# Kinetic theory of acoustic-like modes in nonextensive pair plasmas

E. Saberian11affiliation: Visiting Researcher in Department of Physics,
Faculty of Basic Sciences, Azarbaijan Shahid Madani University, P.O.Box:
53714-161, Tabriz, Iran<EMAIL_ADDRESS>Department of Physics,
Faculty of Basic Sciences, University of Neyshabur, P.O.Box: 91136-599,
Neyshabur, Iran  A. Esfandyari-Kalejahi  Department of Physics, Faculty of
Basic Sciences, Azarbaijan Shahid Madani University, P.O.Box: 53714-161,
Tabriz, Iran<EMAIL_ADDRESS>
###### Abstract

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 equation 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 [W. Oohara, D. Date, and R. Hatakeyama, 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 ($q\rightarrow 1$), where the acoustic modes
have only the Landau damping and no growth.

Pair plasmas: Kinetic theory of plasma waves: Waves, oscillations, and
instabilities in plasmas

##  1 Introduction

Studying the pair plasmas has been an important challenge for many plasma
physicists in two past decades. As we know, the difference between the
electron and ion masses in an ordinary electron-ion plasma (in general, multi-
component plasma with both light and heavy particles) gives rise to different
time-space scales which are used to simplify the analysis of low- and high-
frequency modes. Such time-space parity disappears when studying a pure pair
plasma which consisting of only positive- and negative-charged particles with
an equal mass, because the mobility of the particles in the electromagnetic
fields is the same. Pair plasmas consisting of electrons and positrons have
attracted an especial of interest because of their significant applications in
astrophysics. In fact, electron-positron plasmas play an important role in the
physics of a number of astrophysical situations such as active galactic nuclei
(Begelman et al., 1984; Miller & Witta, 1987), pulsar and neutron star
magnetosphere (Goldreich & Julian, 1969; Max & Perkins, 1972; Michel, 1982),
solar atmosphere (Tandberg-Hansen & Emslie, 1988), accretion disk (Orsoz et
al., 1997), black holes (Daniel & Tajima, 1998), the early universe (Misner et
al., 1973; Gibbons et al., 1983) and many others. For example, the detection
of circularly polarized radio emission from the jets of the archtypal quasar
3C297, indicates that electron-positron pairs are an important component of
the jet plasma (Wardle et al., 1998). Similar detections in other radio
sources suggest that, in general, extragalactic radio jets are composed mainly
of an electron-positron plasma (Wardle et al., 1998). Furthermore, it has been
suggested that the creation of electron-positron plasma in pulsars is
essentially by energetic collisions between particles which are accelerated as
a result of electric and magnetic fields in such systems (Sturrock, 1971;
Michel, 1982, 1991). On the other hand, the successful achievements for
creation of the electron-positron plasmas in laboratories have been frequently
reported in the scientific literatures (Gibson et al., 1960; Gahn et al.,
2000; Pedersen et al., 2003; Helander & Ward, 2003; Amoretti et al., 2003;
Pedersen et al., 2004; Chen et al., 2009). In this regard, many authors have
concentrated on the relativistic electron-positron plasmas (Berezhiani et al.,
1993; Verheest, 1996; Gedalin et al., 1998; Keston et al., 2003; Muoz, 2004;
Laing & Diver, 2006) because of its occurrence in astrophysics and
encountering with positron as an antimatter in high-energy physics. However,
there are many experiments that confirm the possibility of nonrelativistic
electron-positron plasmas in laboratory (Trivelpiece, 1972; Boehmer et al.,
1995). It has been observed that the annihilation time of electron-positron
pairs in typical experiments is often long compared with typical confinement
times (Surko & Murphy, 1990), showing that the lifetime of electron-positron
pairs in the plasma is much longer than the characteristic time scales of
typical oscillations. The long lifetime of electron-positron pairs against
pair annihilation indicates that many collective modes can occur and propagate
in an electron-positron plasma.

Although pair plasmas consisting of electrons and positrons have been
experimentally produced, however, because of fast annihilation and the
formation of positronium atoms and also low densities in typical electron-
positron experiments, the identification of collective modes in such
experiments is practically very difficult. To resolve this problem, one may
experimentally deal with a pure pair-ion plasma instead of a pure electron-
positron plasma for identification of the collective modes. An appropriate
experimental method has been developed by Oohara and Hatakeyama (Oohara &
Hatakeyama, 2003) for the generation of pure pair-ion plasmas consisting only
positive and negative ions with equal masses by using fullerenes
$\mathrm{C}_{60}^{-}$ and $\mathrm{C}_{60}^{+}$. The fullerenes are molecules
containing 60 carbon atoms in a very regular geometric arrangement, and so a
fullerene pair plasma is physically akin to the an electron-positron plasma,
without having to worry about fast annihilation. By drastically improving the
pair-ion plasma source in order to excite effectively the collective modes,
Oohara _et al._ (Oohara et al., 2005) have experimentally examined the
electrostatic modes propagating along the magnetic-field lines in a fullerene
pair plasma.

In exploring the electrostatic modes in a pair plasma, most of authors have
merely studied the high frequency Langmuir-type oscillation in a pure
electron-positron plasma (Tsytovich & Wharton, 1978; Iwamoto, 1993; Zank &
Greaves, 1995; Verheest, 2005) or in a pure pair-ion plasma (Vranjes & Poedts,
2005) via the theoretical studies. Particularly, Iwamato (Iwamoto, 1993) and
Veranjes and Poedts (Vranjes & Poedts, 2005) have studied the longitudinal
modes in a pair plasma only in the case in which the phase velocity of the
wave is much larger than the thermal velocity of the particles which leads to
the Langmuir-type waves. However, in the experiment of Oohara _et al._ (Oohara
et al., 2005), three kinds of electrostatic modes have been observed from the
obtained dispersion curves: a relatively low-frequency band with nearly
constant group velocity (the acoustic waves), an intermediate-frequency
backward-like mode (to our knowledge, with the lack of a satisfactory
theoretical explanation), and the Langmuir-type waves in a relatively high-
frequency band. This experiment indicates that in a pure pair plasma, besides
of the Langmuir-type waves, the acoustic-like modes are possible in practice.
There, Oohara _et al._ (Oohara et al., 2005) have briefly discussed some
aspects of their experimental results by using a theoretical two-fluid model.
Here, our goal is to investigate the possibility and the properties of the
intriguing acoustic-like modes in a pair plasma (in both symmetric and
asymmetric cases) by using a kinetic theory model and to argue some properties
of these modes in a subtler manner. It is to be noted that the only asymmetry
in a pure pair plasma may arise from a difference in temperatures of species.
Physically, the temperature-asymmetry in a pair plasma may arise from the
typical experimental procedure in which a pair plasma is produced in the
laboratory. For example, an effective technique for creating an electron-
positron plasma in the laboratory is as follows: at first, we may obtain a
positron plasma through scattering from a buffer gas into a panning trap
(Surko et al., 1989; Greaves et al., 1994); using this technique, the
positrons can be stored at densities of order $10^{7}cm^{-3}$ and lifetime of
order $10^{3}sec$ in the recent experiments. Then, an electron-positron plasma
with sufficient stability can be produced by injecting a low-energy electron
beam into the positrons (Greaves & Surko, 1995; Liang et al., 1998; Greaves &
Surko, 2001; Surko & Greaves, 2004). For our purpose, we assume that the phase
speed of the acoustic-like modes lies in the vicinity of the thermal
velocities of the species (in fact, between the thermal velocities of the two
species). The situation is somewhat similar to the case in which the
possibility and properties of the electron-acoustic waves in a two-temperature
(cold and hot) electron plasma is examined (Defler & Simonen, 1969; Watanabe &
Taniuti, 1977; Dubouloz et al., 1991; Kakad et al., 2007; Amour et al., 2012).

It is often observed that the physical distribution of particles in space
plasmas as well as in laboratory plasmas are not exactly Maxwellian and
particles show deviations from the thermal distribution (Huang & Driscoll,
1994; Liu et al., 1994). Presence of nonthermal particles in space plasmas has
been widely confirmed by many spacecraft measurements (Montgomery et al.,
1968; Feldman et al., 1975; Maksimovic et al., 1997; Zouganelis, 2008). In
many cases, the velocity distributions show non-Maxwellian tails decreasing as
a power-law distribution in particle speed. Several models for phase space
plasma distributions with superthermal wings or other deviations from purely
Maxwellian behavior have become rather popular in recent years, like the so-
called kappa ($\kappa$) distribution which was introduced initially by
Vasyliunas in 1968 (Vasyliunas, 1968) for describing plasmas out of the
thermal equilibrium such as the magnetosphere environments and the Solar winds
(Maksimovic et al., 1997), or the nonthermal model advanced by Cairns _et al._
in 1995 (Cairns et al., 1995) which was introduced at first for an explanation
of the solitary electrostatic structures involving density depletions that
have been observed in the upper ionosphere in the auroral zone by the Freja
satellite (Dovner et al., 1994), and also the nonextensive model which go
under the name of Tsallis. In the following we want to briefly review the
formalism of the Tsallis model and to argue why it is preferred, rather than
that of the Cairns and kappa model.

