Degree Of Nonextensivity Research Articles

Impact of nonextensivity on the transport coefficients of a magnetized hot and dense QCD matter

We have studied the impact of the nonextensivity on the transport coefficients related to charge and heat in thermal QCD. For this purpose, the electrical (sigma _textrm{el}), Hall (sigma _textrm{H}), thermal (kappa ) and Hall-type thermal (kappa _textrm{H}) conductivities are determined using the kinetic theory approach in association with the nonextensive Tsallis statistical mechanism. The effect of nonextensivity is encoded in the nonextensive Tsallis distribution function, where the deviation of the parameter q from 1 signifies the degree of nonextensivity in the concerned system. The thermal and electrical conductivities are found to increase with the introduction of nonextensivity, which means that the deviation of the medium from thermal equilibrium enhances both charge and heat transports. With the magnetic field, the deviations of sigma _textrm{el}, sigma _textrm{H}, kappa and kappa _textrm{H} from their respective equilibrated values increase, whereas these deviations decrease with the chemical potential. We have also studied how the extent of the nonextensivity modulates the longevity of magnetic field. Present work is further extended to the study of some observables associated with the aforesaid transport phenomena, such as the Knudsen number and the elliptic flow within the nonextensive Tsallis framework.

Analysis of the effects of nonextensivity for a generalized dissipative system in the SU(1,1) coherent states

The characteristics of nonextensivity for a general quantum dissipative oscillatory system in the SU(1,1) coherent states are investigated using the invariant operator method. We consider a deformed Caldirola-Kanai oscillator represented in terms of a parameter q which is a measure of the degree of nonextensivity. The nonextensivity effects on the parametric evolution of the SU(1,1) coherent states are elucidated. We compare our results with those of previous researches and address the advantage of our methodology which adopts the linear invariant operator. In particular, the nonextensive behaviors associated with the fluctuations of canonical variables and the dissipation of quantum energy are analyzed in detail regarding their dependence on q. The properties of SU(1,1) coherent states that we adopt here can be utilized in quantum-information processes such as cloning, swapping, and teleportation of state information.

The ‘wavelet’ entropic index q of non-extensive statistical mechanics and superstatistics

Generalized entropies developed for non-extensive statistical mechanics are derived from the Boltzmann-Gibbs-Shannon entropy by a real number q that is a parameter based on q-calculus; where q is called ‘the entropic index’ and determines the degree of non-extensivity of a system in the interval between 1 and 3. In a very recent study, we introduced a new calculation method of the entropic index q of non-extensive statistical mechanics. In this study, we show the mathematical proof of this calculation method of the entropic index. Firstly, we propose that the number of degrees of freedom, n is proportional to the inverse of the wavelet scale index,n≡1iscale, where iscale is a wavelet based parameter called wavelet scale index that quantitatively measures the non-periodicity of a signal in the interval between 0 and 1. Then, by applying this proposition to the superstatistics approach, we derive the equation that expresses the relationship between the entropic index and the wavelet scale index, q=1+2iscale. Therefore, we name this q-index as the ‘wavelet’ entropic index. Lastly, we calculate the Abe entropy, Landsberg-Vedral entropy and q-dualities of the Tsallis entropy of the Logistic Map and Hennon Map using the ‘wavelet’ entropic index, and based on our results, compare and discuss these generalized entropies.

Ordinary and extraordinary waves in hot nonextensive plasma

In this paper, a kinetic method based on particle velocity distribution function in the context of the nonextensive Tsallis statistics is presented for ordinary and extraordinary waves in a collisionless magnetized hot plasma. The components of the generalized dielectric tensor for perpendicular propagation in hot nonextensive plasma are derived in terms of generalized hypergeometric functions. The components are expressed as functions of different values of the nonextensive parameter q, which quantifies the degree of nonextensivity of the system. The dispersion relations of ordinary and extraordinary waves in hot plasma are obtained in the context of the nonextensive statistics. It is shown that in hot nonextensive plasma the generalized dispersion relations of these waves depend on q parameter, which means that the initial distribution function alters the properties of ordinary and extraordinary waves. For the hot nonextensive plasma, the ordinary and extraordinary wave experience resonances at cyclotron harmonics, whilst the upper hybrid resonance is eliminated due to thermal effects. However, cutoff frequencies shift to higher values as q parameter increases.

