The collisional relaxation rate of kappa-distributed plasma with multiple components

The kappa-distributed fully ionized plasma with collisional interaction is investigated. The Fokker-Planck equation with Rosenbluth potential is employed to describe such a physical system. The results show that the kappa distribution is not a stationary distribution unless the parameter kappa tends to infinity. The general expressions of collisional relaxation rate of multiple-component plasma with kappa distribution are derived and discussed in specific cases in details. For the purpose of visual illustration, we also give those results numerically in figures. All the results show that the parameter kappa plays a significant role in relaxation rate.


I. INTRODUCTION
After introduced by Vasyliunas in 1968 1 , kappa distribution, as a non-equilibrium stationary distribution, has been applied to model plenty of space plasmas successfully, for instance, solar corona, 2,3 solar wind, 4-7 inner heliosheath, 8 the planetary magnetosphere, [9][10][11] and so on. Theoretical investigations based on kappa distribution are also interesting topics, which refer to various properties of plasma deviated from Maxwellian equilibrium, such as the discussions of the Debye length, 12,13 the definition of temperature, 14,15 solitary waves and nonlinear waves in plasmas, [16][17][18] the instabilities of plasma, [19][20][21][22][23][24][25][26] transport properties, [27][28][29][30][31] , and some other works, [32][33][34][35] etc. The kappa velocity distribution is usually written as, 36 where v is the velocity vector, m is the mass, k is the Boltzmann constant, T is the temperature, u is the overall bulk velocity of plasma, B κ is the normalization factor, with the condition κ > 3/2 for the convergence purpose. In the distribution function (1), κ is a parameter which describes a distance away from equilibrium. And in the limitation κ → +∞, the distribution function (1) recovers the Maxwellian distribution. Because of the similarity in mathematical form, it has been considered that the kappa distribution has a very close connection with the q-distribution in nonextensive statistical mechanics (NSM). 14,[37][38][39] It can be shown that the two distributions are equivalent to each other by the appropriate definition of temperature, 39 which means that some properties of NSM could be utilized in the research of kappa-distributed plasma.
As a widely existed nonthermal stationary state in collisionless space plasma, the kappa distribution can be regarded as a steady-state solution of the collisionless Vlasov equation.
In this view, one can derive an equation of κ in a nonisothermal physical system with external electromagnetic field, 40,41 κ − where e is the elementary charge, ϕ c is the Coulomb potential, c is the speed of light and How fast is that relaxation process? All the above questions are still unclear in the present theory of kappa distribution.
Therefore, the present work focuses on the influence of the collision in fully ionized plasma consisted by multiple kappa-distributed components. The paper is organized as follows.
In section 2, we introduce the FP equation with Rosenbluth potential in order to study the collision effects on kappa distribution. In section 3, we examine whether the kappa distribution is the stationary solution of the FP equation and give the relaxation rate by both analytic expressions and numerical results in figures. In section 4, we make the conclusions and the discussions. At last, some detailed calculations are written in Appendix.

II. KINETIC EQUATION
We consider a fully ionized plasma consisted by multiple components in which the interaction between particles is Coulomb collision. In such a kind of plasma, small angle Coulomb scatterings are more probably occurred than a large angle one, while a large deflection is consisted by a series of small angle scattering. So the collisional effects of plasma can be described by the FP equation with Rosenbluth potential in which the collision term is given by, 48,49 where f α is the velocity distribution function of the α-species particle, the summation of β on the right side is over all kinds of particles in plasma, Γ αβ is a factor which reads with the number density n β and the charge q β of the β constituent, the charge q α and the mass m α of the α one, and the scattering factor ln Λ. The Rosenbluth potential h αβ and g αβ is defined as 48,49 h with the distribution function f β (v ′ ) for β-species particle. The Eq.(4) can also be expressed in a general form of FP equation as where S α is the current of probability, where H α (v) is the dynamical friction vector and ← → G α (v) is the dynamical diffusion tensor, and For a stationary state f α (v), the FP collision term (∂f α /∂t) c should equal zero for all possible velocity v; while for non-stationary it represents the relaxation rate of initial distribution

RELAXATION RATE
In order to test whether the kappa distribution is the stationary solution of FP equation, we need to substitute Eq.(1) into Eq.(4). Generally speaking, one can assume that each kind of particles in plasma follows the kappa distribution with different temperatures and κ-indexes, where the subscript β stands for the physical quantities of β component in plasma. And the bulk velocity u is omitted in the calculations, because on the right side of the collision term (4) the partial derivative is only with respect to v, or in other words, ∂u/∂v = 0. Based on these simplifications, we firstly calculate the Rosenbluth potential. The function h αβ (v) has been obtained, 28,50 where kT β ] is an abbreviation, and 2 F 1 (a, b; c, z) is hypergeometric function defined by, 51 The function g αβ (v) can also be derived by substituting the kappa distribution (12) into the integral (7) (details in Appendix A), The distribution function f α denotes a stationary state if and only if the partial derivative vanishes, namely (∂f α /∂t) c = 0 for arbitrary v. For such a complicated equation (19), it is difficult to find whether there are some special values of κ leading to the zero FP collision term. So several cases would be investigated in the next subsections.
However, if the FP collision term (∂f α /∂t) c = 0, which means f α is not a stationary state and will relax to the equilibrium, then we can study the rate of the relaxation. For the purpose of characterization of the relaxation rate, we adopt the absolute rate of change (∂f α /∂t) c as well as the relative rate of change, which describes the relative change of the distribution function in unit time and can be also regarded as the inverse of relaxation time.

