Envelope excitations in nonextensive plasmas with warm-ions

In this work we study the modulational instability of plasmas with q-entropy electrons and warm ions, using the hydrodynamic approach. A nonlinear Schrödinger equation (NLSE), governing the dynamics of envelope excitations in the plasma, is obtained by using the conventional multiscales method. Investigation of the modulational instability of the nonextensive plasmas reveals that the criteria for propagation of bright/dark envelope excitations in such plasmas are significantly affected by value of the nonextensivity parameter, q, and the fractional ion-temperature, σ  . In particular, by setting σ≠0$\sigma \ne 0$, a new region of modulational instability appears, indicating that the study of modulation instability in the cold-ion limit (σ=0$\sigma =0$) is completely different from that of warm ions. The study of the growth-rate and rogue-wave amplitudes in terms of different plasma parameters, reveals that their magnitude is of different scales for two ranges of the nonextensivity parameters, q>1$q>1$ and q<1$q<1$.     

On generalized Leibniz triangles and q-Gaussians

Generalized Leibniz triangles have been used in nonextensive statistical mechanics as theoretical models that yield q  -Gaussians (q<1$q<1$) as attractors. We study such triangles from a probability point of view. Our results show that one can get any distribution on [0,1]$[0,1]$ (or any distribution that has a compact support, after a linear transform) from such triangles, including q  -Gaussians with q<1$q<1$. Next we propose conceptual models that are triangular arrays of row-wise exchangeable random variables and yield q  -Gaussians for q<1$q<1$ and q?1$q?1$ as attractors, via laws of large numbers and central limit theorems, respectively.     

Nonextensivity measure for earthquake networks

Studying earthquakes and the associated geodynamic processes based on the complex network theory enables us to learn about the universal features of the earthquake phenomenon. In addition, we can determine new indices for identification of regions geophysically. It was found that earthquake networks are scale free and its degree distribution obeys the power law. Here we claim that the q$q$-exponential function is better than power law model for fitting the degree distribution. We also study the behavior of q$q$ parameter (nonextensivity measure) with respect to resolution. It was previously asserted in Eur. Phys. J. B (2012) 85: 23; that the topological characteristics of earthquake networks are dependent on each other for large values of the resolution. A peak in the plot of q$q$ against resolution determines the beginning of the assertion range.     

Generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy

We provide generalized entanglement constraints in multi-qubit systems in terms of Tsallis entropy. Using quantum Tsallis entropy of order q$q$, we first provide a generalized monogamy inequality of multi-qubit entanglement for q=2$q=2$ or 3$3$. This generalization encapsulates the multi-qubit CKW-type inequality as a special case. We further provide a generalized polygamy inequality of multi-qubit entanglement in terms of Tsallis-q$q$ entropy for 1≤q≤2$1\le q\le 2$ or 3≤q≤4$3\le q\le 4$, which also contains the multi-qubit polygamy inequality as a special case.     

Some properties of generalized Fisher information in the context of nonextensive thermostatistics

We present two extended forms of Fisher information that fit well in the context of nonextensive thermostatistics. We show that there exists an interplay between these generalized Fisher information, the generalized q$q$-Gaussian distributions and the q$q$-entropies. The minimum of the generalized Fisher information among distributions with a fixed moment, or with a fixed q$q$-entropy is attained, in both cases, by a generalized q$q$-Gaussian distribution. This complements the fact that the q$q$-Gaussians maximize the q$q$-entropies subject to a moment constraint, and yields new variational characterizations of the generalizedq$q$-Gaussians. We show that the generalized Fisher information naturally pop up in the expression of the time derivative of the q$q$-entropies, for distributions satisfying a certain nonlinear heat equation. This result includes as a particular case the classical de Bruijn identity. Then we study further properties of the generalized Fisher information and of their minimization. We show that, though non additive, the generalized Fisher information of a combined system is upper bounded. In the case of mixing, we show that the generalized Fisher information is convex for q≥1$q\ge 1$. Finally, we show that the minimization of the generalized Fisher information subject to moment constraints satisfies a Legendre structure analog to the Legendre structure of thermodynamics.     

