pith. sign in

arxiv: 2511.04950 · v3 · pith:R727PLWAnew · submitted 2025-11-07 · ❄️ cond-mat.stat-mech

Recursive entropy in thermodynamics: expounding the statistical-physics basis of the zentropy approach

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

Pith reviewed 2026-05-21 20:18 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords recursive entropycoarse-grainingHelmholtz free energypartition functionstatistical mechanicsthermodynamicsmultiscaleemergent behavior
0
0 comments X

The pith

Maximizing entropy in recursive form derives the Helmholtz energy and partition function while treating temperature-dependent states as coarse-grained configurations.

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

The paper shows that entropy's recursive property, known in information theory, can be used in thermodynamics to divide systems into the most useful groups of configurations. Maximizing entropy in this recursive way recovers the usual expressions for Helmholtz free energy and the partition function. It also gives a clear meaning to states that appear to depend on temperature: they are simply the coarse-grained groups. The distinction between physical laws and statistical counting is kept sharp throughout. Examples with spin arrangements in magnets and nearest-neighbor shells in liquids illustrate how probability shifts among groups produce emergent thermodynamic behavior.

Core claim

The recursive property allows coarse-graining thermodynamic systems into the most useful groups. Deriving the Helmholtz energy and partition function by maximizing entropy in its recursive form clarifies the thermodynamic meaning of so-called states that depend on temperature as coarse-grained configurations, and maintains a clear distinction between the physical and statistical aspects of statistical mechanics.

What carries the argument

Maximization of recursive entropy over chosen groupings of configurations, which directly produces the Helmholtz free energy and the partition function.

If this is right

  • The method enables physically meaningful coarse-graining in any system that admits a natural hierarchical grouping of states.
  • Emergent thermodynamic behavior arises solely from redistribution of probabilities among the chosen configurations.
  • The framework remains exact while handling multiple scales in thermodynamics and statistical mechanics.
  • Physical and statistical contributions stay clearly separated at every step.

Where Pith is reading between the lines

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

  • The same recursive-maximization logic could be applied to other systems with natural groupings, such as polymer conformations or alloy short-range order, to test whether temperature dependence emerges without added parameters.
  • Links may exist to renormalization ideas, where each level of grouping corresponds to a successive coarse-graining step.
  • Direct comparison with molecular-dynamics trajectories on the same model systems would provide a concrete check on whether the selected groupings are indeed the most useful ones.

Load-bearing premise

The chosen groupings of states, such as spin arrangements or nearest-neighbor environments, are the most useful ones so that recursive entropy maximization alone produces the correct temperature dependence without extra physical input.

What would settle it

A numerical test on a simple Ising magnet or Lennard-Jones liquid that applies a different but plausible grouping and checks whether the resulting free-energy curve still matches known simulation or experimental results.

Figures

Figures reproduced from arXiv: 2511.04950 by Luke Allen Myers, Nigel Lee En Hew, Shun-Li Shang, Zi-Kui Liu.

Figure 1
Figure 1. Figure 1: Probability tree diagrams illustrating (a) the ungrouped and (b) grouped scenarios. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

The recursive property of entropy is well known in information theory; however, the concept is underutilized in thermodynamics, despite being the field where the concept of entropy originated. The zentropy approach is built on this idea, and it has emerged as a useful framework for describing thermodynamic systems across multiple scales, yet its statistical-physics foundation has not been fully articulated. In this work, we establish that foundation by showing that the recursive property allows us to coarse-grain thermodynamic systems into the most useful groups, and deriving the Helmholtz energy and partition function by maximizing entropy in its recursive form. This derivation clarifies the thermodynamic meaning of so-called "states that depend on temperature" as coarse-grained configurations, and maintains a clear distinction between the physical and statistical aspects of statistical mechanics. We then illustrate the usefulness of the approach through two representative applications: magnetic materials, where configurations are defined by spin arrangements, and liquids, where configurations are defined by nearest-neighbor environments. In both cases, the framework enables physically meaningful coarse-graining and captures emergent behavior arising from probability redistribution among configurations. These results position zentropy as an exact and flexible multiscale framework for thermodynamics and statistical mechanics, particularly for systems that admit a natural hierarchical grouping of states.

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 paper claims to establish the statistical-physics foundation of the zentropy approach by showing that the recursive property of entropy permits coarse-graining thermodynamic systems into the most useful groups. It derives the Helmholtz free energy and partition function via maximization of the recursive entropy S = −k ∑_c p_c ln p_c + ∑_c p_c S_c subject to the usual constraints, yielding p_c ∝ exp[−β(E_c − T S_c)] and thus F_c = E_c − T S_c with Z = ∑ exp(−β F_c). This is said to clarify temperature-dependent states as coarse-grained configurations. The framework is illustrated on magnetic materials (spin-arrangement groupings) and liquids (nearest-neighbor environments), positioning zentropy as an exact multiscale thermodynamic tool.

