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VOLUME 76, NUMBER 19

PH YSIC AL R E VIE W LETTE RS

6M

AY

1996

Pair Correlation Function and Nonlinear Kinetic Equation for a Spatially Uniform Polarizable Nonideal Plasma
V. V. Belyi,1 Yu. A. Kukharenko,2 and J. Wallenborn
1 3

3

IZMIRAN, Troitsk, Moscow region, 142092, Russia 2 UITPRAN, Moscow, Russia Physique Statistique, Plamsas et Optique non LinÈaire, UniversitÈ Libre de Bruxelles, C.P. 231, 1050 Bruxelles, Belgium (Received 27 October 1995)

Taking into account the first non-Markovian correction to the Balescu-Lenard equation, we have derived an expression for the pair correlation function and a nonlinear kinetic equation valid for a nonideal polarized classical plasma. This last equation allows for the description of the correlational energy evolution and shows the global conservation of energy with dynamical polarization. [S0031-9007(96)00077-4]
PACS numbers: 52.25.Dg, 05.20.Dd

The importance of the polarization effects on plasmas in the kinetic regime has been recognized for a long time. A first attempt to include these effects in the linearized collisional integral was made by Gasirowicz, Neuman, and Riddell [1]. Later, the nonlinear equation for weakly coupled polarizable plasma was derived by Balescu [2] with the help of Prigogine's diagram techniques [3] and by Lenard [4] who solved the Bogoliubov equation [5] for the pair correlation function in the plasma approximation. These results were generalized to the quantum case by Konstantinov and Perel [6] and by Wyeld and Pines [7]. Kadanoff and Baym [8] and Klimontovich [9] have noticed that the Balescu-Lenard ( BL) equation takes into account the polarization of the system only in the collision integral while the thermodynamics corresponds to the ideal gas; the dissipative and nondissipative phenomena are not treated on an equal footing. They have shown that this discrepancy can be avoided if non-Markovian effects are taken into account. This means that Bogoliubov's condition [5] of the total synchronization of all correlation functions with the one-particle distribution function (df ) must be omitted. Klimontovich [10] wrote the system of equations for the one-particle df and the pair correlations for the electric field and for the charge density, but he did not solve the equation for the pair correlation function and thus did not obtain a closed kinetic equation. On the other hand, RÈsibois [11] and Dorfman and Cohen [12] have formally derived the fully non-Markovian generalization of the BL equation, but their results are not easily tractable in a practical case. In particular, they did not give the explicit expression for the correlation energy. In this Letter we solve, in the plasma approximation, the equation for the pair correlation function considering the first non-Markovian correction. In this way, we obtain a nonlinear kinetic equation which generalizes the BL equation for weakly nonideal plasma. This equation, which includes the dynamical screening of the interaction 3554 0031 - 9007 96 76(19) 3554(4) $10.00

potential, describes correctly the conservation of the total energy in a nontrivial way. The non-Markovian correction, which is responsible for the offset of the variation of kinetic energy by the variation of potential energy, is of order g 2 , where g is the plasma parameter, while the BL collision integral, which is the Markovian contribution in the plasma approximation, is of order g . The BL approximation is thus consistent with the conservation of kinetic energy without potential energy. However, Markovian contributions of order g 2 , which also conserve kinetic energy, are not included in the plasma approximation. We do not consider these contributions which give rise to a small quantitative correction of the system relaxation time; we focus only on the important qualitative effect related to the potential energy balance. The importance of nonlocal effects (in space and time) for the description of dense fluids has been recognized since the early works of Akcasu and Duderstadt [13] and Forster and Martin [14] on memory function formalism for the correlations of phase space density fluctuations. However, this formalism corresponds to the linearized kinetic theory near equilibrium. Far from equilibrium, memory effects are important when the relaxation time becomes of the same order of magnitude as the duration of a collision, that is in the case of dense systems in which the evolution of the potential energy must be considered. Recently, papers on non-Markovian nonlinear kinetic equations were published. They treat the first density correction to the Uehling-Uhlenbeck equation [15] and the case of the quantum Landau equation [16] which accounts for potential energy in lowest order only. Here, we present a kinetic equation that shows the global energy conservation in an approximation which, at equilibrium, is equivalent to the Debye-HÝckel one. We consider a spatially uniform weakly nonideal multicomponent plasma. Out of equilibrium, its evolution © 1996 The American Physical Society


