On relaxation phenomena in a two-component plasma

The relaxation of temperatures and velocities of the components of a quasi-equilibrium two-component homogeneous completely ionized plasma is investigated on the basis of a generalization of the Chapman-Enskog method applied to the Landau kinetic equation. The generalization is based on the functional hypothesis in order to account for the presence of kinetic modes of the system. In the approximation of a small difference of the component temperatures and velocities, it is shown that relaxation really exists (the relaxation rates are positive). The proof is based on the arguments that are valid for an arbitrary two-component system. The equations describing the temperature and velocity kinetic modes of the system are investigated in a perturbation theory in the square root of the small electron-to-ion mass ratio. The equations of each order of this perturbation theory are solved with the help of the Sonine polynomial expansion. Corrections to the known Landau results related to the distribution functions of the plasma and relaxation rates are obtained. The hydrodynamic theory based on these results should take into account a violation of local equilibrium in the presence of relaxation processes.


Introduction
In his known paper [1] Landau obtained a kinetic equation for a two-component fully ionized electron-ion plasma. This equation is widely used for investigation of plasma kinetics (see, for example, [2][3][4][5][6]). Of course, it describes the situation in the plasma approximately. The Landau equation takes into account only the short-range part of the Coulomb interaction because the Coulomb potential is artificially cut in the collision integral at the Debye radius. This can be done exactly using the Balescu-Lenard equation. In the Landau collision integral, the Coulomb potential is also cut at small distances where this potential is big and the situation needs special attention. This was done in an exact consideration by Rukhadze and Silin [7]. A comparison of the mentioned theories shows that the Landau kinetic equation describes the effects of the short-range part of the Coulomb interaction in plasma with a logarithmic accuracy (see the discussion of the mentioned results in [2]). The long-range part of the Coulomb interaction can be taken into account by the Vlasov term which describes the one-particle effects of a self-consistent field [2]. In homogeneous states, there is no self-consistent field, and one can investigate these states only on the basis of the Landau kinetic equation.
On the basis of his equation [1] Landau investigated the case in which the components are spatially homogeneous equilibrium subsystems with different temperatures T a (t ) (a = e,i) and the temperature relaxation is observed. This phenomenon is of great interest because of its fundamental importance for applications in plasma theory and condensed matter physics in general. Temperature and velocity relaxation in two-component systems is observed in many systems and breaks local equilibrium in them. Even for spatially uniform states, the problem of calculating the distribution function in the presence of relaxation is considered to be a complicated one [6] because of the lack of a small parameter. Among the existing applications, it is worth to mention the plasma hydrodynamics with account of the relaxation processes (two fluid hydrodynamics) [8,9], electron-phonon hydrodynamic phenomena in metals and semiconductors [10], magnon-phonon hydrodynamic phenomena in ferromagnetic materials [11], relaxation of the hot spot [12], etc.
In the present paper, the relaxation of temperatures T a (t ) and velocities υ an (t ) of the plasma compo- are investigated (on the basis of the ideas of paper [1] the velocity relaxation was studied, for example, in [13]). In (1.1), T, υ n are the equilibrium temperature and velocity of the plasma; τ T , τ u are the relaxation times. Our consideration is based on the Chapman-Enskog method generalized to account for the relaxation phenomena (we use the term "relaxation phenomena" in the narrow sense of the word as nonequilibrium processes that can be observed in spatially uniform states). Such a theory should describe kinetic modes of the system. In the recent statistical mechanics, the problem of investigating the effect of kinetic modes on the behavior of nonequilibrium systems is considered to be very important (see, for instance, review [14]). The generalization is based on the idea of the Bogoliubov functional hypothesis (see, for example, [15,16]), which describes the structure of the nonequilibrium distribution function at the times under consideration (1. 2) The function f ap (T e (t , f 0 ), υ e (t , f 0 )) contains asymptotic values of the parameters So, at times t ≫ τ 0 , the distribution function depends on the time and the initial state of the system f ap0 only through parameters that describe the state of the system (the reduced description parameters). Here, τ 0 is some characteristic time which is chosen to precede the end of relaxation processes. In fact, only the set of reduced description parameters depends on τ 0 , but the value of τ 0 depends on the initial state of the system f ap0 . Arrows in the functional hypothesis (1.2) and definitions (1.3) show that their right-hand sides are a result of the natural evolution of the system. The function f ap (T e (t , f 0 ), υ e (t , f 0 )) is the asymptotic limit of the distribution function f ap (t ), and it exactly satisfies the kinetic equation. The asymptotic distribution f ap (T e , υ e ) does not depend on the initial state of the system f ap0 . These statements are the basic ideas concerning the functional hypothesis applied by us for a generalization of the Chapman-Enskog method. The reduced description parameters in (1.2) do not include the ion temperature and velocity due to the energy and momentum conservation laws in the spatially uniform states under consideration. They are functions of the electron temperature and velocity (in what follows we do not show the dependence of the reduced description parameters on the initial distribution function f ap0 ). The Landau approximation [1] (and the corresponding approximation of [13]) can be written in the form The use of the Maxwell distribution for a system consisting of interacting equilibrium subsystems (quasi-equilibrium state) is quite attractive from the physical point of view. The corresponding local distribution gives the local equilibrium approximation for the description of spatially nonuniform states. This simple idea by Landau is a basis of many investigations. For example, plasma hydrodynamics was investigated in [8,9] on the basis of the Landau approximation. The local equilibrium approximation is a basis of investigations in the book [17] devoted to transport processes in a multicomponent plasma.
In the present paper, the problem of correction of the assumption (1.4) is considered as a very important one, and the distribution function f ap (T e , υ e ) is calculated in a perturbation theory in a small difference of the component temperatures and velocities (let the corresponding small parameter be λ). In terms of the theory of hydrodynamic states, this means that in the plasma in the presence of relaxation, the local equilibrium is violated. Note, that in spatially inhomogeneous states, relaxation in a two-component system was studied in [18]. However, the authors of [18] did not obtain these results because they described the system by the energy densities of the components. The idea of considering the relaxation processes in the system at their end was proposed in our paper [19] and presented at the conferences QEDSP 2011 and MECO 38. It is worth noting that this idea can be also used in the Grad method. As is known [20], the disadvantage of this method is the absence of a small parameter. The above-mentioned parameter λ can be selected for a small parameter in the Grad method. Further development of this approach was presented by us in [21].
The final results of the present paper are given in an additional perturbation theory in the small mass ratio σ ≡ m e /m i , and the integral equations of the theory in each order in σ are solved using the method of truncated Sonine polynomial expansion. The paper is organized as follows. In section 2, the basic definitions and equations of the theory are presented. In section 3, a generalized Chapman-Enskog method is developed and integral equations for the distribution functions are obtained. In section 4, these equations are solved in a σ perturbation theory with the help of the truncated Sonine polynomial expansion method.

