Normal solution to the Enskog-Landau kinetic equation. Boundary conditions method A.E.Kobryn, I.P.Omelyan, M.V.Tokarchuk 9 9 Institute for Condensed Matter Physics 9 1 Ukrainian National Academy of Sciences, n 1 Svientsitskii St., UA–290011 Lviv–11, Ukraine a J 5 1 Abstract ] h Nonstationary and nonequilibrium processes are considered on the basis of an p Enskog-Landau kinetic equation using a boundary conditions method. A nonsta- - m tionary solution of this equation is found in the pair collision approximation. This s solution takes into account explicitly the influence of long-range interactions. New a l terms to the transport coefficients are identified. An application of the boundary p conditions method to hydrodynamic description of fast processes is discussed. . s c i s Key words: Nonequilibrium process, kinetic equation, transport coefficients y PACS: 05.60.+w, 05.70.Ln, 05.20.Dd, 52.25.Dg, 52.25.Fi h p [ 1 v The development of methods to construct a theory for nonequilibrium pro- 0 cesses in dense gases, liquids and plasmas is an important direction in the 2 0 modern theoretical physics. Moreover, the construction of kinetic equations 1 for such classical and quantum systems still remains to be a major problem 0 9 in the kinetic theory. It is complicated additionally in the case of dense gases, 9 liquids and plasmas, where kinetics and hydrodynamics are closely connected / s and should be considered simultaneously [1–5]. c i s y An approach for construction of kinetic equations from the first principles of h statistical mechanics, namely from the Liouville equation, has been developed p in [6,7]. Another approach for obtaining kinetic equations fordense systems, : v which is based on ideas of papers [6,7], has also been proposed [1] and gener- i X alized in [2,3]. Here, the formulation of modified boundary conditions for the r a BBGKY hierarchy is used taking into account corrections connected with lo- cal conservation laws. On the basis of this sequential approach, a new Enskog- Landau kinetic equation has been obtained for an one-component system of charged hard spheres. There is a considerable interest of an application of this kinetic equation for description of transport processes in dense systems of Preprint submitted to Elsevier Preprint 2 February 2008 charged particles as well as in ion melts and electrolyte solutions. The normal solution and transport coefficients for this equation have been found in the pa- per [8] using the Chapman-Enskog method. The same approach has been used for a many-component system of charged hard spheres in the paper [9] with more detailed calculations for a two-component system as well. At the same time, as is well known, the Chapman-Enskog method allows one to find the transport coefficients in a stationary case only. Similar drawbacks are peculiar to the Grad method [10,11] which is oftenly used to solve kinetic equations next to the Chapman-Enskog method. Inthispaper,theEnskog-Landaukineticequationforasystemofchargedhard spheresisinvestigated. Tofindthenormalsolutioninanonstationarycase,the so-called boundary conditions method is used, which has been introduced in [12,13]. As a result, transport coefficients equations in the nonstationary case are written. A limiting case of