Landauer's Principle as a Criterion for Thermodynamic Consistency in Generalized Black Hole Entropies

Ahmad Sheykhi; Fatemeh Sadeghi

REVIEW 3 major objections 5 minor 47 references

Landauer's Principle as a Criterion for Thermodynamic Consistency in Generalized Black Hole Entropies

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that Landauer's principle — the minimum thermodynamic cost of erasing one bit — can serve as a consistency test for generalized black hole entropy formulas, classifying them into temperature-constrained…

desk verdict Systematic but uneven application of Landauer's principle to generalized entropies; the Rényi section has real errors and the temperature choice for the Landauer bound needs explicit defense. read the letter →

arxiv 2607.24132 v1 pith:3YIRHBTH submitted 2026-07-27 gr-qc

Fatemeh Sadeghi , Ahmad Sheykhi This is my paper · ORCID

classification gr-qc MSC 83C5794A17 PACS 04.70.-s04.70.Dy

keywords Landauer'sprincipleblackholeentropyareaquantizationHawkingevaporationgeneralizedentropiesKaniadakisRényithermodynamicconsistency

topics Quantum Gravity

open problems Quantum Gravity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to show that a single information-theoretic inequality — Landauer's requirement that erasing one bit dissipate at least $k_B T \ln 2$ of heat — can serve as a physical criterion for deciding which generalized black hole entropy formulas are thermodynamically consistent. If correct, this would give a principled way to choose among competing entropy proposals purely from information cost. The claimed payoff is a classification: some entropy models are forced into restricted mass or temperature ranges, some into restricted parameter ranges, and Kaniadakis entropy is ruled out entirely except in the Bekenstein-Hawking limit. The same step also yields the horizon area quantization parameter and area spectrum for every surviving model, with a level-dependent spacing for most generalized entropies. The Bekenstein-Hawking case saturates the bound, reproducing the area spacing $\gamma = 4\ln 2$.

What carries the argument

The load-bearing object is the Landauer inequality $\Delta M \geq k_B T \ln 2$, combined with the identification of each downward Hawking transition with the erasure of one bit, so that $\Delta S = k_B \ln 2$. The argument uses the Bekenstein-Mukhanov area quantization $A_n = \gamma l_P^2 n$ and the unmodified Schwarzschild temperature $T = 1/(8\pi G k_B M)$. For each entropy model $S(M)$, the paper differentiates to obtain $\Delta S = S'(M)\Delta M$, imposes the Landauer inequality, and then either solves for allowed temperature, mass, or entropy parameters, or equates the entropy change to $k_B\ln 2$ to obtain $\gamma$ and the area spectrum.

What would settle it

Compute the thermodynamic temperature $T = dM/dS$ for each generalized entropy model instead of using the Schwarzschild formula, and re-check whether $\Delta M \geq k_B T \ln 2$ holds. For Kaniadakis entropy this means testing whether a nonzero-parameter regime can satisfy the bound once the temperature is model-dependent; if such a regime exists, the paper's exclusion is an artifact of its fixed temperature.

Watch

Extended reading notes

Core claim

The paper's central claim is that Landauer's inequality $\Delta M \geq k_B T \ln 2$, applied to a Schwarzschild black hole that loses one bit of information per Hawking transition between quantized area levels, is a valid thermodynamic-consistency criterion for generalized horizon entropies. Under this criterion, the Bekenstein-Hawking entropy exactly saturates the bound and yields $\gamma = 4\ln 2$ in the area spectrum $A_n = \gamma l_P^2 n$, matching the Bekenstein-Mukhanov counting with $k=2$. The paper further claims that mass-to-horizon and corrected Bekenstein-Hawking entropies survive only under temperature or mass constraints; that Rényi, Sharma-Mittal, loop-quantum-gravity, and hypergeometric entropies survive only for restricted parameter values; and that Kaniadakis entropy is incompatible for any nonzero deformation parameter $\kappa$, because the factor $\cosh(\kappa S_{BH}/k_B) \geq 1$ drives the emitted energy per one-bit step below the Landauer bound.

Load-bearing premise

The analysis evaluates every generalized entropy with the unmodified Schwarzschild Hawking temperature $T = 1/(8\pi G k_B M)$; if the true environment temperature is instead $dM/dS$ for the modified entropy, the derived constraints and the exclusion of Kaniadakis entropy do not follow.

