Low frequency twisted waves in a self-gravitating nonextensive complex plasma

The effects of dust–dust self-gravitational force and nonextensive characteristics of plasma species on the low frequency twisted waves owing to the helical wave structure in complex (dusty) plasmas are analyzed. The electrons and ions of the plasma are modelled by nonextensive q-distribution function while massive dust particles are Maxwellian distributed. The self-gravitational effects are incorporated in the Vlasov equation of kinetic theory where perturbed distribution function, electrostatic and gravitational potentials are expressed with Laguerre–Gauss functions. The governing equations of kinetic theory are solved together under paraxial approximations. The dispersion relations and damping rates of twisted dust-acoustic waves (TDAWs) are obtained for two situations; (a) super-extensivity (q < 1) and (b) sub-extensivity (q > 1). The effects of self-gravity, nonextensivity and twist parameter significantly modified the basic features of dust-acoustic waves. This study contributes to our understanding of the complex dynamics of TDAWs in interstellar dust clouds, considering the interplay of self-gravity, nonextensivity, and helical phase structures. The obtained theoretical and numerical results provide valuable insights into the behavior of these waves and offer a foundation for further investigations in this field. However, understanding of the topic can be enhanced through a combination of theoretical models, numerical simulations and observational data.


Introduction
It is worth noting that an exchange of energy and momentum occurs from electromagnetic radiations (laser beams) to matter by its interaction [1]. Allen et al [2] theoretically predicted the well-behaved orbital angular momentum (OAM) states for the Laguerre-Gaussian (LG) light beams. The theme of OAM endowed to coherent excitations has explored novel characteristics of plasmas. The inverse Faraday effect [3], higher harmonic generation [4], particle trapping/transportation [5], nonlinear wave coupling [6], etc are few among the possible paradigms attributed to OAM states involving the collective behavior of ionized matter. Generation of twisted laser beams having finite OAM states is now possible in a laboratory [7][8][9], providing the basis for many scientific applications [10,11]. Moreover, observations have shown many space and astrophysical environments, where OAM can be attributed to plasma twisted motion. The co-rotating stellar and solar winds and spinning regions of the Jovian planets best describe such situations [12]. The formation of helical field structures in solar wind and coronal plasma is often observed where the vortex motions are produced by the interaction of solar wind pressure with Earth's magnetosphere [13]. Extensive research has been carried out to examine the impact of OAM involving nonlinear modes in diverse plasma compositions. In this context, using the fluid equations it has been shown in [14] that the LG potentials and associated angular momentum density for ion-acoustic phonons significantly alter with the angular mode number and beam waist. Intriguingly, the OAM correction to plasma modes evolving helical wavefronts with spiraling field lines relative to wave propagation are termed as the twisted modes. Many research articles [6,[15][16][17][18][19][20] have focused the onset of twisted waves and associated instabilities in laboratory and space plasmas.
Renyi [21] was the first to recognize that charged particles in space plasmas deviate from thermodynamic equilibrium and are therefore described by a nonextensive q-distribution. Later on, many theoretical and experimental studies showed that Boltzmann-Gibbs (BG) statistics is inappropriate for description of the nonextensive systems, i.e. systems exhibiting long range interactions. Due to this discrepancy of the BG statistics, Tsallis [22] presented a new idea of the Tsallis statistics by incorporating all the necessary informations of nonextensive phenomenon. It is to mention that in the super(sub)-extensivity limit when nonextensive parameter q < (>)1, the total entropy of a system is greater (less) than the sum of the entropies of its components. Thus the q-distribution for nonextensive components of a plasma reduces into the Maxwell-Boltzmann distribution function when q → 1. Importantly, the nonextensive plasma occurs in stellar polytropes [23], electron plasma columns [24], and in the galactic clusters [25]. The dispersion relation of damped and undamped electrostatic potentials [26], examined in nonextensive plasmas exactly matches to experimentally observed modes: it confirms the presence of q-distributed particulate.
Dynamical behavior of the charged dust grains observed by Voyager 1 and 2 has successfully shown several indications of collective effects due to gravitational and electromagnetic forces in interplanetary space. Later, the dust particulates have been seen [27][28][29] in interstellar, circumstellar and molecular clouds under the influence of both gravitational and electromagnetic forces which confirmed early reported results. Furthermore, dust grains have also been found causing the contamination on the substrates and wafers [30] in laboratory plasmas. These particles are also found [31] in the Hall thrusters with a size of 1-50 µm and mass of orders 10 −10 -10 −15 kg. Recently, plasma instability in the presence of stationary dust particles has been noted under the influence of recombination [32]. It is important to control the transportation of dust and to develop vigorous understanding of dynamical behavior of dust grains subjected to the action of electric and gravitational forces in laboratory and space dusty plasma environments. It should be noted that dust-dust, dust-planet or dust-satellite interactions are major causes of the production of gravitational forces in dusty plasmas. However, the gravitational effects become significant in regimes where the gravitational forces between dust particles dominate over other forces, such as thermal or electromagnetic forces [33]. In present study, the dust-dust interactions are considered and dust particles are controlled by the combined effects of electric and gravitational forces.
In this work, we present the analytical solution of plasma response function of twisted dust-acoustic waves (TDAWs) in a self-gravitating dusty plasma using kinetic approach for plasma species. In such waves, there are two velocity components; one is the axial component or z-component (v z ) that propagates along the axis whereas the second component appears due to the helical phase structure of the wave, i.e. azimuthal component (v ϕ ). We apply LG wave solutions to the Vlasov-Poisson set of equations and obtain a modified plasma response function, which is numerically analyzed for dispersion characteristics and decay rate. We specifically emphasized on the contribution of finite OAM states and nonextensive parameter in modifying the dispersion relation and damping rate of TDAWs. The present investigation has relevance to the interstellar dust clouds, consisting of massive dust grains whose dynamics are altered by the force of gravity.

