ON RELAXATION PROCESSES IN A COMPLETELY IONIZED PLASMA

Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. The investigation is based on linear kinetic equation obtained by us early from the Landau kinetic equation. Therefore long-range electron-ion Coulomb interaction is consequentially described. The research of the model is based on spectral theory of the collision integral operator. This operator is symmetric and positively defined one. Its eigenvectors are chosen in the form of symmetric irreducible tensors which describe kinetic modes of the system. The corresponding eigenvalues are relaxation coefficients and define the relaxation times of the system. It is established that scalar and vector eigenfunctions describe evolution of electron energy and momentum densities (vector and scalar system modes). By this way in the present paper exact close set of equations for the densities valid for all times is obtained. Further, it is assumed that their relaxation times are much more than relaxation times of all other modes. In this case there exists a characteristic time such, that at corresponding larger times the evolution of the system is reduced described by asymptotic values of the densities. At the reduced description electron distribution function depends on time only through asymptotic densities and they satisfy a closed set of equations. In our previous paper this result was proved in the absence of an external electric field and exact nonequilibrium distribution function was found. Here it is proved that this reduced description takes also place for small homogeneous external electric field. This can be considered as a justification of the Bogolyubov idea of the functional hypothesis for the relaxation processes in the plasma. The proof is done in the first approximation of the perturbation theory in the field. However, its idea is true in all orders in the field. Electron mobility in the plasma, its conductivity and phenomenon of equilibrium temperature difference of electrons and ions are discussed in exact theory and approximately analyzed. With this end in view, following our previous paper, approximate solution of the spectral problem is discussed by the method of truncated expansion of the eigenfunctions in series of the Sonine polynomials. In one-polynomial approximation it is shown that nonequilibrium electron distribution function at the end of relaxation processes can be approximated by the Maxwell distribution function. This result is a justification of the Lorentz–Landau assumption in their theory of nonequilibrium processes in plasma. The temperature and velocity relaxation coefficients were calculated by us early in oneand two-polynomial approximation.


INTRODUCTION
This paper is devoted to the investigation of relaxation processes in completely ionized plasma. It is meant nonequilibrium processes that can be observed in spatially homogenous states of a system. Near the equilibrium they describe the so-called kinetic modes of the considered system. Taking into account relaxation processes in theory of spatially non-uniform states is the next step after their investigation for spatially uniform nonequilibrium states. From a different point of view taking into account relaxation processes is extension of set of parameters that describe nonequilibrium state (reduced description parameters). This is the main trend in theory of nonequilibrium processes. Some important examples are given by the extended irreversible thermodynamics [1], a theory with nonequilibrium correlations of the standard reduced description parameters as additional independent ones (see, for example, [2]), a theory of nonequilibrium states in the vicinity of the standard ones (see, for example, [3]).
In the present paper plasma is considered in the generalized Lorentz model, in which electron-electron interaction is neglected and the ion subsystem is assumed to be an equilibrium ideal gas. In the standard Lorentz model [4] the ion subsystem is a system of hard spheres in the rest. The generalized Lorentz model is based on the Landau kinetic equation [5] (see also in [6]) and, therefore, takes into account peculiarities of the Coulomb interaction. It was introduced in our paper [7]. The same model is discussed in [8] on the base of the Boltzmann kinetic equation.
In plasma states with different component temperatures their relaxation is observed. For the first time the problem of equalizing the electron and ion component temperatures in plasma was investigated by Landau [5] (the component velocity relaxation is considered analogously in [9]). His research was based on the mentioned kinetic equation [5] and shows that temperature relaxation process is slow one because big difference electron and ion masses. In his investigations (see additional examples in [10]) he assumed without proof that the plasma components quickly become equilibrium and are described by the Maxwell distribution functions. This assumption belongs to Lorentz and was introduced by him in his theory of transport phenomena in metals [4]. Fundamental investigation of the plasma hydrodynamics on the same basis belongs to Braginsky [11]. Similar problem for electron-phonon two-component system was discussed by Bogolyubov and Bogolyubov (Jr.) in their research [12] on the polaron theory. They

