Logarithmic Entropy Research Articles

Quantum tunneling of Park black hole in IR modified Ho $$\check{\mathrm{r}}$$ ava gravity with cosmological constant

In this paper, we study the quantum tunneling of non-asymptotically flat Park black hole in IR modified Hořava gravity, as well as its thermodynamical stability. In order to calculate the quantum tunneling more comprehensively, Kraus–Parikh–Wilczek method and Hamilton–Jacoby method are used together. The results show that two methods give us the same logarithmic modified entropy, namely \(S = (\alpha - \Lambda _W) A/4\alpha + \pi /\alpha \ln A/4\). This kind of logarithmic entropy is explained well by the effect of self-gravitation in quantum tunneling picture. At tow that the thermodynamics is stable for small case (\(r_+ r_3\)) where \(r_3\) is the critical position of Park solution, which is concordant with asymptotically flat case shown by Kehagias–Sfetsos (Phys. Lett. B 678:127, 2009).

The logarithmic entropy formula for the linear heat equation on Riemannian manifolds

In this paper we introduce a new logarithmic entropy functional for the linear heat equation on complete Riemannian manifolds and prove that it is monotone decreasing on complete Riemannian manifolds with nonnegative Ricci curvature. Our results are simpler version, without Ricci flow, of R.-G. Ye’s recent result (arXiv:math.DG/0708.2008). As an application, we apply the monotonicity of the logarithmic entropy functional of heat kernels to characterize Euclidean space.

Logarithmic entropy corrected holographic dark energy with nonminimal kinetic coupling

In this study, we have considered a cosmological model with nonminimal kinetic coupling terms and investigated its cosmological implications with respect to logarithmic entropy corrected holographic dark energy (LECHDE). The correspondence between LECHDE in flat Friedmann–Robertson–Walker cosmology and the phantom dark energy model is also examined with the aim of interpreting the current universe acceleration.

Secondary structure formation of homopolymeric single-stranded nucleic acids including force and loop entropy: Implications for DNA hybridization

Loops are essential secondary structure elements in folded DNA and RNA molecules and proliferate close to the melting transition. Using a theory for nucleic acid secondary structures that accounts for the logarithmic entropy -c ln m for a loop of length m, we study homopolymeric single-stranded nucleic acid chains under external force and varying temperature. In the thermodynamic limit of a long strand, the chain displays a phase transition between a low-temperature/low-force compact (folded) structure and a high-temperature/high-force molten (unfolded) structure. The influence of c on phase diagrams, critical exponents, melting, and force extension curves is derived analytically. For vanishing pulling force, only for the limited range of loop exponents 2 < c ≲ 2.479 a melting transition is possible; for c ≤ 2 the chain is always in the folded phase and for 2.479 ≲ c always in the unfolded phase. A force-induced melting transition with singular behavior is possible for all loop exponents c < 2.479 and can be observed experimentally by single-molecule force spectroscopy. These findings have implications for the hybridization or denaturation of double-stranded nucleic acids. The Poland-Scheraga model for nucleic acid duplex melting does not allow base pairing between nucleotides on the same strand in denatured regions of the double strand. If the sequence allows these intra-strand base pairs, we show that for a realistic loop exponent c ≈ 2.1 pronounced secondary structures appear inside the single strands. This leads to a lower melting temperature of the duplex than predicted by the Poland-Scheraga model. Further, these secondary structures renormalize the effective loop exponent [Formula: see text], which characterizes the weight of a denatured region of the double strand, and thus affect universal aspects of the duplex melting transition.

