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arxiv: 2604.24948 · v1 · submitted 2026-04-27 · ❄️ cond-mat.stat-mech

On the Mathematics of Information-Thermodynamics

Dallin Fisher , Qi-Jun Hong This is my paper

Pith reviewed 2026-05-07 17:38 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords entropyinformation theorystatistical mechanicsideal gasharmonic oscillatorShannon entropyresidual mappingthermodynamics
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The pith

Thermodynamic entropy equals the Shannon entropy of residual mappings between decorrelated microstates for ideal gas and harmonic oscillator.

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

The paper establishes that thermodynamic entropy can be obtained directly as the information content captured in a mapping between two uncorrelated microstates. For the classical ideal gas and the one-dimensional harmonic oscillator, the compressibility of this residual mapping object reproduces the exact entropy value known from the canonical partition function. The conditional entropy of the mapping with respect to an uncorrelated state also matches the ensemble entropy. This consistency shows that entropy behaves as an information quantity encoded in the geometry of state-to-state relations. If the same construction holds more broadly, entropy estimation becomes possible from sampled configurations alone.

Core claim

The asdf method reformulates entropy estimation as the Shannon entropy of a residual mapping distribution defined between two decorrelated microstates. For the classical ideal gas and the one-dimensional harmonic oscillator, this entropy reproduces the exact thermodynamic entropy, and the conditional entropy of the residual mapping object coincides with the ensemble entropy derived from the canonical partition function.

What carries the argument

The residual mapping distribution between two decorrelated microstates, whose Shannon entropy equals the thermodynamic entropy of the system.

Load-bearing premise

That the residual mapping distribution between two decorrelated microstates can be defined so its Shannon entropy directly equals the thermodynamic entropy without system-specific adjustments.

What would settle it

A direct computation for the ideal gas showing that the Shannon entropy of the defined residual mapping fails to equal the known Sackur-Tetrode entropy would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.24948 by Dallin Fisher, Qi-Jun Hong.

Figure 1
Figure 1. Figure 1: FIG. 1: Heatmap showing the discrepancy between the view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The entropy of a microstate view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The information relayed from sender to receiver view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Distribution of vector magnitudes view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Marginal coordinate distributions view at source ↗
read the original abstract

We present a validation of the asdf method, an information-theoretic framework for computing thermodynamic entropy from molecular configurations. The method reformulates entropy estimation as the Shannon entropy of a residual mapping distribution defined between two decorrelated microstates. We demonstrate analytically that for the closed-form Hamiltonians with known solutions, the classical ideal gas and the one-dimensional harmonic oscillator's entropy obtained from the compressibility of the residual mapping object reproduces the exact thermodynamic entropy. In each case, the conditional entropy of the residual mapping object with respect to an uncorrelated microstate is shown to coincide with the ensemble entropy derived from the canonical partition function. These results establish consistency between the asdf formalism and classical statistical mechanics for analytically solvable systems. We further discuss how the framework generalizes to interacting Hamiltonians. The analysis supports the interpretation of thermodynamic entropy as an information measure encoded geometrically in inter-microstate mappings and motivates application of the method to complex condensed phases.

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 introduces the 'asdf method,' an information-theoretic framework that reformulates thermodynamic entropy estimation as the Shannon entropy (via compressibility) of a residual mapping distribution defined between two decorrelated microstates. It analytically demonstrates that, for the classical ideal gas and the one-dimensional harmonic oscillator, the conditional entropy of this residual mapping object coincides with the ensemble entropy derived from the canonical partition function. The work further discusses generalization of the approach to interacting Hamiltonians.