A. Plasma with one component
In the first place we consider a single-component plasma such as a pure electron plasma.
In this case, the collision term (19) can be simplified as while the probability current is given by, For convenience, the FP collision term and the probability current can be rewritten respectively as, where the relation 51 of hypergeometric function has been used, The FP collision term will be analyzed in three situations at κ α = +∞, 3/2 and other values, because the parameter κ α ∈ (3/2, +∞).

The limit κ α → +∞
For κ α → +∞, after taking the limitations one derives and in the same way This is obvious because when κ → +∞ the kappa distribution recovers the Maxwellian one, which is the equilibrium state.
When κ α → 3/2, similarly we need to calculate some limitations. The kappa distribution f α and some hypergeometric functions in the above equations under the limit κ α → 3/2 are work out directly, where another relation 51 has been used for evaluating the limit of hypergeometric function, After taking these results back into Eq. (21), we obtain and the relative relaxation rate, Therefore, in this case, the kappa distribution cannot be regarded as a stationary state.

Other values of κ α
For other values of κ α , the FP collision term (21) is a function of v, so it cannot be zero for arbitrary v when κ α = ∞, which means that the kappa distribution is not the stationary solution of the FP equation for single-component plasma. We here take the collisional pure electron plasma as an example in order to show the FP collision term (21) for different values of κ e in Figs.1 and 2, where the typical temperature and density of number in pure electron plasma are set as, T e = 10 4 K, n e = 10 8 cm −3 . And additionally we let the scatter factor ln Λ = 1.
In Fig.1(a) we find that the relaxation rate (∂f α /∂t) c hardly varies when the speed is from 10 −2 cm/s to 10 7 cm/s, while decreases significantly when in the range of 10 7 cm/s − 10 9 cm/s in the case of κ e = 1.6, 2, and 5. It is indicated that the collisions make stronger influences on the distribution of low energy particles than that of high energy particles. The missing fragment of the curves in the neighborhood of v = 10 7 cm/s in Fig.1(a) is the positive part of (∂f e /∂t) c . Thus, Fig.1(b) is drawn to exhibit this fragment more clearly in a relative small range of speed. In Fig.1 the negative relaxation rate, namely (∂f α /∂t) c < 0, implies that the distribution decreases as the increment of the time and vice verse. As we know, the kappa distribution comparing with the Maxwellian one has larger values in small-velocity and superthermal range, which is displayed in a schematic diagram Fig.3. Therefore, when the relaxation rate of the kappa distribution is negative in small-velocity and superthermal range as shown in Figs.1 and 2, we may think that the kappa distribution evolves "towards" Maxwellian distribution at initial time t = 0. In addition, the larger the kappa parameter is, the smaller the relaxation rate (∂f α /∂t) c becomes. When the kappa parameter approach the infinity, (∂f α /∂t) c vanishes since the distribution reduces to the Maxwellian. The relative relaxation rate (∂f α /∂t) c /f α in Fig.2 displays the same behaviors as the absolute rate of change.

B. Plasma with two components
We focus on electron-ion plasma in this part, which is a kind of two-component plasma.
The FP collision term of electrons for such plasmas is divided into two parts,   , −A κ,e v 2 − κ e + 2 κ e 2 F 1 1,    which means the electron-ion collision term is the key point to discuss in this subsection.
1. The limit κ e → +∞ and κ i → +∞ After taking the limit of κ e → +∞ and κ i → +∞, one yields that the FP collision term approach zero if T e = T i as expected, which means the Maxwellian-distributed plasma is in the equilibrium as is known to all.

The limit
In this case, the electron-ion collision term turns to be

The limit κ i → +∞
When the ion distribution approaches the equilibrium, the FP collision term becomes,

IV. CONCLUSIONS
In this work, we study fully ionized collisional plasma with kappa distribution by employing the FP equation with Rosenbluth potential. The results show that the kappa distribution cannot be a stationary unless the kappa parameter tends to infinity. Then we give the general analytical expression of the relaxation rate of kappa distribution (19). In order to analyze the relaxation rate more clearly, we discuss it in two special cases, i.e. , single-component  In electron-proton plasma, we focus on the relaxation rate contributed from the electronion collision, because the contribution from electron-electron collision has been studied in  From these results, we can make the conclusions of the direction of the relaxation at initial time t = 0. First, the kappa-distributed electron evolves "towards" the Maxwellian distribution due to the collision with itself. Second, the direction of the electron relaxation due to the collision with ions are complicated. When colliding with kappa-distributed ions, the electrons are decelerated. When colliding with Maxwellian-distributed ions, kappadistributed election with small enough κ e -index evolves "towards" Maxwellian distribution such as the case of κ e = 1.6 while with other values of κ e the electrons are decelerated as exhibited in Fig.6 and 7. All these results illustrate that the kappa parameter significantly effects the relaxation rate due to the collision. where θ is the angle between w and v. After working out the integral with respect to θ, one has g αβ (v) = πB κ,β κ β vA κ,β ∞ 0

V. ACKNOWLEDGEMENTS
which can be rearranged as by employing the representation of hypergeometric function (14).