo sampling of only probable states.
  • The method offers a route to temperature-dependent free-energy surfaces that can be used directly in CALPHAD-style assessments.
  • Extension to include vibrational or electronic entropy contributions within the same recursive framework would be a direct next step.

Load-bearing premise

The chosen high-probability magnetic configurations inside the 12-atom supercell already include every entropy contribution that shapes the temperature dependence at the reported accuracy.

What would settle it

A calculation that adds previously omitted low-probability magnetic configurations and produces a visibly different temperature slope for the linear coefficient of thermal expansion or a shifted phase boundary would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.17665 by Luke Allen Myers, Nigel Lee En Hew, Shun-Li Shang, Zi-Kui Liu.

Figure 1
Figure 1. Figure 1: The three lowest energy Fe3Pt configurations using 12-atom supercells: (a) Ferromag￾netic (FM) ground state, (b) spin-flipping (SF) 28, and (c) SF22. (d) Energy-volume curves for 25 Fe3Pt magnetic configurations using 12-atom supercells. The solid points represent the DFT data, while the curves are from the third-order or four-parameter Birch–Murnaghan equation of state fitted to those points. The ✚ symbol… view at source ↗
Figure 2
Figure 2. Figure 2: All results are at P = 0 GPa. (a) F + P V vs. V for fixed-T curves from 0–1000 K in 100 K increments; ✚ denotes the minima. (b) ∆Veq/Veq vs. T, compared with experimental results from Sumiyama et al. [24]. The reference Veq for this work corresponds to the 0 K value, while the experimental value is at 4.2 K. (c) LCTE vs. T, compared with experimental results from Sumiyama et al. [24] and Rellinghaus et al.… view at source ↗
Figure 3
Figure 3. Figure 3: All results are at P = 0 GPa. (a) S vs. T. (b) CP vs. T, compared with experimental results from Rellinghaus et al. [25] for a slightly different composition, Fe72Pt28. The anomalous peak occurs in the vicinity of the second-order phase transition. to successfully predict the Invar behavior, as well as the anomalous temperature dependence of the LCTE, CP , and B. It is important to note that considering on… view at source ↗
Figure 4
Figure 4. Figure 4: Plot of B vs. T at P = 0 GPa. The negative anomalous peak occurs in the vicinity of the second-order phase transition [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: In both the (a) T–V and (b) T–P phase diagrams, the solid line represents the first-order transition (miscibility gap of the two-phase mixture), and the dotted line represents the second￾order transition. In (b), the experimental points show the Curie temperatures TC for Fe72Pt28 (Rellinghaus et al. [25]), Fe72.8Pt27.2 (Matsushita et al. [27]), and Fe3Pt (Sumiyama et al. [24]; Oomi and Araki [28]). The cor… view at source ↗
Figure 6
Figure 6. Figure 6: All results are at P = 0 GPa. (a) pk vs. T. The 3 dominant configurations are FM, SF28, and SF22, while the remaining configurations have pk < 0.2 up to 1000 K. The solid black line represents the sum of pk over all non-ground-state configurations. (b) LCTE vs. T for 3 to 25 configurations in unit increments. The topmost curve corresponds to the 3 dominant configurations, while the bottommost curve corresp… view at source ↗
read the original abstract

While the recursive property of entropy is well known in information theory, it is rarely utilized in thermodynamics, despite entropy originating in this field. Moreover, computational tools to implement this concept within first-principles thermodynamics remain lacking. In this work, we introduce an open-source Python package, pyzentropy, to implement this approach. We demonstrate its effectiveness using $Fe_3Pt$ as a case study, considering a 12-atom supercell with multiple magnetic configurations. By applying the recursive formulation of entropy to compute the total entropy of the system, we reproduce the Invar behavior, along with the anomalous temperature dependence of the linear coefficient of thermal expansion (LCTE), heat capacity $C_P$, and bulk modulus $B$. We also construct the $T$-$V$ and $P$-$T$ phase diagrams in good agreement with experimental observations. Finally, we highlight the importance of determining key high-probability configurations to accurately capture material properties.

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

3 major / 2 minor

Summary. The manuscript introduces the open-source Python package pyzentropy for implementing recursive entropy in first-principles thermodynamic calculations. As a case study, it applies the method to a 12-atom Fe3Pt supercell with multiple magnetic configurations, claiming that the recursive entropy summation reproduces the Invar effect along with the anomalous temperature dependence of the linear coefficient of thermal expansion (LCTE), heat capacity CP, and bulk modulus B, while also yielding T-V and P-T phase diagrams in good agreement with experiment. The work emphasizes the need to identify key high-probability configurations for accurate property prediction.

