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16 pages, 327 KB

Open AccessArticle

Toward a Tripartite Taxonomy of Entropy in Physics

by Antoine Druilhe

Entropy 2026, 28(6), 704; https://doi.org/10.3390/e28060704 - 18 Jun 2026

The term “entropy” denotes several mathematically distinct quantities across modern physics, including thermodynamic, statistical, quantum-informational, and geometric notions that are often conflated in foundational discussions. We propose an operational distinction among three such quantities: a geometric capacity entropy

S_{capacity}

proportional to a [...] Read more.

The term “entropy” denotes several mathematically distinct quantities across modern physics, including thermodynamic, statistical, quantum-informational, and geometric notions that are often conflated in foundational discussions. We propose an operational distinction among three such quantities: a geometric capacity entropy

S_{capacity}

proportional to a region’s bounding area, a microscopic content entropy

S_{content}

given by the fine-grained von Neumann entropy of the reduced state, and a thermodynamic entropy

S_{thermo}

corresponding to the observer-relative ignorance that remains after accessible information is accounted for. We argue that keeping these quantities distinct is not merely terminological: within this framework, the second law of thermodynamics can be formulated as a typical consequence of unitary dynamics combined with bounded observational access, rather than as an independent postulate. The distinction also clarifies which entropy enters established results such as the Bekenstein–Hawking entropy of black holes and the Clausius relation in Jacobson’s thermodynamic derivation of Einstein’s equations. The proposed framework is conceptual and does not modify established physical theories; it is intended as a useful clarification for informational approaches to physics. Full article

(This article belongs to the Section Quantum Information)

20 pages, 1826 KB

Open AccessArticle

Entropy, Information, and the Curvature of Spacetime in the Informational Second Law

by Florian Neukart, Eike Marx and Valerii Vinokur

Information 2026, 17(2), 169; https://doi.org/10.3390/info17020169 - 6 Feb 2026

Viewed by 1266

Abstract

We develop an informational extension of spacetime thermodynamics in which local entropy production is coupled to spacetime curvature within an effective covariant framework. Spacetime is modeled as a continuum limit of finite-capacity information registers, giving rise to a coarse-grained entropy field whose gradients [...] Read more.

We develop an informational extension of spacetime thermodynamics in which local entropy production is coupled to spacetime curvature within an effective covariant framework. Spacetime is modeled as a continuum limit of finite-capacity information registers, giving rise to a coarse-grained entropy field whose gradients define an informational flux. Within a nonminimally coupled scalar–tensor formulation, the resulting field equations imply that the local divergence of this flux is sourced by the Ricci scalar, establishing a direct relation between curvature and entropy production. The corresponding integral form links cumulative entropy generation to the integrated spacetime curvature over a causal region. In stationary limits, the framework reproduces the Bekenstein–Hawking entropy of horizons, while in homogeneous expanding cosmologies it yields monotonic entropy growth consistent with the observed arrow of time. The construction remains compatible with unitarity at the microscopic level and with holographic entropy bounds in the stationary limit. Numerical solutions in flat FLRW backgrounds are used as consistency checks of the coupled evolution equations and confirm the expected curvature–entropy behavior across cosmological epochs. Overall, the results provide a thermodynamically consistent interpretation of curvature as a geometric source of irreversible information flow, without modifying the underlying gravitational field equations. Full article

(This article belongs to the Section Information Theory and Methodology)

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18 pages, 957 KB

Open AccessArticle

Landauer Bound in the Context of Minimal Physical Principles: Meaning, Experimental Verification, Controversies and Perspectives

by Edward Bormashenko

Entropy 2024, 26(5), 423; https://doi.org/10.3390/e26050423 - 15 May 2024

Cited by 15 | Viewed by 10335

Abstract

The physical roots, interpretation, controversies, and precise meaning of the Landauer principle are surveyed. The Landauer principle is a physical principle defining the lower theoretical limit of energy consumption necessary for computation. It states that an irreversible change in information stored in a [...] Read more.

The physical roots, interpretation, controversies, and precise meaning of the Landauer principle are surveyed. The Landauer principle is a physical principle defining the lower theoretical limit of energy consumption necessary for computation. It states that an irreversible change in information stored in a computer, such as merging two computational paths, dissipates a minimum amount of heat

k_{B} T l n 2

per a bit of information to its surroundings. The Landauer principle is discussed in the context of fundamental physical limiting principles, such as the Abbe diffraction limit, the Margolus–Levitin limit, and the Bekenstein limit. Synthesis of the Landauer bound with the Abbe, Margolus–Levitin, and Bekenstein limits yields the minimal time of computation, which scales as

τ_{m i n} ~ \frac{h}{k_{B} T}

. Decreasing the temperature of a thermal bath will decrease the energy consumption of a single computation, but in parallel, it will slow the computation. The Landauer principle bridges John Archibald Wheeler’s “it from bit” paradigm and thermodynamics. Experimental verifications of the Landauer principle are surveyed. The interrelation between thermodynamic and logical irreversibility is addressed. Generalization of the Landauer principle to quantum and non-equilibrium systems is addressed. The Landauer principle represents the powerful heuristic principle bridging physics, information theory, and computer engineering. Full article