Inverse bremsstrahlung absorption in laser-fusion plasma

Inverse bremsstrahlung absorption (IBA) is the most efficient absorption mechanism in laser-fusion plasma. IBA is the process in which an electron absorbs a photon while colliding with an ion or with another electron. IBA of laser energy in homogeneous and unmagnetized plasma has been studied using the Fokker–Planck equation. The isotropic function is considered as a q-nonextensive electrons distribution function. By considering a circular-polarized laser, kinetic theory and spherical coordinates, the first anisotropic function is calculated. A pulsed laser is considered, and the effect of the physical parameters such as temperature and q (q is a parameter quantifying the degree of nonextensivity) has been studied on the absorption value. Later, the results of continuous laser and pulsed laser have been compared. The calculations of IBA performed for a variety of q. According to our calculations, the IBA value in neighboring of the critical density layer for continuous laser is higher than pulsed laser. The final results show the IBA value increases with increase in $$q$$ parameter and oppositely it decreases with temperature increasing.

Relativistic effects on dust grain charging in non-equilibrium dusty plasma with relativistic Tsallis distribution

The relativistic effects on the dust grain charging are investigated by calculating the relativistic cross section by the orbit-limited motion (OLM) theory and relativistic Tsallis distribution function for currents carried by ions and electrons in nonequilibrium plasma. The dust grain electrical potential equation is derived in the dust grain charging process, and this nonlinear equation is solved by the numerical method. We indicate that the behavior of nonextensivity is in contrast to ions and electrons, and it is found that the density of species in plasma has the most influence on the degree of nonextensivity. We show that the degree of nonextensivity of the electron and the ion has significant effects on the relationship between dust grain electrical potential and dust density. As another result, it is indicated that by increasing the dust density, the dust particles will be seen as a component from ordinary multi-ionic plasma that shows collective behavior. By comparing the results in relativistic and nonrelativistic regimes, it is indicated that in the relativistic regime, the density of dust is limited, and the electrical potential of dust grains also varies, whereas in the nonrelativistic regime, the dust density is not limited, and the dust grain electrical potential tends to be constant in small amounts of dust density.

Study of the q-Gaussian distribution with the scale index and calculating entropy by normalized inner scalogram

In this paper, we discuss a method based on wavelet analysis for the study of the q-index of the Gaussian distribution. We derive q-index from the scale index, iscale, using the expression; q≡1+2iscale where iscale is a wavelet based tool for measuring the degree of aperiodicity of a dynamical system in the range of 0≤iscale≤1. We show that this expression gives consistent results with the numerical approach of q-Gaussian distribution which determines the degree of non-extensivity of a dynamical system in the range of 1<q<3. We also suggest a new entropy calculation method based on the normalized inner scalogram for studying the chaotic characteristics of nonlinear dynamical systems.

Dissipative properties and isothermal compressibility of hot and dense hadron gas using non-extensive statistics

We evaluate the transport properties such as shear viscosity (eta ), bulk viscosity (zeta ) and their ratios over entropy density (s) for hadronic matter using relativistic non-extensive Boltzmann transport equation (NBTE) in relaxation time approximation (RTA). In NBTE, we argue that the system far from equilibrium may not reach to an equilibrium described by extensive (Boltzmann-Gibbs (BG)) statistics but to a q-equilibrium defined by Tsallis non-extensive statistics after subsequent evolution, where q denotes the degree of non-extensivity. We observe that eta /s and zeta /s decrease rapidly with temperature (T) for various q-values. As q increases, the magnitudes of eta /s and zeta /s decrease with T. We also show the upper mass cutoff dependence of these ratios for a particular q and find that they decrease with the increase in mass cutoff of hadrons. Further, we present the first estimation of isothermal compressibility (kappa _T) using non-extensive Tsallis statistics at finite baryon chemical potential (mu _B). It is observed that, kappa _T changes significantly with the degree of non-extensivity. We also study the squared speed of sound (c_{s}^{2}) as a function of temperature at finite baryon chemical potential for various q and upper mass cutoffs. It is noticed that there is a strong impact of q and mass cutoff on the behaviour of c_{s}^{2}.