From a statistical point of view, there are numerous studies indicating the
breakdown of the Boltzmann-Gibbs (BG) statistics for description of many
systems with long-range interactions, long-time memories and fractal space-
time structures (see, e.g., Landsberg (1984); Tsallis et al. (1995); Tsallis
(1995, 1999)). Generally, the standard BG extensive thermo-statistics
constitutes a powerful tool when microscopic interactions and memories are
short ranged and the environment is an Euclidean space-time, a continuous and
differentiable manifold. Basically, systems subject to the long-range
interactions and correlations and long-time memories are related to the
nonextensive statistics where the standard BG statistics and its Maxwellian
distribution do not apply. The plasma environments in the astrophysical and
laboratorial systems are obviously subject to spatial and temporal long-range
interactions evolving in a non-Euclidean space-time that make their behavior
nonextensive. A suitable generalization of the Boltzmann-Gibbs-Shannon (BGS)
entropy for statistical equilibrium was first proposed by Reyni (Reyni, 1955)
and subsequently by Tsallis (Tsallis, 1988, 1994), preserving the usual
properties of positivity, equiprobability and irreversibility, but suitably
extending the standard extensivity or additivity of the entropy to
nonextensivity.

The nonextensive generalization of the BGS entropy which proposed by Tsallis
in 1988 (Tsallis, 1988, 1994) is given by the following expression:

$S_{q}=k_{B}\frac{1-\sum_{i}p_{i}^{q}}{q-1},$ (1)

where $k_{B}$ is the standard Boltzmann constant, $\\{p_{i}\\}$ denotes the
probabilities of the microstate configurations and $q$ is a real parameter
quantifying the degree of nonextensivity. The most distinctive feature of
$S_{q}$ is its pseudoadditivity. Given a composite system $A+B$, constituted
by two subsystems $A$ and $B$, which are independent in the sense of
factorizability of the joint microstate probabilities, the Tsallis entropy of
the composite system $A+B$ satisfies
$S_{q}(A+B)=S_{q}(A)+S_{q}(B)+(1-q)S_{q}(A)S_{q}(B)$. In the limit of
$q\rightarrow 1$, $S_{q}$ reduces to the celebrated logarithmic Boltzmann-
Gibbs entropy $S=-k_{B}\sum_{i}p_{i}\ln p_{i}$, and the usual additivity of
entropy is recovered. Hence, $|1-q|$ is a measure of the lack of extensivity
of the system. There are numerous evidences exhibiting that the nonextensive
statistics, arising from $S_{q}$, is a better framework for describing many
physical systems such as the galaxy clusters (Lavagno et al., 1998), the
plasmas (Boghosian, 1996; Tsallis & de Souza, 1997), the turbulent systems
(Arimitsu & Arimitsu, 2000; Beck, 2001; Beck et al., 2001), and so on, in
which the system shows a nonextensive behavior as a result of long-range
interactions and correlations. The experimental results in such systems
display a non-Maxwellian velocity distribution for the particles (Huang &
Driscoll, 1994; Liu et al., 1994). The functional form of the velocity
distribution in the Tsallis formalism may be derived through a nonextensive
generalization of the Maxwell ansatz (Silva et al., 1998), or through the
maximizing Tsallis’ entropy under the constraints imposed by normalization and
the energy mean value (Curado, 1999; Abe, 1999). Furthermore, from a
nonextensive generalization of the “molecular chaos hypothesis”, it is shown
that the equilibrium $q$-nonextensive distribution is a natural consequence of
the _H_ theorem (Lima et al., 2001).

It is to be noted that the empirically derived kappa distribution function in
space plasmas is equivalent to the $q$-distribution function in Tsallis
nonextensive formalism, in the sense that the spectrum of the velocity
distribution function in both models show the similar behavior and, in fact,
both the kappa distribution and the Tsallis $q$-nonextensive distribution
describe deviations from the thermal distribution. Particularly, Leubner in
2002 (Leubner, 2002) showed that the distributions very close to the kappa
distributions are a consequence of the generalized entropy favored by the
nonextensive statistics, and proposed a link between the Tsallis nonextensive
formalism and the kappa distribution functions. In fact, relating the
parameter $q$ to $\kappa$ by formal transformation $\kappa=1/(1-q)$ (Leubner,
2002) provides the missing link between the $q$-nonextensive distribution and
the $\kappa$-distribution function favored in space plasma physics, leading to
a required theoretical justification for the use of $\kappa$-distributions
from fundamental physics. Furthermore, Livadiotis and McComas in 2009
(Livadiotis & McComas, 2009) examined how kappa distributions arise naturally
from the Tsallis statistical mechanics. On the other hand, the nonthermal
distribution function introduced by Cairns et al. (Cairns et al., 1995) is a
proposal function to model an electron distribution with a population of
energetic particles. It is especially appropriate for describing the nonlinear
propagation of large amplitude electrostatic excitations such as solitary
waves and double layers which are very common in the magnetosphere. However,
the lack of a statistical foundation behind this proposal function is clearly
seen, leading to less attention to it rather than the kappa function and the
Tsallis distribution. Anyway, the $q$-nonextensive formalism, with a powerful
thermo-statistics foundation and numerous experimental evidences, may cover
many features of the other nonthermal models and provide a good justification
for its preference over the other models. It has considerably extended both
statistical mechanics formalism and its range of applicability. The interested
reader may refer to the Refs. (Plastino, 2004; Abe & Okamato, 2001; Gell-Mann
& Tsallis, 2004; Tsallis, 2009) where the significance, historical background,
physical motivations, foundations and applications of the nonextensive thermo-
statistics have been discussed in detail.

The problem of waves, Landau damping and instabilities in typical plasmas have
been investigated by some authors in the framework of the Tsallis nonextensive
statistics (Lima et al., 2000; Silva et al., 2005; Valentini, 2005; Liyan &
Jiulin, 2008; Saberian & Esfandyari-Kalejahi, 2013). Particularly, it is to be
noted that the physical state described by the $q$-nonextensive distribution
in the Tsallis’s statistics is not exactly the thermodynamic equilibrium
(Liyan & Jiulin, 2008). In fact, the deviation of $q$ from unity quantifies
the degree of inhomogeneity of the temperature $T$ via the formula
$k_{B}\nabla T+(1-q)Q_{\alpha}\nabla\phi=0$ (Du, 2004), where $Q_{\alpha}$
denotes the electric charge of specie $\alpha$, and $\phi$ is the
electrostatic potential. In other words, the nonextensive statistics describes
a system that have been evolved from a nonequilibrium stationary state with
inhomogeneous temperature which contains a number of nonthermal particles.

In the present work, we attempt to investigate the possibility of the
acoustic-like modes in a field-free and collisionless pair plasma (electron-
positron or pair-ion) and to discuss the damping and instability of modes in
the context of the Tsallis’ nonextensive statistics. In Sec. 2, a kinetic
theory model based on the linearized Vlasov and Poisson’s equations is applied
for deriving the dielectric function ($D(k,\omega)$) for longitudinal waves in
an unmagnetized pair plasma. We then find the solutions of $D(k,\omega)=0$ for
the acoustic-like waves with the constraint of weak damping or growth by
considering a $q$-nonextensive distribution for stationary state of the
plasma, as demonstrated in Sec. 3. The dispersion relation, Landau damping and
instability of the acoustic-like modes are discussed in Sec. 4. Finally, a
summary of our results is given in Sec. 5.

##  2 The model equations

In this section, we present a brief review of kinetic equations for describing
the electrostatic collective modes specialized to a pair plasma (electron-
positron or pair-ion) with the constraint of weak damping or growth.

