Nonextensive Parameter Research Articles

Modified Friedmann equations from Tsallis entropy

It was shown by Tsallis and Cirto that thermodynamical entropy of a gravitational system such as black hole must be generalized to the non-additive entropy, which is given by Sh=γAβ, where A is the horizon area and β is the nonextensive parameter [1]. In this paper, by taking the entropy associated with the apparent horizon of the Friedmann–Robertson–Walker (FRW) Universe in the form of Tsallis entropy, and assuming the first law of thermodynamics, dE=ThdSh+WdV, holds on the apparent horizon, we are able to derive the corresponding Friedmann equations describing the dynamics of the universe with any spatial curvature. We also examine the time evolution of the total entropy and show that the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon. Then, modifying the emergence proposal of gravity proposed by Padmanabhan and calculating the difference between the surface degrees of freedom and the bulk degrees of freedom in a region of space, we again arrive at the modified Friedmann equation of the FRW Universe with any spatial curvature which is the same as one obtained from the first law of thermodynamics. We also study the cosmological consequences of Tsallis cosmology. Interestingly enough, we find that this model can explain simultaneously the late time acceleration in the universe filled with pressureless matter without invoking dark energy, as well as the early deceleration. Besides, the age problem can be circumvented automatically for an accelerated universe and is estimated larger than 3/2 age of the universe in standard cosmology. Taking β=2/5, we find the age of the universe ranges as 13.12 Gyr <t0<16.32 Gyr, which is consistent with recent observations. Finally, using the Jeans's analysis, we comment, in brief, on the density perturbation in the context of Tsallis cosmology and found that the growth of energy differs compared to the standard cosmology.

Modulational instability by strong Langmuir waves in q-plasma

Based on the full Zakharov equations, the dispersion relation of the wave-wave interaction by strong Langmuir waves in nonextensive plasma is obtained. The dispersion equation were analyzed with numerical method and the results indicate that both the wave number range and maximum growth rate of the modulational instability by strong Langmuir waves will enhance with the nonextensive parameter q increasing. Moreover, an analytic study of dispersion equations in two special and important cases, the modulation instability induced by transverse perturbations and longitudinal perturbations, is presented. The analysis shows that the growth rate induced by transverse perturbations and that done by longitudinal perturbations are equal when the nonextensive parameter q is the same. Comparison of the calculated modulational instability with the corresponding analytical solutions shows that they are in good agreement. This research provides a clearer physical picture of the modulational instability in nonextensive plasma.

Nonlinear behavior of plasma: Connection with nonextensive statistics

Nonextensive hydrodynamic equations and Zakharov equations are derived by moment equation and two time-scale methods, respectively. The conserved quantities and nonlinear entity collapse scalar law are obtained, from which we find that the conservation energy is relevant to the nonextensive parameter but momentum as well as angular momentum and the number of plasmon are not affected by the nonextensivity of system. The self-similar collapse solution of nonextensive Zakharov equations is also presented. Furthermore, we demonstrate that the nonlinear entity collapse scalar law is relevant to the nonextensive parameter and especially it allows the existence of three dimensional stable and one dimensional collapse nonlinear entity, which is significantly different from the case of Maxwellian distribution. In the extensive limit, all the results obtained in Maxwellian framework are reproduced.

Kinetic theory of ion acoustic waves in a [formula omitted]-nonextensive distributed ions and electrons plasma

Using a kinetic theoretical approach, the characteristics of ion acoustic waves (IAWs) have been studied in a plasma whose electron and ion velocity distributions are modeled by a three dimensional q- nonextensive distribution function. It is found that the results of the one dimensional velocity distribution function are not accurate when compared to the results of three dimensional distribution function. Numerical solution of the dispersion relation for the ion acoustic wave has been obtained to study the effect of nonextensivity parameter q on the dispersion relation, the damping rate and the range of the values of the normalized wavenumber (kλDe) for which the ion acoustic waves are weakly damped. It is also found that the change in the value of the electron nonextensivity parameter qe has a weak effect on the dispersion relation, the damping rate and the range of the values of the normalized wavenumber (kλDe) for which the ion acoustic waves are weakly damped. However, change in the value of the ion nonextensivity parameter qi has very strong effect on the dispersion relation, the damping rate and the range of the values of the normalized wavenumber (kλDe) for which the ion acoustic waves are weakly damped.

