Nonextensivity Index Research Articles

Tsallis statistics and generalized uncertainty principle

It has been argued that non-Gaussian statistics provide a natural framework to investigate semiclassical effects in the context of Planck-scale deformations of the Heisenberg uncertainty relation. Here we substantiate this point by considering the Unruh effect as a specific playground. By working in the realm of quantum field theory, we reformulate the derivation of the modified Unruh effect from the generalized uncertainty principle (GUP) in the language of the nonextensive Tsallis thermostatistics. We find a nontrivial monotonic relation between the nonextensivity index q and the GUP deformation parameter beta , which generalizes an earlier result obtained in quantum mechanics. We then extend our analysis to black hole thermodynamics. We preliminarily discuss our outcome in the broader context of an effective description of Planck-scale gravitational physics based on Tsallis theory.

The effect of non-extensive particles on slowing down and diffusion of a particle's beam in the plasma

The slowing down as well as the deflection time of test particles in the plasma is studied in the non-extensive statistics. The relevant relations are derived using Fokker Planck equation. It is remarked that the slowing down and deflection times modify considerably in the non-extensive statistics in comparison with Boltzmann Gibbs one. It is found that by decreasing non-extensivity index q (1/3<q≤1 which corresponds to plasma with excess super extensive particles), both the slowing down and deflection times will be increased. Also, for q≥1, i.e., the sub-extensive particles, the same results are obtained by decreasing q. Additionally, the effects of non-extensive distributed particles on the electrical conductivity and diffusion coefficient of plasma are studied. It is shown that plasmas with smaller qs are better conductors in both 1/3<q≤1 and q≥1. In addition, it is observed that by increasing q, Dreicer field will increase in both super-extensive and sub-extensive particles. Moreover, it is found that the diffusion coefficient across a magnetic field is decreased by decreasing q. Furthermore, our results reduce to the solutions of Maxwellian plasma at the extensive limit q → 1. This research will be helpful in understanding the relaxation times and transport properties of fusion and astrophysical plasmas.

Arbitrary amplitude electron acoustic waves in a magnetized nonextensive plasma

Arbitrary amplitude electron acoustic (EA) solitary waves in a magnetized nonextensive plasma comprising of cool fluid electrons, hot nonextensive electrons, and immobile ions are investigated. The linear dispersion properties of EA waves are discussed. We find that the electron nonextensivity reduces the phase velocities of both modes in the linear regime: similarly the nonextensive electron population leads to decrease of the EA wave frequency. The Sagdeev pseudopotential analysis shows that an energy-like equation describes the nonlinear evolution of EA solitary waves in the present model. The effects of the obliqueness, electron nonextensivity, hot electron temperature, and electron population are incorporated in the study of the existence domain of solitary waves and the soliton characteristics. It is shown that the boundary values of the permitted Mach number decreases with the nonextensive electron population, as well as with the electron nonextensivity index, q. It is also found that an increase in the electron nonextensivity index results in an increase of the soliton amplitude. A comparison with the Vikong Satellite observations in the dayside auroral zone is also taken into account.

Langmuir oscillations in a nonextensive electron-positron plasma

The Langmuir oscillations, Landau damping, and growing unstable modes in an electron-positron (EP) plasma are studied by using the Vlasov and Poisson's equations in the context of the Tsallis's nonextensive statistics. Logically, the properties of the Langmuir oscillations in a nonextensive EP plasma are remarkably modified in comparison with that of discussed in the Boltzmann-Gibbs statistics, i.e., the Maxwellian plasmas, because of the system under consideration is essentially a plasma system in a nonequilibrium stationary state with inhomogeneous temperature. It is found that by decreasing the nonextensivity index q which corresponds to a plasma with excess superthermal particles, the phase velocity of the Langmuir waves increases. In particular, depend on the degree of nonextensivity, both of damped and growing oscillations are predicted in a collisionless EP plasma, arise from a resonance phenomena between the wave and the nonthermal particles of the system. Here, the mechanism leads to the unstable modes is established in the context of the nonextensive formalism yet the damping mechanism is the same developed by Landau. Furthermore, our results have the flexibility to reduce to the solutions of an equilibrium Maxwellian EP plasma (extensive limit q→1), in which the Langmuir waves are only the Landau damped modes.

Formula omitted]-thermostatistics of noise-driven Van der Pol oscillator

We study the noise-driven Van der Pol oscillator whose dissipative term yields stationary distributions of the q-exponential form. The thermostatistical observables such as internal energy and temperature are also calculated. Since this model depends on a control parameter which acts as a coupling parameter between the system and environment, the noise-driven Van der Pol oscillator also serves as a nice model where the interplay of both nonextensivity and open character of the system can be considered. We observe that the model behaves as if it is coupled to a finite heat bath for small values of control parameter. As the control parameter increases, the effect of the linear friction coefficient disappears, and the stationary distribution becomes a q-Gaussian, thereby breaking the coupling with the finite heat bath. Finally, we remark that negative temperature values are not observed contrary to an earlier analysis of a similar model. The elucidation of this point is commented in the framework of the misidentification of the correct form of the q-exponentials and the nonextensivity index q.