Significance. If the central derivation holds, the work supplies a transparent link between recursive entropy maximization and standard thermodynamic potentials, explicitly crediting the formal step that explains why coarse-grained configurations acquire effective temperature dependence through their internal entropies S_c. This strengthens the conceptual basis for hierarchical modeling in statistical mechanics without introducing extraneous parameters at the maximization stage.

major comments (1)
  1. [Derivation section (recursive entropy maximization and resulting partition function)] The derivation of p_c ∝ exp[−β(E_c − T S_c)] and F_c = E_c − T S_c is formally valid and correctly identifies coarse-grained states as the origin of apparent temperature dependence. However, the load-bearing claim that recursive maximization alone recovers the observed thermodynamic temperature dependence rests on the unproven assertion that the chosen groupings (spin arrangements, nearest-neighbor environments) permit parameter-free evaluation of the intra-configuration entropies S_c from a lower-level physical model. No section demonstrates that the recursive procedure itself selects or justifies these groupings independently of system-specific knowledge.
minor comments (1)
  1. [Abstract and §1] The abstract and introduction repeatedly use the phrase 'most useful groups' without an explicit operational criterion; adding a short paragraph defining 'useful' in terms of scale separation or emergent observables would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for acknowledging the formal validity of the derivation linking recursive entropy maximization to the partition function and effective free energies. We address the single major comment below.

read point-by-point responses

  1. Referee: The derivation of p_c ∝ exp[−β(E_c − T S_c)] and F_c = E_c − T S_c is formally valid and correctly identifies coarse-grained states as the origin of apparent temperature dependence. However, the load-bearing claim that recursive maximization alone recovers the observed thermodynamic temperature dependence rests on the unproven assertion that the chosen groupings (spin arrangements, nearest-neighbor environments) permit parameter-free evaluation of the intra-configuration entropies S_c from a lower-level physical model. No section demonstrates that the recursive procedure itself selects or justifies these groupings independently of system-specific knowledge.

    Authors: We agree that the recursive maximization procedure itself does not select or justify the groupings of states. The specific coarse-grained configurations (e.g., spin arrangements in magnets or nearest-neighbor environments in liquids) are identified on the basis of physical insight into the relevant degrees of freedom, as is standard in multiscale modeling. Once the groupings are so defined, the recursive entropy maximization then supplies the statistical weights and yields parameter-free intra-configuration entropies S_c from the underlying lower-level model, producing the temperature-dependent effective free energies F_c. This distinction between the physical choice of groupings and the subsequent statistical maximization is implicit in the applications but was not stated explicitly. We will revise the derivation section to clarify this point and to emphasize that the framework is exact for any physically motivated partitioning, without claiming that maximization alone discovers the groupings. revision: yes

Circularity Check

0 steps flagged

Recursive entropy maximization yields effective partition function without reducing to input by construction

full rationale

The paper performs a standard constrained maximization of the recursive entropy S = −k ∑ p_c ln p_c + ∑ p_c S_c subject to fixed total energy and normalization. Lagrange multipliers directly produce p_c ∝ exp[−β(E_c − T S_c)], from which F_c = E_c − T S_c and Z = ∑ exp(−β F_c) follow algebraically. This is a valid re-derivation of the coarse-grained Boltzmann factor; the intra-configuration entropies S_c are supplied as external inputs from a lower-scale physical model, which the framework explicitly treats as given rather than derived. No load-bearing step equates a claimed prediction to a fitted parameter or to a self-citation whose validity is presupposed. The choice of groupings (spin arrangements, nearest-neighbor shells) is presented as a modeling decision based on physical insight for each application, not as an output of the recursive procedure itself. The derivation therefore remains self-contained against external benchmarks and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the recursive property of entropy being applicable to thermodynamic coarse-graining and on the assumption that maximizing that form yields the correct partition function.

axioms (1)
  • domain assumption The recursive property of entropy permits coarse-graining of thermodynamic states into useful groups
    Invoked to derive Helmholtz energy and partition function from entropy maximization

pith-pipeline@v0.9.0 · 5758 in / 1232 out tokens · 31061 ms · 2026-05-21T20:18:30.378757+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    cond-mat.mtrl-sci 2026-04 unverdicted novelty 6.0

    pyzentropy implements recursive entropy to compute total system entropy from first-principles supercell calculations, reproducing Invar behavior and experimental phase diagrams for Fe3Pt.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · cited by 1 Pith paper

  1. [1]

    Multiscale entropy and its implications to critical phenomena, emergent behaviors, and information,