Logarithmic entropy of Kehagias–Sfetsos black hole with self-gravitation in asymptotically flat IR modified Hořava gravity

Motivated by recent logarithmic entropy of Hořava–Lifshitz gravity, we investigate Hawking radiation for Kehagias–Sfetsos black hole from tunneling perspective. After considering the effect of self-gravitation, we calculate the emission rate and entropy of quantum tunneling by using Kraus–Parikh–Wilczek method. Meanwhile, both massless and massive particles are considered in this Letter. Interestingly, two types tunneling particles have the same emission rate Γ and entropy Sb whose analytical formulae are Γ=exp[π(rin2−rout2)/2+π/αlnrin/rout] and Sb=A/4+π/αln(A/4), respectively. Here, α is the Hořava–Lifshitz field parameter. The results show that the logarithmic entropy of Hořava–Lifshitz gravity could be explained well by the self-gravitation, which is totally different from other methods. The study of this semiclassical tunneling process may shed light on understanding the Hořava–Lifshitz gravity.

Functional polymorphic engines: formalisation, implementation and use cases

With regards to the known shortcomings suffered by form-based detection, an increasing number of antivirus products considers behavioral detection. Following this trend, form-based mutations could become function-based with the apparition of functional polymorphism: a third generation of mutation mechanism, specially designed to address behavioral detection. In effect, a same global behavior or purpose (replication, propagation, residency, etc.) can be achieved through different functional solutions, thus leaving space for possible mutations. Whereas actual form-based mutation techniques mainly modify the code structure of malware, functional mutations modify the code functionality, and more particularly the resulting interaction scheme with the operating system and other software. These functional mutations could not be achieved without reaching a semantic level of interpretation, higher than actual techniques remaining purely syntactic. Drawing a parallel, this article underlines the consequent relation existing between functional polymorphic engines and compilers. By studying the associated mutation properties, we prove that these engines exhibit logarithmic entropy and result in a NP-complete complexity for behavioral detection. The implementation of a prototype is finally addressed as well as its possible use for antivirus testing and software protection.

A QUALITATIVE STUDY OF LINEAR DRIFT-DIFFUSION EQUATIONS WITH TIME-DEPENDENT OR DEGENERATE COEFFICIENTS

This paper is concerned with entropy methods for linear drift-diffusion equations with explicitly time-dependent or degenerate coefficients. Our goal is to establish a list of various qualitative properties of the solutions. The motivation for this study comes from a model for molecular motors, the so-called Brownian ratchet, and from a nonlinear equation arising in traffic flow models, for which complex long time dynamics occurs. General results are out of the scope of this paper, but we deal with several examples corresponding to most of the expected behaviors of the solutions. We first prove a contraction property for general entropies which is a useful tool for uniqueness and for the convergence to some large time asymptotic solutions which may depend on time. Then we focus on power law and logarithmic relative entropies. When the diffusion term is of the type ∇(|x|α∇·), we prove that the inequality relating the entropy with the entropy production term is a Hardy–Poincaré type inequality, that we establish. Here we assume that α ∈ (0,2] and the limit case α = 2 appears as a threshold for the method. As a consequence, we obtain an exponential decay of the relative entropies. In the case of time-periodic coefficients, we prove the existence of a unique time-periodic solution which attracts all other solutions. The case of a degenerate diffusion coefficient taking the form |x|α with α &gt; 2 is also studied. The Gibbs state exhibits a non-integrable singularity. In this case concentration phenomena may occur, but we conjecture that an additional time-dependence restores the smoothness of the asymptotic solution.

Option price calibration from Rényi entropy

The calibration of the risk-neutral density function for the future asset price, based on the maximisation of the entropy measure of Renyi, is proposed. Whilst the conventional approach based on the use of logarithmic entropy measure fails to produce the observed power-law distribution when calibrated against option prices, the approach outlined here is shown to produce the desired form of the distribution. Procedures for the maximisation of the Renyi entropy under constraints are outlined in detail, and a number of interesting properties of the resulting power-law distributions are also derived. The result is applied to efficiently evaluate prices of path-independent derivatives.