the stationary process is considered. A brief comparison of the obtained transport coefficients with those known previously from the Chapman-Enskog method is given. Let us consider the Enskog-Landau kinetic equation for a one-component sys- tem of charged hard spheres [8]: ∂ ∂ +v f (x ;t) = I (x ;t)+I (x ;t)+I (x ;t), (1) 1 1 1 E 1 MF 1 L 1 (∂t ∂r1) where f (x ;t) is the one-particle distribution function. The right-hand side 1 1 of this equation is the so-called generalized Enskog-Landau collision integral, where each termcan beconsidered as aseparate collision integral. Their struc- ture are as follows: I (x ;t) is the collision integral of the Enskog theory RET [14]: E 1 I (x ;t) = σ2 drˆ dv Θ(rˆ g)(rˆ g) (2) E 1 12 2 12 12 × Z ′ ′ g (r ,r +rˆ σ;t)f (r ,v ;t)f (r +rˆ σ,v ;t) 2 1 1 12 1 1 1 1 1 12 2 − n g (r ,r rˆ σ;t)f (r ,v ;t)f (r rˆ σ,v ;t) , 2 1 1 12 1 1 1 1 1 12 2 − − o where σ is a hard sphere diameter, g denotes a vector of relative velocity for two particles, rˆ is a unit vector along the direction between centres of 12 particles 1 and 2, ′ v = v +rˆ (rˆ g), g = v v , 1 1 12 12 · 2 − 1 v′ = v rˆ (rˆ g), rˆ = r −1r ; 2 2 − 12 12 · 12 | 12| 12 2 I (x ;t) is the collision integral of the kinetic mean field theory KMFT MF 1 [15,16]: 1 ∂Φl( r ) ∂ I (x ;t) = dx | 12| g (r ,r ;t)f (x ;t)f (x ;t), (3) MF 1 m 2 ∂r ∂v 2 1 2 1 1 1 2 Z 1 1 where Φl( r ) is a long-range part of the interparticle interaction potential; 12 | | I (x ;t) is generalized Landau collision integral [2,8]: L 1 1 ∂ ∂Φl( r ) I (x ;t) = dx g (r ,r ;t) | 12| (4) L 1 m2∂v1 Z 2 2 1 2 " ∂r12 #× 0 ∂Φl( r +gt′ ) ∂ ∂ ′ 12 dt | | f (x ;t)f (x ;t). −Z∞ ∂r12 (∂v1 − ∂v2) 1 1 1 2   It is necessary to note that the quasiequilibrium binary correlation function g 2 takesintoaccount thefullinteractionpotential(hardcorepartpluslong-range Coulomb tail). One of a major problem at the correct derivation and solution of kinetic equa- tions is their consistency with local conservation laws of particle density (or mass), momentum, total energy and substantiation of hydrodynamic equa- tions and incomprehensible calculation of transport coefficients via molecular parameters. These conservation laws for classical systems in general have the structure as in [17]. To find a solution of the Enskog-Landau kinetic equation (1) using one or another method, there is necessary to take the advantage of local conserva- tion laws in corresponding approximations. So doing the expressions for ki- netic coefficients will be defined through densities for momentum flow tensor ↔ Π (r;t) and energy flow vector jE(r;t) on the basis of solution f1(x1;t) and corresponding approximations for g (r ,r ;t). As far as we find the solution 2 1 2 that corresponds to linear hydrodynamical transport processes by gradients of ↔ thermodynamical parameters, densities of momentum flow tensor Π (r;t) and energy flow vector j (r;t) could be determined immediately with the help of E kinetic equation(1) without generalformulasfrom[17].Tothis enditisconve- nient similarly to [2], to introduce the following hydrodynamical parameters: density n(r ;t) (or mass density ρ(r ;t)), hydrodynamical velocity V (r ;t) 1 1 1 and density of kinetic energy ω (r ;t). Multiplying initial kinetic equation (1) k 1 by hydrodynamical parameters