We consider a spatially uniform field-free pair plasma at the equilibrium
state. If at a given time $t=0$ a small amount of charge is displaced in the
plasma, the initial perturbation may be described by

$f_{\alpha}(t=0)=f_{0,\alpha}(\vec{v})+f_{1,\alpha}(\vec{x},\vec{v},t=0),\ \ \
\ f_{1,\alpha}\ll f_{0,\alpha},$

where $f_{0,\alpha}$ corresponds to the unperturbed and time-independent
stationary distribution and $f_{1,\alpha}$ is the corresponding perturbation
about the equilibrium state. Here, $\alpha$ stands for electrons and positrons
($\alpha=e^{\pm}$) or fullerene pairs ($\alpha=\mathrm{C}_{60}^{\pm}$). We
assume that the perturbation is electrostatic and the displacement of charge
gives rise to a perturbed electric but no magnetic field. With this
assumption, the time development of $f_{1,\alpha}(\vec{x},\vec{v},t)$ is given
by the solution of the linearized Vlasov and Poisson’s equations as follows
(Landau, 1946; Krall & Trivelpiece, 1973):

$\frac{\partial f_{1,-}}{\partial t}+\vec{v}\cdot\frac{\partial
f_{1,-}}{\partial\vec{x}}+\frac{e}{m}\nabla\phi_{1}\cdot\frac{\partial
f_{0,-}}{\partial\vec{v}}\;=\;0,$ (2) $\frac{\partial f_{1,+}}{\partial
t}+\vec{v}\cdot\frac{\partial
f_{1,+}}{\partial\vec{x}}-\frac{e}{m}\nabla\phi_{1}\cdot\frac{\partial
f_{0,+}}{\partial\vec{v}}\;=\;0,$ (3) $\nabla^{2}\phi_{1}=4\pi
n_{0}e\int(f_{1,-}-f_{1,+})\,\mathrm{d}\vec{v},$ (4)

where $e$, $m$ and $n$ denote, respectively, the absolute charge, mass and
number density of the pairs and $\phi_{1}$ is the electrostatic potential
produced by the perturbation. Here, we have labeled the distribution function
of negative and positive pairs with the subscripts $\pm$. This set of
linearized equations for perturbed quantities may be solved simultaneously to
investigate the plasma properties for the time intervals shorter than the
binary collision times. Specially, we can study the properties of the plasma
waves whose oscillations period are much less than a binary collision time.
The standard technique for simultaneously solving the differential equations
(2)-(4) is the method of integral transforms, as developed for the first time
by Landau in the case of an ordinary electron-ion plasma (Landau, 1946; Krall
& Trivelpiece, 1973). Another simplified method of solving the Vlasov-
Poisson’s equations for the longitudinal waves, with the frequency $\omega$
and the wave vector $\vec{k}$, is to assume that the solution has the form

$\begin{array}[]{l}f_{1,\alpha}(\vec{x},\vec{v},t)=f_{1,\alpha}(\vec{v})e^{i(\vec{k}\cdot\vec{x}-\omega
t)},\ \ \ \alpha=e^{\pm}\ \ \mathrm{or}\ \ \mathrm{C}_{60}^{\pm},\\\
\phi_{1}(\vec{x},t)=\phi_{1}e^{i(\vec{k}\cdot\vec{x}-\omega t)}.\end{array}$
(5)

Without loss of the generality, we consider the $x$-axis to be along the
direction of the wave vector $\vec{k}$, and let $v_{x}=u$. Then, by applying
the Eq. (5) and solving the Eqs. (2)-(4) we find the dispersion relation for
longitudinal waves in a pair plasma as follows

$D(k,\omega)=1-\frac{4\pi
n_{0}e^{2}}{mk^{2}}\int\frac{\frac{\partial}{\partial
u}(f_{0,-}(u)+f_{0,+}(u))}{u-\frac{\omega}{k}}\,\mathrm{d}u=0,$ (6)

where $D(k,\omega)$ is the dielectric function of a field-free pair plasma for
the longitudinal oscillations. We then can investigate the response of the
pair plasma to an arbitrary perturbation via the response dielectric function
$D(k,\omega)$. In general, the frequency $\omega$ which satisfies the
dispersion relation $D(k,\omega)=0$ is complex, i.e.,
$\omega=\omega_{r}+i\omega_{i}$. However, in many cases $Re[\omega(k)]\gg
Im[\omega(k)]$, and the plasma responds to the perturbation a long time after
the initial disturbance with oscillations at a range of the well-defined
frequencies. These are the normal modes of the plasma, in the sense that they
are the nontransient response of the plasma to an initial perturbation. We can
determine the normal modes of the plasma via the dispersion relation
$D[k,\omega(k)]=0$, which gives the frequency of the plasma waves as a
function of the wave number $k$ or vice versa. It should be further mentioned
that when we solve the Vlasov and Poisson’s equations as an initial valve
problem, here via $f_{0,-}+f_{0,+}$, it is possible to obtain the solutions
with negative or positive values of $\omega_{i}$, corresponding to the damped
or growing waves, respectively. This can be explicitly seen from the
electrostatic potential associated with the wave number $k$ of the excitation
as follows:

$\phi_{1}(x,t)=\phi_{1}e^{i(kx-\omega_{r}t)}e^{\omega_{i}t},$ (7)

where a solution with negative $\omega_{i}$ displays a damped wave, while the
solution with positive one corresponds to an unstable mode.

When the damping or growth is weak we can expand the velocity integral in Eq.
(6) around $\omega=\omega_{i}$ to find the zeros of $D(k,\omega)$. The
dielectric function $D(k,\omega)$ is in general a complex function and thus
the dispersion relation can be written as follows:

$D(k,\omega_{r}+i\omega_{i})=D_{r}(k,\omega_{r}+i\omega_{i})+iD_{i}(k,\omega_{r}+i\omega_{i})=0,$
(8)

where $D_{r}$ and $D_{i}$ are the real and imaginary parts of the dielectric
function. Since we want to consider the weakly damped or growing waves, i.e.,
$\omega_{i}\ll\omega_{r}$, the Eq. (8) can be Taylor expanded in the small
quantity $\omega_{i}$ as follows:

$D_{r}(k,\omega_{r})+i\omega_{i}\frac{\partial
D_{r}(k,\omega_{r})}{\partial\omega_{r}}+i[D_{i}(k,\omega_{r})+i\omega_{i}\frac{\partial
D_{i}(k,\omega_{r})}{\partial\omega_{r}}],$ (9)

where $D_{r}$ and $D_{i}$ read

$D_{r}(k,\omega_{r})=1-\frac{4\pi
n_{0}e^{2}}{mk^{2}}P.V.\int\frac{\frac{\partial}{\partial
u}(f_{0,-}(u)+f_{0,+}(u))}{u-\frac{\omega_{r}}{k}}\,\mathrm{d}u,$ (10)
$D_{i}(k,\omega_{r})=-\pi(\frac{4\pi
n_{0}e^{2}}{mk^{2}})[\frac{\partial}{\partial
u}(f_{0,-}(u)+f_{0,+}(u))]_{u=\frac{\omega_{r}}{k}}.$ (11)

Here, we have made the analytic continuation of the velocity integral of the
Eq. (6) over $u$, along the real axis, which passes under the pole at
$u=\frac{\omega}{k}$ with the constraint of weakly damped waves, where
$P.V.\int$ denotes the Cauchy principal value. With the assumption
$\omega_{i}\ll\omega_{r}$, by balancing the real and imaginary parts of the
Eq.(9) and neglecting the terms of order
$(\frac{\omega_{i}}{\omega_{r}})^{2}$, we find that $\omega_{r}$ and
$\omega_{i}$ can be computed, respectively, from the relations

$\displaystyle D_{r}(k,\omega_{r})=0,$ (12a)
$\displaystyle\omega_{i}=-\frac{D_{i}(k,\omega_{r})}{{\partial
D_{r}(k,\omega_{r})}/{\partial\omega_{r}}}.$ (12b)

##  3 Acoustic modes with nonextensive stationary state

Now, we want to obtain the formalism and some features of the acoustic-like
modes in a pair (electron-positron or pair-ion) plasma in the context of the
Tsallis nonextensive statistics. For this purpose we assume that the
stationary state of the plasma obeys the $q$-nonextensive distribution
function, instead of a Maxwellian one, which merely describes a fully
equilibrium plasma. The $q$-nonextensive distribution function of stationary
state for species $\alpha$ in one-dimension is given by (Silva et al., 1998;
Curado, 1999; Abe, 1999; Lima et al., 2001)

$f_{0\alpha}(u)=A_{\alpha,q}[1-(q-1)\frac{m_{\alpha}u^{2}}{2k_{B}T_{\alpha}}]^{\frac{1}{q-1}},$
(13)

where $m_{\alpha}$ and $T_{\alpha}$ are, respectively, the mass and
temperature of species $\alpha$ ($\alpha=e^{\pm}\ \mathrm{or}\
\mathrm{C}_{60}^{\pm}$) and $k_{B}$ is the standard Boltzmann constant. The
normalization constant $A_{\alpha,q}$ can be written as