VOLUME 76, NUMBER 19

PH YSIC AL R E VIE W LETTE RS fa p1 , t t Ja p1 , t 2

6M

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1996

can be described by a kinetic equation which is derived from the Bogoliubov-Born-Green-Kirwood-Yvon ( BBGKY ) hierarchy by making the so-called plasma approximation: triple correlation function as well as interaction between correlated particles are neglected. In this way, the set of equations for the one-particle df fa pa , t and for the pair correlation function gab pa , pb , ra , rb , t is closed. In Fourier space, they take the form 1 i k ? v1 2 v t ik ? F
ab

i XZ d k d p2 Fab k 8p 3 b 3k? gab p1 , p2 , k, t , p1 (1)

where Ja p1 , t is the collision integral and where gab p1 , p2 , k, t is the solution of the equation


2

gab p1 , p2 , k, t 2 i k ?
bc

X fb p2 , t F p2 c

k

Z

d p3 gac

Z X fa p1 , t Fac k d p3 gbc p2 , p3 , k, t 1 p1 c ! fa p1 , t fb p2 , t . p1 , p3 , k, t i Fab k k ? 2 p1 p2

(2)

k 4p ea eb k 2 is the Fourier transform of the Coulomb potential and v1 p1 ma is the velocity of particle 1. At this stage it is useful to introduce compact notations. Let us define a scalar product as XZ d p1 d p2 Aab p1 , p2 Bab p1 , p2 . (3) A?B
ab

and rewrite Eq. (2): where, for instance, Gt
aba 0 b
0

t

LI ? 1T t ? g t
p

At ,
0 d p2 2 p2
0

(4)

0 0 p1 , p2 , p1 , p2

2i k ?

1 ik ?

p2

1

fa p1 , t F

aa

0

kd

bb

0

fb p2 , t Fbb 0 k d

aa

0 d p1 2 p1 .

(5)

We solve Eq. (4) by using the so-called Green's function method [17] which should not be confused with the quantum field theory techniques. Taking into account the fast decay of initial correlations, we write gt Z
t

t 2 L? R0 t 2 t or t 2 L d t 2 t
0

0

d t 2 t0 I
2 R0 1 t 2 t 0 ,

(9)

(10)

dt 0 R t , t
0

0

? A t0 ,

(6)

2`

where we have introduced the inverse Green's function 2 R0 1 t 2 t 0 . Order one of Eq. (8) is t 2 L? R1 t , t
0

where the Green's function R t , t obeys the equation t 2 LI ? 1G t ? R t , t
0

1 C mt ? R0 t 2 t

0

0,

(11)

d t 2 t0 I .

(7)

When G t does not depend on time, R t , t 0 takes the simple form R t 2 t 0 . In order to obtain a Markovian kinetic equation, it is assumed that G t (i.e., the one particle df ) is a constant on the fast characteristic time scale of R t , t 0 and R t , t 0 is replaced by R t 2 t 0 . When non-Markovian effects are considered, the time dependence of R t , t 0 is more subtle, as we now show. Let us write G t G mt , where m is a parameter which specifies the distribution function time scale, and consider G mt 2 G 0 as a perturbation. Equation (7) C mt is then written as t 2 L ? R t , t with L
0

or using Eq. (10) Z` R1 t , t 0 2 dt1 R0 t 2 t1 ? C mt
2`

1

? R 0 t1 2 t 0 . (12)

Finally, by a change of integration variable t1 ! t1 1 t 0 , we obtain Z` R1 t , t 0 2 dt1 R0 t 2 t 0 2 t1 ? C mt1 1 mt 0
2`

? R0 t

1

R1 t 2 t 0 , mt 0 .

(13)

The procedure is easily iterated to any order in perturbation. We thus have R t, t and Eq. (6) becomes 3555
0

1 C mt ? R t , t

0

d t 2 t0 I

(8)

R t 2 t 0 , mt

0

(14)

LI 2 G 0 . At order zero, Eq. (8) becomes


VOLUME 76, NUMBER 19 gt Z
0 `

PH YSIC AL R E VIE W LETTE RS ?A mt2t . (15) g
ab

6M g
0 ab

AY

1996

dt R t, m t 2 t

p1 , p2 , k, t

p1 , p2 , k, t 1 g

1 ab

p1 , p2 , k, t , (20)

