Consistent description of kinetics and hydrodynamics of dusty plasma
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p CONSISTENT DESCRIPTION OF KINETICS ANDHYDRODYNAMICS OF DUSTY PLASMA
B. Markiv
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,1 Svientsitskii Str., 79011 Lviv, Ukraine
M. Tokarchuk
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,1 Svientsitskii Str., 79011 Lviv, Ukraine andNational University “Lviv Polytechnic”,12 Bandera Str., 79013 Lviv, Ukraine
Abstract
A consistent statistical description of kinetics and hydrodynamics of dusty plasma is proposedbased on the Zubarev nonequilibrium statistical operator method. For the case of partial dynamicsthe nonequilibrium statistical operator and the generalized transport equations for a consistentdescription of kinetics of dust particles and hydrodynamics of electrons, ions and neutral atomsare obtained. In the approximation of weakly nonequilibrium process a spectrum of collectiveexcitations of dusty plasma is investigated in the hydrodynamic limit.
PACS numbers: 52.27.Lw . INTRODUCTION The study of nonequilibrium properties of dusty plasma is relevant in the fields of con-trolled thermonuclear fusion, nuclear reactors, plasma-dust structures [1–3], low-temperatureplasma [4, 5] etc. The difficulties in describing such systems with inherent processes of self-organization and structuring are related with a large asymmetry in size, mass and chargeof components (electrons, ions, neutral atoms and dust particles), effect of neutral particlesionization as well as with adsorption of neutral particles onto the surface of dust particles.In review [6] an important analysis of specific properties of dusty plasma including changesin charge of dust particles under the influence of electrons and ions flow is given in compar-ison with the classical (electron-ion) plasma. Herewith, some features of dusty plasma aremanifested. For example, the one is that the own energy of dust particles can be changed byplasma flows and another is that in the potential electric field, force acting on dust particlesis not potential, therefore, vortex dust motions can be excited. The charge of dust particlesis not fixed and depends on the plasma flows on their surface. Herewith, convergence orremoval of dust particles leads to the changes in plasma flows. This, in turn, leads to aconsistent change in charge of dust particles and to a possible change of sign of interactionenergy, in particular, to attraction between negatively charged dust grains [6, 9] what, inturn, leads to the formation of dust plasma crystals [2, 4, 6–8] This is provided by the plasmaflows which serve as a mechanism of accumulation of positive charge of ions between thenegatively charged dust particles.Complex electromagnetic processes of charging and recharging of dust grains, which en-tail the appearance of dust particles with a large electric charge, as well as electron-ionictransport processes demand the development of the theory of effective screening potentialsof interaction between particles [10, 11]. In such a system kinetic and hydrodynamic elec-tromagnetic processes should be described consistently. Keeping this in mind the kinetictheory of dusty plasma was developed be various authors [12–25].In particular, a statistical theory of dusty plasma was formulated in Ref. [14], wherethe microscopic equations for phase densities and BBGKY hierarchy for the nonequilibriumdistribution functions of electrons, ions, neutral atoms and dust particles were obtained. Onthis basis the kinetic theory of electromagnetic fluctuations in dusty plasma was developedin Ref. [26] and it was shown that such a description leads to the dependence of the ef-2ective cross section of dust charging in wave vector. This reflects the influence of plasmainhomogeneity effects. One of the main problems consist in the investigation of charge fluc-tuations in dusty plasma. Important results in this field were obtained in Refs. [27–29]. Inparticular, in [29] charge fluctuations in dusty plasma were studied by means of the Brow-nian dynamics computer simulations within the drift-diffusion approximation. Influence ofemission on charge and effective potential of dust grain in plasma at different intensities ofexternal source of ionization were investigated using numerical methods [30] as well. Thekinetics of atom-ion scattering processes in dusty plasma was studied in a recent paper [31]based on the model collision integrals.A number of papers by Tsitovich and De Angelis [20–25] were devoted to kinetic descrip-tion of dusty plasma by means of the Bogolubov-Klimontovich approach. Here, dusty plasmais considered as a system of interacting ions, electrons, dust particles in the surrounding ofneutral component (atoms), which is not taken into account explicitly. A special feature ofsuch an approach consist in considering plasma flows onto dust particles using the chargeof dust particles as additional variable [20]. Thus, ddt q = I ext + P a I a , where P a I a are theflows of plasma particles onto the surface of grains and I ext includes flows of photoelectronand secondary electron emission from their surface. The investigations of electrons and ionsdistribution in dusty plasma is actively carried out for the description of long-range interac-tions and memory effects within the nonclassical theory [32–34] based on Tsallis and Renyientropies.The study of time correlation functions and transport coefficients, in particular, viscos-ity, heat conductivity, thermodiffusion, as well as ionic and electronic conductivity of dustyplasma being an open spatially inhomogeneous system [35] is another important problem.Since dusty plasma is characterized by large asymmetry in size, charge and mass of thecomponents, obviously, the dynamics of each component in transport processes is corre-sponding. That is to be expected that the time correlation functions “density-density”,“momentum-momentum” as well as transport coefficients of viscosity and heat conductivityfor each component, ionic and electron conductivity have its inherent time and spatial be-havior. This behavior latter is affected by kinetic processes related to the dynamics of dustparticles.In the present work, we propose a consistent description of kinetics and hydrodynam-ics of dusty plasma by means of the Zubarev nonequilibrium statistical operator (NSO)3ethod [36, 37]. We consider description of the system in terms of both partial and con-servative dynamic variables without taking into account processes of ionization and atomsadsorption onto grains surface. In the second section the nonequilibrium statistical operatoris built and based on it the set of transport equations for a consistent description of kinet-ics of dust particles and hydrodynamics of electrons, ions and neutral atoms was obtained.In Sec. III we consider the case of weakly nonequilibrium processes, there the system ofequations for time correlation functions was received and analyzed. In Sec. IV by meansof the perturbation theory for collective excitations [38, 39] a spectrum of collective modesin the system is investigated in the hydrodynamic limit. The obtained results are shortlyconcluded in Sec V. II. NONEQUILIBRIUM STATISTICAL OPERATOR AND TRANSPORT EQUA-TIONS