Characterization of trabecular bone structure from high-resolution magnetic resonance images using fuzzy logic

The purpose of this work was to apply fuzzy logic image processing techniques to characterize the trabecular bone structure with high-resolution magnetic resonance images. Fifteen ex vivo high-resolution magnetic resonance images of specimens of human radii at 1.5 T and 12 in vivo high-resolution magnetic resonance images of the calcanei of peri- and postmenopausal women at 3 T were obtained. Soft segmentation using fuzzy clustering was applied to MR data to obtain fuzzy bone volume fraction maps, which were then analyzed with three-dimensional (3D) fuzzy geometrical parameters and measures of fuzziness. Geometrical parameters included fuzzy perimeter and fuzzy compactness, while measures of fuzziness included linear index of fuzziness, quadratic index of fuzziness, logarithmic fuzzy entropy, and exponential fuzzy entropy. Fuzzy parameters were validated at 1.5 T with 3D structural parameters computed from microcomputed tomography images, which allow the observation of true trabecular bone structure and with apparent MR structural indexes at 1.5 T and 3 T. The validation was statistically performed with the Pearson correlation coefficient as well as with the Bland-Altman method. Bone volume fraction correlation values (r) were up to .99 (P<.001) with good agreements based on Bland-Altman analysis showing that fuzzy clustering is a valid technique to quantify this parameter. Measures of fuzziness also showed consistent correlations to trabecular number parameters (r>.85; P<.001) and good agreements based on Bland-Altman analysis, suggesting that the level of fuzziness in high-resolution magnetic resonance images could be related to the trabecular bone structure.

Entanglement Entropy in a Boundary Impurity Model

Boundary impurities are known to dramatically alter certain bulk properties of (1+1)-dimensional strongly correlated systems. The entanglement entropy of a zero temperature Luttinger liquid bisected by a single impurity is computed using a novel finite size scaling or bosonization scheme. For a Luttinger liquid of length 2L and UV cutoff epsilon, the boundary impurity correction (deltaSimp) to the logarithmic entanglement entropy (Sent proportional, variant lnL/epsilon scales as deltaSimp approximately yrlnL/epsilon, where yr is the renormalized backscattering coupling constant. In this way, the entanglement entropy within a region is related to scattering through the region's boundary. In the repulsive case (g<1), deltaSimp diverges (negatively) suggesting that the entropy vanishes. Our results are consistent with the recent conjecture that entanglement entropy decreases irreversibly along renormalization group flow.

A note on the long time behavior for the drift-diffusion-Poisson system

In this note we analyze the long time behavior of a drift-diffusion-Poisson system with a symmetric definite positive diffusion matrix, subject to Dirichlet boundary conditions. This system models the transport of electrons in semiconductor or plasma devices. By using a quadratic relative entropy obtained by keeping the lowest order term of the logarithmic relative entropy, we prove the exponential convergence to the equilibrium. To cite this article: N. Ben Abdallah et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Integrated imaging with a centrally obscured logarithmic asphere

We study an integrated computational imaging system which incorporates a centrally obscured logarithmic asphere lens and rapid maximum entropy processing. A careful study of imaging for a two-point object at various distances is presented to demonstrate diffraction limited resolution and a tenfold increase in the depth of field. Also the overall transfer function is defined and measured. This system is well suited to high-quality photography where one requires excellent resolution and good contrast with little or no artifacts.

Discrete Approximation of the Cahn-Hilliard/Allen-Cahn system with logarithmic entropy

We propose a numerical scheme for the Cahn-Hilliard/Allen-Cahn system with logarithmic nonlinearity, based on the finite element method. We show its well-posedeness and convergence of the numerical solution to the weak solution of the original system. We point out some properties of a regularized numerical solution.

Logarithmic Sobolev Inequalities, Matrix Models and Free Entropy

We give two applications of logarithmic Sobolev inequalities to matrix models and free probability. We also provide a new characterization of semi-circular systems through a Poincare-type inequality.

Entropy methods for kinetic models of traffic flow

In these notes we first introduce logarithmic entropy methods for time-dependent drift-diffusion equations and then consider a kinetic model of Vlasov-Fokker-Planck type for traffic flows. In the spatially homogeneous case the model reduces to a special type of nonlinear driftdiffusion equation which may permit the existence of several stationary states corresponding to the same density. Then we define general convex entropies and prove a convergence result for large times to steady states, even if more than one exists in the considered range of parameters, provided that some entropy estimates are uniformly bounded.