Landau damping of dust acoustic waves in the presence of hybrid nonthermal nonextensive electrons

Based on the kinetic theory, Landau damping of dust acoustic waves (DAWs) propagating in a dusty plasma composed of hybrid nonthermal nonextensive distributed electrons, Maxwellian distributed ions and negatively charged dust grains is investigated using Vlasov-Poisson’s equations. The characteristics of the DAWs Landau damping are discussed. It is found that the wave frequency increases by decreasing (increasing) the value of nonextensive (nonthermal) parameter, $q$ ( $\alpha $ ). It is recognized that $\alpha $ plays a significant role in observing damping or growing DAW oscillations. For small values of $\alpha $ , damping modes have been observed until reaching a certain value of $\alpha $ at which $\omega _{i}$ vanishes, then a growing mode appears in the case of superextensive electrons. However, only damping DAW modes are observed in case of subextensive electrons. The present study is useful in the space situations where such distribution exists.

Analytical Solutions and Parametric Studies of the Schamel Equation for Two Different Ion-Acoustic Waves in Plasmas

A numerical investigation of the properties of two different ion-acoustic solitons has been performed by using analytical solutions of the Schamel equation. A new type of the analytical solution of the Schamel equation that describes soliton propagation with a negative phase velocity has been found for the first time. This new type of the solution has been applied to investigate the physical properties of two different plasmas and to understand the effect of nonextensivity and the effect of trapped electrons on ion-acoustic waves in a superthermal plasma. It is shown that the soliton amplitude and width depend on the nonextensivity parameter, superthermality of the electron distribution, and characteristic trapping parameter.

Examining nonextensive statistics in relativistic heavy-ion collisions

We show in detailed numerical solutions of the nonlinear Fokker-Planck equation (FPE), which has been associated with nonextensive $q$ statistics, that the available data on rapidity distributions for stopping in relativistic heavy-ion collisions cannot be reproduced with any permitted value of the nonextensivity parameter ($1&lt;q&lt;1.5$). This casts doubt on the nonextensivity concept that is widely used in relativistic heavy-ion physics.

Compressive and rarefactive double layers in non-uniform plasma with q $q$ -nonextensive distributed electrons

Electrostatic solitary waves and double layers (DLs) formed by the coupled ion acoustic (IA) and drift waves have been investigated in non-uniform plasma using \(q\)-nonextensive distribution function for the electrons and assuming ions to be cold \(T_{i}< T_{e}\). It is found that both compressive and rarefactive nonlinear structures (solitary waves and DLs) are possible in such a system. The steeper gradients are supportive for compressive solitary (and double layers) and destructive for rarefactive ones. The \(q\)-nonextensivity parameter \(q\) and the magnitudes of gradient scale lengths of density and temperature have significant effects on the amplitude of the double layers (and double layers) as well as on the speed of these structures. This theoretical model is general which has been applied here to the \(F\)-region ionosphere for illustration.

Study of non-extensive entropy of bound polaron in monolayer graphene

An electron is coupled to the longitudinal acoustic phonon on the surface of the graphene with Coulomb impurity has been considered. The ground state energy of the polaron has been obtained using the variational method. Tsallis entropy variations have been studied with magnetic field, temperature, Coulomb bound parameter and the charge for different substances such as SiC, HfO2, h-BN, and SiO2. It is found that the entropy increases with enhancing temperature and Coulomb bound parameter. But, it decreases with enhancing the charge. The entropy increases with decreasing the non-extensive parameter q for all substances at constant Coulomb bound parameter.

Optimal investment problem under non-extensive statistical mechanics

The optimal investment problem in which the asset price process is modeled by the non-extensive statistical mechanics is studied in this paper. By the methods of deterministic control and the dynamic programming, we obtain the optimal strategy with logarithmic utility function, power utility function and quadratic utility function, respectively. Moreover, the numerical results indicate that the optimal investment strategy is affected by the non-extensive parameter, the proportion invested in the risky asset decreases as the wealth increases under quadratic utility function, but it remains unchanged under power utility and logarithmic utility function.