$A_{\alpha,q}=L_{q}\sqrt{\frac{m_{\alpha}}{2\pi k_{B}T_{\alpha}}},$ (14)

where the dimensionless $q$-dependent coefficient $L_{q}$ reeds

$\displaystyle
L_{q}=\frac{\Gamma(\frac{1}{1-q})}{\Gamma(\frac{1}{1-q}-\frac{1}{2})}\sqrt{1-q},\
\ \ \ \mathrm{for}\ \ -1<q\leq 1$ (15a) $\displaystyle
L_{q}=(\frac{1+q}{2})\frac{\Gamma(\frac{1}{2}+\frac{1}{q-1})}{\Gamma(\frac{1}{q-1})}\sqrt{q-1}.\
\ \ \ \mathrm{for}\ \ q\geq 1$ (15b)

One may examine that for $q>1$ , the $q$-distribution function (13) exhibits a
thermal cutoff, which limits the velocity of particles to the values
$u<u_{max}$, where $u_{max}=\sqrt{\frac{2k_{B}T_{\alpha}}{m_{\alpha}(q-1)}}$.
For these values of the parameter $q$ we have $S_{q>1}(A+B)<S(A)+S(B)$
referred to the _subextensivity_. This thermal cutoff is absent when $q<1$ ,
that is, the velocity of particles is unbounded for these values of the
parameter $q$. In this case, we have $S_{q<1}(A+B)>S(A)+S(B)$ referred to the
_superextensivity_. Moreover, the $q$-nonextensive distribution (13) is
unnormalizable for the values of the $q<-1$. Furthermore, the parameter $q$
may be further restricted by the other physical requirements, such as finite
total number of particles and consideration of the energy equipartition for
contribution of the total mean energy of the system. Interestingly, in the
extensive limit $q\rightarrow 1$ where $S(A+B)=S(A)+S(B)$, and by using the
formula $lim_{\mid
z\mid\rightarrow\infty}z^{-a}[\frac{\Gamma(a+z)}{\Gamma(z)}]=1$ (Abramowitz &
Stegun, 1972), the distribution function (13) reduces to the standard Maxwell-
Boltzmann distribution $f_{0\alpha}(u)=\sqrt{\frac{m_{\alpha}}{2\pi
k_{B}T_{\alpha}}}e^{-\frac{m_{\alpha}u^{2}}{2k_{B}T_{\alpha}}}$. In Fig. 1, we
have depicted schematically the nonthermal behavior of the distribution
function (13) for some values of the spectral index $q$ in which the velocity
$u$ and the distribution function $f(u)$ have normalized by the standard
thermal speed $v_{th}=\sqrt{\frac{2k_{B}T}{m}}$ and $\sqrt{\frac{m}{2\pi
k_{B}T}}$, respectively. We can see that in the case of a superextensive
distribution with $q<1$ [Fig. 1(a)], comparing with the Maxwellian limit
(solid curve), there are more particles with the velocities faster than the
thermal speed $v_{th}$. These are the so-called superthermal particles and we
can see that the $q$-distribution with $q<1$ behave like the $\kappa$
distribution, the same as that introduced for the first time by Vasyliunas in
1968 to describe the space plasmas far from the thermal equilibrium
(Vasyliunas, 1968). In fact, in a superthermal plasma modeled by a
$\kappa$-like distribution (here, the cases in which $q<1$), the particles
have distributed in a wider spectrum of the velocities, in comparison with a
Maxwellian plasma. In other words, the low values of the spectral index $q$
correspond to a large fraction of superthermal particle populations in the
plasma. On the other hand, in the case of a subextensive distribution with
$q>1$ [Fig. 1(b)], comparing with the Maxwellian limit (solid curve), there is
a large fraction of particles with the velocities slower than the thermal
speed $v_{th}$. Moreover, for these values of the parameter $q$, we can
explicitly see the mentioned thermal cutoff which limits the velocity of
particles. In fact, the $q$-nonextensive distributions with $q>1$ are suitable
for describing the systems containing a large number of low speed particles.

The phase velocity of the acoustic modes in a pair plasma lies between the
thermal velocities of the pairs. Here, we assume that $T_{+}<T_{-}$ and
therefore the phase velocity of the acoustic waves lies in the frequency band
$v_{th,+}<v_{\phi}<v_{th,-}$, where $v_{\phi}=\frac{\omega_{r}}{k}$ and
$v_{th,\pm}=(\frac{k_{B}T_{\pm}}{m})^{\frac{1}{2}}$, respectively, denote the
phase velocity of the wave and thermal speed of the pairs. It is to be noted
that because of the symmetry involved in a pair plasma, the other case in
which $T_{-}<T_{+}$ is physically identical to our assumption here. Moreover,
it is reminded that because of the same dynamics of the species in a pure pair
plasma, we do not make a considerable difference in temperatures of the pairs,
but we assume that it is finite and small. As we mentioned earlier, we may
postulate physically that this finite temperature-asymmetry in a pair plasma
may arise from the typical experimental procedure in which the pair plasma is
produced in the laboratory (Greaves & Surko, 1995; Liang et al., 1998; Greaves
& Surko, 2001; Surko & Greaves, 2004).

With $v_{th,+}<v_{\phi}<v_{th,-}$, the Cauchy principal value of Eq. (10) for
the terms that are involving $f_{0,-}$ and $f_{0,+}$ may be evaluated by an
expanding in $u$ as follows:

$\displaystyle\int^{+u_{max}}_{-u_{max}}\frac{\frac{\partial}{\partial
u}f_{0,-}(u)}{u-\frac{\omega_{r}}{k}}\,\mathrm{d}u=\int^{+u_{max}}_{-u_{max}}\frac{\partial
f_{0,-}(u)}{\partial
u}(\frac{1}{u}+\frac{1}{u^{2}}\frac{\omega_{r}}{k}+\frac{1}{u^{3}}\frac{\omega_{r}^{2}}{k^{2}}+...)\,\mathrm{d}u,$
(16a) $\displaystyle\int^{+u_{max}}_{-u_{max}}\frac{\frac{\partial}{\partial
u}f_{0,+}(u)}{u-\frac{\omega_{r}}{k}}\,\mathrm{d}u=-\frac{k}{\omega_{r}}\int^{+u_{max}}_{-u_{max}}\frac{\partial
f_{0,+}(u)}{\partial
u}(1+\frac{k}{\omega_{r}}u+\frac{k^{2}}{\omega_{r}^{2}}u^{2}+\frac{k^{3}}{\omega_{r}^{3}}u^{3}+...)\,\mathrm{d}u.$
(16b)

Here, in order to include both cases $q<1$ (superextensivity) and $q>1$
(subextensivity), we have denoted the integration limits in Eq. (16) by $\pm
u_{max}$. In fact, as discussed earlier, the integration limits are unbounded,
i.e., $\pm u_{max}=\pm\infty$, when $q<1$, and they are given by the
$q$-dependent thermal cutoff $\pm
u_{max}=\pm\sqrt{\frac{2k_{B}T_{\alpha}}{m_{\alpha}(q-1)}}$ when $q>1$.

With the $q$-nonextensive distribution given in Eq. (13), noting that
$f_{0\alpha}(u)$ is an even function with argument $u$ and $\frac{\partial
f_{0\alpha}}{\partial u}$ is an odd function, we may calculate the real part
of the dielectric function in Eq. (10) as follows:

$D_{r}(k,\omega_{r})=1+\frac{4\pi
n_{0}e^{2}}{mk^{2}}\frac{1}{v_{th,-}^{2}}(\frac{1+q}{2})-\frac{4\pi
n_{0}e^{2}}{m\omega_{r}^{2}}[1+3(\frac{2}{3q-1})\frac{k^{2}}{\omega_{r}^{2}}v_{th,+}^{2}].$
(17)

The integrals in Eq. (16) are computed by parts and there, we have calculated
the average values of $u^{2}$ as follows:

$<u^{2}>=\int^{+u_{max}}_{-u_{max}}u^{2}f_{\alpha
0}(u)\,\mathrm{d}u=\frac{2}{3q-1}\frac{k_{B}T_{\alpha}}{m_{\alpha}},$ (18)

which requires that the parameter $q$ must restrict to the values of
$q>\frac{1}{3}$. Note that for the values of $q$ equal or lower than the
critical value $q_{c}=\frac{1}{3}$, the mean value of $u^{2}$ diverges.
Therefore, we see that the parameter $q$ for the case $q<1$ is further
restricted to the values $\frac{1}{3}<q<1$, in order that the physical
requirement of energy equipartition is preserved. We emphasize that our
results here are valid both for the case $\frac{1}{3}<q<1$ where the value of
$u_{max}$ is unbounded and also in the case $q>1$ in which $u_{max}$ is given
by the thermal cutoff
$u_{max}=\sqrt{\frac{2k_{B}T_{\alpha}}{m_{\alpha}(q-1)}}$. Note that in both
cases the integrals in Eq. (16) are evaluated by limits that are symmetric
across the origin. The interested reader may easily check the validity of Eqs.
(17) and (18) for all allowed values of $q$. Furthermore, in the extensive
limit $q\rightarrow 1$, Eq. (18) reduces to the familiar energy equipartition
theorem for each degree of freedom in the BG statistics as
$<\frac{1}{2}m_{\alpha}u^{2}>=\frac{1}{2}k_{B}T_{\alpha}$.