Basic equations of the theory
A two-component fully ionized electron-ion plasma can be described by the Landau kinetic equation [1]. In the considered case of spatially uniform states, the component distribution function f ap satisfies the equation with the standard expression for the collision integral I ap ( f ) where L is the Coulomb logarithm (the subscripts a, b, c, . . . = e, i denote the electron and ion components). The quantities m a , e a are the particle masses and charges e e = −e, e i = ze, where e is the elementary electric charge and z is the ion charge number. The particle number density n a , temperature T a and velocity υ an of the components are defined using the standard formulas [3,15] n a = d 3 p f ap , π an = m a n a υ an = d 3 p f ap p n , ε a = 3 2 n a T a + 1 2 m a n a υ a 2 = d 3 p f ap ε ap , (2.3) where π an and ε a are the momentum and energy densities of the components. In this paper, the temperature is measured in energy units.
Let us introduce the quantities υ n and T as: π n = a π an = υ n a m a n a , ε = a ε a = 3 2 T a n a + 1 2 υ 2 a m a n a , (2.4) where π n , ε are the total momentum and the total energy densities, respectively. These quantities do not depend on time because of the relations , (2.5) where the functions

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The particle densities n a do not depend on time because d 3 p I ap ( f ) = 0. which should be considered as parameters describing the relaxation in the system because the deviations of the ion temperature T i and velocity υ in are also expressed in terms of τ, u n Here, it was taken into account that in spatially uniform states, the charge neutrality condition n i ≡ n, n e = zn is satisfied. Expressions (2.10) justify the functional hypothesis in the form (1.2) that contains only electron variables.

Generalization of the Chapman-Enskog method
The generalization of the Chapman-Enskog method presented here is based on the functional hypothesis (1.2), which can be written in the form suitable for our consideration of the relaxation processes at their end. Then, substitution of (2.3) into (2.5) with account for (2.9) leads at t ≫ τ 0 to the closed-form time equations for the parameters τ, u n Q e ( f (τ(t ), u(t ))).

