Ensemble properties and molecular dynamics of unstable systems

The Hertel-Thirring cell model for unstable systems (of purely attractive particles) is solved in the canonical ensemble for arbitrary dimensions. The differences between the phase transitions found in the canonical and in the microcanonical ensemble are discussed. The cluster phase (with a complete collapse in the ground state) exhibits the nonextensive character of the cell model. The results of the cell model are compared with molecular-dynamics simulations of a one-dimensional model with a rectangular-well pair potential. The simulations support the relevance of the cell model to characterize basic properties of gravitational systems.     

Landau damping of ion-acoustic waves with simultaneous effects of non-extensivity and non-thermality in the presence of hybrid Cairns-Tsallis distributed electrons

The dispersion properties and Landau damping rate of ion-acoustic waves (IAWs) with the hybrid Cairns-Tsallis distributed (CTD) electrons and Maxwellian ions are investigated using the plasma kinetic model based on Vlasov-Poisson's equations. For both super-extensive (q < 1) and sub-extensive (q > 1) plasmas, the dielectric response function, real frequency, and Landau damping rate of IAWs are derived. By taking the effect of θ_{i, e} (ion-to-electron temperature ratio) into account, it is found that with the increase of ion temperature, the real frequency and wave dispersion effects increase as well (for both super-extensive and sub-extensive cases). Exploring the properties of the Landau damping rate of IAWs with the simultaneous presence of non-thermal parameter α and non-extensive parameter q, a comparison of numerical and analytical results is presented. It is found that in different ranges of θ_{e, i} (electron-to-ion temperature ratio), on decreasing the values of the non-extensive parameter and increasing values of the non-thermal parameter, the weak damping rate is observed (vice versa) in super-extensive or super-thermal plasma, although the trend of the damping rate in sub-thermal plasma is similar (as in the case of super-thermal plasma) but is less weak. It is further revealed that the damping rate of IAWs in thermal plasmas (Maxwellian) is stronger than the damping rate of IAWs in the case of non-thermal plasmas (CTD). The current study is applicable to provide deep insight and further allow the exploration of electrostatic plasma modes in different space and laboratory plasma environments where the hybrid CTD plasma exists.     

Effects of axial magnetic field and nonextensivity on small amplitude dust acoustic solitary waves in dusty plasma

The nonlinear propagation of small amplitude dust‐acoustic (DA) solitary waves in magnetized dusty plasma consisting of negatively charged mobile dust fluid, and Boltzmann‐distributed electrons and ions with two distinct temperatures following a q‐nonextensive distribution are investigated. In this article, a number of nonlinear equations, namely, the Korteweg–de‐Vries (K‐dV) equations, have been derived by employing the reductive perturbation technique that is valid for a small but finite amplitude limit. The effects of nonextensivity of ions with two distinct temperatures and dust concentration on the amplitude and width of DA solitary waves are investigated theoretically. It is observed that both the nonextensive and low‐temperatures ions significantly modify the basic properties and polarities of DA solitary waves. It is shown that both positive and negative potential DA solitons occur in this case. The implications of these results to some astrophysical environments and space plasmas (e.g., stellar polytropes, peculiar velocity distributions of galaxies, and collisionless thermal plasma), and laboratory dusty plasma systems are briefly mentioned.     

Effect of nonextensivity on the characteristics of supersolitons in a two-temperature electron plasma

Ion-acoustic supersolitons are investigated in an unmagnetized two-temperature electron plasma comprising cold fluid ions, hot nonextensive electrons, and cool Maxwellian electrons by using the Sagdeev pseudopotential technique. Existence domain of positive polarity supersolitons in terms of Mach number is computed, which is found to exist for Mach numbers beyond the existence of positive double layers. The domain of existence of supersolitons diminishes with the decrease of the nonextensive parameter (q ). The amplitude and width of the supersolitons are dependent on the cool-to-hot electron temperature ratio (τ ), cool electron density (f ), and nonextensive parameter (q ). The increase of cool electron density increases the amplitude of the supersolitons. For fixed values of f , q _, and Mach numbers, the decrease of τ exhibits more distinct wiggles in the electric fields of supersolitons. The present work may be helpful for further study of supersolitons in the auroral plasma.     