It is to be noted that the $q$ distribution given in Eq. (13) describes the
stationary state of the species $\alpha$ in the framework of the Tsallis
nonextensive formalism. The value of the spectral index $q$ is a measure that
determines the slope of the energy spectrum of the nonthermal particles and
measures the deviation from the standard thermal distribution (which is
recovered at the limit $q\rightarrow 1$). The value of the spectral index $q$
is determined as a result of long-range interactions and correlations of the
whole system. Therefore, a distinction between the pairs in $q$ can be or not,
depend on the physics of the system under consideration. Here, following El-
Tantawy _et al._ (El-Tantawy et al., 2012), we make no distinction between the
pairs in $q$.

The solution of the equation $D_{r}(k,\omega_{r})=0$ may yield the dispersion
relation for the acoustic modes in a nonextensive pair plasma as follows:

$\omega_{r}^{2}=k^{2}c_{s}^{2}[\frac{1}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+3(\frac{2}{3q-1})\sigma],$
(19)

where we have defined the sound-speed of the acoustic-like modes as
$c_{s}={(\frac{k_{B}T_{-}}{m})}^{\frac{1}{2}}$. Here,
$\sigma=\frac{T_{+}}{T_{-}}$ is the fractional temperature of positive to
negative species and $\lambda_{D}$ is the Debye screening length and is given
in a charge-neutral pair plasma by

$\lambda_{D}^{-2}=\frac{4\pi
n_{0}e^{2}}{k_{B}}(\frac{1}{T_{-}}+\frac{1}{T_{+}}).$ (20)

By definition of the the natural oscillation frequency in a charge-neutral
pair plasma as $\omega_{p}=(\frac{8\pi n_{0}e^{2}}{m})^{\frac{1}{2}}$
(Saberian & Esfandyari-Kalejahi, 2013), it is convenient to rewrite the linear
dispersion relation for the later references as follows:

$(\frac{\omega_{r}}{\omega_{p}})^{2}=(k\lambda_{D})^{2}[\frac{\frac{1}{2}(1+\frac{1}{\sigma})}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+3(\frac{1}{3q-1})(1+\sigma)].$
(21)

On the other hand, by using the Eq. (11) and applying the $q$-nonextensive
distribution function (13), it is straightforward to obtain the imaginary part
of the dielectric function as follows:

$D_{i}(k,\omega_{r})=L_{q}\frac{\sqrt{\pi}}{k^{3}\lambda_{D}^{3}(1+\frac{1}{\sigma})^{\frac{3}{2}}}\frac{\omega_{r}}{\omega_{p}}\\{[1-(q-1)\frac{\omega_{r}^{2}}{k^{2}\lambda_{D}^{2}\omega_{p}^{2}(1+\frac{1}{\sigma})}]^{\frac{2-q}{q-1}}+\frac{1}{\sigma^{\frac{3}{2}}}[1-(q-1)\frac{\omega_{r}^{2}}{k^{2}\lambda_{D}^{2}\omega_{p}^{2}(1+\sigma)}]^{\frac{2-q}{q-1}}\\}.$
(22)

By $D_{r}(k,\omega_{r})$ and $D_{i}(k,\omega_{r})$ given in Eqs. (17) and
(22), we may obtain the explicit solution of the imaginary part of the
frequency by using the relation (12b), noting that both $k\lambda_{D}$ and
$\frac{\omega_{i}}{\omega_{r}}$ are assumed small. The result is as follows:

$\displaystyle\omega_{i}=-\sqrt{\frac{\pi}{8}}\omega_{r}L_{q}(\frac{1}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+3(\frac{2}{3q-1})\sigma)^{\frac{3}{2}}\times$
$\displaystyle\\{[1-(q-1)(\frac{\frac{1}{2}}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+\frac{3}{2}(\frac{2}{3q-1})\sigma)]^{\frac{2-q}{q-1}}+$
$\displaystyle\frac{1}{\sigma^{\frac{3}{2}}}[1-(q-1)(\frac{\frac{1}{2\sigma}}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+\frac{3}{2}(\frac{2}{3q-1}))]^{\frac{2-q}{q-1}}\\},$
(23)

where $L_{q}$ is that given in Eq. (19).

Note that in deriving the solutions (19) and (23) for the acoustic-like modes
in a pair plasma, we have considered the condition $k\lambda_{D}\ll 1$ which
indicates the regions with weak damping or growth (long wavelength limit).
Moreover, the values of the parameter $\sigma$ (the fractional temperature of
the species) must be considered at the vicinity of unit, in order that a
suitable compatibility with the physical circumstances is preserved.

In the extensive limit $q\rightarrow 1$, our results reduce to the solutions
for the acoustic-like modes in a Maxwellian pair plasma as follows:

$\omega_{r}^{2}=k^{2}c_{s}^{2}[\frac{1}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+3\sigma]$
(24)

$\frac{\omega_{i}}{\omega_{r}}=-\sqrt{\frac{\pi}{8}}(\frac{1}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+3\sigma)^{\frac{3}{2}}\\{e^{-(\frac{\frac{1}{2}}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+\frac{3}{2}\sigma)}+\frac{1}{\sigma^{\frac{3}{2}}}e^{-(\frac{\frac{1}{2\sigma}}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+\frac{3}{2})}\\}$
(25)

Note that in the extensive limit, the acoustic waves have only the (Landau)
damping and no growth, because of the negative value of the imaginary part of
the frequency, provided by Eq. (25). Furthermore, in the symmetric case
$\sigma\rightarrow 1$, the dispersion relation of the acoustic waves in pair
or pair-ion plasmas given in Eq.(24), reduce to Eq.(12) of Ref. (Kaladze et
al., 2012). One basic feature of our work is the inclusion of the
nonextensivity of the system, which is essentially as a result of the long-
range Coulombic interactions of the charge particles in the plasma. The
nonextensivity of the system is determined by the spectral index $q$ and may
lead to positive or negative $\omega_{i}$ in Eq. (23). Therefore, depending on
the nonextensivity of the plasma, both the damped and growing acoustic modes
may be happened in a pair plasma.

##  4 Discussion

###  4.1 Dispersion relation

The solutions (21) and (23) describe the acoustic-like modes in a nonextensive
electron-positron plasma or pair-ion plasma at the limit of long wavelengths
confirmed by $k\lambda_{D}\ll 1$. In Fig. (2a) we have plotted the dispersion
relation of acoustic modes for some values of the nonextensivity index $q$. In
the represented graph, the solid curve corresponds to the extensive limit
$q=1$ and the other ones show the deviations from the Maxwellian limit.

It is seen that for a given wavelength, the phase velocity of the acoustic
modes increases with decreasing the value of $q$. The physical description can
be discussed in the context of the nonextensive statistics as follows. As
mentioned earlier, the $q$-distribution function with $q<1$, comparing with
the Maxwellian one ($q=1$), indicates the systems with more superthermal
particles, i.e., particles with the speed faster than the thermal speed
$v_{th}=\sqrt{\frac{2k_{B}T}{m}}$ (superextensivity). On the other hand, the
$q$-distribution with $q>1$ is suitable to describe systems containing a large
number of low-speed particles (subextensivity). However, because of the long-
range nature of Coulombic interactions in plasma environments and the presence
of many superthermal particles in such systems, confirmed by many
astrophysical measurements (Montgomery et al., 1968; Feldman et al., 1975;
Maksimovic et al., 1997; Zouganelis, 2008), a $q$-distribution with $q<1$ is
strongly suggested for the real plasma systems or superthermal plasmas. It is
obvious that in a plasma with more superthermal particles ($q<1$), the phase
velocity of the acoustic-like modes should be larger than the case with lack
of superthemal particles ($q>1$), in agreement with our results here.