Solitary wave solution and dynamic transition of dust ion acoustic waves in a collisional nonextensive dusty plasma with ionization effect

The solitary wave solution and dynamic transition of dust ion acoustic waves (DIAWs) are studied in an unmagnetized collisional dusty plasma, which consists of negatively charged dust fluid, positively charged ions, q-nonextensive electrons, and background neutral particles, considering the ionization effect, ion loss, ion–neutral, ion–dust, and dust–neutral collisions. Using reductive perturbation technique (RPT) the damped Korteweg–de Vries (DKdV) equation and the damped modified Korteweg–de Vries (DMKdV) equation are derived. Using momentum conservation law, dust ion acoustic solitary wave solutions (DIASWSs) are obtained for the DKdV and MDKdV equations. Effects of nonextensive parameter (q) and ion–dust collisional frequency (νid) are presented on the DIASWSs. Effects of ion–dust collisional frequency (νid), ion–neutral collisional frequency (νin) and ion loss rate (νl) on DIAWs are investigated considering an external periodic perturbation through three-dimensional phase portrait and time-series analysis. It is found that νid, νin and νl significantly affect on the dynamics of the perturbed DIAWs and play a pivotal role in the transition from quasiperiodic and chaotic motions to periodic motion.

Electron acoustic waves in a plasma with a q-nonextensive distribution of electrons

Using a kinetic theoretical approach, the characteristics of electron acoustic waves (EAWs) have been studied in a plasma whose electron velocity distributions are modeled by a q-nonextensive distribution function. Analytical expressions have been derived to show the dependence of the phase velocity of EAWs on nonextensive parameter q for different ranges of wavelengths of EAWs. These expressions show that in all three regions (i.e., long wavelength, intermediate wavelength, and short wavelength regions), the dispersion of EAWs depends on the nonextensive parameter q. The numerical solution of the dispersion relation has been obtained to identify strongly and weakly damped regions of the EAWs. The results show that the range of wavenumbers for which EAWs are weakly damped depends strongly on the nonextensive parameter q. The smaller values of nonextensive parameter q mean more nonthermal electrons and hence the narrow range of the weakly damped region of EAWs. In the limit q→1, the results of the Maxwellian distribution function can be obtained from the results of the q nonextensive distribution function.

Modulational instability: Conservation laws and bright soliton solution of ion-acoustic waves in electron-positron-ion-dust plasmas

We consider the nonlinear propagation of non-planar (cylindrical and spherical) ion-acoustic (IA) envelope solitary waves in an unmagnetized plasma consisting of electron-positron-ion-dust plasma with two-electron temperature distributions in the context of the non-extensive statistics. The basic set of fluid equations is reduced to the modified nonlinear Schrodinger (MNLS) equation in cylindrical and spherical geometry by using the reductive perturbation method (RPM). It is found that the nature of the modulational instabilities would be significantly modified due to the effects of the non-extensive and other plasma parameters as well as cylindrical and spherical geometry. Conservation laws of the MNLS equation are obtained by Lie symmetry and multiplier method. A new exact solution (envelope bright soliton) is obtained by the extended homogeneous balance method. Finally, we study the results of this article.

Rogue Waves in Multi-Ion Cometary Plasmas

The effect of pair ions on the formation of rogue waves in a six-component plasma composed of two hot and one colder electron component, hot ions, and pair ions is studied. The kappa distribution, which provides an unambiguous replacement for a Maxwellian distribution in space plasmas, is connected with nonextensive statistical mechanics and provides a continuous energy spectrum. Hence, the colder and one component of the hotter electrons is modeled by kappa distributions and the other hot electron component, by a q-nonextensive distribution. It is found that the rogue wave amplitude is different for various pair-ion components. The magnitude, however, increases with increasing spectral index and nonextensive parameter q. These results may be useful in understanding the basic characteristics of rogue waves in cometary plasmas.

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.

The effect of heat radiation on the evolution of the Tsallis entropy in self-gravitating systems and plasmas

The evolution of the Tsallis entropy in self-gravitating systems and plasmas is studied in this letter, which is determined by two factors. The first factor is the change of the microstate number of systems, whose spontaneous increase leads to the entropy's increase, consistent with the standard text book. The second is the evolution of the nonextensive parameter, whose evolution rate to time is opposite to the one of entropy. We find the correlation between heat radiation and time evolution of the nonextensive parameter in the self-gravitating systems and plasmas. In such systems, the emission of radiation heat leads to the increase of the parameter while the absorption of radiation heat results in the decrease of this parameter. This is consistent with the inference derived from the Clausius' definition of entropy. In order to evolve to the current state, the solar corona should absorb a large amount of radiation heat, which might be originated from the energy released by solar flare. The magnetic connection probably plays a role in the conversion of energy. A correct dynamics theory of magnetic connection should explain how the energy conversion is achieved.