Plasma response function with finite OAM
We assume the linear propagation of low frequency (ω ≪ kv ti , kv te ) TDAWs in an electrostatic self-gravitating dusty plasma whose constituents are nonextensive q−distributed electrons and ions while Maxwellian distributed dust grains. The plasma under consideration obeys the charge-neutrality condition; n e0 − n i0 + Z d0 n d0 = 0, with n e0 (n i0 ) being the equilibrium electron (ion) density and Z d0 denotes the dust charging state. Let us consider both the static electric and magnetic fields be zero, i.e. E 0 = B 0 = 0. Furthermore, the dust grains are taken as identical in mass, charge (negative) and size. In our present case, the electrons and positive ions obey the following q-nonextensive distribution function , for 0 < q ⩽ 1 and C q = [(1 + q) n s0 /2 for q ⩾ 1, where q is a parameter representing the deviation of distribution from Maxwellian distribution while v ts is thermal speed of the s−species. It is interesting to note that the q-distribution reduces to the Maxwellian distribution in the limit q → 1. Therefore, q-distribution is the more generalized distribution. For such a system, the dynamics of TDAWs accounting for finite OAM states can be examined by the following kinetic equations in a normalized form; ( and where  (3) and (4) with n s =´f s1 (r, v s , t)dv identify as charge and mass density fluctuations, which may result in the identification of electrostatic and gravitational potentials having dust plasma and dust Jeans oscillation frequencies ω pd with w(z) being the beam waist. The azimuthal (l) and radial (p) mode numbers corresponding the LG modes are respectively, define as l(= 0, ±1, ±2 . . .) and p (= 0, 1, 2 . . .). With these considerations, we have solved equation (2) in cylindrical coordinates which leads to the derivation of perturbed distribution function where It should be noted that the usual wave vector k used in equation (5) of [34] is now replaced by an effective wave vector k eff . Making use of equation (5) with LG profiles of electrostatic and gravitational potentials, equations (3) and (4) can be simplified into a generalized dispersion relation for twisted electrostatic modes in a self-gravitating dusty plasma, as where, the plasma dielectric constants along with the coupling parameter (Γ) come out as with I s =´k eff ·∂v s fs0 ω−k eff ·vs dv s . Equation (6) shows both the plasma and gravitational dielectric constants and exactly agrees with the previous work [35] in the hydrodynamic perspectives. The solution of equation (6) with the nonextensive q−distribution function (1) for electrons and ions and Maxwellian ] dust grains yield the following plasma response function with finite OAM states in a self-gravitating dusty plasma 2 )+ξzsZ(ξzs)+ξφsZ(ξφs) ] .
Equation (6a) is obtained under the limits |k r | , |k z | ≪ |lk φ |, with Debye length and Jeans length are respectively defined as λ Ds and λ Js . The axial Z (ξ zs ) and helical Z (ξ φ s ) plasma dispersion functions have the usual definitions [36]. It is important to mention here that helically phase-varying fields (electric and gravitational) not only modify the dispersive characteristics of waves but also alter the resonance conditions. Furthermore, the azimuthal component of velocity (v φ ) of plasma species deals with a vortical motion around the z-axis and only exists in the case of variant phase. Note that, the absence of azimuthal velocity (v φ = 0) and azimuthal wave number (lk φ = 0) reduced the equation (6a) to the straight propagating plane DA waves.
To proceed further, we employed the asymptotic expansion series of plasma dispersion functions for small and large arguments and performed some mathematical steps on equation (6a). Resultantly we obtain; where χ j shows the dielectric susceptibilities of electrons and ions while χ d is the dielectric susceptibility of dust grains. These susceptibilities can be expressed as and with where α(= k/lk φ ) is a parameter indicating the twist in a wave. It is important to note that we have neglected the gravitational effects associated with the electrons and ions owing to their smaller mass as compared to massive charged dust grains. Furthermore, we separate the real part of dielectric constant, equation (7) and set it equal to zero for longitudinal low frequency TDAWs to yield the following result represents the TDA speed with effective Debye length the ratio of dust to ion density. In the absence of dust self-gravitational effects, i.e. ∆ = 0, we simply retrieve the linear dispersion of the twisted DA mode [16]. If we assume l → 0, leading to α → ∞, we ultimately get the real wave frequency for DAWs in the following form Furthermore, under an approximation ω 2 Jd > [ , the TDAWs become unstable leading to the twisted Jeans instability.