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Alexander I. Sokolovsky, Sergey A. Sokolovsky, et al considered solution of the kinetic equation for polarons interacting with equilibrium phonon system using the Maxwell distribution with macroscopic velocity as a good approximation for the polaron distribution function. In fact the mentioned assumption is unfair because the Maxwell distributions for electrons and ions with different temperatures and macroscopic velocities are not solution of kinetic equations for all models of the plasma dynamics. Therefore, the main problem of the theory is to find the main approximation for electron and ion distribution functions of plasma with two component temperatures and velocities. This problem is related to the absence of a small parameter in the theory of relaxation. An approach to solution of this problem was proposed by us with the idea to investigate relaxation processes in the vicinity of standard described nonequilibrium processes (equilibrium states included) [3]. An important example is our investigation of the two-fluid plasma hydrodynamics in the vicinity to one-fluid one [13] (see also a review [14]).
Our previous investigations of nonequilibrium processes in plasma (for example, in [13][14][15]16]) are primarily based on the Bogolyubov idea of the functional hypothesis (its consistent and complete discussion see in [6] ap ap x t x t . The Bogolyubov idea of the functional hypothesis is basis of his method of the reduced description of nonequilibrium systems. In these terms according above discussion the main problem of the relaxation phenomena in plasma investigation is to find the distribution function f ( , ) ap x  in spatially uniform states.
Contrary to our mentioned papers [13][14][15][16], which are based on the Bogolyubov method of the reduced description on nonequilibrium states, the present paper develops kinetics of the system through elaborating the spectral theory of the collision integral operator without assumption that relaxation processes are considered at its completion. This is possible because the plasma is considered in the generalized Lorentz model [7] in which kinetic equation for electrons is a linear one and ions form an equilibrium system. In this approach the relaxation phenomena in plasma are discussed for spatially uniform states in our papers [17,18] and exact distribution function is found in the terms of scalar and vector eigenfunction p A , p n B p of the collision integral operator. These eigenfunction are calculated by the method of truncated expansion in the Sonine polynomial series. The paper [17] discusses this problem for the case of the presence constant small spatially homogeneous external electric field with some simplifying assumptions.
The presented paper provides a consequence investigation of the relaxation processes in plasma at small electric field. Spectral theory of the collision integral operator is discussed in the terms of eigenfunctions that are irreducible tensors.
The paper is constructed as it follows. In the section "Basic equations of the theory" the generalized Lorentz model is formulated following to [7] and basics of spectral theory of the collision integral operator are presented. The section "Evolution of energy and momentum densities of the electron system" discusses dynamics of the densities. The next section "Reduced description of the system by energy and momentum densities" investigates long time evolution of the system and predicts equilibrium state of the system. The section "Approximate calculation of the main quantities of the theory" discusses approximate solution of the spectral problem for the collision integral operator.

BASIC EQUATIONS OF THE THEORY
This paper is devoted to the study of relaxation processes in spatially homogeneous completely ionized plasma in the presence of small external electric field. The plasma is considered in the generalized Lorentz model in which the ion subsystem is assumed to be in equilibrium and in the state of the macroscopic rest with the temperature 0 T . The model is based on the Landau kinetic equation [5] and was introduced in our paper [7]. The electron kinetic equation of the model has the form

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with collision integral given by the formula Here L is the Coulomb logarithm, e  is charge of an electron, ze is charge of an ion, n E is homogeneous constant electric field. Hereafter electron and ion equilibrium distribution functions are written as It is convenient to conduct the research of the considered system using the collision integral operatorK given by the formulas [ (hereafter arbitrary real functions are defined by p a , p b , …). In the term of scalar product 3 ( , ) linear operator K is a symmetric and positively defined one. Therefore, its eigenfunctions ip g and eigenvalues i have the properties ( i b are normalization constants). In this paper eigenfunctions ip g are chosen in the form of symmetric irreducible tensors multiplied by a function of the momentum modulus. The simplest of them are given by expressions Convolution of arbitrary two indices of each tensor Instead of distribution function f p it is convenient to introduce new one p g and rewrite the kinetic equation (3) in (12) to simplify investigation of the small electric field case. Solution of this equation can be sought, following to our paper [17], in the form of a series in eigenfunctions ip g of the operator K ( ) ( )

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Alexander I. Sokolovsky, Sergey A. Sokolovsky, et al In the absence of the electric field this solution is given by relation where the coefficients 0 i c are defined by initial value of the distribution function f ( 0) p t  . Each term in this formula describes a relaxation (kinetic) mode of the system (hereafter (0) a denotes contribution to a quantity a in the absence of the electric field). Eigenvalues i  are called the relaxation coefficients and define the relaxation times i  of the system.

