Research

292 KiB

Open AccessArticle

Coarse Graining Shannon and von Neumann Entropies

by Ana Alonso-Serrano and Matt Visser

Entropy 2017, 19(5), 207; https://0-doi-org.brum.beds.ac.uk/10.3390/e19050207 - 03 May 2017

Cited by 14 | Viewed by 6627

The nature of coarse graining is intuitively “obvious”, but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some [...] Read more.

The nature of coarse graining is intuitively “obvious”, but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes an arbitrary system, with specified initial entropy S, and that monotonically and controllably drives the entropy to its maximum value. This does not have to be a physical process, in fact for some purposes it is better to deal with a gedanken-process, since then it is more obvious how the “hidden information” is hiding in the fine-grain correlations that one is simply agreeing not to look at. We shall present several simple mathematically well-defined and easy to work with conceptual models for coarse graining. We shall consider both the classical Shannon and quantum von Neumann entropies, including models based on quantum decoherence, and analyse the entropy flow in some detail. When coarse graining the quantum von Neumann entropy, we find it extremely useful to introduce an adaptation of Hawking’s super-scattering matrix. These explicit models that we shall construct allow us to quantify and keep clear track of the entropy that appears when coarse graining the system and the information that can be hidden in unobserved correlations (while not the focus of the current article, in the long run, these considerations are of interest when addressing the black hole information puzzle). Full article

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

390 KiB

Open AccessArticle

Spacetime Topology and the Laws of Black Hole-Soliton Mechanics

by Hari K. Kunduri

Entropy 2017, 19(1), 35; https://0-doi-org.brum.beds.ac.uk/10.3390/e19010035 - 17 Jan 2017

Viewed by 3556

Abstract

The domain of outer communication of an asymptotically flat spactime must be simply connected. In five dimensions, this still allows for the possibility of an arbitrary number of 2-cycles supported by magnetic flux carried by Maxwell fields. As a result, stationary, asymptotically flat, [...] Read more.

The domain of outer communication of an asymptotically flat spactime must be simply connected. In five dimensions, this still allows for the possibility of an arbitrary number of 2-cycles supported by magnetic flux carried by Maxwell fields. As a result, stationary, asymptotically flat, horizonless solutions—“gravitational solitons”—may exist with non-vanishing mass, charge, and angular momenta. These gravitational solutions satisfy a Smarr-like relation, as well as a first law of mechanics. Furthermore, the presence of solitons leads to new terms in the well-known first law of black hole mechanics for spacetimes containing black hole horizons and non-trivial topology in the exterior region. I outline the derivation of these results and consider an explicit example in five-dimensional supergravity. Full article

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

6128 KiB

Open AccessArticle

Static Einstein–Maxwell Magnetic Solitons and Black Holes in an Odd Dimensional AdS Spacetime

by Jose Luis Blázquez-Salcedo, Jutta Kunz, Francisco Navarro-Lérida and Eugen Radu

Entropy 2016, 18(12), 438; https://0-doi-org.brum.beds.ac.uk/10.3390/e18120438 - 08 Dec 2016

Cited by 11 | Viewed by 4318

Abstract

We construct a new class of Einstein–Maxwell static solutions with a magnetic field in D-dimensions (with

D \geq 5

an odd number), approaching at infinity a globally Anti-de Sitter (AdS) spacetime. In addition to the mass, the new solutions possess an extra-parameter [...] Read more.

We construct a new class of Einstein–Maxwell static solutions with a magnetic field in D-dimensions (with

D \geq 5

an odd number), approaching at infinity a globally Anti-de Sitter (AdS) spacetime. In addition to the mass, the new solutions possess an extra-parameter associated with a non-zero magnitude of the magnetic potential at infinity. Some of the black holes possess a non-trivial zero-horizon size limit, which corresponds to a solitonic deformation of the AdS background. Full article

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

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225 KiB

Open AccessArticle

Simple Harmonic Oscillator Canonical Ensemble Model for Tunneling Radiation of Black Hole

by Jinbo Yang, Tangmei He and Jingyi Zhang

Entropy 2016, 18(11), 415; https://0-doi-org.brum.beds.ac.uk/10.3390/e18110415 - 23 Nov 2016

Cited by 1 | Viewed by 4812

Abstract

A simple harmonic oscillator canonical ensemble model for Schwarzchild black hole quantum tunneling radiation is proposed in this paper. Firstly, the equivalence between canonical ensemble model and Parikh–Wilczek’s tunneling method is introduced. Then, radiated massless particles are considered as a collection of simple [...] Read more.