Damping rate of TDAWs for super-extensive case (q < 1)
For smaller values of the nonextensive parameter q, i.e. q < 1, the number of the energetic particles in the system increases. This increase in the energetic particles affects the tail of the distribution function. Here, we will assume two situations for the investigation of DAWs.

Case-I: for zero OAM states (l = 0)
Here we take the case of zero azimuthal mode number, i.e. l = 0 and ignore the effect of OAM. The standard procedure [37] of obtaining the damping rate leads to the following form where, δ (= n d0 /n i0 ) shows the negative dust concentration. Equation (13) shows the plane wave solution of damping rate for DAWs significantly modified by self-gravity.

Case-II: for finite OAM states (l ̸ = 0)
In this subsection, we take the contributions of finite OAM states via non-zero values of azimuthal mode number (l ̸ = 0) in the derivation of damping rate. For this, using the standard method, we can obtain the damping rate for the TDAWs in a self-gravitating dusty plasma as and R = exp , while the parameter β is defined as . In the limit, q → 1, equation (14) exactly coincides with the result of [38]. Furthermore, for l → 0, i.e, α → ∞, the expression of equation (14) leads to the damping rate involving the plane DA waves (equation (13)).

Damping rate of TDAWs for sub-extensive case (q > 1)
For q > 1, the number of the energetic particles decreases which affects the top of the distribution function. Furthermore, it should be noted that for q > 1, there exist a thermal cut off for q-distribution, i.e. v c = v ts √ 1 q−1 , which restricts the velocity of particles to the limit v s < v c . The thermal cut off does not exist for q < 1 which shows that the velocity of the particles is unbounded in this limit.