Let us consider dusty plasma as a system of N e electrons, N a atoms of species a , N i ions of species i and N d dust particles that interact. Hamiltonian of such a multicomponentsystem can be presented in the form: H ( t ) = H e + H i + H a + H d + X α = e,i,d N α X s =1 Z s eϕ ( ~r s ; t ) , (2.1)where H α = N α X s =1 ( p αs ) m α + N d X s =1 Z s e a s + 12 X γ N α ,N γ X s = s ′ V αγ ( | ~r ss ′ | ) (2.2)is the Hamiltonian of electron ( e ), ionic ( i ), atom ( a ) and dusty ( d ) subsystems, respectively. Z s e a s is the internal energy of grains, Z s = d is their valency and a s = d is their size; ~p αs = ~p ′ αs − δ α Z α ec ~A ( ~r ; t ) ,δ α = , α = i, e, d ;0 , α = a .Here, ~p α is the impulse of particle α = e, i, a, d , ~A ( ~r ; t ) is the vector potential and ϕ ( ~r ; t ) isthe scalar potential of electromagnetic field induced by charged particles. Expressions for4otentials of interaction V αα as well as for interspecies energy of interaction V ei , V ea , V ed , V ia , V id , V ad are presented in [4].The nonequilibrium state of such a multicomponent system of charged and neutral par-ticles is related with the dynamics of each component as well as with interaction betweenthem. Different components can remain in kinetic or hydrodynamic, stationary or nonsta-tionary state. Herewith, we can say about quasineutrality of a dusty plasma. Let us considerthe nonequilibrium state of the system when subsystem of electrons, ions and neutral atomsis in hydrodynamic state and dust particles whose mass and charge can change remain inkinetic state. The parameters of dusty plasma can be the following: densities of ions andelectrons n i ≈ n e ≈ ÷ cm − , density of atoms n a ≈ ÷ cm − , density of dustparticles n d ≈ ÷ cm − , temperature of electrons T e ≈ ÷ T i /T e = 10 − ,as well as the charge Z d ≈ · ÷ and size a ≈ ÷ µ m of dust particles. Sincedusty plasma is composed of particles with large asymmetry in masses and charges the char-acteristic times of appropriate subsystems will differ significantly. It is important to notethe features of dusty plasma related to the fact that large charge of dust particles (whichis not constant and depends on plasma flows on the grain surface) lead to the emergenceof collective plasma flows, influence of which is comparable or even larger than influenceof electrostatic field [6]. Beside this, the internal energy of grains Z d e a d exceeds much theirkinetic energy and average energy of their interaction. In the potential electric field, forcesacting on dust particles are not potential: [ ~ ∇ × Z d ~E ] = [( ~ ∇ Z d ) × ~E ] = 0 and vortex motionsare observed in the majority of experiments. The interaction of dust particles can be strongenough in distances much more than screening length, and the electrostatic potential of grainabsorbing a plasma flow decays as r at large distance. At certain conditions this providesthe emergence of dust-plasma crystals and other dissipative self-organized structures.That is why as a reduced-description parameters of such a nonequilibrium state it is suit-able to chose partial variables: averaged values of densities of particles number, momentumand energy of electrons, ions and atoms for the description of hydrodynamic state h ˆ n α ( ~r ) i t , h ˆ ~ α ( ~r ) i t , h ˆ ε α ( ~r ) i t . (2.3)Here, we introduce the notation h ... i t = R d Γ N ...̺ ( t ).ˆ n α ( ~r ) = N α X s =1 δ ( ~r − ~r s ) (2.4)5s the microscopic number density of particles of species α = { e, i, a } ;ˆ ~ α ( ~r ) = N α X s =1 ~p s δ ( ~r − ~r s ) (2.5)is the microscopic momentum density of particles of species α ;ˆ ε α ( ~r ) = N α X s =1 p s m α + Z s e a s + 12 N α X s ′ = s =1 Φ αα ( | ~r s − ~r s ′ | ) ! δ ( ~r − ~r s ) (2.6)is the microscopic energy density of particles of species α .For the description of kinetics of dust particles we can chose the nonequilibrium distri-bution function h ˆ n d ( ~r, ~p, Z ) i t , (2.7)where ˆ n d ( ~r, ~p, Z ) = N d X s =1 δ ( ~r − ~r s ) δ ( ~p − m s ~v s ) δ ( Z − Z s ) (2.8)is the microscopic phase density of charged grains, which takes into account changes in massand charge of particle in the charging/recharging processes. The set of partial dynamicvariables allow us to study processes of kinetic origin, for example, relaxation processesrelated to the difference of partial temperatures of subsystems. This is especially importantbeyond the hydrodynamic regime. Such relaxation processes cannot be investigated usingthe set of conservative variables.Motions of electrons, ions and charged grains induce corresponding electromagnetic fields,which, in turn, cause processes of polarization and change dielectric properties of whole thesystem. Thus, besides (2.3), (2.7), as an additional parameters of the reduced descriptionshould be the averaged value of electric and magnetic fields h ˆ ~E ( ~r ) i t , h ˆ ~B ( ~r ) i t induced byelectrons, ions and charged dust particles as well as the their inductions h ˆ ~D ( ~r ) i t , h ˆ ~H ( ~r ) i t ,which satisfy the averaged Maxwell equations: ~ ∇ · h ˆ ~B ( ~r ) i t = 0 , (2.9) ~ ∇ · h ˆ ~D ( ~r ) i t = X i h ˆ n i ( ~r ) i t Z i e + e h ˆ n e ( ~r ) i t + Z dZ Z d~p ( Ze ) h ˆ n d ( ~r, ~p, Z d ) i t , (2.10)6 ∇ × h ˆ ~E ( ~r ) i t + ∂∂t h ˆ ~B ( ~r ) i t = 0 , (2.11) ~ ∇ × h ˆ ~H ( ~r ) i t − ∂∂t h ˆ ~D ( ~r ) i t = em e h ˆ ~ e ( ~r ) i t (2.12)+ X i Z i em i h ˆ ~ i ( ~r ) i t + Z dZ Z d~p ~pm d ( Ze ) h ˆ n d ( ~r, ~p, Z d ) i t . The electric end magnetic fields connected with the scalar and the vector potentials inducedby the charged particles: h ˆ ~E ( ~r ) i t = − ~ ∇ · ϕ ( ~r ; t ) − ∂∂t ~A ( ~r ; t ) , (2.13) h ˆ ~B ( ~r ) i t = ~ ∇ × ~A ( ~r ; t ) . (2.14)As we can see from the structure of equations (2.9)–(2.12), the parameters of reduced de-scription (2.3), (2.7) are not independent of h ˆ ~E ( ~r ) i t , h ˆ ~B ( ~r ) i t , h ˆ ~D ( ~r ) i t , h ˆ ~H ( ~r ) i t . Contrary,field-particle transport processes are interconnected and should be considered consistently.Moreover, coordination of hydrodynamics of electrons, ions and atoms with kinetics of dustparticles occurs not only through the generalized transport equations, but also in the forma-tion of electromagnetic field (2.10), (2.12) [40]. Beside this, known integral relations between h ˆ ~D ( ~r ) i t , h ˆ ~E ( ~r ) i t and h ˆ ~B ( ~r ) i t , h ˆ ~H ( ~r ) i t determine spatially inhomogeneous dielectric function ε ( ~r, ~r ′ ; t, t ′ ) and magnetization χ ( ~r, ~r ′ ; t, t ′ ) describing polarization processes in the systemAveraged values of densities of particles number, momentum and energy of electrons, ions,atoms and phase density of charged grains number (2.3), (2.7) as well as averaged values offields (2.9)–(2.12) are calculated using nonequilibrium statistical operator ̺ ( t ) that satisfiesthe corresponding Liouville equation. In order to find ̺ ( t ) we make use of the Zubarev NSOmethod [36]. This method allows