(3.2)
According to the basic idea of the reduced description method, the distribution function f ap (τ(t ), u(t )) exactly satisfies the kinetic equation (2.1) for times t ≫ τ 0 . This leads to the following integro-differential equation for the function f ap (τ, u) One should add to this equation the definition of the parameters u n and τ given in terms of the distribution function f ap (τ, u) where the component temperatures and velocities T a (τ, u), υ an (τ, u) as functions of τ, u are defined by the formulas (3.5)

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In the present paper, relaxation processes in the system are investigated at their end. The corresponding small parameter λ can be introduced by estimates The solution of equation (3.3) with additional conditions (3.4) is found in the form of a series in λ The further calculation needs only the assumption that u n , τ ∼ λ (λ ≪ 1). In fact, estimates (3.6) follow from the requirement | f (1) ap | ≪ f (0) ap and the expression for f (1) ap obtained below. In the main approximation, equations (3.3), (3.4) give the Maxwell distribution with the equilibrium temperature This is true because the distribution w ap meets additional conditions (3.4) it does not depend on τ, u n and is an equilibrium distribution I ap (w) = 0 (hereafter for an arbitrary function g p , the notation is used).
In view of rotational invariance, the solution of equation (3.3) in the first order in λ has the structure where A a (x), B a (x) are some scalar functions. Note that in the Landau approximation (1.4), these functions are given by the relations Substitution of (3.11) into (2.6) gives the right-hand sides R en ( f (τ, u)), Q e ( f (τ, u)) of equations (3.2) in the first order in λ where the notations are introduced (these formulas are written in terms of the integral brackets {g p , h p } ab defined by (A.1) in the appendix). Substitution of expressions (3.13) into (3.2) gives the evolution equations for the parame- which describe their relaxation in the main approximation. Thus, the quantities λ T , λ u are the relaxation rates, and τ T ≡ λ −1 T , τ u ≡ λ −1 u are the corresponding relaxation times for the temperature and velocity.

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Both sides of equation (3.16) contain the unknown function f (1) ap . So, our choice of the small parameter λ of the theory leads to a generalization of the Chapman-Enskog method too. Note that the left-hand sides of similar equations for the standard hydrodynamic state which is investigated on the basis of the Chapman-Enskog method contain only the known functions.

Substitution of expression (3.11) into this equation leads to the integral equations for the functions
A a (βε ap ) and B a (βε ap ): Here,K ab is the linearized collision operator defined in the appendix by formulas (A.2). Additional conditions to equations (3.18) follow from (3.4) and (3.11), and they can be written in the form Equations (3.18), (3.19) are the main equations of the developed theory that will be analyzed in the next part of this paper.
Integral equations (3.18) show that A a (βε ap ), B a (βε ap )p n are eigenfunctions and λ T , λ u are the corresponding eigenvalues for the linearized collision operator of the kinetic equation under consideration. They describe the temperature and velocity kinetic modes of the system. It is important to emphasize that formulas (3.14) are consequences of integral equations (3.18) and the additional conditions (3.19). Therefore, equations (3.18) with additional conditions (3.19) can be solved without taking into account the expressions (3.14). However, it may be useful to simplify the calculation.
The quantities λ T , λ u are positive due to the identities following from integral equations (3.18) and the definition of the total integral brackets {g p , h p } (A.6) in the appendix. These brackets have the important property, {g p , g p } 0 which completes the proof. It is well known from the kinetic theory that the brackets {g p , h p } reflect the general properties of kinetic equations, which lead to entropy growth and the principle of kinetic coefficients symmetry (see, for example, [2,22]). Thus, the developed theory shows the presence of temperature and velocity relaxation in an arbitrary two-component system for the case of small deviations of the component temperatures and velocities from their equilibrium values.