    Z.-K. Liu, B. Li, and H. Lin, “Multiscale entropy and its implications to critical phenomena, emergent behaviors, and information,”Journal of Phase Equilibria and Diffusion, vol. 40, no. 4, pp. 508–521, Jun. 2019,issn: 1863-7345.doi: 10.1007/s11669- 019-00736-w

    work page doi:10.1007/s11669- 2019

  2. [2]

    Quantifying the degree of disorder and associated phenomena in materials through zentropy: Illustrated with invar fe3pt,

    S.-L. Shang, Y. Wang, and Z.-K. Liu, “Quantifying the degree of disorder and associated phenomena in materials through zentropy: Illustrated with invar fe3pt,”Scripta Mate- rialia, vol. 225, p. 115 164, Mar. 2023,issn: 1359-6462.doi: 10.1016/j.scriptamat. 2022.115164

    work page doi:10.1016/j.scriptamat 2023

  3. [3]

    Predicting phase transitions in pbtio3 using zentropy through quasiharmonic phonon calculations,

    N. L. E. Hew, S. -L. Shang, and Z.-K. Liu, “Predicting phase transitions in pbtio3 using zentropy through quasiharmonic phonon calculations,”Physical Review B, vol. 110, no. 18, p. 184 103, Nov. 2024,issn: 2469-9969.doi:10.1103/physrevb.110.184103

    work page doi:10.1103/physrevb.110.184103 2024

  4. [4]

    Generalized approach for rapid entropy calculation of liquids and solids,

    Q.-J. Hong and Z. -K. Liu, “Generalized approach for rapid entropy calculation of liquids and solids,”Physical Review Research, vol. 7, no. 1, p. l012030, Feb. 2025,issn: 2643-1564.doi:10.1103/physrevresearch.7.l012030

    work page doi:10.1103/physrevresearch.7.l012030 2025

  5. [5]

    Zentropy theory for positive and negative thermal expansion,

    Z.-K. Liu, Y. Wang, and S.-L. Shang, “Zentropy theory for positive and negative thermal expansion,”Journal of Phase Equilibria and Diffusion, vol. 43, no. 6, pp. 598–605, Feb. 2022,issn: 1863-7345.doi:10.1007/s11669-022-00942-z

    work page doi:10.1007/s11669-022-00942-z 2022

  6. [6]

    Thermodynamic fluctuations between magnetic states from first-principles phonon calculations: The case of bcc fe,

    S.-L. Shang, Y. Wang, and Z. -K. Liu, “Thermodynamic fluctuations between magnetic states from first-principles phonon calculations: The case of bcc fe,”Physical Review B, vol. 82, no. 1, p. 014 425, Jul. 2010,issn: 1550-235X.doi: 10.1103/physrevb.82. 014425

    work page doi:10.1103/physrevb.82 2010

  7. [7]

    Magnetic thermodynamics of fcc ni from first-principles partition function approach,

    S.-L. Shang, J. E. Saal, Z.-G. Mei, Y. Wang, and Z.-K. Liu, “Magnetic thermodynamics of fcc ni from first-principles partition function approach,”Journal of Applied Physics, vol. 108, no. 12, Dec. 2010,issn: 1089-7550.doi:10.1063/1.3524480

    work page doi:10.1063/1.3524480 2010

  8. [8]

    Liu, N.L.E

    Z.-K. Liu, N. L. E. Hew, and S. -L. Shang, “Zentropy theory for accurate prediction of free energy, volume, and thermal expansion without fitting parameters,”Microstructures, vol. 4, no. 1, Jan. 2024,issn: 2770-2995.doi:10.20517/microstructures.2023.56 15

    work page doi:10.20517/microstructures.2023.56 2024

  9. [9]

    Achieving accurate entropy and melting point by ab initio molecular dynamics and zentropy theory: Application to fluoride and chloride molten salts,

    S.-L. Shang et al., “Achieving accurate entropy and melting point by ab initio molecular dynamics and zentropy theory: Application to fluoride and chloride molten salts,” Journal of Molecular Liquids, vol. 438, p. 128 651, Nov. 2025,issn: 0167-7322.doi: 10.1016/j.molliq.2025.128651

    work page doi:10.1016/j.molliq.2025.128651 2025

  10. [10]

    S. Wang, S. -L. Shang, Z. -K. Liu, and W. Hao,Zenn: A thermodynamics-inspired computational framework for heterogeneous data-driven modeling, 2025.doi: 10.48550/ ARXIV.2505.09851

    work page arXiv 2025

  11. [11]

    The principles of statistical mechanics,

    R. C. Tolman, “The principles of statistical mechanics,” in 1938

    work page 1938

  12. [12]

    Shannon, Communication theory of secrecy systems, The Bell System Technical Journal 28 (1949) 656 –715