Editorial extensions

If this is right

The Bekenstein-Mukhanov quantization constant $k=2$ follows from Landauer saturation, so the two independent quantization schemes agree on $\gamma = 4\ln 2$.
For mass-to-horizon and corrected Bekenstein-Hawking entropies, the Landauer inequality restricts which black hole masses or temperatures are thermodynamically admissible.
For Rényi, Sharma-Mittal, loop-quantum-gravity, and hypergeometric entropies, the same inequality fixes the sign or range of the free parameters, with Rényi entropy further restricted to $0 \leq \nu < 1$ by the positivity of $\gamma$.
For every compatible model, the relative area spacing $\Delta A_n/A_n$ vanishes as $n \to \infty$, so the discrete area spectrum becomes effectively continuous at macroscopic scales.
Kaniadakis entropy, if the criterion is accepted, cannot describe a one-bit Hawking evaporation step for any $\kappa \neq 0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the whole classification inherits the assumption that the environment temperature is the unmodified Schwarzschild Hawking temperature; if generalized entropies imply a different thermodynamic temperature $dM/dS$, the constraints and the Kaniadakis exclusion could shift.
Editorial inference: the paper's ordering — first impose $\Delta M \geq k_B T \ln 2$, then set $\Delta S = k_B \ln 2$ — reverses the sign of the allowed Rényi parameter relative to earlier work, suggesting that other entropy models might change classification under the same reordering.
Editorial inference: the same test could be carried over to rotating or charged black holes, where the mass-temperature relation differs, to see whether the compatibility classes remain stable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I've read arXiv:2607.24132. It applies Landauer's principle to a large family of generalized black hole entropies and sorts them into three categories: constraints on Hawking temperature/mass, constraints on entropy parameters, and outright incompatibility (Kaniadakis). That classification is new relative to Ref. [9], and most of the algebra is straightforward and correct. The Bekenstein-Mukhanov comparison is clean, and the mass-to-horizon, Sharma-Mittal, LQG, and hypergeometric sections are workmanlike. The Kaniadakis exclusion follows directly from cosh(x) ≥ 1, and if you accept the Landauer criterion, it's a neat result.

The problems are concentrated in the Rényi section, which the paper specifically advertises as correcting Ref. [9]. Equation (71) for the area quantization parameter is wrong: it diverges as ν → 0 instead of giving 4 ln 2, and it doesn't match Fig. 1. The positivity analysis in Eqs. (72)-(74) is internally contradictory — for 0 ≤ ν < 1/2 the numerator of (71) is positive, not non-positive, and the inequality n < 1/(2ν−1) has no solutions in that range. The correct formula is γ = 4(2^ν−1)/[ν(1+n(1−2^ν))], which has the right limit and a different positivity condition. The Rényi section needs to be redone.

The deeper issue is the temperature identification. The paper plugs the standard Schwarzschild Hawking temperature (22) into the Landauer inequality for every model. That's defensible if T is the radiation temperature: Hawking radiation far away is at T_H, and Landauer's bound concerns the environment temperature. But the paper never says this explicitly. If instead one uses the effective thermodynamic temperature dM/dS, which a modified entropy implies through the first law, then every model saturates the bound and the Kaniadakis incompatibility disappears. The paper must state which T is meant and defend the choice. It also should acknowledge that keeping T_H while modifying S means those models do not satisfy the usual first law dM = T dS.

The literature coverage is adequate, with relevant references to the entropy models and Landauer literature. Who is this for? People working on generalized entropies and black hole information theory. It's a useful survey, but I wouldn't rely on the Rényi results or on the three-way classification until the temperature question is answered. It deserves peer review, not a desk reject, but the referee should be told to check the arithmetic carefully. After revision, this could be a solid, modest contribution.

Referee Report

3 major / 5 minor

Summary. The paper proposes to use Landauer's principle, in the form of the inequality ΔM ≥ k_B T ln 2, as a criterion for whether a proposed black hole entropy model is thermodynamically consistent. The physical picture is that each Hawking evaporation step corresponds to the erasure of one bit of horizon information. For the Bekenstein-Hawking entropy, the paper shows that the emitted energy per step saturates the bound. For a collection of generalized entropies (mass-to-horizon, corrected Bekenstein-Hawking, Rényi, Sharma-Mittal, loop-quantum-gravity, hypergeometric, and Kaniadakis), it derives constraints either on the black hole mass/temperature or on the free parameters of the entropy model, and it derives the associated area quantization parameter and relative area spectrum for the compatible models. The Kaniadakis entropy is found to be incompatible with the Landauer criterion for any nonzero deformation parameter.