Case-I: for zero OAM states (l = 0)
For sub-extensive plasmas, the damping rate of dust-acoustic waves in a self-gravitating dusty plasma exhibiting planar wave solution (l = 0) takes the following form with all parameters having the usual meanings. Equation (16) represents the damping of DAWs for sub-extensive case, i.e. q > 1 under the influence of self-gravitational and nonextensive effects in a dusty plasma.

Case-II: for finite OAM states (l ̸ = 0)
The inclusion of different states of OAM ( l = ±1, ±2, ±3 . . .) via dimensionless twist parameter α(= k/lk φ ) significantly modify the damping rate of DAWs. The result is where X and R are defined in the same manner as in equation (14). This expression is obtained by assuming the helical structures of electric field lines in a self gravitating nonextensive dusty plasma.

Damping rate of TDAWs for Maxwellian distributed plasmas
As we already discussed that nonextensive q-distribution can be reduced to the Maxwellian distribution in the limit q → 1, therefore, the damping rate of TDAWs can be easily calculated as Equation (18) represents the damping rate of TDAWs in a self-gravitating Maxwellian distributed dusty plasma. Further discussion on the analytical results will be presented in the next section.

Results and discussion
In this section, we present the numerical analysis of the results for a typical nonextensive distributed self-gravitating dusty plasma whose parameters are the representative of interstellar dust clouds. We have noticed that the kinetic dispersion relation in the present study is formally like the plane wave solution. However, the twisted profile of DAWs exhibits the new resonance conditions due to OAM states. In addition, the modified wave frequency and damping rates are found due to the azimuthal velocity components in distribution functions. The results are presented for two different cases; (a) super-extensive (q < 1) and sub-extensive (q > 1) case. It is important to mention that dust-dust self-gravitation has significant dependence on mass, number density, size distribution and composition of the dust particles. For the numerical analysis of the results, we have normalized the expressions of real wave frequency and damping rates for both cases with the dust plasma parameters. For this purpose, following normalization scheme is used; ) ] < 1 for interstellar dust clouds to exhibit the self-gravitational effects.
The real frequency of TDAWs shows different behavior in the absence and in presence of self-gravitational effects. The gravitational effects appear in the limit ω pd > ω Jd to make the necessary condition ∆ < 1 valid. The two dimensional plots (figure 1) of the real wave frequency are presented at three different values of self-gravitation parameter µ, i.e. plane with red grid lines at ω Jd = 0 (showing no dust self-gravitational force), plane with blue grid lines at µ and plane with yellow grid lines at ω Jd /ω pd = 0.5. Note that the presence of dust self-gravitational force reduces the twisted wave frequency to a significant level. Furthermore, the effect of various states of OAM (l = 0, ±1, ±2 . . .) on the wave frequency is also depicted in figure 1, where l = 0 shows the absence of OAM while l = ±1, ±2 . . . implies finite contribution of OAM. For l → 0, the dimensionless twist parameter α → ∞, which gives the results of plane propagating waves whereas l → ∞ results in α → 0 leading to the generation of phase singularity. It is important to note that twist parameter α(= k/lk φ ) reveals the helical phase structures controlled by azimuthal index, where the handedness is linked to the signature of l and is usually chosen by convention. Here, we observed the symmetrical distribution of planes for wave frequency at either side of the singular point while solution blows up at singular point. Note that, the parameter α = 1 (i.e. k = lk φ ) identifies the equal contributions of planar and non-planar effects on the TDAWs. The observations indicate that wave frequency is more at smaller values of α parameter [near the singular point] indicating that twisted effects significantly enhanced the wave frequency.
Like wave frequency, the damping rate of TDAWs is also strongly influenced by the addition of OAM and same is depicted in figure 2. The contour plots indicate that the damping rate reduces as longitudinal wave number (k) increases in comparison with the azimuthal wave number (lk φ ) in the limit α < 1. More damping rate indicates that more energy is transferred from the wave to the resonant plasma particles. It implies that there are more wave-particle interactions for smaller values of twist parameter, ultimately leading to the loss of significant amount of wave energy. Moreover, the damping rates for both super-extensive and sub-extensive plasmas show the same trend, however, the magnitude is quite different. It can be seen that the damping rate for TDAWs in sub-extensive self-gravitating dusty plasma is more than that of in super-extensive plasma.
