Nonextensive Parameter Research Articles

Ion acoustic solitary waves in magnetized anisotropic nonextensive plasmas

Abstract The nonlinear propagation of ion-acoustic (IA) electrostatic solitary waves (SWs) is studied in a magnetized electron–ion (e–i) plasma in the presence of pressure anisotropy with electrons following Tsallis distribution. The Korteweg–de Vries (KdV) type equation is derived by employing the reductive perturbation method (RPM) and its solitary wave (SW) solution is determined and analyzed. The effect of nonextensive parameter q, parallel component of anisotropic ion pressure p 1, perpendicular component of anisotropic ion pressure p 2, obliqueness angle θ, and magnetic field strength Ω on the characteristics of SW structures is investigated. The present investigation could be useful in space and astrophysical plasma systems.

Charged particles energization during magnetic reconnection in the Earth’s magnetosphere by double layers: an analytical approach

ABSTRACT Charged particles energization at the magnetic reconnection sites in the Earth’s magnetosphere is a subject of active debate in space plasma, specifically in magnetopause and magnetotail regions. In the proximity of the reconnection region, Magnetospheric Multiscale (MMS) and other satellite missions have confirmed the existence of electron acceleration ranging from 100 keV to MeV. There is also evidence of cold ions acceleration in the separatrix region of the Earth’s magnetopause. An important candidate responsible for charged particles acceleration is believed to be the double layer (DL) structures, and which were first observed by the Cluster satellite in the separatrix region of the Earth’s magnetosphere. Moreover, the MMS mission also observed parallel electric field fluctuations up to 100 mV/m carried by DL at the magnetic reconnection site in the Earth’s magnetosphere. Motivated by the reported literature, we investigate the DL and its associated electric field at the magnetopause magnetic reconnection site by using a simple analytical model in which the DLs associated with kinetic Alfvén waves are studied in a low- $\beta$ electron-ion plasma with Tsallis distributed electrons and fluid ions using the Sagdeev potential technique. The non-extensive parameter (superthermality) of the Tsallis distribution has a pronounced effect on the DL’s strength (as defined by the potential drop) and its associated parallel electric field. Our results are consistent with the earlier studies on DLs in Earth’s magnetosphere. Additionally, we discuss the implications of our findings on charged particle energization in the separatrix region of the Earth’s magnetopause during magnetic reconnection.

Theoretical self-consistency in nonextensive statistical mechanics with parameter transformation

Self-consistency in nonextensive statistical mechanics is studied as a recourse to parameter transformation, where different nonextensive parameters are presented for various theoretical branches. The unification between the first and third choices of the average definition and that between the normal and escort distributions are both examined. The problem of parameter inversion in the generalized H theorem is also investigated. The inconsistency between the statistical ensemble pressure and molecular dynamics pressure can be eliminated. This work also verifies the equivalence of physical temperature and gravitational temperature in nonextensive statistical mechanics. In these parameter transformations, the Tsallis entropy form is observed to remain invariant.

Solid-liquid phase transition and heat engine in an asymptotically flat Schwarzschild black hole via the Rényi extended phase space approach

Recently, it has been found that, with the R\'enyi statistics, the asymptotically flat Schwarzschild black hole can be in thermal equilibrium with infinite heat reservoir at a fixed temperature when its event horizon radius is larger than the characteristic length scale ${L}_{\ensuremath{\lambda}}=1/\sqrt{\ensuremath{\pi}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ is the nonextensivity parameter. In the R\'enyi extended phase space with the $PdV$ work term, an off shell free energy in the canonical ensemble with the thermodynamic volume as an order parameter is considered to identify a first-order Hawking-Page (HP) phase transition as a solid-liquid phase transition. It has the latent heat of fusion from solid (corresponding to thermal radiation) to liquid (corresponding to black hole) in the form of $\ensuremath{\sim}1/\sqrt{\ensuremath{\lambda}}$; this is evident in the absence of the HP phase transition in the case of an asymptotically flat Schwarzschild black hole from the Gibbs-Boltzmann statistics ($\ensuremath{\lambda}=0$). Moreover, we investigate the generalized second law of black hole thermodynamics (GSL) in R\'enyi statistics by considering the black hole as a working substance in a heat engine. Interestingly, an efficiency $\ensuremath{\eta}$ of the black hole in a Carnot cycle takes the form ${\ensuremath{\eta}}_{c}=1\ensuremath{-}{T}_{\mathrm{C}}/{T}_{\mathrm{H}}$. This confirms the validity of the GSL in the R\'enyi extended phase space.