Tsallis’ maximum entropy ansatz leading to exact analytical time dependent wave packet solutions of a nonlinear Schrödinger equation

Tsallis maximum entropy distributions provide useful tools for the study of a wide range of scenarios in mathematics, physics, and other fields. Here we apply a Tsallis maximum entropy ansatz, the q$q$-Gaussian, to obtain time dependent wave-packet solutions to a nonlinear Schrödinger equation recently advanced by Nobre, Rego-Monteiro and Tsallis (NRT) [F.D. Nobre, M.A. Rego-Monteiro, C. Tsallis, Phys. Rev. Lett. 106 (2011) 140601]. The NRT nonlinear equation admits plane wave-like solutions (q$q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the q$q$-generalized thermostatistical formalism, is characterized by a parameter q$q$ and in the limit q→1$q\to 1$ reduces to the standard, linear Schrödinger equation. The q$q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known q$q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the q→1$q\to 1$ limit the q$q$-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schrödinger equation. In the present work we also show that there are other families of nonlinear Schrödinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.     

Strange attractor in the Potts spin glass on hierarchical lattices

The spin-glass q-state Potts model on d  -dimensional diamond hierarchical lattices is investigated by an exact real space renormalization group scheme. Above a critical dimension _dl(q)$d_l(q)$ for q>2$q>2$, the coupling constants probability distribution flows to a low-temperature strange attractor   or to the high-temperature paramagnetic fixed point, according to the temperature is below or above the critical temperature _Tc(q,d)$T_c(q,d)$. The strange attractor was investigated considering four initial different distributions for q=3$q=3$ and d=5$d=5$ presenting strong robustness in shape and temperature interval suggesting a condensed phase with algebraic decay.     

Properties of reachable sets in the sub-Lorentzian geometry

The aim of this paper is to develop local theory of future timelike, nonspacelike and null reachable sets from a given point _q0$q_0$ in the sub-Lorentzian geometry. In particular, we prove that if U$U$ is a normal neighbourhood of _q0$q_0$ then the three reachable sets, computed relative to U$U$, have identical interiors and boundaries with respect to U$U$. Further, among other things, we show that for Lorentzian metrics on contact distributions on R²ⁿ⁺¹${\mathbb{R}}^{2n+1}$, n≥1$n\ge 1$, the boundary of reachable sets from _q0$q_0$ is, in a neighbourhood of _q0$q_0$, made up of null future directed curves starting from _q0$q_0$. Every such curve has only a finite number of non-smooth points; smooth pieces of every such curve are Hamiltonian geodesics. For general sub-Lorentzian structures, contrary to the Lorentzian case, timelike curves may appear on the boundary. It turns out that such curves are always Goh curves. We also generalize a classical result on null Lorentzian geodesics: every null future directed Hamiltonian sub-Lorentzian geodesic initiating at _q0$q_0$ is contained, at least to a certain moment of time, in the boundary of the reachable set from _q0$q_0$.     

Nonextensivity and entropy of astrophysical sources

We study the X-ray intensities of 142 light curves of cataclysmic variables, galaxies, pulsars, supernova remnants and other X-ray sources present in the public data collected by the instrument All Sky Monitor on board the satellite Rossi X-ray Timing Explorer. We show that the X-ray light curves coming from astrophysical systems obey Tsallis’s q$q$-Gaussian distribution as probability density. This fact strongly suggests that these astrophysical systems behave in a non-extensive manner. Furthermore, the q$q$ entropic indices for these systems were obtained and they provide an indication of the nonextensivity degree of each of these astrophysical systems. The q$q$-value increases for systems if the Tsallis entropy decreases.     

Quantum Bohmian model for financial market

We apply methods of quantum mechanics for mathematical modeling of price dynamics at the financial market. The Hamiltonian formalism on the price/price-change phase space describes the classical-like evolution of prices. This classical dynamics of prices is determined by “hard” conditions (natural resources, industrial production, services and so on). These conditions are mathematically described by the classical financial potential V(q),$V(q),$ where q=(_q1,…,_qn)$q=(q_1,\dots ,q_n)$ is the vector of prices of various shares. But the information exchange and market psychology play important (and sometimes determining) role in price dynamics. We propose to describe such behavioral financial factors by using the pilot wave (Bohmian) model of quantum mechanics. The theory of financial behavioral waves takes into account the market psychology. The real trajectories of prices are determined (through the financial analogue of the second Newton law) by two financial potentials: classical-like V(q)$V(q)$ (“hard” market conditions) and quantum-like U(q)$U(q)$ (behavioral market conditions).     