Collision of dust ion acoustic multisolitons in a non-extensive plasma using Hirota bilinear method

The collision of two, four, and six dust ion acoustic solitons (DIASs) in an unmagnetized non-extensive plasma is studied. The dispersion characteristics are analyzed. Using the extended Poincaré-Lighthill-Kue method, two different Korteweg–de Vries (KdV) equations are derived for the colliding DIASs. The Hirota's direct bilinear method is applied to get the multisoliton solutions for the derived KdV equations. It is found that the non-extensive parameter for electrons plays a significant role in the collision process, in such a way that the amplitude of the solitons increases as the non-extensive parameter increases.

The effects of variable dust size and charge on dust acoustic waves propagating in a hybrid Cairns–Tsallis complex plasma

The propagation characteristics of dust acoustic waves (DAWs) in a dusty plasma consisting of variable size dust grains, hybrid Cairns–Tsallis-distributed electrons, and nonthermal ions are studied. The charging of the dust grains is described by the orbital-motion-limited theory and the size of the dust grains obeys the power law dust size distribution. To describe the nonlinear propagation of the DAWs, a Zakharov–Kuznetsov equation is derived using a reductive perturbation method. It is found that the nonthermal and nonextensive parameters influence the main properties of DAWs. Moreover, our results reveal that the rarefactive waves can propagate mainly in the proposed plasma model while compressive waves can be detected for a very small range of the distribution parameters of plasma species, and the DAWs are faster and wider for smaller size dust grains. Applications of the present results to dusty plasma observations are briefly discussed.

Effects of the non-extensive parameter on the propagation of ion acoustic waves in five-component cometary plasma system

Some important evolution nonlinear partial differential equations are derived using the reductive perturbation method for unmagnetized collisionless system of five component plasma. This plasma system is a multi-ion contains negatively and positively charged Oxygen ions (heavy ions), positive Hydrogen ions (lighter ions), hot electrons from solar origin and colder electrons from cometary origin. The positive Hydrogen ion and the two types of electrons obey \(q\)-non-extensive distributions. The derived equations have three types of ion acoustic waves, which are soliton waves, shock waves and kink waves. The effects of the non-extensive parameters for the hot electrons, the colder electrons and the Hydrogen ions on the propagation of the envelope waves are studied. The compressive and rarefactive shapes of the three envelope waves appear in this system for the first order of the power of the nonlinearity strength with different values of non-extensive parameters. For the second order, the strength of nonlinearity will increase and the compressive type of the envelope wave only appears.

Rail crack monitoring based on Tsallis synchrosqueezed wavelet entropy of acoustic emission signals: A field study

This article focuses on the rail crack monitoring using acoustic emission technique in the field typically with complex cracking conditions and high operational noise. A novel crack monitoring strategy based on Tsallis synchrosqueezed wavelet entropy was developed, where synchrosqueezed wavelet transform was introduced to explore the time–frequency characteristics of acoustic emission signals and Tsallis entropy was adopted to quantify the local variation of acoustic emission wavelet coefficients more accurately. The mother wavelet of synchrosqueezed wavelet transform and three key parameters of time-Tsallis synchrosqueezed wavelet entropy, including characteristic frequency band, non-extensive parameter, and time window length, were appropriately determined. The performance of the strategy was validated through field tests with an incipient rail crack and trains running at operating speeds. Time-Tsallis synchrosqueezed wavelet entropy efficiently detected and located the crack by extracting the crack-related transients in acoustic emission signals that were easily submerged in the operational noise. Synchrosqueezed wavelet transform further helped to analyze the mechanisms of these crack-related transients, which were distinguished to be either crack propagation or impact. The experimental results demonstrated that the crack monitoring strategy proposed is able to detect both surface and internal rail cracks even in the noisy environment, highlighting its potential for field applications.