The impact of non-extensive parameter on the damping rate of TDAWs in a self-gravitating dusty plasma is categorically presented in figure 3 where various curves are plotted at different values of non-extensive parameter q for both (a) super-extensive and (b) sub-extensive plasmas. It is customary that the non-extensive character of q-distribution reduces to the Maxwellian distribution for q → 1, i.e. either for q approaches to unity from smaller values (q < 1) or from larger values (q > 1). Keeping in view the same fact, it can be easily observed from the trends of various curves that damping rate enhances for super-extensive (q < 1) plasmas while it reduces for sub-extensive (q > 1) plasmas as q approaches to unity. However, the rate of enhancement of the damping rate with q parameter is almost ten times smaller for super-extensive plasmas ( figure 3(a)) as compared to the rate of reduction for sub-extensive case ( figure 3(b)) indicating that more energy exchange will occur during the wave particle interactions for later.
In figure 4, we have presented the comparison of the damping rate of TDAWs for super-extensive and sub-extensive plasmas in the absence and in the presence of self-gravitational parameter . Here, we are interested to show that how damping rate is affected by the variation of Jeans to plasma frequency ratio µ. The curves of figures 4(a) and (b) with black solid lines represent µ = ω Jd = 0, showing zero self-gravitational force among the dust particles, whereas curves with   red dashed lines at µ = ω Jd /ω pd = 0.007 identify the presence of gravitational effects. It is noticed that in the absence of self-gravitational effects, the resonance plasma species absorb more energy from TDAWs during the wave particle interactions leading to the more damped wave in a self gravitating nonextensive dusty plasma. Moreover, the TDAW exhibits more damping in sub-extensive plasmas as compared to the super-extensive plasmas.  It can be seen that equations (14) and (17) are significantly influenced by the inclusion of finite OAM states (l ̸ = 0) leading to the modification of damping rate of TDAWs. Note that smaller values of azimuthal quantum number l, i.e. l → 0 corresponds to the planar wave solutions while as quantum number increases, twisted effects become prominent. It should be noted that l = 0 indicates the absence (zero contribution) of OAM states in a plasma under consideration. The impact of OAM via twist parameter α on the damping rate of TDAWs in a self-gravitating dusty plasma is presented in figure 5. It is observed that damping rate increases with twist parameter showing that TDAWs exhibit comparatively small damping than planar dust-acoustic waves. This is due to the reason that resonant particles gain less energy from the twisted waves as compared to that of planar waves during the wave particle interactions.
In summary, our theoretical study focused on investigating the modifications that occur in the propagation of TDAWs in the presence of self-gravitational effects, nonextensivity, and twist parameters related to the helical phase structure. The self-gravity effect was incorporated into the Vlasov equation, while a well-known kinetic model of plasma physics was utilized with Laguerre-Gaussian type solutions. The governing equations of the kinetic model were solved under paraxial approximations. The obtained analytical results were then subjected to numerical analysis, considering specific physical parameter values relevant to interstellar dust clouds composed of massive dust grains. The parameters of dust clouds satisfy the necessary condition, i.e. ∆ < 1 to exhibit the self-gravitational effects. The analysis demonstrated that self-gravitational effects, nonextensivity, and the twist parameter exerted significant influences on the dispersion relation and damping rates of the TDAWs. Although the study of self-gravitating TDAWs offers valuable insights, there are some limitations to consider; our study is based on the kinetic theory which may not fully capture the complexity of interstellar dust clouds. Real dust clouds can exhibit variations in composition, density, and other physical properties that can affect wave behavior differently than predicted by idealized models. The analysis based on the combination of theoretical models, numerical simulations and observational data may predict more information about the topic.

Data availability statements
All data that support the findings of this study are included within the article (and any supplementary files).