Effective thermodynamical system of Schwarzschild–de Sitter black holes from Rényi statistics

It has been known that the Schwarzschild-de Sitter (Sch-dS) black hole may not be in thermal equilibrium and also be found to be thermodynamically unstable in the standard black hole thermodynamics. In the present work, we investigate the possibility to realize the thermodynamical stability of the Sch-dS black hole as an effective system by using the R\'{e}nyi statistics, which includes the non-extensive nature of black holes. Our results indicate that the non-extensivity allows the black hole to be thermodynamically stable which gives rise to the lower bound on the non-extensive parameter. By comparing the results to ones in the separated system approach, we find that the effective temperature is always smaller than the black hole horizon temperature and the thermodynamically stable black hole in effective approach is always larger than one in separated approach at a certain temperature. There exists only the zeroth-order phase transition from the the hot gas phase to the black hole phase for the effective system while it is possible to have the transition of both the zeroth order and the first order for the separated system.

Positron nonextensivity effect on the propagation of dust ion acoustic Gardner waves

Abstract Propagation of dust ion-acoustic (DIA) Gardner wave in a dusty electron–positron–ion (e–p–i) plasma is investigated. This plasma consists of q-distributed electrons and positrons, warm ions, and dust grains. The effects of the electron nonextensivity, positron nonextensivity, and fractional parameter on the properties of DIA Gardner wave are investigated. Space fractional Gardner (SFG) equation is derived using the semi inverse technique. An efficient modified G′/G-expansion method is presented to solve the SFG equation. It is found that the amplitude of the DIA Gardner wave increases with an increase in space fractional parameter β $\left(\beta \right)$ and spatial parameter ζ $\left(\zeta \right)$ . On other hands, the DIA Gardner wave shape can be modulated using the space fractional parameter β $\left(\beta \right)$ . Our results may help understand the astrophysical environments such as star magnetospheres, solar flares, and galactic nuclei.

Identified particle spectra in Pb–Pb, Xe–Xe and p–Pb collisions with the Tsallis blast-wave model

We investigate the hadron transverse momentum (p T) spectra in Pb–Pb (Pb–Pb, Xe–Xe, p–Pb) collisions at 2.76 (5.02, 5.44, 5.02) TeV in the framework of the Tsallis blast wave model with a linear transverse velocity profile and a constant velocity profile. In this model, the Tsallis temperature (T), the average radial flow velocity (⟨β⟩) and the degree of non-equilibrium (q) of the system are common for all hadrons when a combined fit is performed to the p T spectra of different particles at a given centrality. It is found that the model can describe the particle spectra well up to 3 GeV c −1. For both profiles, the transverse flow velocity decreases from central to peripheral collisions while the non-extensive parameter q exhibits the opposite behavior, indicating a more rapid expansion and reduced off-equilibrium of the system in more central collisions. Moreover, we observe that, in central collisions, ⟨β⟩ and q (T) from the fit with the linear profile are smaller (is slightly larger) than those (that) from the constant profile, while in peripheral collisions ⟨β⟩, T and q from the linear profile are compatible with those from the constant profile. We also derived and discussed the relation between the Tsallis temperature and the thermal temperature. In addition, to check whether a scenario of an early freeze-out of strange particles exists at the Large Hadron Collider, the particle spectra are investigated by grouping them into strange and non-strange hadrons. The combined fit gives an insight on the degree of non-equilibrium, the radial flow and the Tsallis temperature of the system at the kinetic decoupling. It provides a comparison between the results at different energies in the same collision system and the results in different collision systems at the same or similar energy.

Innovation for measuring the distribution function with nonextensive single electric probe

In the field of plasma diagnosis, the measurement of the distribution function is significant because the distribution function is the basis for the use of plasma kinetic theory and it is the prerequisite for analyzing many physical phenomena, such as Landau damping (wave-particle resonance phenomenon) and ion sheath. Theoretical analysis and a large number of experiments have proved that plasma components do not obey Boltzmann–Gibbs statistics and can be well described by nonextensive statistical mechanics. The field of nonextensive electric probe has also made great progress, and the invention of the nonextensive single electric probes has developed and strengthened the power of plasma diagnostics. The nonextensive electric probe can not only measure the electron nonextensive parameter of plasma that cannot be measured by traditional probes but can also measure more accurate plasma parameters that can also be measured by traditional probes, such as Te, Φp, ne, Φf, and αqFe. However, diagnosing the plasma distribution function by the nonextensive electric probe has not been thoroughly and systematically analyzed and discussed. Here, we show the measurement of the plasma distribution function with a nonextensive single electric probe. This work expands the diagnostic capabilities of the nonextensive single electric probe. We utilize the nonextensive single electric probe theory to analyze the experimental data points of the I–V curve, measure the plasma electron distribution function fvx, and display the distribution curve (figure f-vx), and we also measure the plasma parameters of qFe, Te, Φp, ne, Φf, αqFe, etc. The proposed method provides a new approach to the diagnosis of the plasma distribution function and contributes to a more accurate and comprehensive grasp of plasma, which creates better conditions for us to take advantage of plasma. These initial results illustrate the potential of the nonextensive electric probe in the field of plasma diagnosis and, more generally, in accelerating the progress of fusion-energy science and helping to understand complex physical systems.