us to write down ̺ ( t ) in a general form: ̺ ( t ) = ̺ q ( t ) − t Z −∞ e ε ( t ′ − t ) T ( t, t ′ )(1 − P q ( t ′ )) iL N ̺ q ( t ′ ) dt ′ , (2.15)where iL N is the Liouville operator corresponding to the Hamiltonian (2.1); T ( t, t ′ ) =exp n − R t ′ t (1 − P q ( t ′′ )) iL N dt ′′ o is the generalized evolution operator with regard to pro-jection (1 − P q ( t ′′ )); P q ( t ′′ ) is the generalized Kawasaki-Gunton projection operator, whosestructure depends on a form of a quasiequilibrium statistical operator ̺ q ( t ). Within the7SO method, ̺ q ( t ) is derived from the extremum of an informational entropy at the fixedvalues of the reduced-description parameters (in our case Eqs. (2.3), (2.7)) including thenormalization condition R d Γ ̺ q ( t ) = 1: ̺ q ( t ) = exp (cid:26) − Φ( t ) (2.16) − X α Z d~rβ α ( ~r ; t ) (cid:20) ˆ ε α ( ~r ) − (cid:18) ~v α ( ~r ; t ) + δ α Z α em α c ~A ( ~r ; t ) (cid:19) · ˆ ~ α ( ~r ) − (cid:16) ν α ( ~r ; t ) − m α v α ( ~r ; t )2 (cid:17) ˆ n α ( ~r ) (cid:21) − Z d~r Z dξa d ( ~r, ξ ; t )ˆ n d ( ~r, ξ ) (cid:27) , where a new variable ξ = ( ~p, Z ) is introduced, Φ( t ) is the Massieu-Planck functional deter-mined from the normalization condition for ̺ q ( t ):Φ( t ) = ln Z d Γ N exp (cid:26) − X α Z d~rβ α ( ~r ; t ) (2.17) × (cid:20) ˆ ε α ( ~r ) − (cid:18) ~v α ( ~r ; t ) + δ α Z α em α c ~A ( ~r ; t ) (cid:19) · ˆ ~ α ( ~r ) − (cid:16) ν α ( ~r ; t ) − m α v α ( ~r ; t )2 (cid:17) ˆ n α ( ~r ) (cid:21) − Z d~r Z dξa d ( ~r, ξ ; t )ˆ n d ( ~r, ξ ) (cid:27) ; β α ( ~r ; t ) is the inverse local temperature for subsystem α ; ~v α ( ~r ; t ) is the hydrodynamic velocityof component α ; ν α ( ~r ; t ) = µ α ( ~r ; t ) + Z α eϕ ( ~r ; t ); (2.18) ν α ( ~r ; t ) is the electrochemical potential of ions and electrons (at α = i and α = e , respec-tively), ϕ ( ~r ; t ) is the electric potential, whose gradient determine a longitudinal componentof the electric field induced by ions, electrons and dust grains: ~E l ( ~r ; t ) = − ~ ∇ ϕ ( ~r ; t ) = h ˆ ~E ( ~r ) i t ; (2.19) µ α ( ~r ; t ) is the local chemical potential of component α . The local thermodynamic parameters { β α ( ~r ; t ), ~v α ( ~r ; t ), ν α ( ~r ; t ) } are determined from the self-consistency conditions: h ˆ ε ′ α ( ~r ) i t = h ˆ ε ′ α ( ~r ) i tq , h ˆ ~ α ( ~r ) i t = h ˆ ~ α ( ~r ) i tq , h ˆ n α ( ~r ) i t = h ˆ n α ( ~r ) i tq , (2.20)where ˆ ε ′ α ( ~r ) = ˆ ε α ( ~r ) − (cid:16) ~v α ( ~r ; t ) + δ α Z α em α c ~A ( ~r ; t ) (cid:17) · ˆ ~ α ( ~r ) + m α V α ( ~r ; t )2 ˆ n α ( ~r ) is the local value ofenergy in the reference frame moving together with a system element of the mass velocity ~v α ( ~r ; t ). These parameters satisfy the corresponding generalized thermodynamic relations.8he parameter a d ( ~r, ~p ; Z ; t ) is conjugated to h ˆ n d ( ~r, ~p, Z ) i t and is determined from the self-consistency condition h ˆ n d ( ~r, ~p ; Z ) i t = h ˆ n d ( ~r, ~p ; Z ) i tq . (2.21)Taking into account a structure of ̺ q ( t ) the generalized projection operator P q ( t ) can bepresented in the form: P ( t ) ρ = (cid:18) ̺ q ( t ) − X α X l Z d~r δ̺ q ( t ) δ h ˆ b αl ( ~r ) i t (2.22) − Z d~r Z d~p Z dZ δ̺ q ( t ) δ h ˆ n d ( ~r, ~p ; Z ) i t (cid:19) Z d Γ N ρ + X α X l Z d~r δ̺ q ( t ) δ h ˆ b αl ( ~r ) i t Z d Γ N ˆ b αl ( ~r ) ρ + Z d~r Z dξ δ̺ q ( t ) δ h ˆ n d ( ~r, ξ ) i t Z d Γ N ˆ n d ( ~r, ξ ) ρ, where l=1,2,3, ˆ b α = ˆ ε α ( ~r ), ˆ b α = ˆ ~ α ( ~r ), ˆ b α = ˆ n α ( ~r ). The operator P q ( t ) possesses thefollowing properties: P q ( t ) ̺ ( t ) = ̺ q ( t ), P q ( t ) P q ( t ′ ) = P q ( t ), P q ( t ) ̺ q ( t ′ ) = ̺ q ( t ). Substituting(2.16) into (2.15) we obtain the nonequilibrium statistical operator of dusty plasma: ̺ ( t ) = ̺ q ( t ) + X α Z d~r ′ t Z −∞ e ε ( t ′ − t ) T ( t, t ′ ) I αε ( ~r ′ ; t ′ ) β α ( ~r ′ ; t ′ ) ̺ q ( t ′ ) dt ′ − X α Z d~r ′ t Z −∞ e ε ( t ′ − t ) T ( t, t ′ ) I α ( ~r ′ ; t ′ ) β k ( ~r ′ ; t ′ ) (2.23) × (cid:18) ~v α ( ~r ′ ; t ′ ) + δ α Z α em α c ~A ( ~r ′ ; t ′ ) (cid:19) ̺ q ( t ′ ) dt ′ + Z d~r ′ Z dξ ′ t Z −∞ e ε ( t ′ − t ) T ( t, t ′ ) I dn ( ~r ′ , ξ ′ ; t ′ ) a d ( ~r ′ , ξ ′ ; t ′ ) ̺ q ( t ′ ) dt ′ , where I αε ( ~r ; t ) = (1 − P ( t )) iL N ˆ ε α ( ~r ) (2.24)is the generalized flow of energy density; I α ( ~r ; t ) = (1 − P ( t )) iL N ˆ ~ α ( ~r ) (2.25)is the generalized flow of momentum density; I dn ( ~r, ξ ; t ) = (1 − P ( t )) iL N ˆ n d ( ~r, ξ ) (2.26)9s the generalized flow of microscopic density of dust particles.In Eqs. (2.24)–(2.26) the generalized Mori projection operator P ( t ) has the followingstructure: P ( t ) ˆ A = h ˆ A i tq + X α X l Z d~r δ h ˆ A i tq δ h ˆ b αl ( ~r ) i t (ˆ b αl (cid:0) ~r ) − h ˆ b αl ( ~r ) i t (cid:1) (2.27)+ Z d~r Z ξ δ h ˆ A i tq δ h ˆ n d ( ~r, ξ ) i t (cid:0) ˆ n d ( ~r, ξ ) − h ˆ n d ( ~r, ξ ) i t (cid:1) . Herewith, P ( t ) P ( t ′ ) = P ( t ), P ( t )(1 − P ( t ′ )) = 0, P ( t )ˆ b αl ( ~r ) = ˆ b αl ( ~r ), P ( t )ˆ n d ( ~r, ~p ; Z ) =ˆ n d ( ~r, ~p ; Z ). Using the NSO ̺ ( t ) we can obtain the set of transport equations for the reduced-description parameters. Let us present it in a matrix form: ∂∂t h ˜ A ( ~x ) i t = h ˜ A ( ~x ) i tq − Z d~x ′ t Z −∞ e ε ( t ′ − t ) ˜ ϕ AA ( ~x, ~x ′ ; t, t ′ ) ˜ F A ( ~x ; t ′ ) dt ′ , (2.28)where ˜ A ( ~x ) = col (ˆ b αl ( ~r ) , ˆ n d ( ~r, ~p ; Z )) = col (ˆ n α ( ~r ) , ˆ ~ α ( ~r ) , ˆ ε α ( ~r ) , ˆ n d ( ~r, ~p ; Z )) and˜ F A ( ~x ; t ′ ) = col (cid:16) − β α ( ~r ; t )( ν α ( ~r ; t ) − m α v α ( ~r ; t ) / − β α ( ~r ; t ) h ~v α ( ~r ; t ) + δ α Z α em α c ~A ( ~r ; t ) i , β α ( ~r ; t ) , a d ( ~r, ~p ; Z ; t ) (cid:17) are the vector-columns;˜ ϕ AA ( ~x, ~x ′ ; t, t ′ ) = h ˜ I ( ~x ; t ) T ( t, t ′ ) ˜ I + ( ~x ′ ; t ′ ) i t ′ q (2.29)= ˜0 ˜0 ˜0 ˜0˜0 ˜ ϕ II ˜ ϕ IIε ˜ ϕ IIdn ˜0 ˜ ϕ IεI ˜ ϕ IεIε ˜ ϕ IεIdn ˜0 ˜ ϕ IdnI ˜ ϕ IdnIε ˜ ϕ IdnIdn is a block matrix within which ˜ ϕ IAIA are the matrices of the generalized transport kernels,namely: ˜ ϕ II ( ~r, ~r ′ , t, t ′ ) = ϕ ee II ϕ ei II ϕ ea II ϕ ie II ϕ ii II ϕ ia II ϕ ae II ϕ ai II ϕ aa II ( r,r ′ ; t,t ′ ) , (2.30)where diagonal elements are the transport kernels determining the generalized viscositycoefficients of electrons, ions and atoms; nondiagonal elements are the transport kernelsdescribing the dissipative correlations between flows of momentum density. Similarly, in the10atrix ˜ ϕ IεIε ( ~r, ~r ′ , t, t ′ ) = ϕ ee IεIε ϕ ei IεIε ϕ ea IεIε ϕ ie IεIε ϕ ii IεIε ϕ ia IεIε ϕ ae IεIε ϕ ai IεIε ϕ aa IεIε ( r,r ′ ; t,t ′ ) (2.31)the diagonal elements are the transport kernels determining the generalized heat conductivitycoefficients of electrons, ions and atoms subsystems; nondiagonal elements are the transportkernels describing the dissipative correlations between flows of energy density of particles.Correspondingly, the transport kernels of matrices ˜ ϕ IεI ( ~r, ~r ′ , t, t ′ ), ˜ ϕ IIε ( ~r, ~r ′ , t, t ′ ) describedissipative correlations between the generalized flows of momentum and energy of electrons,ions and atoms. The transport kernels of matrices ˜ ϕ IIdn ( ~r, ~r ′ , ~p ′ ; Z ; t, t ′ ), ˜ ϕ IdnI ( ~r, ~r ′ , ~p ′ ; Z ; t, t ′ ),˜ ϕ IdnIε ( ~r, ~r ′ , ~p ′ ; Z ; t, t ′ ), ˜ ϕ IεIdn ( ~r, ~r ′ , ~p ′ ; Z ; t, t ′ ) in Eq. (2.29) describe, respectively, dissipativecorrelations between the generalized flows of momentum and energy density (of electrons,ions and atoms) and the generalized flow of microscopic phase density of dust particles.Herewith, ˜ I ( ~x ; t ) = col ( I αn ( ~r ; t ) , I α ( ~r ; t ) , I αε ( ~r ; t ) , I dn ( ~r, ~p ; Z ; t ))is the column-vector,˜ I (+) ( ~x ′ ; t ′ ) = ( I αn ( ~r ′ ; t ′ ) , I α ( ~r ′ ; t ′ ) , I αε ( ~r ′ ; t ′ ) , I dn ( ~r ′ , ~p ′ ; Z ′ ; t ′ ))is the row-vector, and ˜ I ( ~x ′ ; t ′ ) ˜ I (+) ( ~x ′ ; t ′ ) is their scalar product. The transport kernel˜ ϕ IdnIdn ( ~r, ~p ; Z ; ~r ′ , ~p ′ ; Z ′ ; t, t ′ ) describes dissipative correlations between the generalized flowsof microscopic phase density of grains and determines the generalized diffusion coefficient inphase space of grains.Motions of electrons, ions and charged dust particles, according to Maxwell equations(2.9)–(2.12), induce electromagnetic fields. The dissipative transport processes related tothe flows of momentum and energy densities of electrons, ions, atoms and kinetics of chargedgrains that are described by the set of the nonmarkovian transport equations affect on thedissipation of field variables through the right side of equations (2.10), (2.12).In our approach, effects of adsorption/desorbtion of electrons, ions and atomsat the grain surface as well as processes of clustering of dust particles can bedescribed by means of the nonequilibrium correlation functions h ˆ n d ( ~r, ξ )ˆ n α ( ~r ′ ) i t =11 ˆ G dαnn ( ~r, ξ ; ~r ′ ) i t , h ˆ n d ( ~r, ξ )ˆ n d ( ~r ′ , ξ ′ ) i t = h ˆ G ddnn ( ~r, ξ ; ~r ′ , ξ ′ ) i t , where ˆ G dαnn ( ~r, ξ ; ~r ′ ) = ˆ n d ( ~r, ξ )ˆ n α ( ~r ′ )and h ˆ G ddnn ( ~r, ξ ; ~r ′ , ξ ′ ) = ˆ n d ( ~r, ξ )ˆ n d ( ~r ′ , ξ ′ ). Generally speaking, taking into account the factthat electrons are localized on the grain’s surface, such a description should be con-ducted at quantum level. Then, in the abovementioned correlation functions, ˆ n d ( ~r, ξ )and ˆ n α ( ~r ′ ) are the density operators of particles of corresponding species. In particular,ˆ n a ( ~r ) = P N a jν ˆΨ + jν ( ~r ) ˆΨ jν ( ~r ), ˆ n e ( ~r ) = P N a j~s ˆΨ + j~s ( ~r ) ˆΨ j~s ( ~r ), where ˆΨ + jν ( ~r ) and ˆΨ jν ( ~r ) are theoperators of creation and annihilation of atom in the state ν on the grain’s surface, whereasˆΨ + j~s ( ~r ) and ˆΨ j~s ( ~r ) are the operators of creation and annihilation of electron in state withspin ~s . The ion interacting with the electron from the dust particles surface create the atom,which can be in the adsorbed state on the surface or can desorb into ion-electron-atom sub-system until next ionization. From this point of view, a quantum nature of interaction ofions, electron and atoms with the electron surface of grain should be taken into account inthe Hamiltonian (2.1). Similar problems appear in the theory of catalytic processes [42].Then, quasiequilibrium statistical operator (2.16) will have the following form: ̺ q ( t ) = exp (cid:26) − Φ( t ) − X α Z d~rβ α ( ~r ; t ) (cid:20) ˆ ε α ( ~r ) − (cid:18) ~v α ( ~r ; t ) + δ α Z α em α c ~A ( ~r ; t ) (cid:19) · ˆ ~ α ( ~r ) − (cid:16) ν α ( ~r ; t ) − m α v α ( ~r ; t )2 (cid:17) ˆ n α ( ~r ) (cid:21) − Z d~r Z dξa d ( ~r, ξ ; t )ˆ n d ( ~r, ξ ) − X α Z d~r Z d~r ′ Z dξµ dα ( ~r, ξ ; ~r ′ ; t ) ˆ G dαnn ( ~r, ξ ; ~r ′ ) − X α Z d~r Z d~r ′ Z dξ Z dξ ′ µ dd ( ~r, ξ ; ~r ′ , ξ ′ ; t ) ˆ G ddnn ( ~r, ξ ; ~r ′ , ξ ′ ) (cid:27) and the transport equations (2.28) will contain the nonequilibrium correlation functions h ˆ G dαnn ( ~r, ξ ; ~r ′ ) i t , h ˆ G ddnn ( ~r, ξ ; ~r ′ , ξ ′ ) i t . The parameters µ dα ( ~r, ξ ; ~r ′ ; t ), µ dd ( ~r, ξ ; ~r ′ , ξ ′ ; t ) are deter-mined from the corresponding self-consistency conditions: h ˆ G dαnn ( ~r, ξ ; ~r ′ ) i t = h ˆ G dαnn ( ~r, ξ ; ~r ′ ) i tq and h ˆ G ddnn ( ~r, ξ ; ~r ′ , ξ ′ ) i t = h ˆ G ddnn ( ~r, ξ ; ~r ′ , ξ ′ ) i tq and can be defined as spatially-temporal chemicalpotential of corresponding “dimers”. This issue needs a separate study.Beside the statistical theory of dusty plasma based on the BBGKY hierarchy for nonequi-librium distribution functions [10], we proposed a consistent description of kinetic and hydro-dynamic processes with partial contribution from each component by means of the ZubarevNSO method. Based on the obtained system of transport equations (2.28) and averagedMaxwell equations [40] the time correlation functions as well as the generalized transport12oefficients of dusty plasma can be investigated for both weakly and strongly nonequilibriumprocesses. Such a set of equations takes into account consistently kinetic and hydrodynamicnonmarkovian processes as well as mutual influence of dynamics of particles and electro-magnetic field. The investigation of dependencies of the generalized transport coefficients(viscosity, heat conductivity, ionic and electron conductivity) related with transport kernels(2.30), (2.31) on wave vector and frequency remains very important. In this direction mainproblem can be bring to the calculation of generalized transport kernels (2.29), in particular ϕ I dn I dn ( t ), in the kinetic equation for h ˆ n d ( ~r, ~p, Z d ) i t . At small concentrations of dust particles,in the transport kernel ϕ I dn I dn ( t ) an expansion over the grains density can be used. At thesame time, an important issue consist in the investigation of weakly nonequilibrium processesin the system, when gradients of nonequilibrium thermodynamic parameters ˜ F A ( ~x ; t ′ ) aresmall. In the linear approximation in deviations of the nonequilibrium thermodynamic pa-rameters from their equilibrium values, the set of transport equations (2.28) become closed.As it is known, within the framework of the NSO method, the time correlation functionsbuilt on the basic set of dynamic variables of reduced description Eqs. (2.4)–(2.6), (2.8)satisfy the same system of equations. This issue we consider in the following section. III. WEAKLY NONEQUILIBRIUM PROCESSES IN DUSTY PLASMA
Let us now consider the nonequilibrium processes in dusty plasma when the nonequi-librium thermodynamic parameters ˜ F A ( x ; t ) slightly deviate from their equilibrium values˜ F A ( x ; 0). This is equivalent the fact that h ˜ A ( x ) i t slightly deviate from their equilibriumvalues h ˜ A ( x ) i . Here, h . . . i = R d Γ N . . . ̺ , ̺ is the equilibrium statistical operator ofdusty plasma. Then, expanding the quasiequilibrium statistical operator Eq. (2.16) overdeviations δ ˜ F A ( x ; t ) = ˜ F A ( x ; t ) − ˜ F A ( x ; 0) we restrict ourself to the linear approximation ̺ q ( t ) = ̺ (cid:26) − Z dxδ ˜ F A ( x ; t ) ˜ A ( x ) (cid:27) . (3.1)Further, we use Fourier transformation. Then, excluding parameters δ ˜ F A ( ~k ; t ) from Eq.(3.1) by means of the self-consistency conditions Eqs. (2.20)–(2.21), we obtain ̺ q ( t ) = ̺ X ~k Z dx Z dx ′ δ ˜ A ~k ( x ; t ) ˜Φ − AA ( ~k, x, x ′ ) ˜ A ~k ( x ′ ) , (3.2)13here h δ ˜ A ~k ( x ) i t = δ ˜ A ~k ( x ; t ), δ ˜ A ~k ( x ) = ˜ A ~k ( x ) − h ˜ A ~k ( x ) i . Henceforth x = { ~p, Z } andintegration concerns only to a variable describing the grains subsystem. ˜Φ − AA ( ~k, x, x ′ ) is thematrix inverse to the matrix of correlation functions of variables ˜ A ~k ( x ):˜Φ AA ( ~k, x, x ′ ) = h ˜ A ~k ( x ) ˜ A + ~k ( x ′ ) i (3.3)= ˜Φ nn ˜0 ˜0 ˜0˜0 ˜Φ pp ˜0 ˜0˜0 ˜0 ˜Φ hh ˜0˜0 ˜0 ˜0 ˜Φ NN ( ~k,x,x ′ ) . Here, in particular, ˜Φ nn ( ~k ) = Φ eenn Φ einn Φ eann Φ ienn Φ iinn Φ iann Φ aenn Φ ainn Φ aann ( ~k ) (3.4)is the matrix of the equilibrium correlation functions “density-density” for electrons, ionsand neutral atoms Φ αγnn ( ~k ) = h ˆ n α~k ˆ n γ~k i .˜Φ ( ~k ) = Φ ee ii
