Extended relativistic kinetic model composed of the scalar and two vector distribution functions: Application to the spin-electron-acoustic waves

Detailed deterministic derivation of the kinetic equations for the relativistic plasmas is given. Focus is made on the dynamic of one-coordinate distribution functions of various tensor dimensions, but the closed set of kinetic equations is constructed of three functions: the scalar distribution function, the vector distribution function of dipole moment, and the vector distribution function of velocity (or the dipole moment in the momentum space). All two-coordinate distributions functions are discussed as well. They are presented together with their limits existing in the self-consistent field approximation. The dynamics of the small amplitude spin-electron-acoustic waves in the dense degenerate plasmas is are studied within the kinetic model. This work presents the deterministic approach to the derivation and interpretation of the kinetic equations. So, no probability is introduced during the transition from the level of individual particles to the collective functions. Problem of thermalization is not considered, but we can see that the structure of kinetic equations is kept for the system before and after thermalization. Hence, the kinetic equations can be used to approach this item.


I. INTRODUCTION
Kinetic theory is one of fundamental theories of the macroscopic phenomena in various physical systems [1], [2], [3], [4], [5], [6], [7], [8].Particularly, is plays essential role in the plasma physics [9].Kinetic theory requires proper microscopic derivation, where the evolution of the distribution function is traced from the evolution of individual particles.The distribution function shows the relative positions of all particles in the six dimensional phase space and it can be consistently defined in the deterministic way.So, no probability theory is applied for the derivation or interpretation of the kinetic equations or the distribution function.
Such view on the kinetic theory is the extension of the classical hydrodynamics based on the tracing of the microscopic dynamics of individual particles [10], [11].
Classical mechanics gives us dynamics of particles in the 3N-dimensional configurational space.The model of motion of particles is composed of the individual trajectories.Each of them, formally, takes place in its own three dimensional space.this picture of motion is not obvious if we consider the Newtonian form of mechanics, where we can use our own imagination to place all particles in the single three-dimensional space.However, the Lagrangian and Hamiltonian forms of mechanics show the multidimensional structure of the theory more clearly.
Moreover, the construction of the mathematical model out of trajectories (which are one dimensional mathematical objects) distinguish the classical mechanics from the hydrodynamics or the electrodynamics, which are the field theories.
The derivation of the hydrodynamics from the mechanics requires to represent the mechanics in the form of the field theory.So, the mechanics would have the same mathematical structure as the electrodynamics, which is essentially important for the dynamics of charged particles.
Refs. [10] and [11] basically present the field form of the classical mechanics, where the dynamics of particles takes form of the dynamics of material fields.Inevitably, the field form of classical mechanics appears in the form of hydrodynamic equations.These are equations of nonequilibrium hydrodynamics which requires further reduction for the particular forms of the collective motion.
Particularly, it leads to the some generalized form of nonequilibrium thermodynamics.And choosing regime of thermal equilibrium allows to get the first low of thermodynamics from the energy evolution equation.The possibility to consider different regimes, like regimes close to the thermal equilibrium, and regimes out of equilibrium, shows that this approach can be used for the study of the thermalization process in the physical systems considering the dynamics of the systems in terms of the collective variables (material fields).It would require proper truncation of the hydrodynamics out equilibrium considering higher rank material fields to include the mechanism of relaxation in the model.However, it is an open problem which can be addressed within this formulation.
Formulation of the mechanics in terms of the threedimensional material fields leads to the hydrodynamics form of equations.While we consider the relative positions and relative momentums of all particles we go to the distribution of particles in the six dimensional space (the phase space of coordinate and momentum).This form of the classical mechanics is considered in this paper.
If we consider the microscopic evolution of point-like particles we get the following distribution of the particle number in the coordinate space [12] n m (r, t) = N i=1 δ(r − r i (t)), (1) where r i (t) is the coordinate of i-th particle.This approach is suggested by Yu.L. Klimontovich [1], [12].While we use notation n m (r, t) on the left-hand side of equation (1) we should write n m (r, r 1 (t), ..., r N (t)).So, the time dependence of function n m is directly constructed out of the time evolution of coordinates of particles.The parametrization of the physical space is introduced here r.So this form gives us an intermediate step from the mechanics of trajectories to the field form of classical mechanics.