In addition, we have illustrated the temperature-asymmetry effect, via
$\sigma$, on the dispersion relation of acoustic modes in a pair plasma as
shown in Fig. 2(b). There, the solid curve indicates the case in which the
whole plasma is in a common thermal state with $T_{-}=T_{+}$, signifies a
temperature-symmetric pair plasma, and the other curves show deviations from
this symmetric case. We see that the temperature-asymmetry reduces the phase
velocity of the acoustic modes in a pair plasma. However, our kinetic model
confirms that the acoustic-like modes are possible in both symmetric and
asymmetric pair plasmas, depart from a small shift in phase velocity.

It is reminded that in this work we have specialized our study to the low-
frequency band in which $v_{th,+}<v_{\phi}<v_{th,-}$ . Then, the Cauchy
principal value of Eq. (10) is evaluated by an expanding in velocity in the
form of Eq. (16). So, our calculations in this frequency band may lead to the
acoustic modes and not to the Langmuir waves. On the other hand, considering a
high frequency band in which the phase velocity of the wave is much larger
than the thermal velocity of the particles ($v_{\phi}>>v_{th}$) may lead to
the Langmuir-type waves, as studied in Ref. (Saberian & Esfandyari-Kalejahi,
2013). There, the dispersion relation for the Langmuir waves is given by

$\omega_{r}^{2}=\omega_{p}^{2}[1+3(k\lambda_{D})^{2}\frac{2}{3q-1}].$ (26)

However, for comparison of the acoustic modes and the Langmuir waves in a pair
plasma with $T_{+}=T_{-}$, we have depicted both of the acoustic and Langmuir
branches in Fig. 3. From this graph, we see explicitly that the acoustic waves
belong to a low frequency band which tends to zero at the limit $k\rightarrow
0$, while the Langmuir waves occur at high frequencies above $\omega_{p}$.

On the other hand, the experimental data presented by Oohara _et. al_ (Oohara
et al., 2005) confirm the possibility of the acoustic-like modes in a pair
plasma which is compatible with our results here.

###  4.2 Landau damping and unstable modes

In Fig. 4 we have plotted the ratio $\omega_{i}/\omega_{r}$ with respect to
the nonextensivity index $q$ for all allowed values of $q<1$ (referred to
superextensivity) at the limit of long wavelengths (supported by, e.g.,
$k\lambda_{D}=0.1$). It is seen that both of the damped ($\omega_{i}<0$) and
growing ($\omega_{i}>0$) acoustic-like modes are predicted in a nonextensive
pair plasma with $q<1$. Our numerical analysis shows that in the $q$-region
$0.34\lesssim q\lesssim 0.6$ the acoustic modes are unstable, due to the fact
that $\omega$’s have positive imaginary parts and then the associated modes
will grow in time (Eq. (7) is reminded). The mechanism which leads to this
instability may explain as follows. As we expressed earlier, the
$q$-nonextensive distribution with $q<1$ describes a system with a large
number of superthermal particles. So, our solution for the Vlasov and
Poisson’s equations with small values of $q<1$ indicates an evolution which
has started from a stationary state with a large portion of superthermal
particles. The acoustic-like waves may gain energy from these superthermal
particles and results in growing waves in time. In other words, this
instability arises from a stationary state which describes a superthermal
plasma and, in fact, we have obtained a solution for acoustic-like modes in
which the stationary sate of the plasma has started from a non-equilibrium
distribution. However, our results have the flexibility to reduce to the
equilibrium solutions in the limiting case of $q\rightarrow 1$ indicates a
Maxwellian distribution. Furthermore, the acoustic-like modes have Landau
damping in the $q$-region $0.6\lesssim q\lesssim 0.71$ because $\omega$’s have
negative imaginary parts in these degrees of the nonextensivity (see Fig. 4).
The Landau damping is a resonance phenomena between the plasma particles and
the wave, for the particles moving with nearly the phase velocity of the wave
(Landau, 1946; Krall & Trivelpiece, 1973). Noting that the $q$-distribution is
a decreasing function with $u$, there are more particles moving slightly
slower than the wave than the particles moving slightly faster than the wave;
if the slower particles are accelerated by the wave, this must reduce the
energy of the wave, and the wave damps.

It is to be noted that our analysis shows that after $q=0.71$, the curve in
Fig. 4 rises to positive values for a small interval of $q$ and then it
returns to the negative values. The fluctuation of $\omega_{i}$ between the
positive and negative values continues increasingly until to the limiting case
at $q\rightarrow 1$. In fact, the curve in Fig. 4 don’t show a smooth behavior
for the values $0.71<q<1$ and the analysis break down, until to the extensive
limit at $q\rightarrow 1$, where our solutions reduce smoothly to that of a
Maxwellian pair plasma given in Eqs. (24) and (25). This unsmooth behaviour is
because of the existence of the terms $\Gamma(\frac{1}{1-q})$ and
$\Gamma(\frac{1}{1-q}-\frac{1}{2})$ in our formalism supported by $L_{q}$.
Indeed, this behavior is a mathematical consequence and there is not a
physical justification for it. So, we have analyzed the problem in a well-
defined interval of $q$, i.e, $1/3<q<0.71$, as shown in Fig. 4.

We can also investigate the resonance between the plasma particles and the
acoustic modes for the values of $q>1$ (referred to subextensivity). In Fig.
5, the ratio $\omega_{i}/\omega_{r}$ with respect to nonextensivity index $q$
is plotted for the values of $q>1$ at a typical long wavelength
($k\lambda_{D}=0.1$). From this graph, it is seen that the acoustic-like modes
have only (Landau) damping and no growth for these degrees of the
nonextensivity. Furthermore, the damping rate is relatively weak in these
$q$-regions, in comparison with the case of a superthermal plasma ($q<1$). The
reason is that the number of particles participating in the resonance with the
wave is small for a stationary state with $q>1$. Strictly speaking, the slope
of the velocity $q$-distribution function $f_{0\alpha}(u)$ given in Eq. (13)
increases with $q$ and there is even a thermal cutoff in the case of $q>1$
[see Fig. 1(b)]. This corresponds to a weaker resonance with the wave, in
comparison with the case $q<1$.

Our analysis reveals that the acoustic-like modes are unstable in the
$q$-region $0.34\lesssim q\lesssim 0.6$ (high superthermal $q$-region) yet,
they are heavily damped in the $q$-region $0.6\lesssim q\lesssim 0.71$ (less
superthermal $q$-region) and finally, they are relatively weakly damped for
the values of $q>1$ (subextensive region). In Fig. 6, the damping and growing
rates with respect to the wave number are plotted for some values of the
nonextensive index $q$ for three cases of the heavily damped modes [Fig6(a)],
weakly damped modes [Fig6(b)] and growing unstable modes [Fig6(c)]. We see
that for the waves with longer wavelengths the rate of damping (or growth)
becomes weaker. Moreover, our numerical analysis shows that in a pair plasma
the acoustic-like modes have the maximum damping at the vicinity of $q=0.69$
[see Fig6(a)], and they have the maximum growth when the nonextensivity is at
the vicinity of $q=0.55$ [see Fig6(c)]. In addition, we have included the
Maxwellian limit ($q=1$) to the Fig. 6(b) which emphasizes that the acoustic-
like modes in an equilibrium pair plasma are merely landau damped waves.

For completing our discussion, we have examined the temperature-asymmetry
effect, controlled by $\sigma$, on the Landau damping of the acoustic-like
modes in a pair plasma, as plotted in Fig. 7. It is observed that the
temperature-asymmetry in a pure pair plasma decreases the Landau damping. In
other words, for a fixed value of $q$ and at a given wavelength, the Landau
damping of the acoustic waves is maximum when a full symmetry in temperature
of species is established, i.e, when $T_{-}=T_{+}$.