and integrating with respect to v , one can 1 obtain the equations for these parameters in the form: 3 1 d ∂ ρ(r ;t) = V (r ;t), (5) ρ(r ;t)dt 1 −∂r 1 1 1 d ∂ ↔ ρ(r ;t) V (r ;t)= :P (r ;t), (6) 1 dt 1 −∂r 1 1 d ∂ ↔ ∂ ρ(r ;t) w (r ;t)= q(r ;t) P (r ;t) : V (r ;t), (7) 1 dt k 1 −∂r 1 − 1 ∂r 1 1 1 where ↔ ↔k ↔hs ↔mf ↔l P (r ;t) =P (r ;t)+ P (r ;t)+ P (r ;t)+ P (r ;t), (8) 1 1 1 1 1 q(r ;t) = qk(r ;t)+qhs(r ;t)+qmf (r ;t)+ql(r ;t) 1 1 1 1 1 are the total stress tensor and vector of heat flow correspondingly. They have additive structure and contain several terms, each of them is stipulated by the ↔hs influence from one of collision integrals [2,8]: P and qhs by Enskog collision ↔mf ↔l integral (2), P and qmf by collision integral of KMFT (3), P and ql by ↔k Landau collision integral (4), P and qk are pure kinetic contributions only. ↔l P (r ;t) and ql(r ;t) are new terms in the structure of (8) in comparison 1 1 with results of [18]: ↔l P (r ;t) = (9) 1 1 Z4e4 ∂ r r r ∂ ∂ dv v dx 12 12 12 dλ Fl, m Z 1 1∂v1 Z 2 r152 · g (∂v1 − ∂v2)Z 0 ql(r ;t) = (10) 1 1 Z4e4 ∂ r r r ∂ ∂ dv c2 dx 12 12 12 dλ Fl, 2m Z 1 1∂v1 Z 2 r152 · g (∂v1 − ∂v2)Z 0 Fl = g f f [8]. A short comment is needed for (7). First of all equation (7) is 2 1 1 a balance equation for a kinetic part of total energy. To write the conservation law for total energy it is necessary to know also two-particle distribution func- tion f (x ,x ;t) next to one-particle one, because the potential part of total 2 1 2 energy is expressed via f . The Enskog-Landau kinetic equation in “pair col- 2 lision” approximation has been obtained from the BBGKY hierarchy with a modified boundary condition in [2], where the expression for f is also pointed 2 out. An average value for the potential energy and its flow one should be cal- culated onthebasis ofthis expression. Then, adding it to thebalance equation (7), one can obtain the conservation law for total energy. 4 We shall construct a normal solution to the Enskog-Landau kinetic equation (1) using the boundary conditions method [12,13]. Following this method, let us bring into the right-hand side of equation (1) an infinity small source with ε +0: → ∂ ∂ +v f (x ;t) = (11) (∂t 1∂r1) 1 1 (0) I (x ;t)+I (x ;t)+I (x ;t) ε f (x ;t) f (x ;t) , E 1 MF 1 L 1 − 1 1 − 1 1 (cid:16) (cid:17) (0) where f (x ;t) is some already known one-particle distribution function sat- 1 1 isfying equations (5) – (7) for parameters of reduced description of our system. (0) Then the solution can be found in the form f (x ;t) = f (x ;t)+δf (x ;t) 1 1 1 1 1 andsearch ofthenormalsolutionimplies treatment ofthecorrectionδf (x ;t). 1 Substituting f (x ;t) into (11), one can obtain: 1 1 ∂ ∂ D +v +ε δf + f(0) = I f(0) +I (δf)+ (12) (∂t 1∂r1 ) Dt 1 MF 1 MF (cid:16) (cid:17) (0) (0) (0) (0) (0) (0) I f ,f +I f ,f +I f ,δf +I δf,f + E 1 1 L 1 1 E 1 E 1 (cid:16) (0) (cid:17) (cid:16) (0) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) I f ,δf +I δf,f +I (δf,δf)+I (δf,δf). L 1 L 1 E L (cid:16) (cid:17) (cid:16) (cid:17) Conventional signs used in the equation (12) are obvious [2,8,9]. Also the fact was taken into account about I (x ;t), collision integral (3), which is a MF 1 functional of one-particle distribution function only. Terms with the subscript E