Flow Mechanism of a Submerged Jet Impinging on a Free Interface.

We report time-dependent flow characteristics of a turbulent water jet impinging vertically upward upon a water-air interface. Mexican-hat wavelet transform was applied to the time-dependent velocity fluctuations obtained in the region near to the air-water interface in order to analyze spatiotemporal turbulent structure at different time scales. The further understanding of the physical meanings of the wavelet transforms was tried by the analysis using logarithmic expectation and information entropy of information theory. The near-interface turbulent structure is constituted not only by the impingement of the large-scale structure of the approaching turbulent jet but also by the time-dependent deformation of the surface bump. Time-series images of the surface bump visualized by an argon laser sheet technique showed some periodical regulation of the surface bump coming from the large-scale structure of the impinging jet

Weakly nonextensive thermostatistics and the Ising model with long-range interactions

We introduce a nonextensive entropic measure $S_{\chi}$ that grows like $N^{\chi}$, where $N$ is the size of the system under consideration. This kind of nonextensivity arises in a natural way in some $N$-body systems endowed with long-range interactions described by $r^{-\alpha}$ interparticle potentials. The power law (weakly nonextensive) behavior exhibited by $S_{\chi}$ is intermediate between (1) the linear (extensive) regime characterizing the standard Boltzmann-Gibbs entropy and the (2) the exponential law (strongly nonextensive) behavior associated with the Tsallis generalized $q$-entropies. The functional $S_{\chi} $ is parametrized by the real number $\chi \in[1,2]$ in such a way that the standard logarithmic entropy is recovered when $\chi=1$ >. We study the mathematical properties of the new entropy, showing that the basic requirements for a well behaved entropy functional are verified, i.e., $S_{\chi}$ possesses the usual properties of positivity, equiprobability, concavity and irreversibility and verifies Khinchin axioms except the one related to additivity since $S_{\chi}$ is nonextensive. For $1<\chi<2$, the entropy $S_{\chi}$ becomes superadditive in the thermodynamic limit. The present formalism is illustrated by a numerical study of the thermodynamic scaling laws of a ferromagnetic Ising model with long-range interactions.

Quantitative fuzzy measures for threshold selection

Discrimination of Image Qualities for some applications in Computer Vision is very much important. The process of evaluation of image quality is very difficult as it possesses noise which is fuzzy in nature. Threshold selection for object extraction, for a bimodal or multi modal histogram is still a difficult problem. It is proposed in this paper that based on the four fuzzy measures, namely, linear and quadratic index of fuzziness, logarithmic and exponential entropy measures, how the best threshold will be identified and used. A comparative study of four such fuzzy measures for real life images has been carried out and promising results are obtained. Selection of the best threshold is tried out using a neural network (BPN) as it helps for fine tuning.

Monotone Riemannian metrics and relative entropy on noncommutative probability spaces

We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0,∞) satisfying g(1)=0. We show that these convex operator functions can be partitioned into convex subsets, each of which defines a unique symmetrized relative entropy, a unique family (parametrized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including g(w)=−log w, which yields the familiar logarithmic relative entropy. The relative entropies, Riemannian metrics, and geodesic distances obtained by our procedure all contract under completely positive, trace-preserving maps. We then define and study the maximal contraction associated with these quantities.

Field theoretical representation of the Hohenberg-Kohn free energy for fluids.

To go beyond Gaussian approximation to the Hohenberg-Kohn free energy playing the key role in the density functional theory (DFT), the density functional integral representation would be relevant, because the field theoretic approach to perturbative calculations becomes available. Then we first derive the associated Hamiltonian of the density functional, explicitly including the logarithmic entropy term, from the grand partition function expressed by configurational integrals. Moreover, two things are done so that the efficiency of the obtained form may be revealed: we demonstrate that this representation facilitates the field theoretic treatment of the perturbative calculation and, further, compare our perturbative formulation with that of the DFT.