Considering the microscopic distribution (1) different authors applied some form of averaging (see for instance [1], [13]).Sometimes, authors use some unspecified distributions n m (see for instance [1]).Overwise authors use specific form of averaging via some distribution function (see for instance [13], [14]).Anyway, authors try to impose probabilistic interpretation of the procedure of averaging.Let us mention that the aim of averaging is to make the transition to the macroscopic scale by averaging on the physically infinitesimal volume, but usually it is being replaced by the averaging on some distribution function.
Systematic, nonstatistical, definition of the averaging on the physically infinitesimal volume is suggested by L.S. Kuz'menkov [10] (see also [15], [16]) Equation ( 2) contains vector ξ which scanning the physically infinitesimal volume (it is illustrated in equation 1).While the physically infinitesimal volume appears as the ∆-vicinity of each point of space.It is assumed that N is the full number of particles in the system.It is composed of two species: electrons with numbers i ∈ [1, N/2] and protons i ∈ [N/2 + 1, N ].We need to trace the evolution of each species in the system.With no restrictions, we illustrate our derivation on the subsystem of electrons, so subindex e is dropped for all functions describing electrons in equations below n e ≡ n.Similar background is suggested for the kinetic theory.So, the microscopic distribution function is composed of delta functions in the physical coordinate space and the momentum space Transition to the macroscopic level is made similarly to equation (2).We introduce the physically infinitesimal volume in six-dimensional phase space where dη is the element of volume in the momentum space, with ∆p dη integral over ∆ p -vicinity in the momentum space, ∆ ≡ ∆ r is the delta vicinity in the coordinate space.Recent development of this approach can be found in Refs.[17], [18], [19], [20].This discussion is about the field form of the classic mechanics.However, similar insight is imposed on the quantum mechanics.So, the formulation of the manyparticle quantum mechanics in the form of evolution of the material fields is developed [21], [22], [23], [24], [25], [26], [27], [28].
The discussion presented above is focused on the fundamental background of the hydrodynamics and kinetics, since the theoretical part of this paper is based on the development of the kinetic theory.However, the application of the kinetic theory to the high-density magnetized spin polarized degenerate electron gas is also considered.
Particularly, we focus on the spin-electron-acoustic waves (SEAWs).Existence of the SEAWs in the partially spin-polarized degenerate plasmas was theoretically suggested in 2015 [29].They appear as the longitudinal waves of density and electric field, which are propagate parallel to the anisotropy direction in the magnetized electron gas [29], [30].
If we have no spin polarization we find single longitudinal wave in the electron gas under assumption of the motionless ions.It is the Langmuir wave, which corresponds to the collective oscillations of electrons relatively motionless ions.If we include the spin-polarization we also find the SEAWs.The physical picture of dynamics is following.It corresponds to relative motion of elec-trons with different spin projections.The spin polarization manifests itself in different partial concentrations of electrons with the particular spin polarization.Hence, the relative oscillations of electrons with different amplitudes of concentration give the oscillation of small proportion of electrons relatively motionless ions as well [29], [30].The SEAWs show similarity to the spin-plasmons considered in two-dimensional structures [31], [32], [33], [34], [35], [36], [37], [38], [39], [40].Recently, the SEAWs have been considered in the degenerate electron gas [41], while the background of applied model is given in Refs.[16], [42], [43].
Increase of the concentration of electrons n 0e up to values, where the Fermi energy is proportional to the rest energy of electron ε F e = (3π 2 n 0e ) 2/3 h2 /2m e ∼ m e c 2 , gives noticeable change in the dispersion dependence of the SEAWs.It also corresponds to the change of the dispersion dependence of the Langmuir waves.It shows the decrease of the relative frequency of the Langmuir wave and SEAW.Moreover, the relativistic regime shows the growth of the amplitude of the concentration of the SEAWs relatively the amplitude of the Langmuir waves for the chosen value of the electric field in these waves.
This paper is organized as follows.In Sec.II the method of derivation of the kinetic equation from the microscopic motion is demonstrated and the general structure of the kinetic equation is derived.In Sec.III we present an approximate kinetic model, where the contribution of physically infinitesimal volume is considered in minimal regime called the monopole regime.In Sec.IV the selfconsistent field approximation of monopole regime is considered.In Sec.V the multipole approximation in the relativistic kinetic equation is presented as the generalization of the monopole regime considered in Sec.III.In Sec.VI the selfconsistent field approximation of the multipole approximation is presented.In Sec.VII the kinetic equation for the evolution of the dipole moment vector distribution function is obtained.In Sec.VIII the kinetic equation for the distribution function of velocity is derived.In Sec.IX closed set of three kinetic equations is discussed.In Sec.X the spin-electron-acoustic waves in the spin polarized electron gas of high density are considered.In Sec.XI a brief summary of obtained results is presented.