##  5 Conclusions

In this paper, we have studied the acoustic-like modes in a collisionless and
magnetic-field-free pair plasma on the basis of the nonextensive statistics.
We have thereby used a kinetic theory model by employing the Vlasov and
Poisson’s equations to obtain the response dielectric function of the pair
plasma for the electrostatic waves. By using the dielectric function, we have
investigated the acoustic-like modes whose phase speed lies between the
thermal velocities of the species. The resultant dispersion relation in our
study is compatible with the acoustic branch of the experimental data
presented by Oohara _et. al_ (Oohara et al., 2005), in which the electrostatic
waves have been examined in a pure pair-ion plasma. It has been shown that by
decreasing the nonextensivity index $q$ the phase velocity of the acoustic
modes increases, indicating to a plasma with a great deal of superthermal
particles. Our kinetic model confirms the possibility of the acoustic modes in
the case of a temperature-asymmetric and also symmetric pair plasma. However,
it is found that the temperature-asymmetry in a pair plasma reduces the phase
velocity of the acoustic modes. Furthermore, depending on the degree of
nonextensivity of the plasma, both the damped and unstable acoustic modes are
predicted in a collisionless pair plasma, arise from a resonance phenomena
between the wave and nonthermal particles of the plasma. In the case of a
superthemal plasma confirmed by $q<1$ (superextensivity), the heavily damped
and growing unstable modes are predicted, while in the case $q>1$
(subextensivity) the acoustic-like modes have only damping and no growth. The
mechanism that leads to the damping is the same as presented by Landau
(Landau, 1946), arises from a decreasing velocity distribution function, but
the mechanism of instability lies in the heart of the nonextensive formalism.
We have postulated that the concerned instability can be associated with the
presence of superthermal particles (in the case $q<1$), in the sense that in
the process of the resonance they can give energy to the wave and then results
in growing waves in time. This instability disappears in the case $q>1$,
describing a plasma with plenty of the low-speed particles. Additionally, the
damping rate is relatively weak in the case $q>1$, in comparison with the case
of a superthermal plasma ($q<1$) with heavily damped modes. The reason is that
the number of particles participating in the resonance with the wave is small
for a stationary state with $q>1$. Moreover, our analysis indicates that the
temperature-asymmetry in a pure pair plasma decreases the Landau damping of
the acoustic-like modes.

We emphasize that in the present work, we have considered an inhomogeneous
plasma in a nonequilibrium thermal state by considering the $q$-nonextensive
distribution for stationary state of the plasma. However, our solutions reduce
to the ones for a homogeneous and equilibrium pair plasma at the extensive
limit $q\rightarrow 1$, in which the Boltzamnn-Gibbs statistics describe the
plasma state.

It is hoped that the present study would be useful for explanation of the
intriguing low-frequency modes in a pure pair plasma, which are out of the
scope of the plasma fluid theory and the Boltzmann-Gibbs statistics.

## References

  * Abe (1999) Abe, S. 1999, Physica (Amsterdam), 269A, 403
  * Abe & Okamato (2001) Abe, S., & Okamato, Y. 2001, _Nonextensive Statistical Mechanics and Its Applications, Series Lecture Notes in Physics_ , Springer-Verlag, Heidelberg
  * Abramowitz & Stegun (1972) Abramowitz, M. & Stegun, I.A. 1972, _Handbook of Mathematical Functions_ , Dover, New York, 257
  * Amour et al. (2012) Amour, R., Tribeche, M., & Shukla, P.K. 2012, Ap&SS, 338, 287
  * Amoretti et al. (2003) Amoretti, M., et al. 2003, Phys. Rev. Lett., 91, 055001
  * Arimitsu & Arimitsu (2000) Arimitsu, T. & Arimitsu, N. 2000,Phys. Rev. E, 61, 3237
  * Boghosian (1996) Boghosian, B.M. 1996, Phys. Rev. E, 53, 4754
  * Beck (2001) Beck, C. 2001, Phys. Rev. Lett., 87, 180601
  * Beck et al. (2001) Beck, C., Lewis, G.S., Swinney, H.L. 2001, Phys. Rev. E, 63, 035303
  * Begelman et al. (1984) Begelman, M.C., Blandford, R.D., & Rees, M.D. 1984, Rev. Mod. Phys. 56, 255
  * Berezhiani et al. (1993) Berezhiani, V.I., Skarka, V., & Mahajan, S. 1993, Phys. Rev. E, 48, R3252
  * Boehmer et al. (1995) Boehmer, H., Adams, M., & Rynn, N. 1995, Phys. Plasmas, 2, 4369
  * Cairns et al. (1995) Cairns, R.A., Mamum, A.A., Bingham, R., Bostrom, R., Dendy, R.O., Nairn, C.M.C., & Shukla, P.K. 1995, Geophys. Res. Lett., 22, 2709
  * Chen et al. (2009) Chen, H., et al. 2009, Phys. Rev. Lett., 102, 105001
  * Curado (1999) Curado, E.M.F. 1999, Braz. J. Phys., 29, 36
  * Daniel & Tajima (1998) Daniel, J., & Tajima, T. 1998, ApJ, 498, 296
  * Defler & Simonen (1969) Defler, H., & Simonen, T.C. 1969, Phys. Fluids, 12, 260
  * Dovner et al. (1994) Dovner, P.O., Eriksson, A.I., Bostrom, R., & Holbackm B. 1994, Geophys. Res. Lett., 21, 1827
  * Du (2004) Du, J.L. 2004, Phys. Lett. A, 329, 262
  * Dubouloz et al. (1991) Dubouloz, N., Pottelette, R., Malingre, M. & Treumann, R.A. 1991, Geophys. Res. Lett., 18, 155
  * El-Tantawy et al. (2012) El-Tantawy, S.A., Tribeche, M., & Moslem, W.M. 2012, Phys. Plasmas, 19, 032104
  * Feldman et al. (1975) Feldman, W.C., Asbridge, J.R., Bame, S.J., Montgomery, M.D., & Gary, S.P. 1975, J. Geophys. Res., 80, 4181
  * Gahn et al. (2000) Gahn, C., et al. 2000, Appl. Phys. Lett., 77, 2662
  * Gedalin et al. (1998) Gedalin, M., Melrose, D.B., & Gruman, E. 1998, Phys. Rev. E, 57, 3399
  * Gibson et al. (1960) Gibson, G., Jordan, W.C., & Lauer, E.J. 1960, Phys. Rev. Lett., 5, 141
  * Gell-Mann & Tsallis (2004) Gell-Mann, M., & Tsallis, C. 2004, _Nonextensive Entropy - Interdisciplinary Applications_ , Oxford University Press, New York
  * Gibbons et al. (1983) Gibbons,G.W., Hawking, S.W., & Siklos, S. 1983, _The Very Early Universe_ , Cambridge University Press, Cambridge, UK
  * Goldreich & Julian (1969) Goldreich, P., & Julian, W.H. 1969, ApJ, 157, 869
  * Greaves et al. (1994) Greaves, R.G., Tinkle, M.D. & Surko, C.M. 1994, Phys. Plasmas, 1, 1439
  * Greaves & Surko (1995) Greaves, R.G., & Surko, C.M. 1995, Phys. Rev. Lett., 72, 3846
  * Greaves & Surko (2001) Greaves, R.G., Surko, C.M. 2001, AIP Conf. Proc., 606, 10-23; doi: http://dx.doi.org/10.1063/1.1454263.
  * Huang & Driscoll (1994) Huang, X.-P. & Driscoll, C.F. 1994, Phys. Rev. Lett., 72, 2187
  * Helander & Ward (2003) Helander, P., & Ward, D.J. 2003. Phys. Rev. Lett., 90, 135004
  * Iwamoto (1993) Iwamoto, N. 1993, 47, 604
  * Kakad et al. (2007) Kakad, A.P., Singh, S.V., Reddy, R.V., Lakhina, G.S., Tagare, S.G., & Verheest, F. 2007, Phys. Plasmas, 14, 052305
  * Kaladze et al. (2012) Kaladze, T., Mahmood, S., & Ur-Rehman, H. 2012 Phys. Scripta, 86, 035506
  * Keston et al. (2003) Keston, D.A., Laing, E.W., & Diver, D.A. 2003 Phys. Rev. E, 67, 036403
  * Krall & Trivelpiece (1973) Krall, N.A., & Trivelpiece A.W. 1973, _Principles of Plasma Physics_ , McGraw-Hill, Kogakusha
  * Laing & Diver (2006) Laing, E.W., & Diver, D.A. 2006, Phys. Plasmas, 13, 092115
  * Landau (1946) Landau, L.D. 1946, J. Phys. USSR, 10, 25
  * Landsberg (1984) Landsberg, P.T. 1984, J. Stat. Phys., 35, 159
  * Lavagno et al. (1998) Lavagno, A., Kaniadakis, G., Rego-Monteiro, M., Quarati, P., & Tsallis, C. 1998, Astrophys. Lett. Commun., 35/6, 449
  * Leubner (2002) Leubner, M.P. 2002, Ap&SS, 282, 573
  * Liang et al. (1998) Liang, E.P., Wilks, S.C., Tabak, M. 1998, Phys. Rev. Lett., 81, 4887
  * Lima et al. (2000) Lima, J.A.S., Silva, R., & Janilo Santos 2000, Phys. Rev. E, 61, 3260
  * Lima et al. (2001) Lima, J.A.S., Silva, R., & Plastino, A.R. 2001, Phys. Rev. Lett., 86, 2938
  * Liu et al. (1994) Liu, J.M., De Groot, J.S., Matte, J.P., Johnston, T.W. & Drake, R.P. 1994, Phys. Rev. Lett., 72, 2717
  * Livadiotis & McComas (2009) Livadiotis, G., & McComas, D.J. 2009, J. Geophys. Res., 114, A11105
  * Liyan & Jiulin (2008) Liyan, L., & Jiulin, D. 2008, Physica A, 387, 4821
  * Maksimovic et al. (1997) Maksimovic, M., Pierrard, V., & Riley, P. 1997, Geophys. Res. Lett., 24, 1151
  * Maksimovic et al. (1997) Maksimovic, M., Pierrard, V., & Lemaire, J.F. 1997, A&A, 324, 725
  * Max & Perkins (1972) Max, C., & Perkins, F.W. 1972, Phys. Rev. Lett., 29, 1731
  * Michel (1982) Michel, F.C. 1982, Rev. Mod. Phys. 54, 1-66
  * Michel (1991) Michel, F.C. 1991, _Theory of Neutron Star Magnetospheres_ , University of Chicago Press, Chicago
  * Miller & Witta (1987) Miller, H.R., & Witta, P.J. 1987, _in Active Galactic Nuclei_ , Springer, Berlin, 202
  * Misner et al. (1973) Misner, W., Thorne, K.S., & Wheeler, J.A. 1973, _Gravitation_ , Freeman, San Francisco, 763
  * Montgomery et al. (1968) Montgomery, M.D., Bame, S.J., & Hundhausen, A.J. 1968, J. Geophys. Res., 73, 4999
  * Muoz (2004) Muoz, V. 2004, Phys. Plasmas, 11, 3497
  * Oohara & Hatakeyama (2003) Oohara, W., & Hatakeyama, R. 2003, Phys. Rev. Lett., 91, 205005
  * Oohara et al. (2005) Oohara, W., Date, D., & Hatakeyama, R. 2005, Phys. Rev. Lett., 95, 175003
  * Orsoz et al. (1997) Orsoz, J.R., Remillard, R.A., Bailyn, C.D., & McClintock, J. E. 1997, ApJ, 478, L83
  * Pedersen et al. (2003) Pedersen, T.S., Boozer, A.H., Dorland, W., Kremer, J.P., & Schmitt, R. 2003, J. Phys. B: At. Mol. Opt. Phys., 36, 1029
  * Pedersen et al. (2004) Pedersen, T.S., Boozer, A.H., Kremer, J.P., & Lefrancois, R. 2004, Phys. Plasmas, 11, 2377
  * Plastino (2004) Plastino, A. 2004, Physica A, 344, 608
  * Reyni (1955) Reyni, A. 1955, Acta Math. Hungaria 6, 285
  * Saberian & Esfandyari-Kalejahi (2013) Saberian, E., & Esfandyari-Kalejahi, A. 2013, Phys. Rev. E, 87, 053112
  * Silva et al. (1998) Silva Jr., R., Plastino, A.R., & Lima, J.A.S. 1998, Phys. Lett. A, 249, 401
  * Silva et al. (2005) Silva, R., Alcaniz, J.S., & Lima, J.A.S. 2005, Physica A, 356, 509
  * Sturrock (1971) Sturrock, P. A. 1971, ApJ, 164, 529
  * Surko et al. (1989) Surko, C.M., Leventhal, M., & Passner, A. 1989, Phys. Rev. Lett., 62, 901
  * Surko & Murphy (1990) Surko, C.M., & Murphy, T. 1990, Phys. Fluids B, 2, 1372
  * Surko & Greaves (2004) Surko, C.M., & Greaves, R.G. 2004, Phys. Plasmas, 11, 2333
  * Tandberg-Hansen & Emslie (1988) Tandberg-Hansen, E., & Emslie, A.G. 1988, _The physics of solar flares_ , Cambridge University Press, Cambridge, 124
  * Trivelpiece (1972) Trivelpiece, A.W. 1972, Comments Plasma Phys. Controlled Fusion, 1, 57
  * Tsallis (1988) Tsallis, C. 1988, J. Stat. Phys., 52, 479
  * Tsallis (1994) Tsallis, C. 1994, Phys. Lett. A, 195, 329
  * Tsallis & de Souza (1997) Tsallis, C., & de Souza, A.M.C. 1997, Phys. Lett. A, 235, 444
  * Tsallis et al. (1995) Tsallis, C., Sa Barreto, F.C., & Loh, E.D. 1995, Phys. Rev. E, 52, 1447
  * Tsallis (1995) Tsallis, C. 1995, Chase, Chaos. Soliton. Fract., 6, 539
  * Tsallis (1999) Tsallis, C. 1999, Braz. J. Phys., 29, 1
  * Tsallis (2009) Tsallis, C. 2009, _Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World_ , Springer, New York
  * Tsytovich & Wharton (1978) Tsytovich, V. & Wharton, C.B. 1978, Comments Plasma Phys. Controlled Fusion, 4, 91
  * Valentini (2005) Valentini, F. 2005, Phys. Plasmas, 12, 072106
  * Vasyliunas (1968) Vasyliunas, V.M. 1968, J. Geophys. Res., 73, 2839
  * Verheest (1996) Verheest, F. 1996, Phys. Lett. A, 213, 177
  * Verheest (2005) Verheest, F. 2005, Nonlinear Processes Geophys., 12, 569
  * Vranjes & Poedts (2005) Vranjes, J., & Poedts, S. 2005, Plasma Sources Sci. Technol., 14, 485
  * Wardle et al. (1998) Wardle, J.F.C., Homan, D.C., Ojha, R., & Roberts, D.H. 1998, Nature, 395, 457
  * Watanabe & Taniuti (1977) Watanabe, K., & Taniuti, T. 1977, J. Phys. Soc. Jpn., 43, 1819
  * Zank & Greaves (1995) Zank, G.P. & Greaves, R.G. 1995, Phys. Rev. E, 51, 6079
  * Zouganelis (2008) Zouganelis, I. 2008, J. Geophys. Res., 113, A08111