Significance. If the residual mapping can be constructed in a Hamiltonian-agnostic manner, the framework would provide a configuration-based route to entropy that is consistent with classical statistical mechanics for solvable cases and potentially extensible to complex systems. The analytical matches for the ideal gas and oscillator constitute a necessary consistency check, but the overall significance depends on whether the method avoids embedding system-specific knowledge that would limit its generality.

major comments (2)
  1. [Section 2] The definition of the residual mapping distribution (Section 2) must be shown to be independent of the equilibrium measure or known partition function. If the mapping rule implicitly incorporates the phase-space volume or Hamiltonian form used in the canonical ensemble, the reported equality for the ideal gas and oscillator becomes tautological by construction rather than a non-trivial validation.
  2. [Sections 3 and 4] In the analytical demonstrations for the ideal gas (Section 3) and harmonic oscillator (Section 4), the derivations should explicitly display how the compressibility of the residual mapping yields S = k ln Z + <E>/T without presupposing the canonical result. Absent these steps, it is unclear whether the coincidence holds for arbitrary decorrelated microstates or only under the specific mapping chosen for these solvable cases.
minor comments (2)
  1. [Abstract] The abstract refers to 'compressibility of the residual mapping object' without defining the term; the main text should clarify its precise relation to Shannon entropy and any normalization factors.
  2. [Generalization section] The discussion of generalization to interacting Hamiltonians is qualitative; adding a brief outline or pseudocode for applying the mapping to a simple interacting model (e.g., two-particle system) would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Section 2] The definition of the residual mapping distribution (Section 2) must be shown to be independent of the equilibrium measure or known partition function. If the mapping rule implicitly incorporates the phase-space volume or Hamiltonian form used in the canonical ensemble, the reported equality for the ideal gas and oscillator becomes tautological by construction rather than a non-trivial validation.

    Authors: The residual mapping distribution is defined via a fixed, configuration-based procedure that operates directly on the phase-space coordinates of two decorrelated microstates. The rule identifies residuals through a deterministic differencing operation that does not reference the equilibrium probability density or the numerical value of the partition function. We will add an explicit subsection to Section 2 demonstrating this independence for a generic pair of microstates before specializing to the ideal gas or oscillator. This addition will make clear that the subsequent match to the thermodynamic entropy is a consistency result rather than a tautology. revision: yes

  2. Referee: [Sections 3 and 4] In the analytical demonstrations for the ideal gas (Section 3) and harmonic oscillator (Section 4), the derivations should explicitly display how the compressibility of the residual mapping yields S = k ln Z + <E>/T without presupposing the canonical result. Absent these steps, it is unclear whether the coincidence holds for arbitrary decorrelated microstates or only under the specific mapping chosen for these solvable cases.

    Authors: We will expand the derivations in Sections 3 and 4 to include all intermediate steps. Starting from the definition of the residual mapping distribution, we will compute its conditional Shannon entropy explicitly, relate the compressibility to the phase-space volume factor, and arrive at S = k ln Z + <E>/T by direct calculation for each solvable system. The mapping itself is constructed from the decorrelation condition alone and does not presuppose the final entropy expression. We will also note that the result is shown for the class of decorrelated microstate pairs that satisfy the method's definition, which is the setting in which the asdf approach is intended to be applied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analytic validation uses independent canonical partition function benchmarks

full rationale

The paper claims an analytic demonstration that the Shannon entropy (via compressibility) of the residual mapping distribution coincides with the exact ensemble entropy S = k ln Z + ... for the ideal gas and 1D harmonic oscillator. This is presented as a consistency check against the known closed-form partition functions, which are external, independently derived results from classical statistical mechanics. No equations in the provided abstract or description reduce the target entropy to a fitted parameter or self-referential definition; the mapping is used to reproduce known quantities rather than derive them from themselves. The framework is therefore self-contained against external benchmarks for the solvable cases, with generalization to interacting systems left as future discussion rather than a load-bearing claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on standard definitions from information theory and statistical mechanics plus the novel construction of the residual mapping; no free parameters are mentioned, but the mapping itself is an invented construct.

axioms (2)
  • standard math Definition of Shannon entropy for a discrete or continuous probability distribution
    Invoked to equate the entropy of the residual mapping distribution to thermodynamic entropy.
  • domain assumption Existence of decorrelated microstates whose joint distribution allows a well-defined residual mapping
    Required to define the mapping object in the ensemble.
invented entities (1)
  • residual mapping distribution / object no independent evidence
    purpose: To encode thermodynamic entropy as an information-theoretic quantity derived from inter-microstate mappings
    New construct introduced by the asdf method; no independent evidence outside the two validation cases is provided.

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