II. DERIVATION OF THE VLASOV EQUATION TRACING THE MICROSCOPIC MOTION OF PARTICLES
We need to consider the evolution of the distribution function.It requires to present equation of motion of each particle in full details ṗi (t) = F(r i (t), t), where 2 is the relativistic momentum of i-th particle, and F(r i (t), t) is the force acting on i-th particle from the electromagnetic field created by other particles in the system.The equation of motion for each particle appears as the evolution of the momentum under action of the Lorentz force where , and B i = B(r i (t), t), while the external fields are included along with the field of interaction of particles.The electric E i,int and magnetic B i,int fields caused by particles surrounding the i-th Equations ( 7) and (8) allow to introduce the Green function of the retarding electromagnetic interaction [44] We consider the high-temperature plasmas , where electrons (and may be other species) have relativistic temperatures comparable with the rest energy of the electron m e c 2 (of the particle of corresponding species).Obtaining such huge temperatures of the species would lead to high degree of ionization of the atomic objects.Moreover, the two-particle interactions (collisionlike processes) of the high-energy electrons with the ions would change the electron configurations of ions or provide additional ionization.These dynamical processes should contribute in the model.To avoid these complexifications we consider the hydrogen plasmas, where we have electrons and protons only.
To find the equation for evolution of the distribution function (4) we consider the time derivative of this function to obtain the following equation The second (third) term in this equation appears as the result of action of the time derivative on the deltafunction containing the coordinate (the momentum).
The second term in equation ( 10) contains the following function where v i (t) = dr i (t)/dt, δ ri ≡ δ(r + ξ − r i (t)), and δ pi ≡ δ(p + η − p i (t)).For the further representation of this function we apply relativistic expression for the velocity of particle via its momentum , where additional replacement of the momentum p i (t) on p + η can be applied.It gives the representation of function v i (t) : δ ri δ pi .
(12) Here, our goal is the extraction of traditional for the physical kinetics terms v • f .Hence, expression (12) is represented in required form where with with p = m s v/ 1 − v 2 /c 2 and v = pc/ p 2 + m 2 s c 2 .We include representation (13) in equation (10).We also include equations of motion of the individual particles ( 5)- (8) in the last term in equation (10): where we use symbol ∂t instead of the time derivative due to the change of the structure of argument of the Green function G.We replaced r i (t) by r + ξ in the arguments of the external electric field, the external magnetic field, and the Green function using the delta-function δ ri .Initially the action of the time derivative has the following form where δ ′ is the derivative of the delta function on its argument.It is also can be rewritten in the following form For the introduction of ξ ′ see the equations below.
Next, we use representation (13) in terms describing the interaction: where δ r ′ j ≡ δ(r ′ + ξ ′ − r j (t)), and Presented kinetic equation ( 19) contains the deviation of coordinates ξ from the center of the ∆-vicinity r.It also contain the deviation of the velocity ∆v i from the value corresponding to the center of the ∆ p -vicinity in the momentum space v = pc/ p 2 + m 2 s c 2 .This is the intermediate general representation of the kinetic equation which is considered below under some additional assumptions.