00 0 Φ aa ( ~k ) (3.5)is the matrix of the equilibrium correlation functions “momentum-momentum” Φ αγ ( ~k ) = h ˆ ~ α~k ˆ ~ γ~k i . ˜Φ hh ( ~k ) = Φ eehh Φ eihh Φ eahh Φ iehh Φ iihh Φ iahh Φ aehh Φ aihh Φ aahh ( ~k ) (3.6)is the matrix of the equilibrium correlation functions “enthalpy-enthalpy” for electrons, ionsand neutral atoms Φ αγhh ( ~k ) = h ˆ h α~k ˆ h γ~k i .ˆ h α~k = ˆ ε α~k − X γγ ′ h ˆ ε α~k ˆ n γ~k i [Φ − nn ( ~k )] γγ ′ ˆ n γ ′ ~k is the generalized enthalpy of subsystem α .˜Φ NN ( ~k, x, x ′ ) = Φ NN ( ~k, ~p, Z, ~p ′ , Z ′ ) = h ˆ N ~k ( ~p, Z ) ˆ N − ~k ( ~p ′ , Z ′ ) i (3.7)14s the equilibrium correlation function “phase density - phase density” for dust particles.Herewith, the inverse function Φ − NN ( ~k, ~p, Z, ~p ′ , Z ′ ) should be defined from the integral relation Z d~p ′′ Z dZ ′′ Φ NN ( ~k, ~p, Z, ~p ′′ , Z ′′ )Φ − NN ( ~k, ~p ′′ , Z ′′ , ~p ′ , Z ′ ) = (3.8)= δ ( ~p − ~p ′ ) δ ( Z − Z ′ ) . The function Φ − NN ( ~k, ~p ′′ , Z ′′ , ~p ′ , Z ′ ) is calculated in Appendix A. The new kinetic variableˆ N ~k ( ~p, Z ) is orthogonal to the partial hydrodynamic variables, and it appears as a result ofexcluding the thermodynamic parameters:ˆ N ~k ( ~p, Z ) = (1 − P H )ˆ n d~k ( ~p, Z ) . (3.9) P H is the hydrodynamic part of Mori projection operator (3.11).In approximation Eq. (3.2), the nonequilibrium statistical operator (2.23) has the follow-ing form: ̺ ( t ) = ̺ q ( t ) − X ~k Z dx Z dx ′ Z t ∞ e ε ( t ′ − t ) (3.10) × T ( t, t ′ )(1 − P ) iL N ˜ A ~k ( x ) ˜Φ − AA ( ~k, x, x ′ ) δ ˜ A − ~k ( x, t ′ ) ̺ dt ′ , where T ( t, t ′ ) = exp { ( t ′ − t )(1 − P ) iL N } , P is the Mori projection operator for weaklynonequilibrium processes, which acts on dynamic variables as follows: P ˜ B = h ˜ B i + Z dx Z dx ′ h ˜ B ˜ A + ~k ( x ) i ˜Φ − AA ( ~k, x, x ′ ) ˜ A ~k ( x ′ ) (3.11)It possesses the properties P (1 − P ) = 0, P ˜ A ~k ( x ) = ˜ A ~k ( x ). As we can see, the NSO isa functional of the averaged values h ˜ A ~k ( x ) i t and the generalized flows (1 − P ) iL N ˜ A ~k ( x ).Using ̺ ( t ) Eq. (3.10) we can obtain the set of equations for h ˜ A ~k ( x ) i t [37]. We present it ina matrix form: ∂∂t h δ ˜ A ~k ( x ) i t − Z i ˜Ω AA ( ~k, x, x ′ ) h δ ˜ A ~k ( x ′ ) i t dx ′ (3.12)+ Z dx ′ Z t −∞ e ε ( t ′ − t ) ˜ ϕ AA ( ~k, x, x ′ ; t, t ′ ) h δ ˜ A ~k ( x ′ ) i t ′ dt ′ = 0 . i ˜Ω AA ( ~k, x, x ′ ) = Z dx ′′ h ˙˜ A ~k ( x ) ˜ A + ~k ( x ′′ ) i ˜Φ − AA ( ~k, x ′′ , x ′ ) = (3.13)= ˜0 i ˜Ω n ˜0 ˜0 i ˜Ω n ˜0 i ˜Ω h i ˜Ω N ˜0 i ˜Ω h ˜0 ˜0˜0 i ˜Ω N ˜0 i ˜Ω NN ( ~k,x,x ′ ) is the frequency matrix, whose elements describe the static correlations between densities ofparticles number, momentum end energy of each component of dusty plasma.˜ ϕ AA ( ~k, x, x ′ ; t, t ′ ) = ˜0 ˜0 ˜0 ˜0˜0 ˜ ϕ ˜ ϕ h ˜ ϕ N ˜0 ˜ ϕ h ˜ ϕ hh ˜ ϕ hN ˜0 ˜ ϕ N ˜ ϕ Nh ˜ ϕ NN ( ~k,x,x ′ ; t,t ′ ) (3.14)= Z dx ′′ h (1 − P ) ˙˜ A ~k ( x ) T ( t, t ′ )(1 − P ) ˙˜ A + ~k ( x ′′ ) i ˜Φ − AA ( ~k, x ′′ , x ′ )is the matrix, whose elements are the transport kernels (memory functions), describingdissipative processes in dusty plasma, namely, diffusivity, viscosity and heat conductivity.Herewith, ˜ ϕ ( ~k ; t, t ′ ) = ϕ eeI I ϕ eiI I ϕ eaI I ϕ ieI I ϕ iiI I ϕ iaI I ϕ aeI I ϕ aiI I ϕ aaI I ( ~k ; t,t ′ ) (3.15)is the matrix with elements describing viscous processes, in particular, ϕ eeI I , ϕ iiI I and ϕ aaI I define the generalized viscosity coefficients of electron, ionic and atom component. Thenondiagonal elements describe the intercomponent viscous processes.˜ ϕ hh ( ~k ; t, t ′ ) = ϕ eeI h I h ϕ eiI h I h ϕ eaI h I h ϕ ieI h I h ϕ iiI h I h ϕ iaI h I h ϕ aeI h I h ϕ aiI h I h ϕ aaI h I h ( ~k ; t,t ′ ) (3.16)is the matrix with elements describing heat conductivity processes, namely, ϕ eeI h I h , ϕ iiI h I h and ϕ aaI h I h define the generalized heat conductivity coefficients of electron, ionic and atomsubsystems, and the nondiagonal elements describe the intercomponent heat processes.16t can be shown, that within the NSO method [37] the time correlation functions˜Φ AA ( ~k, x, x ′ ; t ) = h ˜ A ~k ( x ; t ) ˜ A + ~k ( x ′ ) i = ˜Φ nn ˜Φ n ˜Φ nh ˜Φ nN ˜Φ n ˜Φ ˜Φ h ˜Φ N ˜Φ hn ˜Φ h ˜Φ hh ˜Φ hN ˜Φ Nn ˜Φ N ˜Φ Nh ˜Φ NN ( ~k,x,x ′ ; t ) (3.17)will also satisfy the equations (3.12): ∂∂t ˜Φ AA ( ~k, x, x ′ ; t ) − Z dx ′′ i ˜Ω AA ( ~k, x, x ′′ ) ˜Φ AA ( ~k, x ′′ , x ′ ; t ) (3.18)+ Z dx ′′ Z t −∞ e ε ( t ′ − t ) ˜ ϕ AA ( ~k, x, x ′′ ; t, t ′ ) ˜Φ AA ( ~k, x ′′ , x ′ ; t ′ ) dt ′ = 0 . In particular, the matrix of the time correlation functions “density-density” ˜Φ nn ( ~k ; t ) isconnected with the matrix of partial dynamic structure factors ˜ S ( ~k ; ω )˜ S ( ~k ; ω ) = 12 π Z ∞−∞ e iωt ˜Φ nn ( ~k ; t ) = S ee S ei S ea S ie S ii S ia S ae S ai S aa ( ~k ; ω ) . (3.19)Applying the Laplace transformation to the matrix equation (3.18) we obtain s ˜Φ AA ( ~k ; s ) − i ˜Ω AA ( ~k ) ˜Φ AA ( ~k ; s ) + ˜ ϕ AA ˜Φ AA ( ~k ; s ) = − ˜Φ AA ( ~k ; 0) , (3.20)where s = ω + iε .It is worth noting that for the variable A ~k ( x ) = ˆ N ~k ( ~p, Z ) in the set of transport equationsan integration over ~p and Z is present s ˜Φ NN ( ~k, ~p, Z, ~p ′ , Z ′ ; s ) (3.21) − Z d~p ′′ Z dZ ′′ i ˜Ω NN ( ~k, ~p, Z, ~p ′′ , Z ′′ ) ˜Φ NN ( ~k, ~p ′′ , Z ′′ , ~p ′ , Z ′ ; s ) − i ˜Ω NA ( ~k, ~p, Z ) ˜Φ AN ( ~k, ~p ′ , Z ′ ; s )+ Z d~p ′′ Z dZ ′′ ˜ ϕ NN ( ~k, ~p, Z, ~p ′′ , Z ′′ ; s ) ˜Φ NN ( ~k, ~p ′′ , Z ′′ , ~p ′ , Z ′ ; s )+ ˜ ϕ NA ( ~k, ~p, Z ; s ) ˜Φ AN ( ~k, ~p ′ , Z ′ ; s ) = − ˜Φ NN ( ~k, ~p, Z, ~p ′ , Z ′ ; s ) . The relation det | zI − i ˜Ω AA ( ~k ; s ) + ˜ ϕ AA ( ~k ; s ) | = 0 determines the collective excita-tions in dusty plasma caused by electron, ionic and atom interactions. Each compo-nent has its own role in sound spreading, in processes of electroconductivity, heat con-ductivity, and polarization processes. It manifests in processes of appearing/disappearing17f ordered structures, in polarization processes in dusty plasma through the partial dy-namics structure factors Eq. (3.19) and the “charge-charge” structure factor for grains S NN ( ~k, Z, Z ′ ; ω ) = R d~p R d~p ′ S NN ( ~k, ~p, Z, ~p ′ , Z ′ ; ω ). IV. COLLECTIVE EXCITATIONS
