Gibbs–Jaynes Entropy Versus Relative Entropy

The maximum entropy formalism developed by Jaynes determines the relevant ensemble in nonequilibrium statistical mechanics by maximising the entropy functional subject to the constraints imposed by the available information. We present an alternative derivation of the relevant ensemble based on the Kullback–Leibler divergence from equilibrium. If the equilibrium ensemble is already known, then calculation of the relevant ensemble is considerably simplified. The constraints must be chosen with care in order to avoid contradictions between the two alternative derivations. The relative entropy functional measures how much a distribution departs from equilibrium. Therefore, it provides a distinct approach to the calculation of statistical ensembles that might be applicable to situations in which the formalism presented by Jaynes performs poorly (such as non-ergodic dynamical systems).     

Relative Entropy of Entanglement of a Kind of Two-Qubit Entangled States

We strictly prove that some block diagonalizable two-qubit entangled state with six non-zero elements reaches its quantum relative entropy entanglement by a separable state having the same matrix structure. The entangled state comprises local filtering result state as a special case.     

Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy

A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a “sandwiched” Rényi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.     

Mixed-Up-Ness or Entropy?

Simple SummaryThe second law of thermodynamics has a mystical appeal in disciplines with tenuous connections to its origins. We hypothesize that many of these appeals instead should be to another principle heretofore unrecognized: the law of mixed-up-ness (LOM). Instead of using a number such as entropy to characterize randomness, non-thermodynamic systems can be arranged in simple diagrams according to their degree of mixed-up-ness. Curiously, the evolution of such systems from degrees of low to high mixed-up-ness is both consistent with and richer than the principle of increasing entropy.AbstractMixed-up-ness can be traced to unpublished notes by Josiah Gibbs. Subsequently, the concept was developed independently, and under somewhat different names, by other investigators. The central idea of mixed-up-ness is that systems states can be organized in a hierarchy by their degree of mixed-up-ness. In its purest form, the organizing principle is independent of thermodynamic and statistical mechanics principles, nor does it imply irreversibility. Yet, Gibbs and subsequent investigators kept entropy as the essential concept in determining system evolution, thus retaining the notion that systems evolve from states of perfect “order” to states of total “disorder”. Nevertheless, increasing mixed-up-ness is consistent with increasing entropy; however, there is no unique one-to-one connection between the two. We illustrate the notion of mixed-up-ness with an application to the permutation function of integer partitions and then formalize the notion of mixed-up-ness as a fundamental hierarchal principle, the law of mixed-up-ness (LOM), for non-thermodynamic systems.     

A Geometric Criterion for Positive Topological Entropy¶II: Homoclinic Tangencies

In a series of important papers [GS1,GS2] Gavrilov and Shilnikov established a topological conjugacy between a surface diffeomorphism having a dissipative hyperbolic periodic point with certain types of quadratic homoclinic tangencies and the full shift on two symbols, thus exhibiting horseshoes near a tangential homoclinic point. In this note, which should be viewed of as an addendum to [BW] we extend this result by showing that such a diffeomorphism with a one-sided isolated homoclinic tangency having any order contact, possible with infinite order contact, possesses a horseshoe near the homoclinic point. Received: 2 March 1999 / Accepted: 14 May 1999     

A New Holographic Entropy Bound from Conformal Field Theory at the Killing Horizon

A new holographic entropy bound is obtained by using conformal field theory at the Killing horizon.The entropy bound is tighter than the well-known bounds,such as the Bekenstein,Bekenstein-Mayo and‘t Hooft bounds.The result shows that the entropy of a system decreases when quantum effects are included.Therefore,the quantum effect will increase the degree of order of the system.     

A New Method for the Analysis of Relative Permeability in Porous Media

By combining three-dimensional digital microtomography techniques with the lattice Boltzmann method,a new methodology is used to analyse the relative permeability of multiphase flow in porous media.The results indicate that the two coupling coefficients K12 and K21 have the same magnitude,therefore the Onsager reciprocity still holds,the results also agree well with the results of pipe flow numerical experiments.     

Kolmogorov–Sinaj Entropy on MV-Algebras

There is given a construction of the entropy of a dynamical system on arbitrary MV-algebra M. If M is the MV-algebra of characteristic functions of a σ-algebra (isomorphic to the σ-algebra), then the construction leads to the Kolmogorov–Sinaj entropy. If M is the MV-algebra (tribe) of fuzzy sets, then the construction coincides with the Maličky modification of the Kolmogorov–Sinaj entropy for fuzzy sets (Maličky and Riečan, 1986; Riečan and Mundici, 2002; Riečan and Neubrunn, 1997).     