The maximization of Tsallis entropy with complete deformed functions and the problem of constraints

By only requiring the q deformed logarithms ( q exponentials) to possess arguments chosen from the entire set of positive real numbers (all real numbers), we show that the q-logarithm ( q exponential) can be written in such a way that its argument varies between 0 and 1 (among negative real numbers) for 1 ⩽ q < 2 , while the interval 0 < q ⩽ 1 corresponds to any real argument greater than 1 (positive real numbers). These two distinct intervals of the nonextensivity index q, also the expressions of the deformed functions associated with them, are related to one another through the relation ( 2 − q ) , which is so far used to obtain the ordinary stationary distributions from the corresponding escort distributions, and vice versa in an almost ad hoc manner. This shows that the escort distributions are only a means of extending the interval of validity of the deformed functions to the one of ordinary, undeformed ones. Moreover, we show that, since the Tsallis entropy is written in terms of the q-logarithm and its argument, being the inverse of microstate probabilities, takes values equal to or greater than 1, the resulting stationary solution is uniquely described by the one obtained from the ordinary constraint. Finally, we observe that even the escort stationary distributions can be obtained through the use of the ordinary averaging procedure if the argument of the q-exponential lies in ( − ∞ , 0 ] . However, this case corresponds to, although related, a different entropy expression than the Tsallis entropy.

Q-Gaussian representation of non-Lorentzian Mössbauer lineshapes

Generalized statistical physics (non-extensive/Tsallis) is being extensively used to study anomalous results in condensed matter physics. Mossbauer line shapes for systems like proteins and glasses show non-Lorentzian behaviour. In this paper we show q-Gaussian distribution can be used to represent non Lorentzian Mossbauer line shapes where q is non-extensivity index.

Majorization relations and disorder in generalized statistics

The theory of majorization is applied to examine the disorder properties of generalized thermal distributions arising in non-extensive statistics. We show that they share with the Boltzmann–Gibbs thermal state the property of becoming more mixed as the temperature increases, implying the increase of any associated disorder measure. We also show that power-law distributions exhibit a second mixing parameter associated with the non-extensivity index. As application, we examine the thermal response of quantum entanglement in a spin system for different statistics.

Majorization properties of generalized thermal distributions

We examine the majorization properties of general thermal-like mixed states depending on a set of parameters. Sufficient conditions which ensure the increase in mixedness, and hence of any associated entropic form, when these parameters are varied, are identified. We then discuss those exhibiting a power law distribution, showing that they can be characterized by two distinct mixing parameters, one associated with temperature and the other with the non-extensivity index q. Illustrative numerical results are also provided.

Density operators that extremize Tsallis entropy and thermal stability effects

Quite general, analytical (both exact and approximate) forms for discrete probability distributions (PDs) that maximize Tsallis entropy for a fixed variance are here investigated. They apply, for instance, in a wide variety of scenarios in which the system is characterized by a series of discrete eigenstates of the Hamiltonian. Using these discrete PDs as “weights” leads to density operators of a rather general character. The present study allows one to vividly exhibit the effects of non-extensivity. Varying Tsallis’ non-extensivity index q one is seen to pass from unstable to stable systems and even to unphysical situations of infinite energy.

Escort Husimi distributions, Fisher information and nonextensivity

We evaluate generalized information measures constructed with Husimi distributions and connect them with the Wehrl entropy, on the one hand, and with thermal uncertainty relations, on the other one. The concept of escort distribution plays a central role in such a study. A new interpretation concerning the meaning of the nonextensivity index q is thereby provided. A physical lower bound for q is also established, together with a “state equation” for q that transforms the escort-Cramer–Rao bound into a thermal uncertainty relation.

Escort Husimi distributions, Fisher information and nonextensivity

We evaluate generalized information measures constructed with Husimi distributions and connect them with the Wehrl entropy, on the one hand, and with thermal uncertainty relations, on the other one. The concept of escort distribution plays a central role in such a study. A new interpretation concerning the meaning of the nonextensivity index q is thereby provided. A physical lower bound for q is also established, together with a “state equation” for q that transforms the escort-Cramer–Rao bound into a thermal uncertainty relation.

A note on non-thermodynamical applications of non-extensive statistics

It is pointed out that the constraint to be imposed to the maximization of the entropy for processes outside the class of thermodynamical systems, is generally not well defined. In fact, any probability distribution can be derived from Jaynes's principle with a suitable choice of the constraint. In the case of Tsallis's non-extensive formalism, this implies that it is not possible to establish any connection between specific non-thermodynamical processes and non-extensive mechanisms and, in particular, to assign any unambiguous non-extensivity index q to those processes.

Q-thermostatistics and the analytical treatment of the ideal Fermi gas

We discuss relevant aspects of the exact q-thermostatistical treatment for an ideal Fermi system. The grand-canonical exact generalized partition function is given for arbitrary values of the nonextensivity index q, and the ensuing statistics is derived. Special attention is paid to the mean occupation numbers of single-particle levels. Limiting instances of interest are discussed in some detail, namely, the thermodynamic limit, considering in particular both the high- and low-temperature regimes, and the approximate results pertaining to the case q∼1 (the conventional Fermi–Dirac statistics corresponds to q=1). We compare our findings with previous Tsallis’ literature.

Equipartition and virial theorems in a nonextensive optimal Lagrange multipliers scenario

We revisit some topics of classical thermostatistics from the perspective of the nonextensive optimal Lagrange multipliers formalism (OLM), a recently introduced technique for dealing with the maximization of Tsallis' information measure. It is shown that equipartition and virial theorems can be reproduced by Tsallis' nonextensive formalism independently of the value of the nonextensivity index.