arenonlocal,thereforeinfurthercalculationsweshouldtaketheirexpansion with respect to the local one-particle distribution function and cut-off this expansion by terms with degrees higher than δf. In the case when terms with subscripts MF and L also mean nonlocal functionals, one should apply mentioned above procedure to them too. Let us combine some terms in (12): (0) (0) I (δf) = I f ,δf +I δf,f linearized nonlocal Enskog colli- E E 1 E 1 (cid:16) (cid:17) (cid:16) (cid:17) sion functional, (0) (0) I (δf) = I f ,δf +I δf,f linearized Landau collision func- L L 1 L 1 (cid:16) (cid:17) (cid:16) (cid:17) tional. (0) Now let us designate L (δf) = I (δf) + I (δf) + I (δf) and introduce t E MF L ′ an operator S(t,t) with the following properties: ∂ ′ ′ ′ S(t,t) = Lt(δf)S(t,t), S(t,t) t′=t = 1. ∂t | 5 ′ Using these properties of operator S(t,t), one can represent equation (12) in an integral form. Having correction δf(x;t) in an integral form, it is easy to cross to itemizing procedure for finding it in corresponding approximation. For example, it can be organized in the following way: t D ∂ δf(k+1)(x ;t) = dt′ e−ε(t−t′)S(t,t′) f(0) v δf(k) + (13) 1 −Z∞ (−Dt 1 − 1∂r1 I f(0),f(0) +I f(0) +I f(0),f(0) +I(1)(δf(k)) , E 1 1 MF 1 L 1 1 E (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) )t′ ′ where subscript t at the bottom of right brase means that integrated expres- ′ sion is a function of t. An additional condition to find δf(x;t) is the evident limit lim δf(x;t) = 0.In order to construct the (k+1)-thapproximation it is t→−∞ necessary to use the fact that δf = 0 and the conservation laws (or equa- |k=0 tions for reduced description parameters) in k-th approximation. To realize this procedure a zeroth approximation for the one-particle distribution func- (0) (0) tion f (x ;t) is needed. In the case of spherical charged particles, f (x ;t) 1 1 1 1 can be chosen as the local-equilibrium Maxwell distribution function m 3/2 mc2(r ;t) f(0)(x ;t) = n(r ;t) exp 1 1 . 1 1 1 2πkT (r1;t)! (−2kT (r1;t)) (0) Let us find a correction to the distribution function f (x ;t) using itemizing 1 1 procedure (13). Calculating and obtaining of conservation laws (5), (6) and equation (7), we should take into account the following relations: g (r ,r ;t) g r ,r ;n(t),β(t) g r ;n(t),β(t) , 2 1 2 2 1 2 2 12 ≡ → (14) (cid:16) (cid:17) (cid:16) (cid:17) F F(0) = g r ;n(t),β(t) f(0)(x ;t)f(0)(r ,v ;t), → 2 12 1 1 1 1 2 (cid:16) (cid:17) where g (r ;n(t),β(t)) is the binary quasiequilibrium correlation function, 2 12 which depends on relative distance between particles. We obtain for stress tensor and heat flow vector: ↔k ↔ P =I Pk, Pk = nkT, (15) ↔hs ↔ 2 P =I Phs, Phs = πn2σ3kTg (σ n,β), (16) 2 3 | ∞ ↔mf ↔ 2 dr P =I Pmf, Pmf = π(nZe)2 g (r n,β), (17) 2 −3 r | Zσ 6 ↔l P = 0, qk = qhs = qmf = ql = 0. (18) ↔ ↔l In these expressions I is the unit tensor, P and ql are equal to zero because the integration between symmetrical limits goes over odd function. As far as ↔k ↔hs ↔mf ↔l calculated components P (r ;t), P (r ;t), P (r ;t) and P (r ;t) (15) – 1 1 1 1 (18) are known, one can write total pressure in the zeroth approximation: ∞ 2 2 dr P = nkT 1+ πnσ3g (σ n,β) π(nZe)2 g (r n,β). 