EVOLUTION OF ENERGY AND MOMENTUM DENSITIES OF THE ELECTRON SYSTEM
In this paper relaxation processes are investigated which related to electron subsystem energy  and momentum n  densities that are defined by the formulas ). (15) In this connection scalar p A and vector p n B p eigenvalues and the corresponding eigenvalues T To these relations the normalization conditions should be added which for the further convenience are chosen in the form According to (13), (16) Taking into account the rotational invariance considerations and conditions (17) gives Note, to explain these identities that tensors are obtained. The rotational invariance considerations, formulas (16), (17), expressions (20) and identities (21) give final form of these equations The obtained equations are valid for all times and without assumption that electric field is small.

REDUCED DESCRIPTION STATE OF THE SYSTEM BY ENERGY AND MOMENTUM DENSITIES
Let us discuss state of the system at long times. Here and in our paper [17] it is assumed that characteristic time 0 exists. Let us prove that at long times the following relation According (1) In our paper [17] under assumption (25) it was proved that in the absence of external electric field function ( , ) p g   is given by exact expression In order to investigate possibility of the reduced description in the presence of electric field let us restrict ourselves by small field. Exact electron distribution function is sought in a series in powers of the field l E f (1 ) First of these equations is due to (16), (31) true. The main contribution (0) p g to function p g with arbitrary coefficients c , l c is chosen in the form (31) that is enough for investigation of the system at (see (14), (25)). The second equation (32) with account for normalization condition (10) give set of equations for functions (1) Here according to (31) after integration by parts the time independent coefficients il are introduced (in [17] this expression wrongly assumed to be constant). Solution of equations (34) has the form (1) (1) 0 Initial conditions (1) (1) 0 for (1) i c are not related to the external field and further will be replaced by zero.
According to (25), (33) expression (36) gives 0 (1) Entering here coefficients are defined by (35). The rotational invariance considerations give Also due to (20), (40) asymptotic values of energy and momentum densities are written as In . Therefore, in this paper the functional hypothesis is proved in the presence of small electric field. For the case of the absence of the field it was proved in our paper [17].
the second of which does not contain the field. The Cauchy problem for these equations can be easily solved. Equilibrium state of the system is described by relations , ( ) An example of this approach is given in our paper [19], where relaxation processes in polaron subsystem of semiconductors are investigated on the basis of equations (16), (17) but with different operatorK and with functions p A , p n B p , which describe relaxation processes close to the equilibrium. The choice of polynomials is suggested by normalization condition which contains the average with the electron Maxwell distribution p w (4). Formulas (48), (49) and normalization conditions (17) give first coefficients in expansions (48) In the s -polynomial approximation it is assumed that only s coefficients in (48) are not equal to zero. The convergence rate of this procedure cannot be estimated because the considered spectral problem does not related to a small parameter.
The further calculations are similar to ones in quantum-mechanical perturbation theory. Equations (16) In paper [13] these values were calculated in the main approximation in the parameter  which is a small one because (here e m , p m are electron and proton masses). As a result our expression for u  coincides with one from [13] but our expression for T  gives one from [13] after the replacement 0 0 ( 1) n n z   . This result is expected because in [13] the dynamics of ions was more fully taken into account. Starting from Landau investigation on the temperature relaxation in the completely ionized plasma [5] it is assumed that electron distribution function in spatially uniform states coincides in the main approximation with the Maxwell one (analogous assumption is made in the velocity relaxation theory [9]). In the terms of our consideration it is confirmed only in one-polynomial approximation at small temperature difference 0 T T  and small velocity n u . Really, according to (4), (50) the Maxwell distribution for electron system with temperature T and macroscopic velocity n u can be rewritten as 0 [1] [1] 0 0 0 0 0