The Effects of Nonextensivity for a General Driven Dissipative Oscillator in the Coherent States

AbstractThe nonextensivity for a general quantum dissipative system driven by an external force is investigated. The dynamics of the system is characterized by a nonextensivity parameter q which indicates the degree of departure from extensivity for the system. The wave function of the system in the coherent state is derived with the aid of the linear invariant operator which obeys the Liouville–von Neumann equation. It is confirmed that, although the corresponding quantum energy dissipates with time like the classical energy, its time behavior is somewhat different from that of the usual dissipative systems depending on the scale of nonextensivity. The rate of such a dissipation becomes slightly larger as q increases. The effects of nonextensivity on other quantum characteristics of the system, such as fluctuations and uncertainty relations for canonical variables, are also analyzed rigorously. It is shown that the authors' results in the nonextensive regime recover to those of the well known standard ones in the limit .

Investigation of dust ion acoustic shock waves in dusty plasma using Cellular Neural Network

The Cellular Neural Network (CNN) is implemented to investigate the features of dust ion acoustic shock waves in a two-fluid model of magnetized dusty plasma. The electrons in this model obey the hybrid Cairns-Tsallis distribution. The reductive perturbation method is used to derive the corresponding Zakharov Kuznetsov Burger (ZKB) equation. Then, the CNN algorithm is integrated with the Finite Difference Method to simulate the ZKB equation with a high accuracy. The obtained solution is approximately identical to the analytical solution which is obtained from the Tanh method. An algorithm to solve ZKB equation using the Finite Difference Method is employed to asses the accuracy of the CNN method. Moreover, it is found that the plasma parameters (viscosity coefficients, cyclotron frequency, nonextensive parameter,…etc.) have significant effects on the shock wave characteristics.

Asymmetric Tsallis distributions for modeling financial market dynamics

Financial markets are highly non-linear and non-equilibrium systems. Earlier works have suggested that the behavior of market returns can be well described within the framework of non-extensive Tsallis statistics or superstatistics. For small time scales (delays), a good fit to the distributions of stock returns is obtained with q-Gaussian distributions, which can be derived either from Tsallis statistics or superstatistics. These distributions are symmetric. However, as the time lag increases, the distributions become increasingly non-symmetric. In this work, we address this problem by considering the data distribution as a linear combination of two independent normalized distributions — one for negative returns and one for positive returns. Each of these two independent distributions are half q-Gaussians with different non-extensivity parameter q and temperature parameter beta. Using this model, we investigate the behavior of stock market returns over time scales from 1 to 80 days. The data covers both the .com bubble and the 2008 crash periods. These investigations show that for all the time lags, the fits to the data distributions are better using asymmetric distributions than symmetric q-Gaussian distributions. The behaviors of the q parameter are quite different for positive and negative returns. For positive returns, q approaches a constant value of 1 after a certain lag, indicating the distributions have reached equilibrium. On the other hand, for negative returns, the q values do not reach a stationary value over the time scales studied. In the present model, the markets show a transition from normal to superdiffusive behavior (a possible phase transition) during the 2008 crash period. Such behavior is not observed with a symmetric q-Gaussian distribution model with q independent of time lag.

