Power distributions for self-gravitating astrophysical systems based on nonextensive Tsallis kinetics

The long-time development of self-gravitating gaseous astrophysical systems (in particular, the evolution of the protoplanet accretion disk) is mainly determined by relatively fast processes of the collision relaxation of particles. However, slower dynamical processes related to force (Newton or Coulomb) interactions between particles should be included (as q-collisions) in the nonextensive kinetic theory as well. In the present paper, we propose a procedure to include the Newton self-gravity potential and the centrifugal potential in the near-equilibrium power-like q-distribution in the phase space, obtained (in the framework of nonextensive statistics) by means of the modified Boltzmann equation averaged with respect to an unnormalized distribution. We show that if the power distribution satisfies the stationary q-kinetic equation, then the said equation imposes clear restrictions on the character of the long-term force field and on the possible dependence of hydrodynamic parameters of the coordinates: it determines those parameters uniquely. We provide a thermodynamic stability criterion for the equilibrium of the nonextensive system. The results allow us to simulate the evolution of gaseous astrophysical systems (in particular, the gravitational stability of rotating protoplanet accretion disks) more adequately.     

What Does the Nonextensive Parameter Stand for in Self-Gravitating Systems?

It is natural important question for us to ask what the nonextensive parameter stands for when Tsallis statistics is applied to the self-gravitating systems. In this paper, some properties of the nonextensive parameter and Tsallis’ equilibrium distribution for the self-gravitating system are discussed in the framework of nonextensive kinetic theory. On the basis of the solid mathematical foundation, the nonextensive parameter can be expressed by a formula with temperature gradient and the gravitational potential and it can be related to the non-isothermal (nonequilibrium stationary state) nature of the systems with long-range interactions. We come to the conclusion that Tsallis’ equilibrium distribution is corresponding to the physical state of self-gravitating system at the hydrostatic equilibrium. PACS numbers 05.20.-y, 95.30.Lz     

Investigation on spectral character of ELF electromagnetic radiations during Leonid 2009 meteor shower

The problem of solitary electron acoustic (EA) wave propagation in a plasma with nonthermal hot electrons featuring the Tsallis distribution is addressed. A physically meaningful nonextensive nonthermal velocity distribution is outlined. It is shown that the effect of the nonthermal electron nonextensivity on EA waves can be quite important. Interestingly, we found that the phase speed of the linear EA mode increases as the entropic index q decreases. This enhancement is weak for q>1, and significant for q<1. For a given nonthermal state, the minimum value of the allowable Mach numbers is lowered as the nonextensive nature of the electrons becomes important. This critical limit is shifted towards higher values as the nonthermal character of the plasma is increased. Moreover, our plasma model supports rarefactive EA solitary waves the main quantities of which depend sensitively on q. This dependency (for q>1) becomes less noticeable as the nonthermal parameter decreases. Nevertheless, decreasing α yields for q<0 a different result, a trend which may be attributed to the functional form of the nonthermal nonextensive distribution. Our study (which is not aimed at putting the ad hoc Cairns distribution onto a more rigorous foundation) suggests that a background electron nonextensivity may influence the EA solitons.     

Effect of q-nonextensive distribution of electrons on dust-ion acoustic solitons

Properties of dust-ion acoustic solitary waves (DIASWs) in dusty plasmas composed of nonextensive electrons, cold fluid ions and stationary dust particles are investigated. The possibility of soliton formation and the effect of nonextensivity of the electron distribution on the soliton characters are studied using the pseudo-potential method. Regions of parameters in which a solitary wave can be propagated in the plasma is analyzed too. It is found that the solitary excitations strongly depend on the electron-ion density ratio (μ), Mach numbers (M) as well as the nonextensive parameter (q). It is shown that the domain of allowed Mach numbers depends drastically on the plasma parameters and especially on the electron nonextensivity. It is found that beyond a threshold value of the nonextensive parameter (q), dust-ion acoustic solitons are admitted.     