Approximate solutions of the main equations of the theory
In this section, equations (3.18), (3.19) are investigated in a σ perturbation theory and the obtained equations in each order in σ are solved with a truncated Sonine polynomial expansion method. The investigation is based on the following estimates of the momentum p n in w ap (4.1) We seek the relaxation rates and the distribution functions in a σ perturbation theory: In the zeroth order in σ, equations (3.18) and additional conditions (3.19) give the equations for the quantities The chosen polynomials S 1/2 m (x) are convenient due to their orthogonality condition which, in particular, gives where the matrix G am,bm ′ is defined by integral bracket em,e1 = 0 for ∀m). Equation (4.9) shows that g (0) em = 0 for m 2 and, therefore, In the first order in σ, equations (3.18) and additional conditions (3.19) give the equations for the quantities A (1) e (βε ep ), Equations (4.12) give zero for λ (1) where the identity (4.5) was taken into account. This expression leads to the equationK  Note that the operatorK (1) ii describes the collisions in a closed ion system [see expression (B.4) in the appendix]. The second formula in (4.12) with (4.14) lead to the equationK (1) ii A (0) i βε ip = 0 and, therefore, i βε ip is a hydrodynamic scalar eigenfunction of the closed ion subsystem and has the structure c 1 + c 2 ε ip . Using additional conditions from (4.12) gives the constants c 1 , c 2 and the result

Calculation of A (1) i (x), A (2) e (x), λ (2) T from equations of the order σ 2
In the second order in σ, equations (3.18) and additional conditions (3.19) give the equations for the quantities A (2) e (βε ep ), The contributions to operatorsK ab entering these equations are given in the appendix.

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On relaxation phenomena in a two-component plasma

The solution of equations (4.16) is found in the form of Sonine polynomial series
In a matrix form, the equations (4.16) with account for (4.11), (4.15) are given by the relations (4.20) where the matrix G am,bm ′ is defined in (4.10). The first formula from (4.20) gives the relations because substitution of (4.8) into (4.10) with account for (4.5) gives The first relation coincides with the expression for λ (2) T from (4.17). The second one is a set of equations for the coefficients g (2) em (m 2). In the one-polynomial approximation, it gives the following expression for the function A (2)   The fourth formula in (4.20) gives the relations where the identities are used. They come from the explicit expressions for the components ofK ab , orthogonality condition (4.7) of the polynomials S 1/2 m (x) and formulas (4.8). The first relation in (4.24) is an expression for λ (2) T which is equivalent to the one from (4.21). The second relation shows that g (1) im ′ = 0 (m 2) and, therefore, (4.25)

Calculation of A (3) e (x), A (2) i (x), λ (3) T from equations of the order σ 3
Let us start with the formula (3.14) for λ T using the expressions forK ab in the appendix, which in the third order in σ gives [see (4.5), (4.14), (4.25)].
In the third order in σ, equations (3.18) and additional conditions (3.19) lead to equations

Calculation of A (3) i (x), λ (4) T from equations of the order σ 4
Let us start with the formula (3.14) for λ T using expressions forK ab in the appendix, which in the fourth order in σ gives [see (4.5), (4.14), (4.28)]. The first and third terms here in λ (4) T correspond to the Landau approximation [1] because they are based on the functions A (0) a (βε ap ) of the Landau approximation. The calculation gives the following expression for λ (4) T λ (4) T = −3z 2 (z + 1)σ 4 Λ − 9 2z 3 (z + 1)σ 4 Λ. (4.30) The first summand coincides with the Spitzer result based on the Landau approximation (see, for example, [23]). The second summand takes place due to our correction A (2) e (βε ep ) of the order σ 2 to the Landau contribution A (0) e (βε ep ) (4.23). In the fourth order in σ, equations (3.18) and additional conditions (3.19) give the equations for A (3) i (βε ip ) The solution of this equation in the one-polynomial approximation gives the following expression (4.33)

The temperature relaxation: results
Finally, the above-described procedure of solution of equations in (3.18), (3.19) which are devoted to the temperature relaxation leads to the formulas A e (βε ep ) = β(βε ep − 3/2) + 3 2z(z + 1)βσ 2 S 1/2 2 (βε ep ) + O(σ 3 ), Integral equations (3.18) for B i (βε ip ) contain on the left-hand side a momentum of the order σ −1 . In the order σ −1 , these equations and additional conditions (3.19) give  In the zero order in σ, equations (3.18) and additional conditions (3.19) with (4.36) give (explicit expressions for the operatorsK ab p s in different orders in σ are given in the appendix).
The second equation in (4.37) shows that  The chosen polynomials S 3/2 m (x) are convenient due to their orthogonality condition which, in particular, gives  (4.42) where the notation where the quantity Λ is defined in (4.17). These expressions coincide with the result obtained in the Landau approximation (3.12) (see, for example, [13]). The theory developed in the present paper allows one to find corrections of higher orders in σ to this result.