QCD chiral phase transition and critical exponents within the nonextensive Polyakov-Nambu-Jona-Lasinio model * *Supported by the National Natural Science Foundation of China (12005192) and the Project funded by China Postdoctoral Science Foundation (2020M672255, 2020TQ0287)

We present a nonextensive version of the Polyakov-Nambu-Jona-Lasinio model that is based on nonextentive statistical mechanics. This new statistics model is characterized by a dimensionless nonextensivity parameter q that accounts for all possible effects violating the assumptions of the Boltzmann-Gibbs (BG) statistics (for , it returns to the BG case). Based on the nonextensive Polyakov-Nambu-Jona-Lasinio model, we discussed the influence of nonextensive effects on the curvature of the phase diagram at and especially on the location of the critical end point (CEP). A new and interesting phenomenon we found is that with an increase in q, the CEP position initially shifts toward the direction of larger chemical potential and lower temperature. However, when q is larger than a critical value , the CEP position moves in the opposite direction. In other words, as q increases, the CEP position moves in the direction of smaller chemical potential and higher temperature. This U-turn phenomenon may be important for the search of CEP in relativistic heavy-ion collisions, in which the validity of BG statistics is questionable due to strong fluctuations and long-range correlations, and nonextensive effects begin to manifest themselves. In addition, we calculated the influence of the nonextensive effects on the critical exponents and found that they remain almost constant with q.

Bose-Einstein condensation of pions in proton–proton collisions at the Large Hadron Collider using non-extensive Tsallis statistics

The possibility of formation of Bose-Einstein Condensation (BEC) is studied in pp collisions at $$\sqrt{s} = 7$$ TeV at the Large Hadron Collider. A thermodynamically consistent non-extensive formulation of the identified hadron transverse momentum distributions is used to estimate the critical temperature required to form BEC of charged pions, which are the most abundant species in a multi-particle production process in hadronic and nuclear collisions. The obtained results have been contrasted with the systems produced in Pb-Pb collisions to have a better understanding. We observe an explicit dependency of BEC critical temperature and number of particles in the pion condensates on the non-extensive parameter q, which is a measure of degree of non-equilibrium – as q decreases, the critical temperature increases and approaches to the critical temperature obtained from Bose-Einstein statistics without non-extensivity. Studies are performed on the final state multiplicity dependence of number of particles in the pion condensates in a wide range of multiplicity covering hadronic and heavy-ion collisions, using the inputs from experimental transverse momentum spectra.

Tsallis eXtropy

The Tsallis entropy has high performance in handling uncertain information, which is the extension of information entropy. The extropy is a complementary dual of entropy, which is a research hot spot. This paper proposed the Tsallis eXtropy and maximum Tsallis eXtropy, which are the complementary of Tsallis entropy and maximum Tsallis entropy respectively. When the non-extensive parameter is equal to 1, then the Tsallis eXtropy and the maximum Tsallis eXtropy will degenerate into information extropy and maximum information extropy respectively. Some meaningful theorems and proofs of the Tsallis eXtropy and the maximum Tsallis eXtropy has been proposed. Numerical examples have been designed to prove the effectiveness of the proposed models in evaluating uncertainties. Comparison application has been applied to prove the superiority of the proposed models than other models. The experimental results have shown that the proposed models are very high effective in measuring unknown issues.

Characteristics of Magnetized Dusty Plasma Sheath With Two Ion Species and q-Nonextensive Electrons

The effect of q-nonextensive electrons, nature, and magnitude of dust charge on magnetized dusty plasma sheath with two ion species has been numerically investigated via fluid model. The initial common speed of ion species at the sheath entrance has been extended, which increases from subsonic to supersonic with the decrease in nonextensive parameter q and ion-neutral drag force as well. The presence of super extensive primary electrons ( q <; 1) has significantly affected the sheath parameters characterizing the dusty plasma sheath: dust charging, potential profile, ion and electron densities profile, ion velocity profile, and dust dynamics. It is found that the negatively charged dust grains accelerate in the region of weak electric field, whereas the fluctuation decreases in the region of strong electric field. A significant change has been observed in dust dynamics with the presence of two different natures of dust charges and its magnitudes. The velocity fluctuations of negatively and positively charged grains are in opposite phase. Furthermore, the comparative study of dust velocity in the presence and absence of dust-neutral collisions has been presented.

Modelling of electronegative collisional warm plasma for plasma-surface interaction process

An electronegative collisional plasma having warm and massive positive ions, non-extensive distributed electrons and Boltzmann distributed negative ions is modelled for the plasma-surface interaction process that is used for the surface nitriding. Specifically the sheath formation is evaluated through the Bohm’s criterion, which is found to be modified, and the variation of the sheath thickness and profiles of the density of plasma species and the net space charge density in the sheath region in addition to the electric potential. The effect of ion temperature, non-extensivity and collisional parameter is examined in greater detail considering the collisional cross-section to obey power-law dependency on the positive ion velocity. The positive ions are found to enter in the sheath region at lower velocities in the collisional plasma compared to the case of collision-less plasma; this velocity sees minuscule reduction with increasing non-extensivity. The increasing ion temperature and collisional parameter lead to the formation of sheath with smaller thickness.