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.     

Ion acoustic solitons in a nonextensive plasma with multi-temperature electrons

The properties of propagation of small amplitude ion acoustic solitary waves (IASWs) are studied in a plasma containing cold fluid ions and multi-temperature electrons (cool and hot electrons) with nonextensive distribution. Korteweg-de Vries (KdV) equation with finite amplitude is derived using a reductive perturbation method. From the solitary solutions of KdV equation, the combined effects of nonextensivity and density ratio are studied on characteristics of ion acoustic (IA) solitary waves. Positive as well as negative polarity solitons exist. Since singularity exists for A=0 so we have also derived modified Korteweg de Vries (mKdV) equation to study the solitonic solution for critical values of physical parameters (q,f,σ). The nonextensivity of electrons (via q) and density ratio of electrons and ions (via f) and temperature ratio (σ) significantly influence the characteristics of ion acoustic solitary structures.     

Analytical predictions of non-Gaussian distribution parameters for stellar plasmas

Stimulated by the recent debate on the physical relevance and on the predictivity of q-Gaussian formalism, we present specific analytical expressions for the parameters characterizing non-Gaussian distributions, such as the nonextensive parameter q, expressions that we have proposed for different physical systems, an important example being plasmas in the stellar cores.     

Ion acoustic solitary waves in a weakly relativistic plasma with <Emphasis Type="Italic">q</Emphasis>-nonextensive electrons and thermal positrons

Nonlinear ion acoustic solitary waves (IASWs) are addressed in a weakly relativistic plasma consisting of cold ion fluid, q-nonextensive electron velocity distribution and Boltzmann distributed positron. The Korteweg-de Vries- (KdV) equation is derived by reductive perturbation method. We investigate the effect of nonextensive electrons on solitary waves in this medium. It is found that only compressive solitons can be appeared in the existence of nonextensive electrons. It is shown that the structure of soliton depend sensitively on the q-nonextensive parameter.     

Ion-acoustic rogue waves in a plasma with a q-nonextensive electron velocity distribution

Ion-acoustic rogue waves (IARWs) are addressed in a two-component plasma with a q-nonextensive electron velocity distribution. A weakly nonlinear analysis is carried out to derive a Korteweg-de Vries (K-dV) equation with a particular emphasis on its application to the IARWs. This K-dV equation is transformed to a nonlinear Schr?dinger equation, provided that the frequency of the carrier wave is much smaller than the ion plasma frequency. Interestingly, it is found that the IARWs may be drastically affected by electron nonextensivity depending on whether the entropic index q is positive or negative. In view of the crucial importance of RWs in space environments, our results should be useful in understanding the basic features of the nonextensive IARGs that may occur in space plasmas.     

Kinetic theory of acoustic-like modes in nonextensive pair plasmas

The low-frequency acoustic-like modes in a pair plasma (electron-positron or pair-ion) is studied by employing a kinetic theory model based on the Vlasov and Poisson’s equations with emphasizing the Tsallis’s nonextensive statistics. The possibility of the acoustic-like modes and their properties in both fully symmetric and temperature-asymmetric cases are examined by studying the dispersion relation, Landau damping and instability of modes. The resultant dispersion relation in this study is compatible with the acoustic branch of the experimental data (Oohara et al. in Phys. Rev. Lett. 95:175003, 2005) in which the electrostatic waves have been examined in a pure pair-ion plasma. Particularly, our study reveals that the occurrence of growing or damped acoustic-like modes depends strongly on the nonextensivity of the system as a measure for describing the long-range Coulombic interactions and correlations in the plasma. The mechanism that leads to the unstable modes lies in the heart of the nonextensive formalism yet, the mechanism of damping is the same developed by Landau. Furthermore, the solutions of acoustic-like waves in an equilibrium Maxwellian pair plasma are recovered in the extensive limit (q→1), where the acoustic modes have only the Landau damping and no growth.     