A considerable asymmetry of dusty plasma in charges and masses makes its descriptionnatural in terms of partial dynamics. However, when the concentration of grains is smalland its dynamic can be considered at diffusion level, and for the case of isothermal plasmawhen temperature of all components is close, we can describe dusty plasma based on theconservative variables. In particular, we can include into the set of the reduced-descriptionparameters the averaged values of Fourier components of partial number densities, totalmomentum and total enthalpy. The dust particles subsystem we will again described bya kinetic variable. The choice of such variables is caused by the fact that further we areinterested in collective excitations of the system in the hydrodynamic limit and such adescription is common for description of multicomponent liquids. Thus,ˆ n α~k = N α X l =1 e − ikr αl , ˆ ~ ~k = X α ˆ ~ α~k = X α N α X l =1 p αl e − ikr αl (4.1)are the Fourier components of particles number density of species α and total momentumdensity, respectively, ˆ h ~k = ˆ ε ~k − X αγ h ˆ ε ~k ˆ n α~k i [Φ − nn ( ~k )] αγ ˆ n γ~k (4.2)are the Fourier components of generalized enthalpy density, andˆ ε ~k = X α ˆ ε α~k = X α N α X l =1 e αl e − ikr αl are the Fourier components of total energy density, where e αl = ( ~p αl ) m α + 12 X γ N γ X j =1 V αγ ( | ~r αl − ~r γj | ) . For these parameters of a reduced description we can obtain a set of transport equationsanalogous to Eq. (3.12). However, now the kinetic variable ˆ N d ( ~k, ~p, Z ) is defines by theexpression similar to Eq. (3.9)ˆ N d ( ~k, ~p, Z ) = (1 − P H ) n d~k ( ~p, Z ) = (1 − P H ) n d~k ( ξ ) , (4.3)18ut the Mori projection operator P H is constructed on the dynamic variables (4.1)–(4.2). Inthis case, the generalized hydrodynamic matrix can be written as follows (herewith, in theintegral term R dξT NN ( ~k, ξ, ξ ′ ) h δ ˆ N d ( ~k, ξ ′ ) i t in the equation for h δ ˆ N d ( ~k, ξ ) i t we make the ap-proximation T NN ( ~k, ξ, ξ ′ ) = T NN ( ~k, ξ, ξ ) δ ( ξ − ξ ′ ) that corresponds to instantaneous transferof impulse and charge, thus, R dξT NN ( ~k, ξ, ξ ′ ) h δ ˆ N d ( ~k, ξ ′ ) i t ≈ T NN ( ~k, ξ, ξ ) h δ ˆ N d ( ~k, ξ ) i t . Dy-namic variable h δ ˆ N d ( ~k, ξ ) i t for the grains is a function of impulse and charge which changein the charging/discharging processes): T ( k ) = ϕ eenn ϕ einn ϕ eann − i Ω en ϕ enh ϕ enN ϕ ienn ϕ iinn ϕ iann − i Ω in ϕ inh ϕ inN ϕ aenn ϕ ainn ϕ aann − i Ω an ϕ anh ϕ anN − i Ω en − i Ω in − i Ω an ϕ − i Ω h − i Ω N ϕ ehn ϕ ihn ϕ ahn − i Ω h ϕ hh ϕ hN ϕ eNn ϕ iNn ϕ aNn − i Ω N ϕ Nh T NN ( ~k,~p,Z ) , where T NN = − i Ω NN + ϕ NN .The calculation of collective modes reduces to finding the eigenvalues of this matrix.Since it is difficult to obtain exact analytical expressions for eigenvalues of matrix (4.4) weuse approximate calculations. Conventionally, in the limit of small k , eigenvalues can befound as a series over wave vector z = z + z k + z k . On the other hand, when certaincross-correlations are small the perturbation theory for collective modes [38, 39] can bedeveloped. Herewith, for the sake of simplification of calculations it is convenience to passfrom variables (4.1)–(4.2) to the completely orthogonal set of variables:ˆ n (1) ~k = (cid:18) ˆ n e~k − S ei ( k ) S ii ( k ) ˆ n i~k − S ea ( k ) S aa ( k ) ˆ n a~k (cid:19) /C ( k ) , (4.4)ˆ n (2) ~k = (cid:18) ˆ n i~k − S ia ( k ) S aa ( k ) ˆ n a~k (cid:19) /C ( k ) , (4.5)ˆ n (3) ~k = ˆ n a~k /C ( k ) , (4.6)ˆ ~J ~k = ˆ ~ ~k / h ˆ ~ ~k ˆ ~ − ~k i / , (4.7)ˆ H ~k = ˆ h ~k / h ˆ h ~k ˆ h − ~k i / , (4.8)where S αγ ( k ) = h ˆ n α~k ˆ n γ − ~k i are the partial structure factors. Constants C α ( k ) should pro-vide the normalization h ˆ n ( α ) ~k ˆ n ( α ) − ~k i = 1. The kinetic variable is orthogonal to this set.Then, the generalized hydrodynamic matrix in new variables in determined by the rela-19ion ˜ T ( k ) = L − ( k ) T ( k ) L ( k ) and now it is symmetric ( L ( k ) denotes the matrix of a lineartransformation): T ( k ) = k D k D k D − ikω (1) nJ k φ (1) nH k φ (1) nN ( ξ ) k D k D k D − ikω (2) nJ k φ (2) nH k φ (2) nN ( ξ ) k D k D k D − ikω (3) nJ k φ (3) nH k φ (3) nN ( ξ ) − ikω (1) Jn − ikω (2) Jn − ikω (3) Jn k D l − ikω JH − ikω JN ( ξ ) k φ (1) Hn k φ (2) Hn k φ (3) Hn − ikω HJ k D H k φ HN ( ξ ) k φ (1) Nn ( ξ ) k φ (2) Nn ( ξ ) k φ (3) Nn ( ξ ) − ikω NJ ( ξ ) k φ NH ( ξ ) − ikω NN ( ξ ) + k D NN ( ξ ) . Here, we extract the dependence of the frequency matrix and the matrix of memory functionson the wave vector in the hydrodynamic limit: i ˜Ω AA ( k ) = ik ˜ ω AA , ˜ ϕ AA ( k ) = k ˜ φ AA . D αγ arethe corresponding diffusion coefficients, φ ( α ) Hn and φ ( α ) nH describe thermodiffusion processes, D l is the longitudinal viscosity coefficient, D H = λ/C V , where C V and λ are the coefficients ofheat capacity and heat conductivity, respectively. D NN is the generalized diffusion coefficientfor dust particles dependent on ξ . Perturbation theory on correlations
According to the perturbation theory for collective modes [38, 39], in order to calculatecollective excitations in the zero approximation we chose the matrix in the following form: T ( k ) = k D k D k D − ikω (3) nJ − ikω (3) Jn k D l k D H
00 0 0 0 0 − ikω NN ( ξ ) + k D NN ( ξ ) . (4.9)Such a choice is caused by the fact that dynamic variable ˆ n is related to the particles numberdensity of neutral atoms (see (4.6)), concentration of which is 5–7 orders of magnitude largerthan ion and electron concentration. Respectively, the neutral component is the most energy-consuming and its contribution to the momentum density is the basic. That is why, the roleof the neutral subsystem in the sound spreading is determinative. Visco-thermal correlations20s well as correlations related to the dust component (due to small concentration) supposedto be small and can be taken into account as a perturbation.Thus, in the zero approximation we obtain the spectrum of collective excitations andcorresponding eigenvectors, namely:– two relaxation modes due to diffusion of plasma particles z (0) D = − k D , x D = (1 , , , , , , (4.10) z (0) D = − k D , x D = (0 , , , , , z (0) s ± = ± ikc − k Γ , x s ± = 1 √ , , ± , , , , (4.12)where c = (cid:16) ω (3) Jn ω (3) nJ (cid:17) / is the isothermal sound velocity, Γ = ( D l + D ) is the sounddamping coefficient in the zero approximation;– relaxation heat mode z (0) H = − k D H , x H = (0 , , , , , z (0) N = ikω NN ( ξ ) − k D NN ( ξ ) , x N = (0 , , , , , . (4.14)The cross-correlation define the perturbation matrix δT ( k ) = k D k D − ikω (1) nJ k φ (1) nH k φ (1) nN ( ξ ) k D k D − ikω (2) nJ k φ (2) nH k φ (2) nN ( ξ ) k D k D k φ (3) nH k φ (3) nN ( ξ ) − ikω (1) Jn − ikω (2) Jn − ikω JH − ikω JN ( ξ ) k φ (1) Hn k φ (2) Hn k φ (3) Hn − ikω HJ k φ HN ( ξ ) k φ (1) Nn ( ξ ) k φ (2) Nn ( ξ ) k φ (3) Nn ( ξ ) − ikω NJ ( ξ ) k φ NH ( ξ ) 0 . Now we can calculate the corrections to collective modes caused by the weak cross-correlations. According to [38, 39], such corrections equal to zero in the first order inperturbation, and in the second order are determined by the formula δz (2) α = X β δ ¯ T ∗ αβ δ ¯ T βα z (0) α − z (0) β , (4.15)21erewith, β runs all possible values but never equals α . δ ¯ T αβ is the perturbation matrix inrepresentation of eigenvectors of matrix T ( k ). δ ¯ T αβ = ( x α δT x β ) = P i,j x ∗ i,α δT ij x j,β , andthe conjugated matrix is determined by the relation δ ¯ T ∗ αβ = P i,j x ∗ i,β