Small amplitude double layers in an electronegative dusty plasma with q-distributed electrons**Project supported by the “Strategic Priority Research Program” of the Chinese Academy of Sciences (Grant No. XDA01020304), the National Natural Science Foundation of China (Grant Nos. 11747306 and 11565022), and the Youth Science and Technology Foundation of Gansu Province, China (Grant No. 1606RJYA263).

The small amplitude dust ion-acoustic double layers in a collisionless four-component unmagnetized dusty plasma system containing nonextensive electrons, inertial negative ions, Maxwellian positive ions, and negatively charged static dust grains are investigated theoretically. Using the pseudo-potential approach and reductive perturbation method, an energy integral equation for the system has been derived and its solution in the form of double layers is obtained. The results appear that the existence regime of the double layer is very sensitive to the plasma parameters, e.g., electron nonextensivity, negative-to-positive ion number density ratio etc. It has been observed that for the selected set of parameters, the system supports rarefactive, (compressive) double layers depending upon the degree of nonextensivity of electrons.

Sheath Properties in Two-Temperature Non-Maxwellian Electron Plasmas

The sheath criterion is modified for a plasma consisting of two-temperature nonextensively distributed electrons and positive ions with finite temperature. Using a two-fluid model, effects of the plasma parameters, such as temperature and density of positive ions and temperature ratio and degree of nonextensivity of both electron species on the allowable velocity of ions at the sheath edge ( $u_{0i}$ ) are investigated. The results show that $u_{0i}$ depends on temperature and density of ions and electron species as well as degree of nonextensivity of cold and hot electrons ( $q_{c}$ and $q_{h}$ ). Using our derived sheath criterion, the effects of temperature of ions and temperature and density of hot electrons on the properties of the sheath region of a superthermal two-temperature electrons plasma are studied numerically.

Characteristics of Electronegative Plasma Sheath with q-Nonextensive Electron Distribution

The characteristics of sheath in a plasma system containing q-nonextensive electrons, cold fluid ions, and Boltzmann-distributed negative ions are investigated. A modified Bohm sheath criterion is derived by using the Sagdeev pseudopotential technique. It is found that the proposed Bohm velocity depends on the degree of nonextensivity (q), negative ion temperature to nonextensive electron temperature ratio (σ), and negative ion density (B). Using the modified Bohm sheath criterion, the sheath characteristics, such as the spatial distribution of the potential, positive ion velocity, and density profile, have been numerically investigated, which clearly shows the effect of negative ions, as well as the nonextensive distribution of electrons. It is found that, as the nonextensivity parameter and the electronegativity increases, the electrostatic sheath potential increases sharply and the sheath width decreases.

Non-extensive statistical analysis on solar activity dependence of magnetospheric dynamics

Tsallis q-Gaussian distributions and associated q-statistics have been used for the last couple of decades to describe non-equilibrium dynamical systems with varying degrees of complexity. In the present study, we use Tsallis non-extensive statistical analysis for a better understanding of magnetospheric dynamics and its relationship with solar activity. The Tsallis' q-triplet, a set of indices (such as qsens, qstat and qrel) used as empirical quantifiers of non-extensivity, has been estimated for magnetospheric proxies such as auroral electrojet (AE) and disturbance storm time (Dst) indices, for a period of 1985–2007. Our results indicate that the degree of non-extensivity of AE index is quite different from that of Dst index in relation with solar activity dependence. We have seen that the values of the q-triplets calculated from Dst index are more solar activity dependent than those computed from AE index. This shows that, other than solar wind exertions, certain complex phenomena of internal origin also modulate the dynamics of geomagnetic fluctuations in the auroral region.

QRNN: $q$ -Generalized Random Neural Network.