Problems of nonextensivity in hadron thermodynamics

Hadron thermodynamics deals with a gas of indistinguishable particles whose mass spectrum is taken to have the formKm ^a e ^bm . A long-standing inconsistency is pointed out, namely the nonextensive nature of the entropy found in some treatments by way of the microcanonical ensemble, which contrasts with the extensive nature found by way of the canonical ensemble. The former result is due to an error. After correction, the two ensembles are found to lead to the same expressions for the thermodynamic quantities ifa –7/2. Some of these expressions are new. Fora<–5/2, the microcanonical approach is used to examine a model in which one particle is appreciably heavier than the rest. However, the resulting entropy is found to be unphysical.     

Generalized algebra within a nonextensive statistics

By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators: addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g. associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element are no more universal within the generalized algebra.     

Statistical entropy of a Schwarzschild black hole

A model describing the internal microstates of particles is used to calculate the statistical entropy of a Schwarzschild black hole. The state of the system is described by a nonextensive entropy function which is superadditive and so fails to be concave. A strict maximum of the entropy does not exist; nonetheless, the entropy increases on merging two such systems.     

Cylindrical and Spherical Electron-Acoustic Shock Waves in Electron-Positron-Ion Plasmas with Nonextensive Electrons and Positrons

Electron-acoustic shock waves (EASWs) in an unmagnetized four-component plasma (containing hot electrons and positrons following the q-nonextensive distribution, cold mobile viscous electron fluid, and immobile positive ions) are studied in nonplanar (cylindrical and spherical) geometry. With the help of the reductive perturbation method, the modified Burgers equation is derived. Analytically, the effects of nonplanar geometry, nonextensivity, relative number density and temperature ratios, and cold electron kinematic viscosity on the basic properties (viz. amplitude, width, speed, etc.) of EASWs are discussed. It is examined that the EASWs in nonplanar geometry significantly differ from those in planar geometry. The results of this investigation can be helpful in understanding the nonlinear features of EASWs in various astrophysical plasmas.     

Nonlinear ion‐acoustic cnoidal wave in electron‐positron‐ion plasma with nonextensive electrons

Effects of plasma nonextensivity on the nonlinear cnoidal ion‐acoustic wave in unmagnetized electron‐positron‐ion plasma have been investigated theoretically. Plasma positrons are taken to be Maxwellian, while the nonextensivity distribution function was used to describe the plasma electrons. The known reductive perturbation method was employed to extract the KdV equation from the basic equations of the model. Sagdeev potential, as well as the cnoidal wave solution of the KdV equation, has been discussed in detail. We have shown that the ion‐acoustic periodic (cnoidal) wave is formed only for values of the strength of nonextensivity (q). The q allowable range is shifted by changing the positron concentration (p) and the temperature ratio of electron to positron (σ). For all of the acceptable values of q, the cnoidal ion‐acoustic wave is compressive. Results show that ion‐acoustic wave is strongly influenced by the electron nonextensivity, the positron concentration, and the temperature ratio of electron to positron. In this work, we have investigated the effects of q, p, and σ on the characteristics of the ion‐acoustic periodic (cnoidal) wave, such as the amplitude, wavelength, and frequency.     

An entropy group and its representation in thermodynamics of nonextensive systems

The Abel entropy group and its matrix representation with the general law of nonextensive entropy composition and quadratic nonlinearity are defined. Four types of matrices for the corresponding parametrical entropies are given and geometries for their measures are determined. New two-parameter entropy is introduced, and a distribution is found for equilibrium thermodynamics of nonextensive systems. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 79–86, February, 2009.