δT ij x j,α . When thenecessary elements δ ¯ T αβ are calculated using Eqs. (4.10)–(4.14) and taking into accountthe zero order results for collective modes Eqs. (4.10)–(4.14), we can calculate the desiredcorrections. For diffusive modes they have the form δz (2) D = − k D D D − D + φ (1) nH φ (1) Hn D − D H + ω (1) Jn ω (1) nJ Γ − D c ! , (4.16) δz (2) D = − k D D D − D + φ (2) nH φ (2) Hn D − D H + ω (2) Jn ω (2) nJ Γ − D c ! ; (4.17)for sound modes we obtain δz (2) s ± = ± ikc (cid:18) ∆2 ∓ ω JN ( ξ ) ω NJ ( ξ )2[ ω NN ( ξ ) ∓ c ] (cid:19) (4.18) − k (cid:26) − D l ∆2 1 c (cid:20) D H ω JH ω HJ + D ω (1) Jn ω (1) nJ + D ω (2) Jn ω (2) nJ + D ω (3) Jn ω (3) nJ (cid:18) − ∆2 (cid:19) (cid:21) + ω JN ( ξ ) ω NJ ( ξ )[Γ − D NN ( ξ )]2[ ω NN ( ξ ) ∓ c ] (cid:27) , ∆ = ω (1) Jn ω (1) nJ + ω (2) Jn ω (2) nJ + ω JH ω HJ c ;the correction for heat mode δz (2) H = − k φ (1) Hn φ (1) nH D H − D + φ (2) Hn φ (2) nH D H − D + ω HJ ω JH Γ − D H c ! ; (4.19)the correction for mode describing dust particles dynamics δz (2) N = ikω NN ( ξ ) ω JN ( ξ ) ω NJ ( ξ ) ω NN ( ξ ) − c (4.20) − k ω JN ( ξ ) ω NJ ( ξ ) [Γ − D NN ( ξ )] [ ω NN ( ξ ) + c ][ ω NN ( ξ ) − c ] . Now we are ready to write down the analytical expressions for collective excitations withcorrections. Thus, diffusive modes have the following form z D = − k D " − ω (1) Jn ω (1) nJ c T + D D D − D + φ (1) nH φ (1) Hn D − D H + Γ ω (1) Jn ω (1) nJ c T ! , (4.21)22 D = − k D " − ω (2) Jn ω (2) nJ c T + D D D − D + φ (2) nH φ (2) Hn D − D H + Γ ω (2) Jn ω (2) nJ c T ! . (4.22)Sound excitations are as follows: z s ± = ± ikc (cid:18) ∓ ω JN ( ξ ) ω NJ ( ξ )2[ ω NN ( ξ ) ∓ c ] (cid:19) (4.23) − k (cid:26) D l (cid:18) − ∆2 (cid:19) c (cid:20) D H ω JH ω HJ + D ω (1) Jn ω (1) nJ + D ω (2) Jn ω (2) nJ + D ω (3) Jn ω (3) nJ (cid:18) − ∆2 (cid:19) (cid:21) + ω JN ( ξ ) ω NJ ( ξ )[Γ − D NN ( ξ )]2[ ω NN ( ξ ) ∓ c ] (cid:27) . Heat mode has the form z (2) H = − k D H (cid:20) − ω HJ ω JH c T (cid:21) + φ (1) Hn φ (1) nH D H − D + φ (2) Hn φ (2) nH D H − D + Γ ω HJ ω JH c T ! . (4.24)For relaxation mode for dust particles we obtain z N = ikω NN ( ξ ) (cid:20) ω JN ( ξ ) ω NJ ( ξ ) ω NN ( ξ ) − c (cid:21) (4.25) − k ( D NN + ω JN ( ξ ) ω NJ ( ξ ) [Γ − D NN ( p )] [ ω NN ( ξ ) + c ][ ω NN ( ξ ) − c ] ) . Analyzing the obtained results we can see that taking into account cross-correlationsslightly modifies the collective modes of the system. Thus, contributions from dynamics ofdust subsystem effect the sound velocity and coefficient of its attenuation in the system. Con-sidering cross-correlations is manifested on the relaxation modes for dust particles. Herewith,dynamics of grains does not effect diffusive and heat modes. Considering cross-correlationswe can take into account the dynamics of all components, in particular dust component, inthe process of sound spreading and attenuation. It is worth noting that sound modes in thesecond order in correlation are now not complex conjugated (due to [ ω NN ( ξ ) ∓ c ] in Eq.(4.22)), and are characterized by the different velocity and sound damping. Since c is theisothermal sound velocity in the neutral component, then, ω NN ( ξ ) can be defined as “sound”for dust subsystem dependent on ξ . Then, the expression [ ω NN ( ξ ) ∓ c ] in the denominatorsof terms for sound velocity and damping strengthens or weakens acoustic excitations. If weneglect the contribution of “sound” for dust ([ ω NN ( ξ ) ∓ c ] = 0) we reproduce two conju-gated collective modes. It is important to note that sound modes and relaxation mode forgrains depend on the value of charge and impulse of dust grains, and, therefore change inthe charging/discharging processes. 23eglecting the contribution from the dust subsystem we can reproduce the well-knownexpression for collective modes in multicomponent system within the hydrodynamic descrip-tion [43]. Within the description in terms of partial dynamics, besides modes mentionedabove, we also obtain relaxation excitations of kinetic type related with interspecies inter-action. V. CONCLUSIONS
In summary, in the present paper we proposed another approach to the description ofsuch a complicated system as dusty plasma based on the Zubarev NSO method. Since thesubsystems of dusty plasma are in different states, such an approach allows us to take intoaccount consistently kinetics of dust particles and hydrodynamics of electrons, ions andneutral atoms. Within the partial dynamics we constructed the nonequilibrium statisticaloperator of duty plasma and using it we obtained the generalized transport equations, whichconsistently take into account kinetic and hydrodynamic processes in the system. For thecase of weakly nonequilibrium processes when the deviations of the nonequilibrium thermo-dynamic parameters from their equilibrium values are small, the set of transport equationsis obtained. Analogous system of equations can be obtained for equilibrium time correlationfunctions built on the dynamic variables of a reduced description. In such an approximationthis set of equations is closed.Alternative to partial variables can serve variables of partial number densities, totalmomentum and total energy (enthalpy) densities. Such a set of variables can be usedwhen investigate a spectrum of collective excitations of isothermal plasma in hydrodynamiclimit. since investigation of collective modes in terms of partial variables is more suitablebeyond the of hydrodynamic region. Using the perturbation theory for collective modesin correlations we found a spectrum of collective excitations of dusty plasma. Herewith,in the second order in perturbation in diffusive and heat modes the terms connected withinteraction of plasma particles and dust component do not appear. Considering the dynamicsof grains is manifested in the renormalization of sound velocity and dumping, and thelatter are different for both modes. Neglecting the correlations with dust subsystem we canreproduce the well-known expression for collective modes in multicomponent system [43].24
PPENDIX A
First, we consider the case of partial dynamic variables. In calculation of correlationfunctionΦ NN ( ~k, x, x ′ ) = h ˆ N ~k ( ~p, Z ) ˆ N − ~k ( ~p ′ , Z ′ ) i = h (1 − P )ˆ n d~k ( ~p, Z )(1 − P )ˆ n d − ~k ( ~p ′ , Z ′ ) i (A.1)the static correlation function equals h ˆ n d~k ( ~p, Z )ˆ n d − ~k ( ~p ′ , Z ′ ) i = nδ ( ~p − ~p ′ ) δ ( Z − Z ′ ) f ( p ) f ( Z )+ n d f ( p ) f ( p ′ ) f ( Z ) f ( Z ′ ) h d ( k ) (A.2)with equilibrium distributions in impulse and charge f ( p ) = (cid:18) β πm (cid:19) / exp (cid:18) − β p m (cid:19) , f ( Z ) = (cid:18) βe πa d (cid:19) / exp (cid:18) − β Z d e a d (cid:19) . (A.3)Taking into account new variable ξ we can write down: h ˆ n d~k ( ξ )ˆ n d − ~k ( ξ ′ ) i = nδ ( ξ − ξ ′ ) f ( ξ ) + n d f ( ξ ) f ( ξ ′ ) h d ( k ) = Φ dnn ( ~k, ξ, ξ ′ ) , (A.4)where h d ( k ) is the correlation function for dust particles related with the direct correlationfunction h d ( k ) = c d ( k ) (cid:2) − n d c d ( k ) (cid:3) − , [Φ dnn ( ~k, ξ, ξ ′ )] − = δ ( ξ − ξ ′ ) n d f ( ξ ′ ) − c d ( k ) . (A.5)Let us calculate the action of the