New bounds for Tsallis parameter in a noncommutative phase–space entropic gravity and nonextensive Friedmann equations

In this paper, we have analyzed the nonextensive Tsallis statistical mechanics in the light of Verlinde’s formalism. We have obtained, with the aid of a noncommutative phase–space entropic gravity, a new bound for Tsallis nonextensive (NE) parameter (TNP) that is clearly different from the ones present in the current literature. We derived the Friedmann equations in a NE scenario. We also obtained here a relation between the gravitational constant and the TNP.     

Modified entropic gravitation in superconductors

Verlinde recently developed a theoretical account of gravitation in terms of an entropic force. The central element in Verlinde’s derivation is information and its relation with entropy through the holographic principle. The application of this approach to the case of superconductors requires to take into account that information associated with superconductor’s quantum vacuum energy is not stored on Planck size surface elements, but in four volume cells with Planck-Einstein size. This has profound consequences on the type of gravitational force generated by the quantum vacuum condensate in superconductors, which is closely related with the cosmological repulsive acceleration responsible for the accelerated expansion of the Universe. Remarkably this new gravitational type force depends on the level of breaking of the weak equivalence principle for cooper pairs in a given superconducting material, which was previously derived by the author starting from similar principles. It is also shown that this new gravitational force can be interpreted as a surface force. The experimental detection of this new repulsive gravitational-type force appears to be challenging.     

On the Origin of Entropic Gravity and Inertia

It was recently suggested that quantum field theory is not fundamental but emerges from the loss of phase space information about matter crossing causal horizons. Possible connections between this formalism and Verlinde’s entropic gravity and Jacobson’s thermodynamic gravity are proposed. The holographic screen in Verlinde’s formalism can be identified as local Rindler horizons and its entropy as that of the bulk fields beyond the horizons. This naturally resolves some issues on entropic gravity. The quantum fluctuation of the fields is the origin of the thermodynamic nature of entropic gravity. It is also suggested that inertia is related to dragging Rindler horizons.     

Quantum Mechanics Emerges from Information Theory Applied to Causal Horizons

It is suggested that quantum mechanics is not fundamental but emerges from classical information theory applied to causal horizons. The path integral quantization and quantum randomness can be derived by considering information loss of fields or particles crossing Rindler horizons for accelerating observers. This implies that information is one of the fundamental roots of all physical phenomena. The connection between this theory and Verlinde’s entropic gravity theory is also investigated.     

Critique of multinomial coefficients method for evaluating Tsallis and Rényi entropies

Oikonomou [Th. Oikonomou, Physica A 386 (2007) 119] has published a calculation which purports to show that the Tsallis and Rényi entropies can be obtained from the generalized multinomial coefficients. In this paper, we prove that the method of generalized multinomial coefficients failed to determine the Tsallis entropy at equilibrium. Moreover, it is shown that Oikonomou’s analysis contains mistakes which led to misleading statements related to the Jaynes principle of maximum entropy, the Tsallis and the Rényi statistics.     

Dilatonic Entropic Force

We show in detail that the entropic force of the static spherically symmetric spacetimes with unusual asymptotics can be calculated through the Verlinde’s arguments. We introduce three different holographic screen candidates, which are first employed thoroughly by Myung and Kim [Phys. Rev. D 81, 105012 (2010)] for Schwarzschild black hole solutions, in order to identify the entropic force arising between a charged dilaton black hole and a test particle. The significance of the dilaton parameter on the entropic force is highlighted, and shown graphically.     

Entropic Corrections to Coulomb’s Law

Two well-known quantum corrections to the area law have been introduced in the literatures, namely, logarithmic and power-law corrections. Logarithmic corrections, arises from loop quantum gravity due to thermal equilibrium fluctuations and quantum fluctuations, while, power-law correction appears in dealing with the entanglement of quantum fields in and out the horizon. Inspired by Verlinde’s argument on the entropic force, and assuming the quantum corrected relation for the entropy, we propose the entropic origin for the Coulomb’s law in this note. Also we investigate the Uehling potential as a radiative correction to Coulomb potential in 1-loop order and show that for some value of distance the entropic corrections of the Coulomb’s law is compatible with the vacuum-polarization correction in QED. So, we derive modified Coulomb’s law as well as the entropy corrected Poisson’s equation which governing the evolution of the scalar potential ϕ. Our study further supports the unification of gravity and electromagnetic interactions based on the holographic principle.     

Can Sobolev Inequality Be Written for Sharma-Mittal Entropy?

In this paper, we focus on Sobolev inequality in the context of Sharma-Mittal entropy. Using this new inequality, generalized entropic uncertainty relation in accordance with Sharma-Mittal entropy is derived and the pseudoadditivity relation has been obtained. This new entropic uncertainty relation has then been applied to physical examples such as one dimensional harmonic oscillator and Pösch-Teller potential. Finally, it has been shown that for certain values of the parameters of Sharma-Mittal measure, the present results reduce to the corresponding results of Shannon, Renyi and Tsallis measures.     

Correct thermodynamic forces in Tsallis thermodynamics: connection with Hill nanothermodynamics

The equivalence between Tsallis thermodynamics and Hill's nanothermodynamics is established. The correct thermodynamic forces in Tsallis thermodynamics are pointed out. Through this connection we also find a general expression for the entropic index q which we illustrate with two physical examples, allowing in both cases to relate q to the underlying dynamics of the Hamiltonian systems.     