III. MONOPOLE APPROXIMATION OF THE KINETIC EQUATION FOR THE SCALAR DISTRIBUTION FUNCTION
To consider the monopole approximation of the kinetic equation we need to neglect contribution of ξ, ξ ′ , η (including ∆v i ), and η ′ (including ∆v ′ j ) in dynamical functions.However, on this step, we present the kinetic equation, where the monopole approximation is considered for the space variables only.Hence, we neglect contribution of ξ, and ξ ′ , but we keep the contribution of η: Transition from equation (19) to equation ( 20) includes the transformation like E(r + ξ, t) ≈ E(r, t) and where the electric field E(r, t) is placed outside of the integral, while the rest is the derivative of the distribution function on the momentum.
Equation (20) allows to introduce the two-particle (or in other terms two-coordinate) distribution functions.To the best of my knowledge, majority of papers and books, including my own works, use notion "two-particle distribution functions".It can be confusing to some extend since the model describes the many-particle systems.Hence, the distribution function and the twoparticle distribution function actually describe the dynamics of whole system.Sometimes, the two-particle distribution functions are called "two-coordinate distribution functions".It looks more logical since it contains two coordinates r and r ′ (or more exactly two coordinates in the phase space {r, p} and {r ′ , p ′ }).
Let us present the following kinetic equation in the monopole approximation both on coordinate and momentum (we do it due to the technical reasons, since we do not want to introduce several two-coordinate distribution functions, but we show the complete picture below): where q s ′ f 2 = q e f 2,ee + q i f 2,ei and is the two-coordinate distribution function.The compact notation is also introduced for the two-coordinate distribution function constructed of double brackets 1 .We use this notation below for other two-coordinate distribution functions.

IV. THE SELFCONSISTENT FIELD APPROXIMATION IN CLASSIC MONOPOLE KINETICS FOR THE EQUATION OF EVOLUTION OF SCALAR DISTRIBUTION FUNCTION
In order to consider the selfconsistent field (the mean-field) approximation we need to split the twocoordinate distribution function f 2,ee (r, r ′ , p, p ′ , t, t ′ ) into the product of two one-coordinate distribution functions Hence, the term containing interaction in equation ( 22) reappears in the following form ) It allows to introduce the scalar and vector potentials of the self-consistent electromagnetic field and Some discussion on the selfconsistent field approximation for the kinetics based on the deterministic microscopic motion of particles is gives in Ref. [45] for the nonrelativistic kinetics and hydrodynamics.So, we find the well-known form of the Vlasov kinetic equation for the system of relativistic particles [46] where E = E ext +E int , B = B ext +B int , the electric field is caused by the distribution of charges in the coordinate space: 27) appears to be coupled with the electromagnetic field:

V. MULTIPOLE APPROXIMATION IN THE RELATIVISTIC KINETIC EQUATION FOR THE SCALAR DISTRIBUTION FUNCTION (THE VLASOV EQUATION)
A. The interaction of particles with the external electromagnetic field The multipole expansion of the electric field E ext (r + ξ, t), the magnetic field B ext (r + ξ, t), and the Green function G(r−r ′ +ξ−ξ ′ , t−t ′ ) leads to appearance of new distribution functions of different tensor ranks.Let us introduce these functions before we find they appearance from the expansion of equation ( 19) on ξ and ξ ′ up to the second order on these vectors.
Firstly, we introduce the vector distribution function of the electric dipole moment (divided by the charge q s ) or the displacement of particles relatively the center of the ∆-vicinity: We also include in our analysis the distribution function of the electric quadrupole moment (divided by the charge q s ) (31) Presented functions contain vector ξ scanning the ∆vicinity.Functions (30) and (31) appear as two examples of the infinite set of distribution functions containing different degrees of vector ξ (the product of different number of projections to construct an element of tensor of corresponding rank).
As it is demonstrated by equations ( 11)-( 14) we can find the velocity of particle under integral defining the distribution function.We can also find the product of several projections of the velocity (of the same particle, in order to get one-coordinate distribution function) or the product of the projection of the velocity on the projections of vector ξ.
One of such distribution functions appearing in our derivation is the distribution function of flux of the electric dipole moment (divided by the charge q s ) It is more useful to extract the velocity corresponding to the center of ∆ p -vicinity in the momentum space, like we make it for v a i (t) in equations ( 11)-( 14).Therefore, function (32) leads to the following distribution function: (33) Let us introduce one more distribution function, which is the third rank tensor.It is the distribution function of flux of the electric quadrupole moment (divided by the charge q s ) (34) For this distribution function, we can also introduce the reduced flux of the electric quadrupole moment on the velocities in the local frame ∆v a i (t): 30)- (34) appear in the Vlasov-like kinetic equation for the scalar distribution function.Firstly, these functions appear at the analysis of the action of the external electromagnetic field on the system of charged particles.Here, we need to consider the fourth term in equation (19).We apply the expansion of the external electromagnetic field on the deviations of coordinate ξ from the center of the ∆-vicinity of point r.This expansion requires slow change of the fields E and B over the diameter 3  √ ∆ of the ∆-vicinity.Particularly, if we consider the electromagnetic wave we assume that the wavelength λ is larger than the diameter 3  √ ∆: λ ≥ 3 √ ∆.For instance, if we consider diameter 3  √ ∆ ≈ 0.1 µm we get estimation on the minimal value of the frequency of the electromagnetic wave of order of ω max ∼ 10 16 s −1 .So, we can consider electromagnetic waves with frequencies up to the frequencies of the visible light, but the ultraviolet radiation is our of the range of applicability of the model.If we deal with the dense medium we can choose the delta-vicinity of the smaller value.So, the range of the ultraviolet radiation can be included in our analysis.
During derivation of the kinetic equations we consider the expansion up to the second order on ξ a : r B(r, t) + .... Hence, the fourth term in equation ( 19) can be rewritten as before the expansion.After described expansion we obtain