2 2 3 | − 3 r | (cid:18) (cid:19) Zσ Calculating expressions in brackets on the right hand side in (13), one can write total expression for correction δf(x ;t) in first approximation: 1 t δf(1)(x ;t) = dt′ e−ε(t−t′)S(t,t′) f(0)(x ;t) (19) 1 − " 1 1 × −Z∞ 2 mc2 5 ∂ 1+ πnσ3g (σ n,β) 1 c lnT (r ;t)+ ((cid:18) 5 2 | (cid:19)"2kT − 2# 1∂r1 1 4 m 1 ↔ ∂ 1+ πnσ3g (σ n,β) c c c2 I : V (r ;t) . (cid:18) 15 2 | (cid:19) kT (cid:20) 1 1 − 3 1 (cid:21) ∂r1 1 )#t′ Terms related to short-range interactions only contribute evidently into the correction inthis approximation. Contrary tothe kinetic theoryof dilute gases particle sizes take part here [6,7,11], where particles are considered as point- like objects. Nevertheless, the influence of both long-range and short-range ′ parts of interactions are also “hidden” in operator S(t,t) (through operator L ). Formally, the expression (19) looks completely the same as the correction t ′ in [18]. But a difference lies in the structure of the operator S(t,t). Having total expression for correction δf(x ;t) in the first approximation (19) 1 one can calculate conservation laws (5), (6) and equation (7) in the same ap- proximation.Therefore,itisnecessary,first,toobtainrelationsfordetermining ↔k1 quantities (7) in which the correction (19) can be engaged. For P (r ;t) we 1 obtain: t P↔k1 (r ;t) =↔I Pk dt′ e−ε(t−t′)Mk(t,t′) ↔S , (20) 1 −−Z∞ (cid:20) (cid:21)t′ where S is a velocity shift tensor, αβ m Mk(t,t′) = dv c c S(t,t′) (21) 1 1 1 5 × Z 7 4 m 1 ↔ f(0)(x ;t) 1+ πnσ3g (σ n,β) c c c2 I (cid:20) 1 1 (cid:18) 15 2 | (cid:19) kT (cid:18) 1 1 − 3 1 (cid:19)(cid:21)t′ is a kernel of kinetic part of the transport equations. ↔hs1 ↔mf 1 ↔l1 For calculating P (r ;t), P (r ;t) and P (r ;t), we have to expand 1 1 1 Fhs, Fmf and Fl on inhomogeneity and deviation δf (x ;t) and keep in the 1 series initial terms only. The expansion for Fmf, Fl reads the same as for Fhs with changing g (σ n,β) g (r n,β), rˆ r −1r , σ r . The 2 2 12 12 12 12 12 | → | → | | → | | calculations show: ↔hs1 ↔hs 4 6 ↔ ↔ P (r ;t) =P n2σ4g (σ n,β)√πmkT S +( V ) I (22) 1 2 −9 | 5 ∇· − (cid:20) (cid:21) t 4 ↔ πnσ3g (σ n,β) dt′ e−ε(t−t′)Mk(t,t′) S , 2 15 | −Z∞ (cid:20) (cid:21)t′ ↔mf 1 ↔ P (r ;t) =I Pmf. (23) 1 A mean field influence into the total stress tensor remains the same as in ↔l1 zeroth approximation. Similar situation arises as to P (r ;t): 1 ↔l1 ↔l P (r ;t) =P (r ;t) = 0. (24) 1 1 Total expression for stress tensor in the first approximation is a sum of (20), (22), (23) and (24): ↔ ↔ 4 6 ↔ ↔ P (r ;t) =I P(r ;t) n2σ4g (σ n,β)√πmkT S +( V ) I 1 1 2 − 9 | 5 ∇· − (cid:20) (cid:21) t 4 ↔ 1+ πnσ3g (σ n,β) dt′ e−ε(t−t′)Mk(t,t′) S . 2 (cid:18) 15 | (cid:19)−Z∞ (cid:20) (cid:21)t′ The calculations for heat flow vectors give: t 1 qk1(r ;t) = dt′ e−ε(t−t′)Lk(t,t′) T , (25) 1 −−Z∞ (cid:20)T∇ (cid:21)t′ 2 πk3T qhs1(r ;t) = n2σ4g (σ n,β) T (r ;t) (26) 1 2 1 −3 | s m ∇ − t 2 1 πnσ3g (σ n,β) dt′ e−ε(t−t′)Lk(t,t′) T , 2 5 | −Z∞ (cid:20)T∇ (cid:21)t′ 8 ql1(r ;t) = ql(r ;t) = 0. (27) 1 1 Here 1 mc2 Lk(t,t′) = dv c 1S(t,t′) (28) 1 1 3 2 × Z 2 mc2 5 f(0)(x ;t) 1+ πnσ3g (σ n,β) 1 c " 1 1 (cid:18) 5 2 | (cid:19) 2kT − 2! 1#t′ is another kernel of kinetic part of transport equations. Total expression for heat flux vector is a sum of (25) – (27): 2 πk3T q(r ;t) = n2σ4g (σ n,β) T (r ;t) 1 2 1 −3 | s m ∇ − t 2 1 1+ πnσ3g (σ n,β) dt′ e−ε(t−t′)Lk(t,t′) T . 