Non-extensive Tsallis Distribution Research Articles

Nonlinear periodic ion-acoustic waves in nonthermal plasmas

A Sagdeev pseudopotential analysis is developed for the propagation of nonlinear periodic ion-acoustic waves in a plasma comprising cold fluid ions and various models of nonthermal electron descriptions. In plasma nonlinear wave studies, whether addressing solitary or periodic modes, the more common nonthermal distributions are the Cairns, kappa, and Tsallis models. A mathematically and physically consistent description incorporates three evident properties: there is conservation per cycle of ion and electron densities in addition to ion flux, the solutions reduce for very small amplitudes to linear waves, and the nonlinear periodic structures are generated by a perturbation of the undisturbed equilibrium. After establishing the corresponding general analytical methodology, a numerical analysis is given, with illustrative graphs, for the nonthermal Cairns, superthermal kappa, and nonextensive Tsallis distributions.

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.

Hadron gas in the presence of a magnetic field using non-extensive statistics: a transition from diamagnetic to paramagnetic system

Non-central heavy-ion collisions at ultra-relativistic energies are unique in producing magnetic fields of the largest strength in the laboratory. Such fields being produced at the early stages of the collision could affect the properties of Quantum Chromodynamics matter formed in the relativistic heavy-ion collisions. The transient magnetic field leaves its reminiscence, which in principle, can affect the thermodynamic and transport properties of the final state dynamics of the system. In this work, we study the thermodynamic properties of a hadron gas in the presence of an external static magnetic field using a thermodynamically consistent non-extensive Tsallis distribution function. Various thermodynamical observables such as energy density (ϵ), entropy density (s), pressure (P) and speed of sound (c s) are studied. Investigation of magnetization (M) is also performed and this analysis reveals an interplay of diamagnetic and paramagnetic nature of the system in the presence of a magnetic field of varying strength. Further, to understand the system dynamics under equilibrium and non-equilibrium conditions, the effect of the non-extensive parameter (q) on the above observables is also studied.

Quasi-Maxwellian Interference Term Functions

It is a well-known fact that the physical processes taking place in nuclear reactors currently operating are addressed based on the Maxwell–Boltzmann distribution. However, for new reactor designs that can be developed, as well as for any operational condition beyond the basis of the project, it may be relevant to consider another type of velocity distribution. In recent papers, for the evaluation of the absorption cross-section, the consequences of replacing the Maxwell–Boltzmann distribution by the non-extensive Tsallis distribution in addition to the Kaniadakis distribution were discussed. To complement these results, this paper deals with the influence of these same two distributions for the resonant scattering, which is described mathematically by the Interference Term Function. That way, two new deformed Interference Term Functions are obtained. To study the range of values where the effect is meaningful, numerical tests are carried out considering several values for the deformation parameters.

Event multiplicity, transverse momentum and energy dependence of charged particle production, and system thermodynamics in pp collisions at the Large Hadron Collider

In the present work, we study the recent collision energy and multiplicity dependence of the charged particle transverse momentum spectra as measured by the ALICE collaboration in pp collisions at and 13 TeV using the non-extensive Tsallis distribution and the Boltzmann–Gibbs blast wave (BGBW) model. A thermodynamically consistent form of the Tsallis distribution is used to extract the kinetic freeze-out parameters from the transverse momentum spectra of charged particles at mid-rapidity. In addition, a comprehensive study of fitting range dependence of transverse momentum spectra on the freeze-out parameters is done using Tsallis statistics. The applicability of BGBW model is verified by fitting the transverse momentum spectra of the bulk part (∼2.5 GeV/c) for both 5.02 and 13 TeV energies and also in different multiplicity classes. The radial flow, is almost independent of collision energy and multiplicity whereas the behavior of kinetic freeze-out temperature significantly depends on multiplicity classes. It is found that the Tsallis distribution generally leads to a better description for the complete transverse momentum spectra whereas the BGBW model explains the bulk part of the system.

A deformed Doppler Broadening Function considering the Tsallis speed distribution

The Doppler Broadening Phenomenon provides a good description for the effects of thermal agitation on the cross-section of the neutron-nucleus interaction and has its theoretical basis on Quantum Mechanics through the Single Level Breit-Wigner Formalism, as well as on Statistical Mechanics, with the Maxwell Boltzmann distribution. In this paper, the consequences of replacing the Maxwell Boltzmann speed distribution by the non-extensive Tsallis distribution in the evaluation of the absorption cross section in three-dimension is discussed. The Tsallis speed distribution introduces a deformation parameter q that measures the deviation from the Maxwellian speed distribution and, by taking the limit q→1, the deformation is removed. In this context, and considering the Tsallis distribution, an integral form is obtained for a deformed Doppler Broadening Function in the scope of the single-level Breit-Wigner formalism and Bethe-Placzek approximations. This deformed Doppler Broadening Function reproduces the well-established conventional Doppler Broadening Function at the limit when q→1 in which the deformation is removed. In order to study the range of values where the effect is significant, numerical tests were carried out considering several values for the q parameter.