B. The interparticle interaction
Let us introduce function Ω which presents a term in the kinetic equation describing the interparticle interaction (the last term in equation ( 19)).We split this term on three parts , where we use short notation for the average on the physically infinitesimal volume, like (37) Here, functions ϕ i,int and A i,int are given by equations (7) and (8).
Let us consider the multipole expansion for each of function Ω j (j=1,2,3).We start this presentation with Ω 1 which can be represented in the following form We make expansion of the Green functions on ξ and ξ ′ .We also expand the velocity of particle v b i on the deviation from the average velocity η.For the Green function Function Ω 1 can be represented in terms of twocoordinate distribution functions: where we use the following notations for the twocoordinate distribution functions ) Potential part of the electric force q s E of the interparticle interaction is presented in terms of the two-coordinate distribution function.Next, we make same representation for the part of the electric force q s E expressed via the vector potential.Therefore, let us demonstrate the explicit form of Ω 2 : Function Ω 2 can be approximately represented in terms of two-coordinate distribution functions after expansion on ξ and ξ ′ : where we use the following notations for the twocoordinate distribution functions and Finally, we present the explicit form of the magnetic part of the interparticle interaction: Here, we make expansion on ξ and ξ ′ and make interpretation of found terms via the two-coordinate distribution functions: where we use no specific notations for the two-coordinate distribution functions since we have two many twocoordinate distribution functions.We have no reason to use special symbol for each of them.We also have following limits for the self-consistent field approximation: and (60) To conclude this section, we point out that we made the generalization of equation ( 22) up to the account of the qudrupole moment distribution function in the multipole expansion while equation ( 22) includes the distribution of charger (and currents as well) in the monopole approximation.Presented generalization is demonstrated by four terms labeled as Ω 0 , Ω 1 , Ω 2 , and Ω 3 .

VI. SELFCONSISTENT FIELD APPROXIMATION IN THE MULTIPOLE APPROXIMATION FOR THE SCALAR DISTRIBUTION FUNCTION EVOLUTION EQUATION
We consider the dynamics of charged particles.Therefore, the mean-field or the self-consistent field approximation gives the major contribution in the dynamics of system.Hence, we consider all two-coordinate distribution functions as the product of the corresponding onecoordinate distribution functions.Therefore, we combine equations ( 36), ( 39), ( 44), (54) represented in the selfconsistent field approximation and find the following equation for the scalar distribution function: where the velocity is extracted from functions j a s , j ab D,s , and j abc Q,s .We also have E = E ext + E int , and The integral terms presenting the sources of the electromagnetic field are written as E int and B int , while these functions obey the Maxwell equations in accordance with the explicitly found integral expressions for the electromagnetic field {E int , B int } via the onecoordinate distribution functions: The selfconsistent field approximation shows that all introduced distribution functions appear as the sources of the electromagnetic field in the Maxwell equations.