2 (cid:18) 5 | (cid:19)−Z∞ (cid:20)T∇ (cid:21)t′ Now we can consider one of the limiting cases, namely, the stationary process, ′ ′ when the operator L does not depend on time, i.e. S(t,t) = exp L (t t) . t t ↔ { − } Some terms in expressions for P (r ;t) and q(r ;t) can acquire simpler form. 1 1 We can compare them with those from the Enskog-Landau kinetic equation for one-component system of charged hard spheres with using the Chapman- Enskog method in the case, when in a long-range part of the collision inte- gral we put g (σ n,β) 1. It should be noted that bulk viscosity has the 2 | → same structure as in the Chapman-Enskog method [8]. But other transport coefficients exhibit some distinctions. The structure for shear viscosity η and thermal conductivity λ is: 4 2 1+ πnσ3g (σ n,β) 3 15 2 | η = æ+2 nkT (cid:26) (cid:27) , (29) 5 I(0)(δf)+I (δf) E L n o 2 2 1+ πnσ3g (σ n,β) 3k 5k 5 2 | λ = æ+ nkT (cid:26) (cid:27) . (30) 2m m I(0)(δf)+I (δf) E L n o (0) Then the problem lies in calculating collision integrals I (δf) and I (δf), E L this means that we should calculate collision integrals (2) (in the zeroth ap- proximation on inhomogeneity) and (4) together in the first approximation 9 on deviation δf, where δf is substituted from (19). The matter of some diffi- culty is that correction (19) in its turn is expressed also via collision integrals (0) ′ I (δf), I (δf), which are in the operator S(t,t). So the first acceptable ap- E L (0) proximation should be that, when correction δf (19) is expressed via I (δf), E ′ ′ ′ (0) I (δf) calculated with δf , where δf = δf at S(t,t) = 1. For I (δf) we L E obtain the results [8], for I (δf) in (29), (30) we can obtain the following: L Z4e4 ∂ r r 1 ∂ ∂ I (δf) = dr dv g (r ,r +r ;t) 12 12 L m2 ∂v1 Z 12 2 2 1 1 12 r152 g (∂v1 − ∂v2) f (x ;t)δf (r +r ,v ;t)+δf (x ;t)f (r +r ,v ;t) , (31) 1 1 1 12 2 1 1 1 12 2 × n o where δf (x;t) is evaluated from (19) with ′ ′ (0) ′ ′ ′ S(t,t) = exp L (t t) = exp [I (δf )+I (δf )](t t) t − E L − n o n o ′ at δf (x;t) = δf (x;t) ′ . This stage of calculations needs an ex- S(t,t) = 1 plicit form for the binary(cid:12) quasiequilibrium correlation function g both on the (cid:12) 2 (cid:12) contact and in r-space. The results of this paper (19), (29) and (30) will coincide completely with those from [8] when in a long-range part of the collision integral (31) one puts g (r ,r + r ) 1 and represents it in Boltzmann-like form. But the 2 1 1 12 ≡ used boundary conditions method has turned out more convenient than the Chapman-Enskog one [2,8]. As was discussed in details in [12] at construct- ing the normal solution for a kinetic equation using the boundary conditions method,timederivatives∂/∂tofhydrodynamicparametersofreduceddescrip- tion do not set to be small. Therefore, the normal solution to this equation could be used for hydrodynamic description of fast processes. References [1] D.N.Zubarev, V.G.Morozov, Teor. Mat. Fiz. 60(1984)270 (in Russian). [2] D.N.Zubarev, V.G.Morozov, I.P.Omelyan, M.V.Tokarchuk, Teor. Mat. Fiz. 87(1991)113 (in Russian). [3] D.N.Zubarev, V.G.Morozov, I.P.Omelyan, M.V.Tokarchuk, Teor. Mat. Fiz. 96(1993)325 (in Russian). [4] Yu.L.Klimontovich, Teor. Mat. Fiz. 92(1992)312 (in Russian). [5] Yu.L.Klimontovich, Phys. Lett. A 170(1992)434. 10