Fundamental constants,entropic gravity and nonextensive equipartition theorem

By using Verlinde’s formalism, we propose that the positive numerical factor, in which Klinkhamer states that it is necessary to define the fundamental length, can be associated to the parameter q$q$ of Tsallis’ nonextensive statistical mechanics.     

Cosmological constant in the Bianchi type-I-modified Brans-Dicke cosmology

In 1961, Brans and Dicke [1] provided an interesting alternative to general relativity based on Mach’s principle. To understand the reasons leading to their field equations, we first consider homogeneous and isotropic cosmological models in the Brans-Dicke theory. Accordingly we start with the Robertson-Walker line element and the energy tensor of a perfect fluid. The scalar field φ is now a function of the cosmic time only. Then we consider spatially homogeneous and anisotropic Bianchi type-I-cosmological solutions of modified Brans-Dicke theory containing barotropic fluid. These have been obtained by imposing a condition on the cosmological parameter Λ(φ). Again we try to focus the meaning of this cosmological term and to relate it to the time coordinate which gives us a collapse singularity or the initial singularity. On the other hand, our solution is a generalization of the solution found by Singh and Singh [2]. As far as we are aware, such solution has not been given earlier.     

Evolving Stochastic Learning Algorithm based on Tsallis entropic index

In this paper, inspired from our previous algorithm, which was based on the theory of Tsallis statistical mechanics, we develop a new evolving stochastic learning algorithm for neural networks. The new algorithm combines deterministic and stochastic search steps by employing a different adaptive stepsize for each network weight, and applies a form of noise that is characterized by the nonextensive entropic index q, regulated by a weight decay term. The behavior of the learning algorithm can be made more stochastic or deterministic depending on the trade off between the temperature T and the q values. This is achieved by introducing a formula that defines a time-dependent relationship between these two important learning parameters. Our experimental study verifies that there are indeed improvements in the convergence speed of this new evolving stochastic learning algorithm, which makes learning faster than using the original Hybrid Learning Scheme (HLS). In addition, experiments are conducted to explore the influence of the entropic index q and temperature T on the convergence speed and stability of the proposed method.     

Phase diagram of a 2D Ising model within a nonextensive approach

In this work we report Monte Carlo simulations of a 2D Ising model, in which the statistics of the Metropolis algorithm is replaced by the nonextensive one. We compute the magnetization and show that phase transitions are present for q ≠ 1. A q - phase diagram (critical temperature vs. the entropic parameter q) is built and exhibits some interesting features, such as phases which are governed by the value of the entropic index q. It is shown that such phases favors some energy levels of magnetization states. It is also shown that the contribution of the Tsallis cutoff is capital to the existence of phase transitions.     

Steepest entropy increase is justified by information theory. The relations between Ziegler’s principle, Onsager’s formalism and Prigogine’s principle

Evidence for Ziegler’s principle of maximum entropy production has been accumulated from different fields such as climatic studies, crystal growth, dynamics of ecosystems and cellular metabolism. However, Ziegler’s principle is still seen with scepticism by the scientific community. The reasons for this scepticism are the absence of an accepted theoretical justification as well as the fact that Ziegler’s principle formulation seems to contradict Prigogine’s principle of minimum entropy production. In this work we aim to provide a theoretical justification for Ziegler’s principle based on information theory, which is at the basis of Gibbs’ formalism for statistical physics. Similar approaches have previously been attempted, however we believe that the justification provided here is simpler and relies in less questionable hypotheses. Once Ziegler’s principle has been formulated as a consequence of information theory, its relations with Onsager’s formulation and Prigogine’s principle are explored.     

Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy

In this work, we analyze two important stochastic processes, the fractional Brownian motion and fractional Gaussian noise, within the framework of the Tsallis permutation entropy. This entropic measure, evaluated after using the Bandt & Pompe method to extract the associated probability distribution, is shown to be a powerful tool to characterize fractal stochastic processes. It allows for a better discrimination of the processes than the Shannon counterpart for appropriate ranges of values of the entropic index. Moreover, we find the optimum value of this entropic index for the stochastic processes under study.     

A scalar-tensor theory of gravity incorporating Mach's principle

A solution to the long-standing problem involving the incorporation of Mach's principle into general relativity is proposed in terms of a new scalar-tensor theory of gravity, which seems to introduce no change in the predictions of general ralativity for any of its existing tests.     

Is computation reversible?

Recent studies have suggested that computation is essentially reversible, provided no information is lost. This is a consequence of Landauer’s principle which only requires energy expenditure and entropy increase for information deletion. In this paper we propose to treat information as being intrinsic to points of non-analyticity, so that the movement of information is always associated with the dissipation of heat. This allows us to construct a theory consistent with causality, and the second law of thermodynamics. Since computation requires the movement of information bits through finite volume gates, energy is dissipated even when information is not destroyed, thus indicating that computation is fundamentally non-reversible.     

Is the universe homogeneous and isotropic in the time when quark-gluon plasma exists?

In this study, quark and strange quark matter which exist in the first seconds of the early Universe have been studied in the context of general relativity to be able to obtain space–time geometry of first seconds of the early Universe. For this purpose, Einstein’s field equations for quark and strange quark matter in the non static spherically symmetric space–time have been solved by using experimental result that anisotropy parameter of quark matter is very small. We have concluded from obtained solutions that the space–time structure of first seconds of the Early Universe is homogeneous and isotropic. Also we have concluded that the color interactions of the quarks may be origin of primordial magnetic field in the early universe.