A simple harmonic oscillator canonical ensemble model for Schwarzchild black hole quantum tunneling radiation is proposed in this paper. Firstly, the equivalence between canonical ensemble model and Parikh–Wilczek’s tunneling method is introduced. Then, radiated massless particles are considered as a collection of simple harmonic oscillators. Based on this model, we treat the black hole as a heat bath to derive the energy flux of the radiation. Finally, we apply the result to estimate the lifespan of a black hole. Full article

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

562 KiB

Open AccessArticle

Thermodynamics of Noncommutative Quantum Kerr Black Holes

by Lenin F. Escamilla-Herrera, Eri A. Mena-Barboza and José Torres-Arenas

Entropy 2016, 18(11), 406; https://0-doi-org.brum.beds.ac.uk/10.3390/e18110406 - 16 Nov 2016

Cited by 7 | Viewed by 3711

Abstract

The thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state and thermodynamic response functions are explicitly given, and the effect of noncommutativity and quantum correction is discussed. It is [...] Read more.

The thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state and thermodynamic response functions are explicitly given, and the effect of noncommutativity and quantum correction is discussed. It is shown that the well-known divergence exhibited in specific heat is not removed by any of these corrections. However, regions of thermodynamic stability are affected by noncommutativity, increasing the available states for which some thermodynamic stability conditions are satisfied. Full article

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

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280 KiB

Open AccessFeature PaperArticle

When Is an Area Law Not an Area Law?

by Anushya Chandran, Chris Laumann and Rafael D. Sorkin

Entropy 2016, 18(7), 240; https://0-doi-org.brum.beds.ac.uk/10.3390/e18070240 - 24 Jun 2016

Cited by 13 | Viewed by 3607

Abstract

Entanglement entropy is typically proportional to area, but sometimes it acquires an additional logarithmic pre-factor. We offer some intuitive explanations for these facts. Full article

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

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267 KiB

Open AccessArticle

Extended First Law for Entanglement Entropy in Lovelock Gravity

by David Kastor, Sourya Ray and Jennie Traschen

Entropy 2016, 18(6), 212; https://0-doi-org.brum.beds.ac.uk/10.3390/e18060212 - 30 May 2016

Cited by 13 | Viewed by 3856

Abstract

The first law for the holographic entanglement entropy of spheres in a boundary CFT (Conformal Field Theory) with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a [...] Read more.

The first law for the holographic entanglement entropy of spheres in a boundary CFT (Conformal Field Theory) with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a family of boundary CFTs. The new contribution to the first law is found to be the product of the variation

δ a

of the “A”-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension

δ a^{*}

to include odd dimensional boundaries, times the ratio

S / a^{*}

. Since

a^{*}

is a measure of the number of degrees of freedom N per unit volume of the boundary CFT, this new term has the form

μ δ N

, where the chemical potential μ is given by the entanglement entropy per degree of freedom. Full article

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

730 KiB

Open AccessArticle

Charged, Rotating Black Objects in Einstein–Maxwell-Dilaton Theory in D ≥ 5

by Burkhard Kleihaus, Jutta Kunz and Eugen Radu

Entropy 2016, 18(5), 187; https://0-doi-org.brum.beds.ac.uk/10.3390/e18050187 - 16 May 2016

Cited by 1 | Viewed by 4000

Abstract

We show that the general framework proposed by Kleihaus et al. (2015) for the study of asymptotically flat vacuum black objects with

k + 1

equal magnitude angular momenta in

D \geq 5

spacetime dimensions (with [...] Read more.

We show that the general framework proposed by Kleihaus et al. (2015) for the study of asymptotically flat vacuum black objects with

k + 1

equal magnitude angular momenta in

D \geq 5

spacetime dimensions (with

0 \leq k \leq [\frac{D - 5}{2}]

) can be extended to the case of Einstein–Maxwell-dilaton (EMd) theory. This framework can describe black holes with spherical horizon topology, the simplest solutions corresponding to a class of electrically charged (dilatonic) Myers–Perry black holes. Balanced charged black objects with

S^{n + 1} \times S^{2 k + 1}

horizon topology can also be studied (with

D = 2 k + n + 4

). Black rings correspond to the case

k = 0

, while the solutions with

k > 0

are black ringoids. The basic properties of EMd solutions are discussed for the special case of a Kaluza–Klein value of the dilaton coupling constant. We argue that all features of these solutions can be derived from those of the vacuum seed configurations. Full article

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

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315 KiB

Open AccessFeature PaperArticle

Pressure and Compressibility of Conformal Field Theories from the AdS/CFT Correspondence

by Brian P. Dolan

Entropy 2016, 18(5), 169; https://0-doi-org.brum.beds.ac.uk/10.3390/e18050169 - 03 May 2016

Cited by 30 | Viewed by 4331

Abstract

The equation of state associated with

N = 4

supersymmetric Yang–Mills in four dimensions, for

S U (N)

in the large N limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background [...] Read more.