VII. EQUATION FOR THE EVOLUTION OF THE DIPOLE MOMENT DISTRIBUTION FUNCTION
Let us repeat the definition of the distribution function of dipole moment If we want to consider the evolution of the distribution function of dipole moment we can consider the time derivative of function ξ a (65).We can consider alternative function r a (t) = r a f (r, p, t) + ξ a .
The time derivative of function (65) has the following structure The second term in this kinetic equation can be represented via functions (32) or (33).While the last term in this equation requires longer discussion similar to the analysis of interaction in the Vlasov equation presented above.
Substituting the time derivative of the momentum ṗb i into equation (66) (equation of motion of each particle is given by equation ( 5))-( 8) we obtain Next, we make expansion on ξ and ξ ′ in order to get the multipole expansion.We keep terms up to the electric quadrupole moment and find: The two-coordinate distribution functions are presented as the double brackets of the corresponding values with no usage of specific notations since part of them are introduced above, while part of them get no specific notation.Here, we make transition to the self-consistent field approximation.To this end, we split the two-coordinate distribution functions on the product of one-coordinate distribution functions (we do it in the same way as it is done above for the equation of evolution of the scalar distribution function) On this stage we can introduce the averaged scalar and vector potentials of the electromagnetic field and corresponding representation of the integral terms within E int and B int .This electromagnetic field obeys the Maxwell equations ( 62), ( 63), (64).
Finally, equation ( 69) is represented in terms of E int and B int :

VIII. EQUATION FOR THE DISTRIBUTION FUNCTION OF VELOCITY
Let us repeat the definition of the considering vector distribution function (11) ) If we consider the evolution of function (71) we would have four terms in the initial form of the kinetic equation However, we can represent the velocity of each particle v i (t) via parameters p and η which do not depend on time (see equation ( 12)) Equation ( 72) can be simplified to equation (72) using the kinetic equation for the scalar distribution function f (r, p, t).
We consider equation ( 73) in more details Next, we represent equation (74) via the multipole expansion Finally, we present equation (75) in the self-consistent field approximation where and (82) Finally, we introduce electric E int and magnetic B int fields caused by the charges of the system and present the kinetic equation for the velocity distribution function in compact form:

IX. ON THE EXTENDED CLOSED SET OF KINETIC EQUATIONS CONTAINING SCALAR AND VECTOR DISTRIBUTION FUNCTIONS
Equations for evolution of f (r, p, t), d a (r, p, t) and j a (r, p, t) (see equations ( 61), ( 70), (83)) give incomplete set of equations.This set requires additional assumptions to make final truncation.
Let us repeat equations ( 61), ( 70), (83) together to make additional analysis of these equations and and FIG. 2: The illustration of the ∆p-vicinity in the momentum space with few numbered particles being in the vicinity.Here, p is the center of the vicinity, η is the vector scanning the vicinity, p i (t) are the momentums of particles, p is the average value of the momentum of particles being in the vicinity, while vector p differs from the center of the vicinity p.
Here we have equations for three distribution functions f , d a , and j a = v a f + F a , but we have a number of other distribution functions like , and v a i (t)v c i (t)ξ f .We need to obtain the closed set of equations at this step.Before we make this truncation, we need to provide discussion of physical picture hidden in these functions.