A new model of agegraphic dark energy

In this note, we propose a new model of agegraphic dark energy based on the Károlyházy relation, where the time scale is chosen to be the conformal time η   of the Friedmann–Robertson–Walker (FRW) universe. We find that in the radiation-dominated epoch, the equation-of-state parameter of the new agegraphic dark energy wq=−1/3$w_q=-1/3$ whereas Ωq=n²a²${\Omega }_q=n^2{a}^2$; in the matter-dominated epoch, wq=−2/3$w_q=-2/3$ whereas Ωq=n²a²/4${\Omega }_q=n^2{a}^2/4$; eventually, the new agegraphic dark energy dominates; in the late time wq→−1$w_q\to -1$ when a→∞$a\to \infty$, and the new agegraphic dark energy mimics a cosmological constant. In every stage, all things are consistent. The confusion in the original agegraphic dark energy model proposed in [R.G. Cai, Phys. Lett. B 657 (2007) 228, arXiv: 0707.4049 [hep-th]] disappears in this new model. Furthermore, Ωq?1${\Omega }_q?1$ is naturally satisfied in both radiation-dominated and matter-dominated epochs where a?1$a?1$. In addition, we further extend the new agegraphic dark energy model by including the interaction between the new agegraphic dark energy and background matter. In this case, we find that wq$w_q$ can cross the phantom divide.     

Comparing emerging and mature markets during times of crises: A non-extensive statistical approach

One of the important issues in finance and economics for both scholars and practitioners is to describe the behavior of markets, especially during times of crises. In this paper, we analyze the behavior of some mature and emerging markets with a Tsallis entropy framework that is a non-extensive statistical approach based on non-linear dynamics. During the past decade, this technique has been successfully applied to a considerable number of complex systems such as stock markets in order to describe the non-Gaussian behavior of these systems. In this approach, there is a parameter q$q$, which is a measure of deviation from Gaussianity, that has proved to be a good index for detecting crises. We investigate the behavior of this parameter in different time scales for the market indices. It could be seen that the specified pattern for q$q$ differs for mature markets with regard to emerging markets. The findings show the robustness of the stated approach in order to follow the market conditions over time. It is obvious that, in times of crises, q$q$ is much greater than in other times. In addition, the response of emerging markets to global events is delayed compared to that of mature markets, and tends to a Gaussian profile on increasing the scale. This approach could be very useful in application to risk and portfolio management in order to detect crises by following the parameter q$q$ in different time scales.     

Relativistic kinetic theory and non-gaussian statistical

The nonextensive statistical mechanics is extended in the special relativity context through a generalization of H$H$-theorem. We show that the Tsallis framework is compatible with the second law of the thermodynamics when the nonadditive effects are consistently introduced on the collisional term of the Boltzmann equation. The proof of the H$H$-theorem follows from using of q$q$-algebra in the generalization of the molecular chaos hypothesis (Stosszahlansatz). A thermodynamic consistency is possible whether the entropic parameter belongs to interval q∈[0,2]$q\in [0,2]$.     

Quasi-additivity of Tsallis entropies and correlated subsystems

We use Beck's quasi-additivity of Tsallis entropies for n   independent subsystems to show that like the case of n=2$n=2$, the entropic index q$q$ approaches 1 by increasing system size. Then, we will generalize that concept to correlated subsystems to find that in the case of correlated subsystems, when system size increases, q$q$ also approaches a value corresponding to the additive case.     

Lifshitz-point correlation length exponents from the large-n expansion

The large-n expansion is applied to the calculation of thermal critical exponents describing the critical behavior of spatially anisotropic d-dimensional systems at m  -axial Lifshitz points. We derive the leading non-trivial 1/n$1/n$ correction for the perpendicular correlation-length exponent _νL2${\nu }_{L2}$ and hence several related thermal exponents to order O(1/n)$O(1/n)$. The results are consistent with known large-n expansions for d  -dimensional critical points and isotropic Lifshitz points, as well as with the second-order epsilon expansion about the upper critical dimension d^?=4+m/2$d^{?}=4+m/2$ for generic m∈[0,d]$m\in [0,d]$. Analytical results are given for the special case d=4$d=4$, m=1$m=1$. For uniaxial Lifshitz points in three dimensions, 1/n$1/n$ coefficients are calculated numerically. The estimates of critical exponents at d=3$d=3$, m=1$m=1$ and n=3$n=3$ are discussed.     

On generalisations of the log-Normal distribution by means of a new product definition in the Kapteyn process

We discuss the modification of the Kapteyn multiplicative process using the q$q$-product of Borges [E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A 340 (2004) 95]. Depending on the value of the index q$q$ a generalisation of the log-Normal distribution is yielded. Namely, the distribution increases the tail for small (when q<1$q<1$) or large (when q>1$q>1$) values of the variable upon analysis. The usual log-Normal distribution is retrieved when q=1$q=1$, which corresponds to the traditional Kapteyn multiplicative process. The main statistical features of this distribution as well as related random number generators and tables of quantiles of the Kolmogorov–Smirnov distance are presented. Finally, we illustrate the validity of this scenario by describing a set of variables of biological and financial origin.