    C. E. Shannon, “A mathematical theory of communication,”Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, Jul. 1948,issn: 0005-8580.doi: 10.1002/j.1538- 7305.1948.tb01338.x

    work page doi:10.1002/j.1538- 1948

  13. [15]

    T. M. Cover and J. A. Thomas,Elements of Information Theory. Wiley, Apr. 2005, isbn: 9780471748823.doi:10.1002/047174882x

    work page doi:10.1002/047174882x 2005

  14. [16]

    Gibbs vs boltzmann entropies,

    E. T. Jaynes, “Gibbs vs boltzmann entropies,”American Journal of Physics, vol. 33, no. 5, pp. 391–398, May 1965,issn: 1943-2909.doi:10.1119/1.1971557

    work page doi:10.1119/1.1971557 1965

  15. [17]

    Geometry of probability distributions,

    I. Bengtsson and K. ˙Zyczkowski, “Geometry of probability distributions,” inGeometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Aug. 2017, ch. 2, pp. 29–62,isbn: 9781107656147.doi: 10.1017/9781139207010

    work page doi:10.1017/9781139207010 2017

  16. [18]

    On the temperature dependence of cooperative relaxation properties in glass-forming liquids,

    G. Adam and J. H. Gibbs, “On the temperature dependence of cooperative relaxation properties in glass-forming liquids,”Journal of Chemical Physics, vol. 43, no. 1, pp. 139– 146, 1965.doi:10.1063/1.1696442

    work page doi:10.1063/1.1696442 1965

  17. [19]

    A critical review of high entropy alloys and related concepts,

    D. B. Miracle and O. N. Senkov, “A critical review of high entropy alloys and related concepts,”Acta Materialia, vol. 122, pp. 448–511, 2017,issn: 1359-6454.doi: 10.1016/ j.actamat.2016.08.081

    work page 2017

  18. [20]

    Nielsen and Isaac L

    M. A. Nielsen and I. L. Chuang, “Entropy and information,” inQuantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, Jun. 2012, ch. 11, pp. 500–527,isbn: 9780511976667.doi:10.1017/cbo9780511976667

    work page doi:10.1017/cbo9780511976667 2012

  19. [21]

    The concept of entropy in the theory of probability,

    A. Y. Khinchin, “The concept of entropy in the theory of probability,”Uspekhi Matem- aticheskikh Nauk, vol. 8, no. 3, pp. 3–20, 1953

    work page 1953

  20. [22]

    On the concept of entropy of a finite probabilistic scheme,

    D. K. Faddeev, “On the concept of entropy of a finite probabilistic scheme,”Uspekhi Matematicheskikh Nauk, vol. 11, no. 1, pp. 227–231, 1956

    work page 1956

  21. [23]

    Why the shannon and hartley entropies are ‘natural’,

    J. Acz´ el, B. Forte, and C. T. Ng, “Why the shannon and hartley entropies are ‘natural’,” Advances in Applied Probability, vol. 6, no. 1, pp. 131–146, Mar. 1974,issn: 1475-6064. doi:10.2307/1426210 16

    work page doi:10.2307/1426210 1974

  22. [24]

    First-principles thermodynamics from phonon and debye model: Application to ni and ni3al,

    S.-L. Shang, Y. Wang, D. Kim, and Z. -K. Liu, “First-principles thermodynamics from phonon and debye model: Application to ni and ni3al,”Computational Materials Science, vol. 47, no. 4, pp. 1040–1048, Feb. 2010,issn: 0927-0256.doi: 10.1016/j.commatsci. 2009.12.006

    work page doi:10.1016/j.commatsci 2010

  23. [25]

    Thermodynamic properties of al, ni, nial, and ni3al from first-principles calculations,

    Y. Wang, Z.-K. Liu, and L. -Q. Chen, “Thermodynamic properties of al, ni, nial, and ni3al from first-principles calculations,”Acta Materialia, vol. 52, no. 9, pp. 2665–2671, May 2004,issn: 1359-6454.doi:10.1016/j.actamat.2004.02.014

    work page doi:10.1016/j.actamat.2004.02.014 2004

  24. [26]

    Coarse- grained protein models and their applications,

    S. Kmiecik, D. Gront, M. Kolinski, L. Wieteska, A. E. Dawid, and A. Kolinski, “Coarse- grained protein models and their applications,”Chemical Reviews, vol. 116, no. 14, pp. 7898–7936, Jun. 2016,issn: 1520-6890.doi:10.1021/acs.chemrev.6b00163

    work page doi:10.1021/acs.chemrev.6b00163 2016

  25. [27]

    F. E. Rosas et al.,Software in the natural world: A computational approach to hierar- chical emergence, 2024.doi:10.48550/ARXIV.2402.09090 17

    work page doi:10.48550/arxiv.2402.09090 2024