Transverse-momentum spectra and nuclear modification factor using Boltzmann Transport Equation with flow in Pb+Pb collisions at $\sqrt{s_{NN}} = 2.76$ s N N = 2 . 76 TeV

In the continuation of our previous work, the transverse momentum ($p_T$) spectra and nuclear modification factor ($R_{AA}$) are derived using relaxation time approximation of Boltzmann Transport Equation (BTE). The initial $p_T$-distribution used to describe $p+p$ collisions has been studied with the pQCD inspired power-law distribution, the Hagedorn's empirical formula and with the Tsallis non-extensive statistical distribution. The non-extensive Tsallis distribution is observed to describe the complete range of the transverse momentum spectra. The Boltzmann-Gibbs Blast Wave (BGBW) distribution is used as the equilibrium distribution in the present formalism, to describe the $p_T$-distribution and nuclear modification factor in nucleus-nucleus collisions. The experimental data for Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV at the Large Hadron Collider at CERN have been analyzed for pions, kaons, protons, $K^{*0}$ and $\phi$. It is observed that the present formalism while explaining the transverse momentum spectra upto 5 GeV/c, explains the nuclear modification factor very well upto 8 GeV/c in $p_T$ for all these particles except for protons. $R_{AA}$ is found to be independent of the degree of non-extensivity, $q_{pp}$ after $p_T \sim$ 8 GeV/c.

Quasi-power Law Ensembles

Quasi-power law ensembles are discussed from the perspective of nonextensive Tsallis distributions characterized by a nonextensive parameter q. A number of possible sources of such distributions are presented in more detail. It is further demonstrated that data suggest that nonextensive parameters deduced from Tsallis distributions functions f (pT ), q1, and from multiplicity distributions (connected with Tsallis entropy), q2, are not identical and that they are connected via q1 + q2 = 2. It is also shown that Tsallis distributions can be obtained directly from Shannon information entropy, provided some special constraints are imposed. They are connected with the type of dynamical processes under consideration (additive or multiplicative). Finally, it is shown how a Tsallis distribution can accommodate the log-oscillating behavior apparently seen in some multiparticle data.

Study of the particle transverse momentum spectra in relativistic heavy ion collisions using the Tsallis statistics

In this work, we have analized the transverse momentum spectra of charged pions produced in Au+Au collisions at different energies using the non-extensive Tsallis distribution. The centrality and rapidity dependence of the non-extensivity parameter, q, is studied. In addition, comparisons with simulated heavy ion collisions data using the UrQMD code will be presented.

Mass-energy radiative transfer and momentum extraction by gravitational wave emission in the collision of two black holes

We examine numerically the head-on collision of two boosted Schwarzschild black holes, in the realm of Robinson-Trautman spacetimes. Characteristic initial data for the system are constructed and the Robinson-Trautman equation is integrated for these data using a numerical code based on the Galerkin-collocation method. The initial data already have a common horizon so that the evolution covers the post-merger regime up to the final configuration, when the gravitational wave emission ceases. In the nonlinear regime gravitational waves are emitted, extracting mass and linear momentum from the system. The final configuration is a boosted Schwarzschild black hole with rest mass larger than the masses of the two individual initial black holes, and with a smaller final boost parameter characterizing the recoil velocity of the remnant. The efficiency $\ensuremath{\Delta}$ of the mass-energy extraction by gravitational waves is evaluated. The points $(\ensuremath{\Delta},y)$, where $y$ is the (normalized) rest mass of the remnant black hole, satisfy a nonextensive Tsallis distribution with entropic index $q\ensuremath{\simeq}1/2$ for $y\ensuremath{\lesssim}12$. Beyond $y\ensuremath{\sim}12$ the experimental points deviate from the distribution function and the efficiency presents an absolute maximum for the case of equally massive individual colliding black holes; the remnant has no recoil in this case. By using the Bondi mass formula we also evaluate the total energy ${E}_{W}$ carried out by gravitational waves as well as the radiative corrections to the efficiency. ${E}_{W}$ increases monotonically with $y$ and the experimental points $({E}_{W},y)$ also satisfy a nonextensive Tsallis distribution but with $q\ensuremath{\simeq}2/3$, up to $y\ensuremath{\sim}14.2$. Beyond this value the experimental points increase faster than the distribution function. For any initial infalling velocity $v$, the distribution of momentum of the remnant exhibits a maximum at ${\ensuremath{\alpha}}_{1}={\ensuremath{\alpha}}_{m}\ensuremath{\simeq}0.667$, where ${\ensuremath{\alpha}}_{1}$ is related to the ratio of pre-merger rest masses, and has a one-to-one correspondence with $y$ for fixed $v$. Two distinct regimes of gravitational wave emission can be characterized according to (i) ${\ensuremath{\alpha}}_{1}<{\ensuremath{\alpha}}_{m}$: bursts of gravitational bremsstrahlung; (ii) ${\ensuremath{\alpha}}_{1}>{\ensuremath{\alpha}}_{m}$: quiescent long time emission of gravitational waves. This picture is also sustained by the analysis of the time behavior of the power emitted ($d{E}_{W}/du$).

Near-thermal equilibrium with Tsallis distributions in heavy-ion collisions

Hadron yields in high-energy heavy-ion collisions have been fitted with thermal models using standard (extensive) statistical distributions. These models give insight into the freeze-out conditions at varying beam energies and lead to a systematic consistent picture of freeze-out conditions at all beam energies. In this paper, we investigate changes to this analysis when the statistical distributions are replaced by non-extensive Tsallis distributions for hadrons. We investigate the particle yields at SPS and RHIC energies and obtain better fits with smaller χ2 for the same hadron data, as applied earlier in the thermal fits for SPS energies but not for RHIC energies.