(This article belongs to the Special Issue The Landauer Principle and Its Implementations in Physics, Chemistry and Biology: Current Status, Critics and Controversies)

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25 pages, 423 KB

Open AccessFeature PaperArticle

String Theory Bounds on the Cosmological Constant, the Higgs Mass, and the Quark and Lepton Masses

by Per Berglund, Tristan Hübsch and Djordje Minic

Symmetry 2023, 15(9), 1660; https://doi.org/10.3390/sym15091660 - 28 Aug 2023

Cited by 6 | Viewed by 3398

Abstract

In this paper, we elaborate on the new understanding of the cosmological constant and the gauge hierarchy problems in the context of string theory in its metastring formulation, based on the concepts of modular spacetime and Born geometry. The interplay of phase space [...] Read more.

In this paper, we elaborate on the new understanding of the cosmological constant and the gauge hierarchy problems in the context of string theory in its metastring formulation, based on the concepts of modular spacetime and Born geometry. The interplay of phase space (and Born geometry), the Bekenstein bound, the mixing between ultraviolet (UV) and infrared (IR) physics and modular invariance in string theory is emphasized. This new viewpoint is fundamentally rooted in quantum contextuality and not in statistical observer bias (anthropic principle). We also discuss the extension of this point of view to the problem of masses of quarks and leptons and their respective mixing matrices. Full article

(This article belongs to the Special Issue Fundamental Aspects of Theoretical Physics - Memorial Issue for Prof. Dr. Weinberg)

7 pages, 243 KB

Open AccessArticle

Black Hole Information Paradox without Hawking Radiation

by Hrvoje Nikolić

Universe 2023, 9(1), 11; https://doi.org/10.3390/universe9010011 - 23 Dec 2022

Cited by 1 | Viewed by 2865

Abstract

By entangling soft massless particles, one can create an arbitrarily large amount of entanglement entropy that carries an arbitrarily small amount of energy. By dropping this entropy into the black hole (b.h.), one can increase the b.h. entropy by an amount that violates [...] Read more.

By entangling soft massless particles, one can create an arbitrarily large amount of entanglement entropy that carries an arbitrarily small amount of energy. By dropping this entropy into the black hole (b.h.), one can increase the b.h. entropy by an amount that violates the Bekenstein bound or any other reasonable bound, leading to a version of the b.h. information paradox that does not involve Hawking radiation. Among the many proposed solutions for the standard b.h. information paradox with Hawking radiation, only a few can also resolve this version without Hawking radiation. The assumption that both versions should be resolved in the same way significantly helps to reduce the number of possible resolutions. Full article

(This article belongs to the Section Compact Objects)

6 pages, 241 KB

Open AccessCommunication

Bekenstein Bound and Non-Commutative Canonical Variables

by Fabio Scardigli

Universe 2022, 8(12), 645; https://doi.org/10.3390/universe8120645 - 5 Dec 2022

Cited by 2 | Viewed by 2780

Abstract

A universal upper limit on the entropy contained in a localized quantum system of a given size and total energy is expressed by the so-called Bekenstein bound. In a previous paper [Buoninfante, L. et al. 2022], on the basis of general thermodynamic arguments, [...] Read more.

A universal upper limit on the entropy contained in a localized quantum system of a given size and total energy is expressed by the so-called Bekenstein bound. In a previous paper [Buoninfante, L. et al. 2022], on the basis of general thermodynamic arguments, and in regimes where the equipartition theorem still holds, the Bekenstein bound has been proved practically equivalent to the Heisenberg uncertainty relation. The smooth transition between the Bekenstein bound and the holographic bound suggests a new pair of canonical non-commutative variables, which could be thought to hold in strong gravity regimes. Full article

(This article belongs to the Special Issue The Quantum & The Gravity)

12 pages, 768 KB

Open AccessArticle

Bekenstein’s Entropy Bound-Particle Horizon Approach to Avoid the Cosmological Singularity

by James R. Powell, Rafael Lopez-Mobilia and Richard A. Matzner

Entropy 2020, 22(7), 795; https://doi.org/10.3390/e22070795 - 21 Jul 2020

Cited by 3 | Viewed by 6369

Abstract

The cosmological singularity of infinite density, temperature, and spacetime curvature is the classical limit of Friedmann’s general relativity solutions extrapolated to the origin of the standard model of cosmology. Jacob Bekenstein suggests that thermodynamics excludes the possibility of such a singularity in a [...] Read more.