Gardner solitons in dusty plasmas with nonextensive ions and two-temperature superthermal electrons

Nonlinear propagation of dust-acoustic waves in an unmagnetized dusty plasma consisting of negatively charged mobile dust, nonextensive ions following nonextensive q-distribution and two distinct temperature superthermal electrons following superthermal kappa distribution each, is investigated by employing lower and higher order nonlinear equations, namely the Korteweg-de-Vries (K-dV), the modified Korteweg-de-Vries (mK-dV) and the Gardner equations. The characteristic features of the hump (positive potential) and dip (negative potential) shaped dust-acoustic (DA) Gardner solitons are found to exist beyond the K-dV limit. The effects of two superthermal temperature electrons and ions nonextensivity on the basic characteristics of DA K-dV, mK-dV and Gardner solitons have also been investigated. It has been found that the DA Gardner solitons exhibit either negative or positive potential solitons only for q<q _c where, q _c is the critical value of the nonextensive parameter q. The possible applications of our results in understanding the localized nonlinear electrostatic structures existing in solar atmosphere, Saturn’s magnetosphere etc. (where the tails of the high energetic particles at different temperatures follow power-law like distribution) are also briefly discussed.     

Dust ion-acoustic shock waves in nonextensive dusty plasma

A parametric survey on the propagation characteristics of the dust ion-acoustic (DIA) shock waves showing the effect of nonextesivity with nonextensive electrons in a dissipative dusty plasma system has been carried out using the reductive perturbation technique. We have considered continuity and momentum equations for inertial ions, q-distributed nonextensive electrons, and stationary charged dust grains, to derive the Burgers equation. It has been found that the basic features of DIA shock waves are significantly modified by the effects of electron nonextensivity and ion kinematic viscosity. Depending on the degree of nonextensivity of electrons, the dust ion-acoustic shock structures exhibit compression and rarefaction. The implications of our results would be useful to understand some astrophysical and cosmological scenarios like stellar polytropes, hadronic matter and quark-gluon plasma, protoneutron stars, dark-matter halos, etc., where effects of nonextensivity can play the significant roles.     

Effect of <Emphasis Type="Italic">q</Emphasis>-nonextensive distribution of electrons on ion acoustic shock waves in dissipative plasma

Ion acoustic shock waves (IASWs) are studied in a plasma consisting of nonextensive electrons and ions. The dissipation is taken into account the kinematic viscosity among the plasma constituents. The Korteweg-de Vries-Burgers (KdV-Burgers) equation is derived by reductive perturbation method. Shock waves are solutions of KdV-Burgers equation. It is shown that acceptable values of q-parameter (where q stands for the electron nonextensive parameter) are more than 3 in a weakly nonlinear analysis. We have found that the amplitude of shock waves decreases by an increasing q-parameter.     

Ion acoustic solitary and shock waves with nonextensive electrons and thermal positrons in nonplanar geometry

The nonlinear wave structures of ion acoustic waves (IAWs) in an unmagnetized plasma consisting of nonextensive electrons and thermal positrons are studied in bounded nonplanar geometry. Using reductive perturbation technique we have derived cylindrical and spherical Korteweg-de Vries-Burgers’ (KdVB) equations for IAWs. The presence of nonextensive q-distributed electrons is shown to influence the solitary and shock waves. Furthermore, in the existence of ion kinematic viscosity, the shock wave structure appears. Also, the effects of nonextensivity of electrons, ion kinematic viscosities, positron concentration on the properties of ion acoustic shock waves (IASWs) are discussed in nonplanar geometry. It is found that both compressive and rarefactive type solitons or shock waves are obtained depending on the plasma parameter.     

Head-on collision of electron acoustic solitary waves in a plasma with nonextensive hot electrons

The head-on collision between two electron-acoustic solitary waves (EASWs) in an unmagnetized plasma is investigated, including a cold electrons fluid, hot electrons, obeying a nonextensive distribution and stationary ions. By using the extended Poincaré-Lighthill–Kuo (PLK) perturbation method, the analytical phase shifts following the head-on collision are derived. The effects of the ratio of the number density of hot electrons to the number density of cold electrons α, and the nonextensive parameter q on the phase shifts are studied. It is found that q and the hot-to-cold electron density ratio significantly modify the phase shifts.     