Calculation of B (1) e (x), B (2)
i (x), λ (1) u from equations of the order σ 1 In the first order in σ, equations (3.18) and additional conditions (3.19) with account for the explicit expressions forK ab p s in the appendix and (4.36) give the equations for B (1) e (βε ep ),

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The equations for B (1) e (βε ep ) in (4.46) in a matrix notation have the form The second equation here has only a trivial solution, therefore, formulas (4.46), (4.47) lead to the expressions  According to (3.12), this expression corresponds to the Landau approximation.

Calculation of B (2) e (x), B (3)
i (x), λ (2) u from equations of the order σ 2 In the second order in σ, equations (3.18) and additional conditions (3.19) with account for the expressions forK ab p s in the appendix and (4.36) give equations for the quantities B (2) e (βε ep ),   The first and the second equations in (4.52) in a matrix form with the notation (4.43) are given by which leads to the relations (2) em,e0 (m 1). (4.56) The second relation here is a set of equations for the coefficients h (2) em , the first equation allows one to calculate λ (2) u . In the one-polynomial approximation, they lead to the expressions for B (2) e (βε ep )  In the third order in σ, equations (3.18) and additional conditions (3.19) with account for the explicit expressions forK ab p s in the appendix and (4.36) give the equations for the quantities B (3) e (βε ep ), λ (3) u ,

The velocity relaxation: results
Finally, the above-described procedure of the solution of the equations in (3.18), (3.19) which are devoted to the velocity relaxation leads to the formulas

Conclusion
The relaxation of the temperatures and velocities of the components of a quasi-equilibrium twocomponent homogeneous fully ionized plasma described by the Landau kinetic equation is investigated.
The Chapman-Enskog method is generalized to take into account the kinetic modes of the system. The generalization was made on the basis of the idea of Bogoliubov functional hypothesis.
In the approximation of a small difference of the component temperatures and velocities it is shown that relaxation really exists (the relaxation rates are positive). This proof is based on the arguments that are valid for an arbitrary two-component system because they rely on the general properties of kinetic equations.

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The integral equations for the functions A a (βε ap ) describing the temperature kinetic mode of the system are solved approximately in a σ = (m e /m i ) 1/2 perturbation theory (i.e., in the small electron-toion mass ratio) up to the fourth order in σ. The equations of each order are solved with the help of a truncated expansion in the Sonine polynomials.
It is shown that the equations for the zero order in σ contributions A (0) a (βε ap ) are exactly solvable and A (0) a (βε ap ) coincide with the distribution functions of the Landau theory [1]. The developed theory gives the main corrections A (2) e (βε ep ), A (3) i (βε ip ) of the orders σ 2 and σ 3 to the functions A (0) a (βε ap ).
The corrections are calculated in the one-polynomial approximation. The temperature relaxation rate λ T is found. The main contribution λ (2) T to λ T coincides with the Landau result. The developed theory corrects the σ 4 term λ (4) T of the Spitzer result for λ T due to the account for our correction A (2) e (βε ep ) to the function A (0) e (βε ep ). The Spitzer contribution to λ (4) T is related to the Landau functions A (0) a (βε ap ) with an additional expansion in σ-powers of the Landau collision integral.
The integral equations for the functions B a (βε ap ) describing the velocity kinetic mode of the system are solved approximately in a σ perturbation theory. The main in σ contributions B (0) e (βε ep ), B (2) i (βε ip ) to B a (βε ap ) are calculated in the one-polynomial approximation. They coincide with the results of the theory [13] based on the Landau approximation. The developed theory gives the main corrections B (2) e (βε ep ), B (4) i (βε ip ) to B (0) e (βε ep ), B (2) i (βε ip ), respectively. The corrections are calculated in the one-polynomial approximation. The velocity relaxation rate λ u is obtained. The principal order in σ contribution λ (0) u calculated in the one-polynomial approximation coincides with the result of the theory [13] based on the Landau approximation. The developed theory gives the correction λ (2) u of the order σ 2 to this result calculated in the one-polynomial approximation too.
The obtained results show that in the hydrodynamic theory of the system, the violation of local equilibrium in the presence of relaxation processes should be taken into account.

A. The integral brackets and the linearized collision operator
The integral bracket {g p , h p } ab for arbitrary functions g p , h p is defined by the formula {g p , h p } ab = − d 3 pd 3 p ′ M ab (p, p ′ )w bp ′ g p h p ′ ,