The cosmological singularity of infinite density, temperature, and spacetime curvature is the classical limit of Friedmann’s general relativity solutions extrapolated to the origin of the standard model of cosmology. Jacob Bekenstein suggests that thermodynamics excludes the possibility of such a singularity in a 1989 paper. We propose a re-examination of his particle horizon approach in the early radiation-dominated universe and verify it as a feasible alternative to the classical inevitability of the singularity. We argue that this minimum-radius particle horizon determined from Bekenstein’s entropy bound, necessarily quantum in nature as a quantum particle horizon (QPH), precludes the singularity, just as quantum mechanics provided the solution for singularities in atomic transitions as radius

r \to 0

. An initial radius of zero can never be attained quantum mechanically. This avoids the spacetime singularity, supporting Bekenstein’s assertion that Friedmann models cannot be extrapolated to the very beginning of the universe but only to a boundary that is ‘something like a particle horizon’. The universe may have begun in a bright flash and quantum flux of radiation and particles at a minimum, irreducible quantum particle horizon rather than at the classical mathematical limit and unrealizable state of an infinite singularity. Full article

(This article belongs to the Section Astrophysics, Cosmology, and Black Holes)

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8 pages, 230 KB

Open AccessArticle

Entropy Balance in the Expanding Universe: A Novel Perspective

by Arturo Tozzi and James F. Peters

Entropy 2019, 21(4), 406; https://doi.org/10.3390/e21040406 - 17 Apr 2019

Cited by 3 | Viewed by 7568

Abstract

We describe cosmic expansion as correlated with the standpoints of local observers’ co-moving horizons. In keeping with relational quantum mechanics, which claims that quantum systems are only meaningful in the context of measurements, we suggest that information gets ergodically “diluted” in our isotropic [...] Read more.

We describe cosmic expansion as correlated with the standpoints of local observers’ co-moving horizons. In keeping with relational quantum mechanics, which claims that quantum systems are only meaningful in the context of measurements, we suggest that information gets ergodically “diluted” in our isotropic and homogeneous expanding Universe, so that an observer detects just a limited amount of the total cosmic bits. The reduced bit perception is due the decreased density of information inside the expanding cosmic volume in which the observer resides. Further, we show that the second law of thermodynamics can be correlated with cosmic expansion through a relational mechanism, because the decrease in information detected by a local observer in an expanding Universe is concomitant with an increase in perceived cosmic thermodynamic entropy, via the Bekenstein bound and the Laudauer principle. Reversing the classical scheme from thermodynamic entropy to information, we suggest that the cosmological constant of the quantum vacuum, which is believed to provoke the current cosmic expansion, could be one of the sources of the perceived increases in thermodynamic entropy. We conclude that entropies, including the entangled entropy of the recently developed framework of quantum computational spacetime, might not describe independent properties, but rather relations among systems and observers. Full article

(This article belongs to the Special Issue Quantum Spacetime and Entanglement Entropy)

37 pages, 301 KB

Open AccessArticle

Universal Property of Quantum Gravity implied by Uniqueness Theorem of Bekenstein-Hawking Entropy

by Hiromi Saida

Entropy 2011, 13(9), 1611-1647; https://doi.org/10.3390/e13091611 - 5 Sep 2011

Cited by 13 | Viewed by 8057

Abstract

This paper consists of three parts. In the first part, we prove that the Bekenstein-Hawking entropy is the unique expression of black hole entropy. Our proof is constructed in the framework of thermodynamics without any statistical discussion. In the second part, intrinsic properties [...] Read more.

This paper consists of three parts. In the first part, we prove that the Bekenstein-Hawking entropy is the unique expression of black hole entropy. Our proof is constructed in the framework of thermodynamics without any statistical discussion. In the second part, intrinsic properties of quantum mechanics are shown, which justify the Boltzmann formula to yield a unique entropy in statistical mechanics. These properties clarify three conditions, one of which is necessary and others are sufficient for the validity of Boltzmann formula. In the third part, by combining the above results, we find a reasonable suggestion from the sufficient conditions that the potential of gravitational interaction among microstates of underlying quantum gravity may not diverge to negative infinity (such as Newtonian gravity) but is bounded below at a finite length scale. In addition to that, from the necessary condition, the interaction has to be repulsive within the finite length scale. The length scale should be Planck size. Thus, quantum gravity may become repulsive at Planck length. Also, a relation of these suggestions with action integral of gravity at semi-classical level is given. These suggestions about quantum gravity are universal in the sense that they are independent of any existing model of quantum gravity. Full article

(This article belongs to the Special Issue Black Hole Thermodynamics)

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10 pages, 158 KB

Open AccessArticle

Entropy Bounds, Holographic Principle and Uncertainty Relation

by M. G. Ivanov and I. V. Volovich

Entropy 2001, 3(2), 66-75; https://doi.org/10.3390/e3020066 - 20 Jun 2001

Cited by 9 | Viewed by 6594

Abstract

A simple derivation of the bound on entropy is given and the holographic principle is discussed. We estimate the number of quantum states inside space region on the base of uncertainty relation. The result is compared with the Bekenstein formula for entropy bound, [...] Read more.

A simple derivation of the bound on entropy is given and the holographic principle is discussed. We estimate the number of quantum states inside space region on the base of uncertainty relation. The result is compared with the Bekenstein formula for entropy bound, which was initially derived from the generalized second law of thermodynamics for black holes. The holographic principle states that the entropy inside a region is bounded by the area of the boundary of that region. This principle can be called the kinematical holographic principle. We argue that it can be derived from the dynamical holographic principle which states that the dynamics of a system in a region should be described by a system which lives on the boundary of the region. This last principle can be valid in general relativity because the ADM hamiltonian reduces to the surface term. Full article

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