Entropy of Baker‘s Transformation

Four theorems about four different kinds of entropies for Baker‘s transformation are presented.The Kolmogorov entropy of Baker‘s transformation is sensitive to the initial flips by the time.The topological entropy of Baker‘s transformation is found to be lok k.The conditions for the state of Baker‘s transformation to be forbidden are also derived.The relations among the Shanonn,Kolmogorov,topological and Boltzmann entropies are discussed in details.     

Horowitz-Strominger Black Hole Entropy Without Brick Wall

A Horowitz-Strominger black hole is discussed through a new equation of state density motivated by the gener-alized uncertainty relation in quantum gravity. There is no burst in the last stage of emission from a Horowitz-Strominger black hole. When the new equation of state density is used to investigate the entropy of bosonic fieldand fermionic field outside the horizon of a static Horowitz-Strominger black hole, the divergence that appearsin the brick-wall model is removed without any cutoff. The entropy proportional to the horizon area is derivedfrom the contribution in the vicinity of the horizon.     

Cosmic Entropy from the Bosonic String?

In Kaluza-Klein models, the compactification ofa high number of extra spatial dimensions generatesentropy in the observable four-dimensional universe. AKaluza–Klein cosmological model recently derived from the bosonic string theory in the limit ofan infinite number of extra dimensions is compared withthe available data from the observations of cosmicmicrowave background anisotropies.     

Stretching Generic Pesin’s Entropy Formula

Journal of Statistical Physics - We prove that Pesin’s entropy formula holds generically within a broad subset of volume-preserving bi-Lipschitz homeomorphisms with respect to the...     

Statistical-Mechanical Entropy of Garfinkle-Horowitz-Strominger Black Hole

Taking WKB approximation to solve the scalar field equation in the Garfinkle-Horowitz-Strominger (GHS) black hole spacetime, we can get the classical momenta. Substituting the classical momenta into state density equation corrected by the generalized uncertainty principle, we will obtain the number of quantum states with energy less than ω. It is convergent in the neighborhood of the horizon. Then, it is used to calculate the statistical-mechanical entropy of the scalar field in the GHS black hole spacetime. The calculation shows that the entropy is proportional to the horizon area.     

Entropy of the Kerr–Sen black hole

We study the entropy of Kerr–Sen black hole of heterotic string theory beyond semiclassical approximations. Applying the properties of exact differentials for three variables to the first law of thermodynamics, we derive the corrections to the entropy of the black hole. The leading (logarithmic) and non-leading corrections to the area law are obtained.     

Black Hole Entropy: Inside or Out?

A trialogue. Ted, Don, and Carlo consider the nature of black hole entropy. Ted and Carlo support the idea that this entropy measures in some sense “the number of black hole microstates that can communicate with the outside world.” Don is critical of this approach, and discussion ensues, focusing on the question of whether the first law of black hole thermodynamics can be understood from a statistical mechanics point of view.     

Statistical Entropy of Horowitz—Strominger Black Hole

The partition functions of bosonic and fermionic fields in Horowitz-Strominger black hole are derived directly by quantum statistical method.Then via the improved brick-wall method (membrane model),the statistical entropy of black hole is obtained.If a proper parameter is chosen in our result,it is found out that the entropy is proportional to the area of horizon.The stripped term and the divergent logarithmic term in the original brick-wall method no longer exist.The difficulty in solving the wave equations of scalar and Dirac fields is avoided.A new neat way of calculating the entropy of various complicated black holes is offered.     

Cosmic Neutrino Time Delay Relative to Photons

By solving the Friedmann-Robertson-Walker (FRW) geodesic equations for a free test particle with finite mass,we extend the widely used time-of-flight delay expression, which is just valid locally in the neighbourhood of our Galaxy, to the cosmic distance scale. If neutrino masses are known, this may provide a potential method to determine a large scale geometry of the Universe.     

Holographic Entropy Bound of a Nonstationary Black Hole

In accordance with the holographic principle, by counting the states of the scalar field just at the event horizon of the Vaidya-Bonner black hole, the holographic entropy bound of the black hole is calculated and the Bekenstein- Hawking formula is obtained, With the generalized uncertainty principle, the divergence of state density at event horizon in the ordinary quantum field theory is removed, With the residue theorem, the integral trouble in the calculation is overcome. The present result is quantitatively tenable and the holographic principle is realized by applying the quantum field theory to the black hole entropy problem. Compared with some previous works, it is suggested that the quantum states contributing to black hole entropy should be restricted on the event horizon.     

Quintessence Contribution to a Schwarzschild Black Hole Entropy

The quintessence contribution to a Schwarzschild black hole entropy is thoroughly investigated by using the improved brick-wall model. It is found that due to the present of the quintessence, there are two horizons, one is the cosmological horizon, and the other is the black hole horizon. By regulating the cut-off factor ε and the thickness of the thin layer 5, we obtain that the entropy of the system is 1/4 of the sum of the areas of the two event horizons.