Figure 1: The nonthermal behavior of the $q$-nonextensive distribution
function and its comparison with the Maxwellian one (solid carve): (a)
Superxtensive distribution with $q<1$ that behave alike the
$\kappa$-distributions for superthermal plasmas. In this case, the particles
have distributed in a wider spectrum of the velocities, in comparison with a
Maxwellian distribution. (b) Subextensive distribution with $q>1$ which is
suitable for describing the systems containing a large number of low-speed
particles. In this case, there is a thermal cutoff which limits the velocity
of particles. Figure 2: The linear dispersion relation of acoustic-like modes
in a pair plasma. (a) The nonextensivity effect on dispersion relation with
$\sigma=0.9$, where the solid curve corresponds to the extensive limit ($q=1$)
and the other ones show the deviations from a Maxwellian pair plasma. (b) The
effect of temperature-asymmetry on dispersion relation with $q=0.7$, where the
solid curve corresponds to a temperature-symmetric pair plasma. Figure 3: The
comparison of the acoustic-like modes and the Lanqmuir waves in a pair plasma
with $T_{+}=T_{-}$. The acoustic waves belong to a low frequency band which
tends to zero at the limit $k\rightarrow 0$, while the Langmuir waves occur in
high frequencies above $\omega_{p}$. Figure 4: The imaginary part of the
frequency with respect to the nonextensivity index for $q<1$, which shows the
$q$-regions for the growing and heavily damped acoustic-like modes. Figure 5:
The imaginary part of the frequency with respect to the nonextensivity
parameter for $q>1$. For this values of the nonextensivity index $q$, the
acoustic-like modes have only damping and no growth. Figure 6: The damping
(growing) rate with respect to the wave number for (a) the heavily damped
modes in the $q$-region $0.6\lesssim q\lesssim 0.71$, (b) the relatively
weakly damped modes in the $q$-region $q>1$, and (c) the growing acoustic
modes in the $q$-region $0.34\lesssim q\lesssim 0.6$, when $\sigma=0.9$. We
have included the Maxwellian limit ($q=1$) to our results which emphasizes
that the acoustic-like modes in an equilibrium pair plasma are merely the
landau damped waves. Figure 7: The effect of temperature-asymmetry on Landau
damping of the acoustic-like modes which indicates that the temperature-
asymmetry in a pure pair plasma decreases the Landau damping rate.