The equation of state associated with

N = 4

supersymmetric Yang–Mills in four dimensions, for

S U (N)

in the large N limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background for the Yang–Mills theory on the boundary in which the cosmological constant is equivalent to a volume. The thermodynamic variable conjugate to the cosmological constant is a pressure, and the

P - V

diagram of the quark-gluon plasma is studied. It is known that there is a critical point where the heat capacity diverges, and this is reflected in the isothermal compressibility. Critical exponents are derived and found to be mean field in the large N limit. The same analysis applied to three- and six-dimensional conformal field theories again yields mean field exponents associated with the compressibility at the critical point. Full article

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

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820 KiB

Open AccessFeature PaperArticle

An Exact Efficiency Formula for Holographic Heat Engines

by Clifford V. Johnson

Entropy 2016, 18(4), 120; https://0-doi-org.brum.beds.ac.uk/10.3390/e18040120 - 31 Mar 2016

Cited by 68 | Viewed by 5037

Abstract

Further consideration is given to the efficiency of a class of black hole heat engines that perform mechanical work via the pdV terms present in the First Law of extended gravitational thermodynamics. It is noted that, when the engine cycle is a rectangle [...] Read more.

Further consideration is given to the efficiency of a class of black hole heat engines that perform mechanical work via the pdV terms present in the First Law of extended gravitational thermodynamics. It is noted that, when the engine cycle is a rectangle with sides parallel to the (p,V) axes, the efficiency can be written simply in terms of the mass of the black hole evaluated at the corners. Since an arbitrary cycle can be approximated to any desired accuracy by a tiling of rectangles, a general geometrical algorithm for computing the efficiency of such a cycle follows. A simple generalization of the algorithm renders it applicable to broader classes of heat engine, even beyond the black hole context. Full article

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

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225 KiB

Open AccessArticle

Generalized Einstein’s Equations from Wald Entropy

by Maulik Parikh and Sudipta Sarkar

Entropy 2016, 18(4), 119; https://0-doi-org.brum.beds.ac.uk/10.3390/e18040119 - 31 Mar 2016

Cited by 42 | Viewed by 4934

Abstract

We derive the gravitational equations of motion of general theories of gravity from thermodynamics applied to a local Rindler horizon through any point in spacetime. Specifically, for a given theory of gravity, we substitute the corresponding Wald entropy into the Clausius relation. Our [...] Read more.

We derive the gravitational equations of motion of general theories of gravity from thermodynamics applied to a local Rindler horizon through any point in spacetime. Specifically, for a given theory of gravity, we substitute the corresponding Wald entropy into the Clausius relation. Our approach works for all diffeomorphism-invariant theories of gravity in which the Lagrangian is a polynomial in the Riemann tensor. Full article

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

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844 KiB

Open AccessArticle

Reentrant Phase Transitions and van der Waals Behaviour for Hairy Black Holes

by Robie A. Hennigar and Robert B. Mann

Entropy 2015, 17(12), 8056-8072; https://0-doi-org.brum.beds.ac.uk/10.3390/e17127862 - 08 Dec 2015

Cited by 91 | Viewed by 5488

Abstract

We study the extended phase space thermodynamics for hairy AdS black hole solutions to Einstein-Maxwell-Λ theory conformally coupled to a scalar field in five dimensions. We find these solutions to exhibit van der Waals behaviour in both the charged/uncharged cases, and reentrant phase [...] Read more.

We study the extended phase space thermodynamics for hairy AdS black hole solutions to Einstein-Maxwell-Λ theory conformally coupled to a scalar field in five dimensions. We find these solutions to exhibit van der Waals behaviour in both the charged/uncharged cases, and reentrant phase transitions in the charged case. This is the first example of reentrant phase transitions in a five dimensional gravitational system which does not include purely gravitational higher curvature corrections. Full article

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

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