A. Kinetic theory of fluctuations
Considering transition from the kinetic theory (the field form of the classical mechanics in the six dimensional space) to the hydrodynamic theory (the field form of the classical mechanics in the three dimensional space) we get the concentration of particles n(r, t) = f (r, p, t)dp, the current of particles j(r, t) = n(r, t)v(r, t) = vf (r, p, t)dp.Difference v − v(r, t) is the measure of chaotic motion, which can be associated with the thermal motion.This interpretation getting closer to the thermodynamic temperature if the system get though the process of thermalization.For the quantum systems we also need to include other than thermal mechanisms for the symmetric distribution of particles over the quantum states in the momentum space.Major contribution is given by the Pauli blocking existing in systems of fermions.
The average velocity vf (r, p, t)dp gives the velocity field.The deviation from the average velocity v − v(r, t) leads to other hydrodynamic functions, like the temperature scalar field, the pressure tensor field, etc.These functions are also smooth functions which are defined on certain scale.It leaves as the question: how fluctuations can appear in our model?Since the fluctuations is the natural phenomenon following from the individual motion of particles governed be the microscopic equations.
Basically, the chaotic motion of particles gives small variation of macroscopic functions due to the continuous unequal exchange of particles between nearest vicinities.Hence, the concentration n(r, t) = f (r, p, t)dp (2), the distribution function f (r, p, t) (4), etc show small variation over time and coordinate.These fluctuations can be considered via the correlations like space correlations n(r, t)n(r ′ , t) or the time correlations n(r, t)n(r, t ′ ), where we introduce additional average on the macroscopic space scale Ā = (1/∆V ) ∆V AdV , or the average over the interval of time ∆T : Ã = (1/∆T ) ∆T Adt.
The distribution function is constructed on a certain scale in the coordinate and momentum space.If we want to trace the deviation from average (particularly in the momentum space) we need to focus our attention on the deviation of momentum of all particles in ∆-vicinity from the middle point of the vicinity η.Parameter ∆v i (t) is also associated with η.If we imagine a ∆-vicinity with 5 particles (it is an imaginary example, for the illustration, see Fig. 2, real vicinity contain a large number of particles) we can count the average momentum p = (1/5) 5 i=1 p i (t) = p and we get a deviation from p (the moment of the center of the vicinity).However, the distribution function f (r, p, t) count all these particles as the particles with momentum p.In other words 5 i=1 η i = 0 (in general η is a parameter scanning the vicinity, but here we introduce particular η i as η i (t) = p i (t) − p the deviation of momentum of particle being in the vicinity from the momentum of the center of vicinity).This example shows that physical nature of fluctuations can be presented via functions η a and η a η b (or ∆v a i (t) and ∆v a i (t)∆v b i (t) ).It explains necessity of the account of these functions in our model.We consider equation for evolution of function ∆v a i (t) , while function ∆v a i (t)∆v b i (t) is assumed to be approximately expressed via functions f (r, p, t) and ∆v a i (t) .Here, we introduce equation of state in order to make truncation of the set of kinetic equations (in addition to the self-consistent field approximation).
In the chosen moment of time t we have ∆v i = 0.However, the fluctuations happen on some time scale τ , which can be associated with the time of propagation of average particle via the ∆-vicinity: τ = 3 √ ∆/v 0 , where the average velocity of the ∆ p -vicinity is v 0 = pc/ p 2 + m 2 s c 2 .We can go further and make averaging of the kinetic equation over this time interval ã = (1/τ ) τ 0 adt.Here, we find ∆v i = 0, but ∆v a i ∆v b i = 0.As an estimation we can choose ∆v i,max

B. Truncation method
Let us consider simplified form of the kinetic equation for the vector distribution function of velocity (76).
Function j a (r, p, t) can be represented via two other functions introduced above j a (r, p, t) = v a • f (r, p, t) +F a (r, p, t).Equation (76) includes function Here we extract the velocity from functions j a s , j ab D,s , v a i (t)v b i (t) , and j abc Q,s to show the Lorentz force in more familiar way.It helps us to make the truncation as well.Finally, we represent i ∆v b i ξ c .Our model includes the multipole moments in the coordinate and momentum space, like d a and F a .The multipole moments are defined on the scale of the ∆vicinity.Therefore, they appear to be small.On the macroscopic scale this value tends to zero.Hence, we can introduce the small parameter ε to indicate the relative value of these functions.We find d a ∼ εf , F a ∼ εf , Q ab ∼ ε 2 f , j ab ∼ ε 2 f , etc.To make the truncation we need to estimate the high-rank tensor distribution functions included in the model The third rank tensors are neglected.Function ∆v a ∆v b is symmetric, so it is reasonable to be proportional to δ ab .However, function j ab D is not symmetric, but we assume that deviations of same projections of the coordinate and momentum are correlated.This correlation is included in the model via nonzero value of j ab D , which can be traced in calculations of any specific problem.Further comparison with experiment can show validity of our assumption.If no correlation is found this function can be considered equal to zero in the future modification of the model.
We start the discussion of equations ( 84), (85), (86) with the first of them (84).We can extract terms proportional to the zero degree of the small parameter ε: which correspond to the well-known Vlasov equation.Next, we present terms proportional to the first degree of the small parameter ε: (88) Other terms in equation (84) correspond to the second and third degree of the small parameter ε.They are neglected since we include the first correction to the Vlasov equation.
Next we consider equation (85).Lowest order on parameter ε is equal to one in this equation.We consider the lowest order and one correction to it.Let us show term existing in the lowest order: and in the next order Let us consider equation (86).Function j a splits on two terms v a f + F a .Obviously, equation (86) contains contribution of terms (87) and (88) multiplied by v a .Hence, all these terms are equation to zero in accordance with the equation of evolution of the scalar distribution function.After this simplification we find that the lowest order of terms in equation ( 86) on the parameter ε is equal to one.It contains the following terms: In the next order on ε we obtain