Obliquely propagating solitary wave structures in nonextensive magneto-rotating plasmas

The propagation of the nonlinear electrostatic ion acoustic solitary wave structures in two component, non relativistic, homogenous, magneto rotating plasma are studied. The inertialess electrons are assumed to follow nonextensive q velocity distribution. Small amplitude reductive perturbation technique is applied to derive Korteweg de Vries (KdV) equation and its analytical solution is presented. The effects of variation of different plasma parameters on propagation characteristics of solitary wave structure in the presence of the Coriolis force are discussed. It is observed that nonextensive parameter q modifies the structure of solitary wave structures in rotating plasmas.     

The Influence of a Mass Spectrum on the Long Term Evolution of a Self-Gravitating System of Softened Particles

Using N-body simulations, we study the effects of the mass spectrum in the evolution of self-gravitating systems of softened point-mass particles. The mass function is described by a power law and the ratio between the maximum and minimum mass is . We showed that the dynamical evolution of the system depends on the mass spectrum: the secular evolution time is longer for flatter mass spectrum. For the steepest mass spectrum, the secular evolution time is of the order of the relaxation time. The mass segregation effects are achieved rapidly and the core-halo structures are formed. The projected number distributions for the systems with mass spectrum change drastically with the evolution while the projected mass distributions are not affected. Velocity dispersion profiles are modified in the sense of heating of the central regions of the systems, while the velocity anisotropy profiles are slightly affected. The consequence of our results on the dynamical evolution of clusters of galaxies is presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

K-dV solitons in a magnetized dusty plasma with two distinct temperature nonextensive electrons and Maxwellian ions

Nonlinear propagation of dust-acoustic (DA) waves in a magnetized dusty plasma, consisting of negatively charged mobile dust, Maxwellian ions and two distinct temperature nonextensive electrons (following nonextensive q-distribution each), has been studied and analyzed by deriving and solving the Korteweg-de-Vries (K-dV) equation. According to the outcomes of the investigation, the basic characteristics of the DA solitary profiles are found to be strongly modified by the external magnetic field, nonextensivity of the electrons and the respective number densities of the two species of electrons. The results of this investigation can be applied in both laboratory and astrophysical plasma scenarios for understanding the basic features of the localized electrostatic dust-acoustic solitary waves (DASWs).     

Korteweg-de Vries equation for ion acoustic soliton with negative ions in the presence of nonextensive electrons

Korteweg-de Vries (KdV) equation for electrostatic ion acoustic wave in a three component plasma containing positive and negative ions along with the nonextensive electrons is derived. Fast and slow ion acoustic modes which propagate with different velocities are excited. The effects of variation of quantities like q (nonextensive parameter), Q (mass ratio of positive to negative ion), μ (electron to positive ion number density ratio), θ _i (positive ion to electron temperature ratio) and θ _n (negative ion to electron temperature ratio) have been presented for fast and slow ion acoustic modes. Both compressive and rarefactive solitons are observed. It is found that the solitary excitations strongly depend on the mass and density ratios of the positive and negative ions as well as on nonextensive electron parameter.     

Why q-Expectation Values Must Be Used In Nonextensive Statistical Mechanics

There is a controversy in the area of nonextensive statistical mechanics regarding the form of the expectation value of a physical quantity. Two definitions have been discussed in the literature: one is the ordinary definition and the other is the normalized q-expectation value associated with the escort distribution. Here, it is proved that the normalized q-expectation value is the correct one to be employed. The Shore-Johnson theorem is used to show that the formalism with the normalized q-expectation value is theoretically consistent with minimum cross entropy principle, whereas the ordinary expectation value has to be excluded.