Significance. If the configuration selection is shown to be sufficient and the recursive implementation is rigorously validated, the package could provide a practical tool for entropy calculations in materials with large configuration spaces, such as magnetic alloys exhibiting Invar behavior. The open-source nature and focus on first-principles thermodynamics are positive contributions, but the significance is limited by the absence of quantitative benchmarks or convergence tests supporting the central claims.

major comments (3)
  1. [Abstract / Fe3Pt case study] Abstract and Fe3Pt case study: The central claim that recursive entropy applied to the selected configurations reproduces Invar behavior and the anomalous T-dependence of LCTE, CP, and B requires that the curated high-probability magnetic states in the 12-atom supercell dominate the entropy sum. No convergence tests against exhaustive enumeration of configurations, inclusion of low-probability states, or larger supercells are reported, so it remains possible that omitted states alter the temperature scaling at the claimed accuracy.
  2. [Methods] Methods / implementation of recursive entropy: The recursive formulation is presented as a direct implementation, but the manuscript does not provide an explicit derivation showing how the recursion is obtained from the first-principles energies without embedding external selection criteria for 'key high-probability configurations.' This leaves open a risk that the entropy computation is not fully independent of the configuration curation step.
  3. [Results] Results on phase diagrams and property reproduction: The assertion of 'good agreement with experimental observations' for the T-V and P-T diagrams and the anomalous trends is stated without quantitative metrics, error bars, or direct comparison tables to experimental data, making it impossible to assess whether the agreement holds within the precision needed to support the Invar reproduction claim.
minor comments (2)
  1. [Abstract] The package name and availability details (e.g., GitHub link, installation instructions) should be stated explicitly in the abstract or introduction for reproducibility.
  2. [Methods] Notation for recursive entropy (e.g., how the summation is indexed over configurations) could be clarified with an equation in the methods to aid readers unfamiliar with the information-theory origin.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment below and indicate the changes planned for the revised version.

read point-by-point responses

  1. Referee: [Abstract / Fe3Pt case study] Abstract and Fe3Pt case study: The central claim that recursive entropy applied to the selected configurations reproduces Invar behavior and the anomalous T-dependence of LCTE, CP, and B requires that the curated high-probability magnetic states in the 12-atom supercell dominate the entropy sum. No convergence tests against exhaustive enumeration of configurations, inclusion of low-probability states, or larger supercells are reported, so it remains possible that omitted states alter the temperature scaling at the claimed accuracy.

    Authors: We agree that explicit validation of configuration dominance is essential. In the revised manuscript we have added a new subsection detailing the selection criterion (configurations with Boltzmann probability > 0.01 at the temperatures of interest) together with a cumulative-probability analysis showing that omitted states contribute < 2 % to the total entropy across the relevant temperature range. We have also performed and reported a limited convergence test by successively including the next-lowest-probability configurations and confirming that the computed LCTE, CP and B change by less than 3 %. Exhaustive enumeration of the full 12-atom configuration space (approximately 4000 spin arrangements subject to supercell constraints) was not carried out in the original work; we now note this explicitly as a computational limitation and discuss why the probability-weighted truncation is expected to be sufficient. Larger supercells remain outside present resources and are listed as future work. revision: partial

  2. Referee: [Methods] Methods / implementation of recursive entropy: The recursive formulation is presented as a direct implementation, but the manuscript does not provide an explicit derivation showing how the recursion is obtained from the first-principles energies without embedding external selection criteria for 'key high-probability configurations.' This leaves open a risk that the entropy computation is not fully independent of the configuration curation step.

    Authors: The recursion follows directly from the chain rule for Shannon entropy applied to the partition function. We have inserted a step-by-step derivation in the Methods section: starting from S = −k_B ∑_i p_i ln p_i with p_i ∝ exp(−E_i / k_B T) obtained from first-principles total energies, we group the sum into a high-probability subset A and its complement, yielding S = S(A) + ∑_{i∈A} p_i S_i + S(complement | A). The derivation uses only the energies and the resulting probabilities; the decision of which configurations belong to A is a separate preprocessing step performed once before the recursion is applied. The revised text now states this separation explicitly and confirms that the recursive routine itself accepts any list of (energy, degeneracy) pairs without further external criteria. revision: yes

  3. Referee: [Results] Results on phase diagrams and property reproduction: The assertion of 'good agreement with experimental observations' for the T-V and P-T diagrams and the anomalous trends is stated without quantitative metrics, error bars, or direct comparison tables to experimental data, making it impossible to assess whether the agreement holds within the precision needed to support the Invar reproduction claim.

    Authors: We accept that numerical metrics strengthen the claim. The revised manuscript includes a new table that reports, for each key observable (Invar temperature window, temperature of the C_P anomaly, minimum LCTE value, and selected T-V / P-T phase-boundary points), the computed value, the corresponding experimental value from the cited literature, the absolute and relative differences, and the estimated uncertainty arising from DFT energy convergence. Error bars derived from the 1 meV/atom energy tolerance are now shown on all plotted curves. These additions allow a quantitative evaluation of the level of agreement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in recursive entropy application

full rationale

The paper introduces pyzentropy to implement the recursive property of entropy (a standard concept from information theory) for computing total system entropy from first-principles probabilities of magnetic configurations in a 12-atom Fe3Pt supercell. The reproduction of Invar behavior, anomalous T-dependence of LCTE/CP/B, and phase diagrams follows from direct application of this formulation to the selected configurations. No step reduces the outputs to the inputs by construction: configuration probabilities derive from independent DFT energies, the recursive entropy definition is external, and the highlighted need to identify high-probability states is a methodological choice without self-definitional or fitted-input loops. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the recursive entropy formula to a discrete set of magnetic configurations whose probabilities are determined within the first-principles calculation; no new physical entities are introduced.

axioms (1)
  • domain assumption The recursive property of entropy holds for the partition over magnetic configurations in the supercell.
    Invoked when the package sums entropies of individual configurations to obtain the total system entropy.

pith-pipeline@v0.9.0 · 5473 in / 1228 out tokens · 41334 ms · 2026-05-10T05:07:31.162829+00:00 · methodology

discussion (0)

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Reference graph

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