X. THE SPIN-ELECTRON-ACOUSTIC WAVES PROPAGATION IN THE SPIN POLARIZED ELECTRON GAS OF HIGH DENSITY
The degenerate macroscopically motionless (being in the equilibrium state) electron gas are described via the Vlasov kinetic equation for each spin projection of electrons [30] Subindex s corresponds to the electrons with the spin-up s =↑ or spin-down s =↓.We consider small perturbations of the equilibrium state f s = f 0s + δf s , where f 0s is the equilibrium distribution function, and δf s is the perturbation of the distribution function, which is chosen as the plane wave in the coordinate space δf s = F s e −ıωt+ıkzz , with the constant amplitude F s .It lead to the linearized kinetic equation where δB = 0 for the longitudinal waves.Equilibrium distribution functions for each subspecies of electrons is chosen as the Fermi step f 0s = θ(p F s − p)/(2πh) 3 , where p F s = (6πn 0s ) 1/3 h is the radius of the Fermi sphere in the momentum space for species s.
We consider the longitudinal waves, hence the perturbation of the electric field is parallel to the direction of the wave propagation k δE.
It leads to the dispersion equation (104) Presented dispersion equation is found for the motionless ions, but we have two species of particles: spin-up electrons and spin-down electrons.Both species of electrons are degenerate, but they have different equilibrium concentrations.So we have nonzero equilibrium spin polarization.The spin polarization is caused by the external magnetic field.However, it is not included in equation (100), since is does not affect the final result (101) for the waves propagating parallel to the magnetic field.
Equation (104) gives two solutions: the Langmuir wave and the SEAW.The SEAW has been studied in different regimes and using different method.It is considered within nonrelativistic hydrodynamics [29], nonrelativistic kinetics [30], and relativistic hydrodynamics [41].Here, we present relativistic kinetic theory of the SEAW.Numerical analysis of equation (104) shows that the dispersion dependencies of the Langmuir wave and the SEAW getting close to each other at the large wave vectors, which leads to an instability in the short-wavelength limit.However, it can be subject of another work specified on the instabilities in degenerate plasmas related to the SEAWs, while this paper is focused on derivation of the extended kinetic model.

XI. CONCLUSION
Detailed derivation of kinetic equations for the relativistic plasmas has been presented.It concludes in the set of three kinetic equations (for each species of particles): the Vlasov kinetic equation for the scalar distribution function, the equation of evolution of the vector distribution function of the electric dipole moment (the fluctuation of local center of mass of the particles being in the physically infinitesimal volume), and the equation of evolution of the vector distribution function of the velocity (the fluctuation of average momentum of the particles being in the physically infinitesimal volume).This closed set of equations appears as the result of truncation of the quasi-infinite set of kinetic equations (number of degrees of freedom is restricted due to the finite number of particles N), which extends via the account of two-, threeand more coordinate distribution functions, and higher multipole moments of one-coordinate (along with two-, three-and more coordinate) distribution functions.
The method of derivation shows the representation of the microscopic motion of the individual particles in terms of the macroscopic distribution functions.It gives representation of deterministic classical mechanics in terms of evolution collective functions.No probabilistic approaches are used for the derivation or interpretation of presented equations and found functions.
Equations have been found for the relativistic plasmas meaning both the large temperatures of the system (large velocities of the chaotic motion) and large velocities of the ordered motion.The motion of particles with the velocities close to the speed of light requires the account of full retarding potentials for the electromagnetic field created by particles.It reflects in the full Maxwell equations obtained in the self-consistent field approximation.The Maxwell equations also include the distribution function of the electric dipole moment, distribution function of the velocity, etc as the sources of the electromagnetic field.
The model is found for the hot plasmas, but it can be used for the degenerate electron gas with the Fermi velocity close to the speed of light.Hence, it is applied to the spin-electron acoustic waves in the extremely dense plasmas.The SEAWs are considered in the regime of account of the scalar distribution function governed by the Vlasov equation.

XII. DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study, which is a purely theoretical one.

FIG. 1 :
FIG.1:The illustration of the ∆-vicinity in the coordinate space.
Equation (100) give expression for the perturbation of the distribution function of electronsδf s = −ıq s (v • δE) ∂f 0s ∂ε 1 ω − k z v z, allows us to calculate the perturbations of concentration of the degenerate electrons δn s = δf s (r, p, t)dp