Significance. If the framework is sound, the paper offers a systematic, information-theoretic classification of generalized black hole entropy models, with parameter constraints that are in principle falsifiable and with explicit predictions for area spectra. The authors are transparent that the one-bit erasure assumption is an input, and they derive the Landauer constraints from the principle rather than building the conclusion into the setup, which is a strength. The unified table and comparative figure are useful for assessing the models side by side. However, the significance is currently limited by several load-bearing algebraic and interpretive problems, most notably the unjustified identification of the temperature in the Landauer bound and errors in the Rényi and Sharma-Mittal derivations.

major comments (3)

[§III, Eq. (24) and Eq. (22)] The paper evaluates the Landauer inequality using the unmodified Schwarzschild Hawking temperature T = 1/(8πG k_B M) for every entropy model, but the mass variation ΔM in each model is computed from the first-law relation ΔS = S'(M)ΔM = k_B ln 2, which by construction gives ΔM = k_B T_th ln 2 with T_th = 1/S'(M). If the temperature in the Landauer bound is meant to be this thermodynamic temperature of the modified entropy, then every model saturates the bound and none of the constraints in Table I follow; if it is meant to be the environment (radiation) temperature, that identification must be stated and defended, because the modified first law does not relate dM to T_H dS. The classification into temperature-constrained, parameter-constrained, and incompatible models depends entirely on this unresolved choice.
[§IV.A, Eqs. (71)–(75)] Eq. (71) for the Rényi area quantization parameter is not the solution of the condition S_{n+1}-S_n = k_B ln 2. With S_n = (k_B/ν) ln(1+νγn/4), the correct solution is γ = 4(2^ν−1)/[ν(2^ν(n+1)−n)], which reduces to γ = 4 ln 2 as ν→0. The published expression instead diverges as −4/[ν(n+1)] in that limit. Consequently, the positivity analysis is internally contradictory: the statement that the numerator of Eq. (71) is always non-positive is false for 0≤ν<1/2, and the claim lim_{ν→0} 1/(2ν−1) = ∞ is incorrect (the limit is −1). Figure 1, which uses ν = 0.006, cannot be produced from Eq. (71) with positive γ, so the Rényi entries in Table I and Fig. 1 are not supported.
[§IV.B, Eq. (79)] The mass variation for the Sharma-Mittal entropy has the wrong exponent. From S_SM = (k_B/ϱ)[(1+ϑS_BH/k_B)^{ϱ/ϑ}−1], the derivative gives ΔM = [1+ϑ/(16πG k_B^2 T^2)]^{(ϑ−ϱ)/ϑ} k_B T ln 2, not [(1−ϱ)/ϑ] as written in Eq. (79). The derivation of the parameter constraint ϑ≥ϱ from Eqs. (80)–(82) relies on the order of the exponent, so the conclusion is not justified by the equation as printed. The correct exponent should be restored and the parameter analysis redone.

minor comments (5)

[General] There are numerous typographical and grammatical errors, including 'R’enyi' instead of 'Rényi' and inconsistent notation for the Sharma-Mittal parameters (ϱ, ϑ in the text but θ in the caption of Fig. 1).
[§II, Eq. (9)] The comparison of the Bekenstein-Mukhanov and Landauer approaches yields k = 2, but the paper does not discuss whether the assumption W_n = k^n with k=2 is compatible with the degeneracy structure of the area spectrum; a brief comment would help.
[§IV.D, Eq. (98)] The derivation of γ for the hypergeometric entropy uses an expansion to first order in ξ and ˜ϵ while neglecting O(ξ˜ϵ) terms; the regime of validity of this approximation should be stated explicitly, especially since the resulting Eq. (98) is then used to draw conclusions about the full parameter space.
[§VI, Fig. 1] The figure caption lists parameter values but does not state which entropy formula is being plotted; for the Rényi curve, the plotted γ appears incompatible with Eq. (71), so the figure should be regenerated from the corrected formulas and the caption should identify the exact expressions used.
[§III.B, Eq. (58)] The inequality T < 1/|α f'(M)| is obtained under the assumption α f'(M) ≤ 0; the paper should state separately that if α f'(M) > 0, no temperature can satisfy the Landauer condition for that model, which is a distinct exclusion case rather than a bound on T.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Landauer inequality is applied as an external criterion to imported entropy formulas; the constraints and Kaniadakis exclusion follow from the stated inequality and no fitted quantity is relabeled as a prediction.

full rationale

The paper's core derivation is not circular. The entropy formulas (Renyi, Sharma-Mittal, LQG, hypergeometric, Kaniadakis, etc.) are taken from prior work as external inputs, and Landauer's principle is then applied as an additional thermodynamic criterion. The mass variations used in the Landauer inequality are obtained by differentiating each imported entropy function with respect to M, and the resulting inequalities are solved for parameter or temperature constraints. None of the claimed constraints are fitted parameters renamed as predictions: the constraints are consequences of imposing Delta M >= k_B T ln 2 on expressions derived from the entropy models. The Bekenstein-Hawking saturation result likewise follows by direct substitution, not by construction. The paper's disagreement with reference [9] over the Renyi parameter branch is an interpretive difference, not a circular reduction. Self-citations to Sheykhi's earlier papers appear only as sources for entropy definitions and correction terms; they are not used to justify the Landauer criterion itself or to exclude alternative entropy models. The potentially weakest step is the identification of the temperature T in inequality (24) with the unmodified Schwarzschild Hawking temperature (22) for all generalized entropies. For a modified S(M), the first law would suggest a model-dependent thermodynamic temperature dM/dS, and with that replacement many constraints, including the Kaniadakis incompatibility, would change. However, this is a physical assumption about the environment temperature and not a circular step: the paper does not define T in terms of the quantities it predicts, nor does it rename a fitted input as an output. Thus the honest circularity finding is that the derivation is self-contained under its stated assumptions, with any concern about the temperature identification belonging to physical correctness rather than circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The framework rests on the quantization postulate, the one-bit erasure assumption, and the choice of Hawking temperature; all entropy formulas and their parameters are imported from prior work. No new entities are introduced.