Electron acoustic solitary waves in unmagnetized nonthermal plasmas

Electron acoustic (EA) solitary waves (SWs) are studied in an unmagnetized plasma consisting of hot electrons (following Cairns-Tsalli distribution), inertial cold electrons, and stationary ions. By employing a reductive perturbation technique (RPT), the nonlinear Korteweg–de Vries (KdV) equation is derived and its SW solution is analyzed. Here, the effects of plasma parameters such as the nonextensivity parameter (q), the nonthermality of electrons (α), and the cold-to-hot electron density ratio (β) are investigated.

Head-on collision of two ion-acoustic solitons in pair-ion plasmas with nonthermal electrons featuring Tsallis distribution

Abstract The head-on collision between two ion-acoustic solitons (IASs) is studied in pair ions plasmas with hybrid Cairns–Tsallis-distributed electrons. The chosen model is inspired from the experimental studies of Ichiki et al. [Phys. Plasmas 8, 4275 (2001)]. The extended Poincaré–Lighthill–Kuo (PLK) method is employed to obtain the phase shift due to the IASs collision. Both analytical and numerical results reveal that the magnitude of the phase shift is significantly affected by the nonthermal and nonextensive parameters (α and q), the number density ratios (μ and υ) as well as the mass ratio σ. For a given mass ratio σ ≃ 0.27 $\sigma \simeq 0.27$ (Ar+, SF 6 − ${\text{SF}}_{6}^{-}$ ), the magnitude of the phase shift Δ Q ( 0 ) ${\Delta}{Q}^{\left(0\right)}$ decreases slightly (increases) with the increase of q (α). The effect of α on Δ Q ( 0 ) ${\Delta}{Q}^{\left(0\right)}$ is more noticeable in the superextensive distribution case (q &lt; 1). As σ increases [ σ ≃ 0.89 $\sigma \simeq 0.89$ (Xe+, SF 6 − ${\text{SF}}_{6}^{-}$ )], the phase shift becomes wider. In other terms, the phase shift was found to be larger under the effect of higher densities of the negative ions. Our findings should be useful for understanding the dynamics of IA solitons’ head-on collision in space environments [namely, D-regions ( H + ${\text{H}}^{+}$ , O 2 − ${\text{O}}_{2}^{-}$ ) and F-regions (H+, H−) of the Earth’s ionosphere] and in laboratory double pair plasmas [namely, fullerene (C+, C−) and laboratory experiment (Ar+, F−)].

Head-on collision of multi-solitons in a magnetized space dusty plasma in the presence of nonextensive polarization force

ABSTRACT In this investigation, the head-on collision of dust-acoustic (DA) multi-solitons in a magnetized space dusty plasma consisting of negative dust, Maxwellian electrons and q-nonextensively distributed ions under the influence of polarization force is studied. The presence of the q-nonextensive ions yields eloquent alteration in the polarization force. An increase in the nonextensive parameter (via q) lead to escalate the polarization parameter. Two-sided KdV equations are obtained by adopting Poincaré–Lighthill–Kuo (PLK) method. Further, the direct Hirota method is employed to carry out multi-solitons solutions of KdV equations. The q-nonextensive polarization force has a great impact on the phase shifts after the interaction of single, double, and triple-(DA) solitons. The combined effects of various physical parameters such as polarization force parameter (via R), nonextensivity (via q) and magnetic field strength (via Ω) on the phase shifts of DA multi-solitons have also been illustrated. It is found that the magnetic field alters the polarization effect which leads to modify the phase shifts. The findings of our investigation may be helpful to explore the interaction of multi-solitons in a magnetized space dusty plasma such as planetary rings and comet tails where nonextensively distributed ions, negative dust and Maxwellian electrons are prevalent.

Oblique propagation of nonlinear ion-acoustic cnoidal waves in magnetized electron–positron–ion plasmas with nonextensive electrons

Theoretical investigation of nonlinear electrostatic ion-acoustic cnoidal waves (IACWs) is presented in magnetized electron–positron–ion plasma with nonextensive electrons and Maxwellian positrons. Using reductive perturbation technique, Korteweg–de Vries equation is derived and its cnoidal wave solution is analyzed. For given plasma parameters, our model supports only positive potential (compressive) IACW structures. The effect of relevant plasma parameters (viz., nonextensive parameter q, positron concentration p, temperature ratio σ, obliqueness l 3) on the characteristics of IACWs is discussed in detail.