Artificial neural networks (ANNs) are widely used in applications with complex decision boundaries. A large number of activation functions have been proposed in the literature to achieve better representations of the observed data. However, only a few works employ Tsallis statistics, which has successfully been applied to various other fields. This paper presents a random neural network (RNN) with q -Gaussian activation functions [ q -generalized RNN (QRNN)] based on Tsallis statistics. The proposed method employs an additional parameter q (called the entropic index) which reflects the degree of nonextensivity. This approach has the flexibility to model complex decision boundaries of different shapes by varying the entropic index. We conduct numerical experiments to analyze the efficiency of QRNN compared with RNNs and several other classical methods. Statistical tests (Wilcoxon and Friedman) are used to validate our results and show that the QRNN performs significantly better than RNNs with different activation functions. In addition, we find that QRNN outperforms many of the compared classical methods, with the exception of support vector machines, in which case it still exhibits a substantial advantage in terms of implementation simplicity and speed.

Effect of q-nonextensive parameter and saturation time on electron density steepening in electron-positron-ion plasmas

The effect of q-nonextensive parameter and saturation time on the electron density steepening in electron-positron-ion plasmas is studied by particle in cell method. Phase space diagrams show that the size of the holes, and consequently, the number of trapped particles strongly depends on the q-parameter and saturation time. Furthermore, the mechanism of the instability and exchange of energy between electron-positron and electric field is explained by the profiles of the energy density. Moreover, it is found that the q-parameter, saturation time, and electron and positron velocities affect the nonlinear evolution of the electron density which leads to the steepening of its structure. The q-nonextensive parameter or degree of nonextensivity is the relation between temperature gradient and potential energy of the system. Therefore, the deviation of q-parameter from unity indicates the degree of inhomogeneity of temperature or deviation from equilibrium. Finally, using the kinetic theory, a generalized q-dispersion relation is presented for electron-positron-ion plasma systems. It is found that the simulation results in the linear regime are in good agreement with the growth rate results obtained by the kinetic theory.

Stimulated Raman scattering in nonextensive statistics

Stimulated Raman scattering (SRS) in a collisionless plasma was analytically and numerically investigated in the context of nonextensive statistics proposed by Tsallis. The generalized wavenumbers and growth rates in both backward and forward scattering were derived as a function of different values of the nonextensive parameter q, which quantified the degree of nonextensivity of the system. It was shown that the increased numbers of superthermal particles and low velocity particles could affect the evolution of stimulated Raman scattering. A relativistic Eulerian Vlasov code for conditions similar to single-hot-spot experiments was used in order to compare the analytical results for the frequency and wavenumber with the quantities obtained from the numerical simulations. A nonextensive q-distribution by the restriction q<1 was used as the initial velocity distribution. The results of numerical simulation showed a strong reflection by SRS. According to the simulation, the peak of the main components in both wavenumber and frequency spectra were found at positions estimated from the phase-matching conditions and the analytical data.

Sheath structure in plasmas with nonextensively distributed electrons and thermal ions

Sheath region of an electropositive plasma consisting of q-nonextensive electrons and singly charged positive ions with finite temperature is modeled. Using Sagdeev's pseudo potential technique to derive the modified sheath formation criterion, it is shown that the velocity of ions at the sheath edge is directly proportional to the ion temperatures and inversely proportional to the degree of nonextensivity of electrons (q-parameter). Using the modified Bohm criterion, effect of degree of nonextensivity of electrons and temperature of positive ions on the characteristics of the sheath region are investigated numerically. It is shown that an increase in the ion temperature gives rise to an increase in the electrostatic potential and the velocity of ions in the sheath regardless of the value of q. Furthermore, it is seen that the sheath width and the density distribution of the charged particles decrease by increasing the temperature of positive ions. In addition, it is found that the positive ion temperature is less effective on the sheath structure for higher values of the q-parameter. Finally, the results obtained for a thermal plasma with nonextensively distributed electrons are compared with the results of a cold plasma with nonextensive electrons and an extensive (Maxwellian) plasma with thermal ions.