free parameters (7)

η, λ (mass-to-horizon)
Inputs from the cited mass-to-horizon entropy model; Landauer condition constrains them rather than fitting them to data.
α, β, rc (corrected entropies) = α=0.6, β=0.6, rc=5.5 in Fig. 1
Inputs from cited corrected entropy models; chosen for illustration in Fig. 1, not fitted.
ν (Rényi) = 0.006 in Fig. 1
Free parameter of Rényi entropy; paper derives 0≤ν<1 from Landauer and positivity, but the derivation is flawed.
ϱ, ϑ (Sharma-Mittal) = ϵ=0.6, ϑ=0.6 in Fig. 1
Free parameters inherited from model; constrained by ϑ≥ϱ and a reality condition.
q (LQG) = ¯ϵ=0.6 in Fig. 1
Non-additivity parameter; Landauer requires q≥1.
σ, ξ (hypergeometric) = ξ=0.003 in Fig. 1
Parameters inherited; Landauer requires σ>0 and ξ≥0.
κ (Kaniadakis)
Deformation parameter; paper shows compatibility only at κ=0.

assumptions (6)

domain assumption Horizon area is quantized as A_n = γ l_P^2 n (Eq. (1)).
Bekenstein-Mukhanov quantization, assumed throughout.
domain assumption Each Hawking evaporation step erases exactly one bit, so ΔS = k_B ln 2 (Eq. (20)).
Links information erasure to entropy decrease; used to derive ΔM and γ.
domain assumption Landauer bound ΔM ≥ k_B T ln 2 (Eq. (24)).
Standard principle applied to black hole emission.
ad hoc to paper Temperature in Landauer bound is the unmodified Schwarzschild Hawking temperature T = 1/(8π G k_B M) (Eq. (22)).
Applied to all generalized models; not derived from the modified entropy's first law.
domain assumption Entropy formulas from cited literature (Rényi, Sharma-Mittal, LQG, hypergeometric, Kaniadakis, etc.).
Taken as given from previous works.
standard math Small-parameter expansions (ϵ, ¯ϵ, ˜ϵ, ξ small) are valid for closed-form γ.
Used to solve for γ in several models; validity conditions not fully stated.

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Cite this review

Pith. "Pith review of Landauer's Principle as a Criterion for Thermodynamic Consistency in Generalized Black Hole Entropies." pith.science (2026). https://pith.science/paper/3YIRHBTH

@misc{pith2026260724132,
  author       = {Pith},
  title        = {Pith review of: Landauer's Principle as a Criterion for Thermodynamic Consistency in Generalized Black Hole Entropies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3YIRHBTH}},
  note         = {Machine review of arXiv:2607.24132}
}

read the original abstract

Under the assumption that black hole horizon area is quantized, each Hawking evaporation step, during which the black hole loses mass and transitions to a lower area level, is interpreted as the erasure of one bit of information. In this paper, by employing Landauer's principle, we test the consistency of various black hole entropy relations with this information-theoretic framework. For the Bekenstein-Hawking entropy, the energy emitted per step saturates the Landauer bound. We extend this analysis to a broad class of generalized entropy models, yielding three distinct outcomes. In the first category, Landauer's principle imposes constraints on the Hawking temperature and, consequently, on the black hole mass. In the second, it restricts the free parameters of the entropy model. The third category, exemplified by Kaniadakis entropy, proves incompatible with Landauer's principle. For all compatible models, we derive the area quantization parameter and the corresponding area spectrum. While this parameter is constant for Bekenstein-Hawking entropy, it becomes level-number-dependent for many generalized models. Nonetheless, the relative spacing between successive area levels vanishes in the classical limit. Our findings point to a deep link between information theory and black hole thermodynamics.

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Figures reproduced from arXiv: 2607.24132 by the authors.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [9]

J. D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole , Phys. Lett. B 360, 7 (1995)

work page 1995

[2] [1]

Christodoulou, Reversible and irreversible transfor- mations in black-hole physics , Phys

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