projection operatorˆ N ~k ( ξ ) = (1 − P )ˆ n d~k ( ξ ) = ˆ n d~k ( ξ ) − P ˆ n d~k ( ξ ) . (A.6) P ˆ n d~k ( ξ ) = X α,γ h ˆ n d~k ( ξ )ˆ n α − ~k i [Φ − nn ( ~k )] αγ ˆ n γ~k + X γ h ˆ n d~k ( ξ )ˆ ~ γ − ~k i [Φ − ( ~k )] γγ ˆ ~ γ~k + X α,γ h ˆ n d~k ( ξ )ˆ h α − ~k i [Φ − hh ( ~k )] αγ ˆ h γ~k = f ( ξ ) X α,γ h ˆ n d~k ˆ n α − ~k i [Φ − nn ( ~k )] αγ ˆ n γ~k + f ( ξ ) X α,γ h ˆ n d~k ˆ h α − ~k i [Φ − hh ( ~k )] αγ ˆ h γ~k (A.7)with partial structure factorsΦ dγ ( k ) = h ˆ n d~k ˆ n γ − ~k i , Φ dγnh ( k ) = h ˆ n d~k ˆ h γ − ~k i . (A.8)25ntroducing the normalized static correlation functions¯Φ dγnn ( ~k ) = X α Φ dαnn [Φ − nn ( ~k )] αγ , ¯Φ dγnh ( ~k ) = X α Φ dαnh [Φ − hh ( ~k )] αγ , (A.9)we obtain P ˆ n d~k ( ξ ) = f ( ξ ) X γ n ¯Φ dγnn ( ~k )ˆ n γ~k + ¯Φ dγnh ( ~k )ˆ h γ~k o ,P ˆ n d − ~k ( ξ ′ ) = f ( ξ ′ ) X γ n ¯Φ dγnn ( ~k )ˆ n γ − ~k + ¯Φ dγnh ( ~k )ˆ h γ − ~k o . (A.10)Further, we calculate contributions into the correlation functionΦ NN ( ~k, ξ, ξ ′ ) = h ˆ N ~k ( ξ ) ˆ N − ~k ( ξ ′ ) i = h ˆ n d~k ( ξ )ˆ n d − ~k ( ξ ′ ) i − h ˆ n d~k ( ξ ) P ˆ n d − ~k ( ξ ′ ) i −h P ˆ n d~k ( ξ )ˆ n d − ~k ( ξ ′ ) i + h P ˆ n d~k ( ξ ) P ˆ n d − ~k ( ξ ′ ) i . (A.11)Taking into account Eq. (A.9) it easily to show that h ˆ n d~k ( ξ ) P ˆ n d − ~k ( ξ ′ ) i = h P ˆ n d~k ( ξ )ˆ n d − ~k ( ξ ′ ) i = h P ˆ n d~k ( ξ ) P ˆ n d − ~k ( ξ ′ ) i = f ( ξ ) f ( ξ ′ ) G ( k ) , (A.12)where G ( k ) = X γ n ¯Φ dγnn ( ~k )Φ γdnn ( ~k ) + ¯Φ dγnh ( ~k )Φ γdnh ( ~k ) o . (A.13)Finally, we obtainΦ NN ( ~k, ξ, ξ ′ ) = n d δ ( ξ − ξ ′ ) f ( ξ ′ ) + f ( ξ ) f ( ξ ′ )[ n d h d ( k ) − G ( k )]= n d δ ( ξ − ξ ′ ) f ( ξ ′ ) + f ( ξ ) f ( ξ ′ ) ¯ G ( k ) , (A.14)where ¯ G ( k ) = n d h d ( k ) − G ( k ) . (A.15)Function inverse to Φ NN ( ~k, ξ, ξ ′ ) we find from the condition Z dξ ′′ Φ − NN ( ~k, ξ, ξ ′′ )Φ NN ( ~k, ξ ′′ , ξ ′ ) = δ ( ξ − ξ ′ ) . (A.16)The final result isΦ − NN ( ~k, ξ, ξ ′ ) = δ ( ξ − ξ ′ ) n d f ( ξ ′ ) − B ( k ) , B ( k ) = n d ¯ G ( k ) n d (cid:2) n d + ¯ G ( k ) (cid:3) . (A.17)In the case of conservative variables, for Φ − NN ( ~k, ξ, ξ ′ ) we obtain similar expression, theonly difference is ¯ G ( k ) replaced with ¯ G c ( k ), where¯ G c ( k ) = n d h d ( k ) − G c ( k ) , G c ( k ) = X γ n (cid:16) Φ γdnn ( ~k ) (cid:17) + (cid:16) Φ γnH ( ~k ) (cid:17) o . (A.18)26 PPENDIX B
Here, we calculate the elements of the generalized hydrodynamic matrix (4.9). Let startwith frequency matrix i Ω AB ( k ) = ik ω AB ( k ), which in new completely orthogonal variables(4.4)–(4.8) is symmetric and whose elements are defined by the relation i Ω AB ( k ) = h ˙ˆ A ~k ˆ B − ~k i . For instance, let us consider i Ω (3) nJ ( k ) = i Ω (3) Jn ( k ) = h ˙ˆ n (3) ~k ˆ ~J − ~k i = h ˙ˆ n a~k ˆ ~ − ~k i h ˆ n a~k ˆ n a − ~k i / h ˆ ~ ~k ˆ ~ − ~k i / = ikm a h ˆ ~ a~k ˆ ~ a − ~k i h ˆ n a~k ˆ n a − ~k i / h ˆ ~ ~k ˆ ~ − ~k i / . (B.19)The static correlation function can be easily calculated h ˆ ~ ~k ˆ ~ − ~k i = X l h ˆ ~ l~k ˆ ~ l − ~k i = 1 β X l N l m l = 1 β ( N a m a + N i m i + N e m e ) , (B.20)where the index l = a, i, e . Taking into account that in our case m a ≃ m i ≫ m e , N a ≫ N i ≃ N e (B.21)we obtain an approximate result h ˆ ~ ~k ˆ ~ − ~k i ≃ N a m a β . (B.22)Thus, we obtain for the following elements: i Ω (3) nJ ( k ) = i Ω (3) Jn ( k ) = − ikm α h ˆ ~ a~k ˆ ~ a − ~k i / h ˆ n a~k ˆ n a − ~k i / = − ik (cid:18) N a βm a S aa (cid:19) / , (B.23) i Ω (2) nJ ( ~k ) = i Ω (2) Jn ( ~k ) = − ik (cid:18) N a βm a C (cid:19) / (cid:18) N i N a − f ia f aa (cid:19) , (B.24) i Ω (1) nJ ( ~k ) = i Ω (1) Jn ( ~k ) = − ik (cid:18) N a βm a C (cid:19) / (cid:18) N e N a − α N i N a − δ (cid:19) , (B.25) i Ω HJ ( ~k ) = i Ω JH ( ~k ) = − ik α P κ T (cid:18) T Vn ¯ mC V (cid:19) / . (B.26)Here, we use the following notations: C = 1 S aa (cid:20) S aa S ee − S ea − ( S aa S ei − S ia S ea ) S aa S ii − S ia (cid:21) ,C = S ii − S ia S aa , C = S aa ,δ = S ia S aa , δ = S aa S ei − S ia S ea S aa S ii − S ia , δ = S ea S aa − δ δ . k φ AB ( ~k ; t, t ′ ) = h I A ( ~k ) T ( t, t ′ ) I B ( ~k ) i and in our case should be calculated in theMarkovian approximation: φ = D aa S − aa , (B.27) φ = C − (cid:2) D ii − δ D ia + δ D aa (cid:3) , (B.28) φ = ( C C ) − / [ D ia − δ D aa ] , (B.29) φ = ( C C ) − / [ D ea − δ D ia − δ D aa ] , (B.30) φ = ( C C ) − / [ D ei − δ D ea − δ D ii + ( δ − δ ) D ia + δ δ D aa ] , (B.31) φ = ( C ) − (cid:2) D ee + δ D ii + δ D aa − δ D ei − δ D ea + 2 δ δ D ia (cid:3) , (B.32)where D ll ′ are the diffusion coefficients.The transport kernels φ ( α ) nH = φ ( α ) Hn = h I ( α ) n T ( t, t ′ ) I H i have the form φ (3) nH = φ (3) Hn = K a (cid:18) k B C V S aa (cid:19) / , (B.33) φ (2) nH = φ (2) Hn = ( K i − δ K a ) (cid:18) k B C V C (cid:19) / , (B.34) φ (1) nH = φ (1) Hn = ( K e − δ K i − δ K a ) (cid:18) k B C V C (cid:19) / (B.35)and are expressed via thermodiffusion coefficients K l . The transport kernels φ JJ = η L ¯ mn , (B.36) φ HH = λT c v (B.37)define coefficients of longitudinal viscosity and heat conductivity.Further, we calculate the correlations between plasma particles and the grains subsystem.For coefficients φ ( α ) Nn we obtain: φ ( α ) Nn ( ξ ) = φ ( α ) nn ( ξ ) − f ( ξ ) X β Φ dβnn φ βα − f ( ξ )Φ dnH φ ( α ) Hn , (B.38)where φ βα are defined by the relations (B.27)–(B.32), φ ( α ) Hn – by Eqs. (B.33)–(B.35). φ ( α ) nn ( ξ )is the diffusion coefficient (in momentum space) constructed similarly to Eqs. (B.27)–(B.32).In analogous way φ ( α ) nN ( ξ ) = Z n φ ( α ) nn ( ξ ′ ) − f ( ξ ′ ) X β Φ dβnn φ βα − f ( ξ ′ )Φ dnH φ ( α ) Hn o Φ − NN ( ξ ′ , ξ ) dξ ′ . (B.39)28he generalized diffusion coefficient for dust particles has the following form φ NN ( ξ, ξ ′ ) = Z n φ nn ( ξ, ξ ′′ ) − f ( ξ ′′ ) X β Φ dβnn φ β ( ξ ) − f ( ξ ) X β Φ dβnn φ β ( ξ ′′ ) (B.40) − f ( ξ ′′ )Φ dnH φ nH ( ξ ) − f ( ξ )Φ dnH φ nH ( ξ ′′ ) + f ( ξ ) f ( ξ ′′ ) X α,β Φ dαnn Φ dβnn φ αβ +2 f ( ξ ) f ( ξ ′′ ) X α Φ dαnn Φ dnH φ αnH + f ( ξ ) f ( ξ ′′ )(Φ dnH ) φ HH o Φ − NN ( ξ ′ , ξ ) dξ ′ , where φ nn ( ξ, ξ ′′ ) is a “pure” diffusion coefficient in momentum space for dust grains.For element i Ω NN = ikω NN we can obtain i Ω NN ( xi, xi ′ ) = − i~k · ~pm d ( f ( ξ ) δ ( ξ − ξ ′ ) f ( ξ ′ ) (B.41)+ n d f ( ξ ) (cid:2) h d ( k ) − B ( k ) (cid:3) − n d f ( ξ ) B ( k ) h d ( k ) ) . The last functions to be calculated are i Ω NJ ( xi ) = − f ( ξ ) "X α Φ dαnn i Ω αnJ + Φ dnH i Ω HJ ,i Ω JN ( xi ) = − "X α Φ dαnn i Ω αnJ + Φ dnH i Ω HJ − n d B ( k ) n d , where elements i Ω nJ and i Ω HJ are defined by the relations Eqs. (B.23)–(B.26). [1] V.N. Tsytovich, J. Winter, Phys.-Usp., 1998, , 815.[2] V.N. Tsytovich, Phys.-Usp., 1997, , 53.[3] Clark R.A., Sheldon R.B. Dusty plasma based fission fragment nuclear reactor. In: 41 stAIAA/ASMA/SAE/ASEE, Joint Propulsion Conf., Exhibit, July 10-15, 2005, Tucson, AZ,p.1-7.[4] V.E. Fortov, A.G. Khrapak, S.A. Khrapak, V.I. Molotkov, O.F. Petrov, Phys.-Usp., 2004, ,447.[5] Fortov V.E., Ivlev A.V., Khrapak S.A., Khrapak A.G., Morfill G.E., Phys. Rep., 2005, ,1.[6] Tsytovich V.N., Phys.-Usp., 2007, , 409.
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