Nonextensive statistics and the sheath criterion in collisional plasmas

The Bohm criterion in an electropositive plasma containing nonextensively distributed electrons and warm ions is investigated by using a steady state two-fluid model. Taking into account the ion-neutral collisions and finite temperature of ions, a modified Bohm criterion is derived which limits both maximum and minimum allowable velocity of ions at the sheath edge (u0i). It is found that the degree of nonextensivity of electrons (q) and temperature of positive ions (Ti) affect only the lower limit of the entrance velocity of ions into the sheath while the degree of ion collisionality (α) influences both lower and upper limits of the ion velocities at the sheath edge. In addition, depending on the value of q, it is shown that the minimum velocity of positive ions at the sheath edge can be greater or smaller than its Maxwellian counterpart. Moreover, it is shown that, depending on the values of α and Ti, the positive ions with subsonic velocity may enter the sheath for either q &amp;gt; 1 or −1 &amp;lt; q &amp;lt; 1. Finally, as a practical application, the density distribution of charged particles in the sheath region is studied for different values of u0i, and it is shown that monotonical reduction of the positive ion density distribution occurs only when the velocity of positive ions at the sheath edge lies between two above mentioned limits.

Electron Bernstein waves in nonextensive statistics

The propagation of electron Bernstein waves in a magnetized plasma is investigated in the context of nonextensive q-distribution of Tsallis statistics. The dispersion relation is expressed as a function of generalized hypergeometric function. It is shown that the generalized dispersion relation significantly depends on the q-parameter, which quantifies the degree of nonextensivity of the system. The reduced number of superthermal particles shifts the Bernstein wave curves to higher wavenumbers. For harmonics whose frequency lies above the upper hybrid frequency, an increase in the value of q increases the maximum frequency and the wavenumber at which the group velocity vanishes. For Bernstein waves which propagate at frequencies lower than the upper hybrid frequency or close to it, diminishing q, or increasing the number of superthermal particles, gives rise to faster frequency fall-off. The generalized electron Bernstein waves are studied in both the strongly and weakly magnetized regimes and for both small and large wavenumbers compared with the Larmor radius. It is found that for the weak magnetic field, the frequency range occupied by the mode spans the complete intraharmonic frequency, and the q value significantly affects the dispersion curves. On the other hand, in strongly magnetized regimes the frequency variation domain is extremely restricted and has little dependence on the q value.

The effects of nonextensivity on quantum dissipation

Nonextensive dynamics for a quantum dissipative system described by a Caldirola-Kanai (CK) Hamiltonian is investigated in SU(1,1) coherent states. To see the effects of nonextensivity, the system is generalized through a modification fulfilled by replacing the ordinary exponential function in the standard CK Hamiltonian with the q-exponential function. We confirmed that the time behavior of the system is somewhat different depending on the value of q which is the degree of nonextensivity. The effects of q on quantum energy dissipation and other parameters are illustrated and discussed in detail.

TIME-DEPENDENT NONEXTENSIVITY ARISING FROM THE ROTATIONAL EVOLUTION OF SOLAR-TYPE STARS

Nonextensive formalism is a generalization of the Boltzmann-Gibbs statistics. In this formalism the entropic index q is a quantity characterizing the degree of nonextensivity, and is interpreted as a parameter of long-memory or long-range interactions between the components of the system. Since its proposition in 1988, this formalism has been applied to investigate a wide variety of natural phenomena. In stellar astrophysics, theoretical distribution function based on nonextensive formalism (q-distributions) has been successfully applied to reproduce the distribution of stellar radial and rotational velocity data. In this paper, we investigate the time variation of the entropic index q obtained from the distribution of rotation, Vsini, for a sample of 254 rotational data for solar-type stars from 11 open clusters aged between 35.5Myr and 2.6Gyr. As a result, we have found an anti-correlation between the entropic index q and the age of clusters, and that the distribution of rotation Vsini for these stars becomes extensive for an age greater than about 170Myr. Assuming that the parameter q is associated with long-memory effects, we suggest that the memory of the initial angular momentum of solar-type stars can be scaled by the entropic index q. We also propose a physical link between the parameter q and the magnetic braking of stellar rotation.