The Markovian limit of Eq. (15) is clearly obtained when t is large enough and when R t , m t 2 t is such that we can made t 2 t t in the integral. In the present Letter, we consider only this Markovian limit and the first non-Markovian correction in Eq. (15). Let us point out that it is not a trivial approximation since it leads to the total energy conservation. We proceed by expanding the rhs of Eq. (15): Z` d t 1 2 t t R t , mt ? A mt . (16) gt
0

0 where gab pa , pb , t is well known result of Markovian theory [17 ­ 19]:

g

0 ab

p1 , p2 , k, t

i pd2 k ? v1 2 k ? v2 3G
ab

p1 , p2 , k, t
p1

(21)

with Gab p1 , p2 , k, t Fab k
1 ´ k?v2 ,k

fb p2 , t k ? Z

3 fa p 1 , t 2 3 fb p 2 , t 2 3d 1d ? and ´ k ? v , k, 1 2 ip 3d
1 1

By introducing the resolvent R z , mt as the Laplace transform of R t , mt , Z` d t exp iz t R t , mt , Im z . 0, R t , mt
0

X
c

1 ´ k?v1 ,k

fa p1 , t k ? dp

p2

ip

Fcc k 3 j´ k?v3 ,k j2

k ? v3 2 k ? v1 k ? v2 2 k ? v3
p1

1

(17) Equation (16) takes the form (since from now on all quantities depend only on the df time scale we put m 1): µ R z, t ? A t . gt lim 1 1 i t (18) z !i ´ z Finally, using the identity we obtain gt
z !i ´ z

3 fc p3 , t k ?
p2

fa p1 , t k (22)

fb p2 , t X
a

Faa k

Z

d p1 fa p1 , t p1 (23)

R z, t

2R z , t ? R z , t ,

k ? v 2 k ? v1 k ?

lim R z , t ? A t 2 i t R z , t ? R z , t ? A t . (19) t constant in time R0 t 2 t 0 is given which are solutions e.g., Refs. [18,19]). straightforward, al-

The solution of Eq. (7) with G is well known. Green's function as a product of two propagators of the linear Vlasov equation (see, The remaining calculations are though tedious. The results are ( g
1 ab

is the dynamical dielectric function. We also used 1 x 2 i0 the conventional notations i pd2 x P 1 x 1 i0 2i pd x 1 i pd x 1 x , 2i pd1 x P , where P means the Cauchy principal part. It can x be shown that expression (21) reduces at equilibrium to the Debye-HÝckel pair correlation function. The non-Markovian correction to the pair correlation function is X
c

p1 , p2 , k, t

i

t

0 i pd2 k ? v1 2 k ? v2 Gab p1 , p2 , k, t 1

Fbc k ´ k?v1 ,k

i pd2 k ? v1 2 k ? v2 k ? X
c Fac k ´ k?v2 ,k

p2

3 fb p 2 , t 3k? 3F
bd p1

Z

0 d p3 i pd2 k ? v1 2 k ? v3 Gac p1 , p3 , k, t 2

i pd2 k ? v1 2 k ? v2 ip
P k?v1 2k?v2

fa p1 , t Z dp h
3

Z

0 d p3 i pd2 k ? v3 2 k ? v2 G

cb

p3 , p2 , k, t 1 fa p1 , t k ?

X
cd

Fac k

k

d2 k?v3 2k?v1 j´ k?v1 ,k j2 cd

1

d2 k?v3 2k?v2 j´ k?v2 ,k j2 )

i

k?

p1

p2

fb p2 , t

Z

0 d p4 i pd2

3 k ? v3 2 k ? v4 G

p3 , p4 , k, t

,

(24)

0 where d2 x is the derivative of d2 x . It is interesting to remark that the last three terms of Eq. (24) have a structure similar to Eq. (21) [with Eq. (22)] in which one-

particle df's have been replaced by correlated functions or in other words, uncorrelated propagators have been replaced by correlated ones.

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VOLUME 76, NUMBER 19

PH YSIC AL R E VIE W LETTE RS

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1996

The correction to the collision integral is obtained from Eq. (1): i XZ 1 d k d p2 Fab k k 23 Ja p1 , t 8p b 1 g p1 , p2 , k, t . (25) p1 ab The kinetic equation (1) including the collision integral (25) can be solved by using Grad's method, for instance. In the case of a two temperature plasma the nonMarkovian contribution results in a correction of the thermodynamic coefficients. The conservation laws are easily verified with the following symmetry property in mind: ? gab p1 , p2 , k, t gba p2 , p1 , 2k, t . (26)

At equilibrium, this expression becomes the DebyeHÝckel energy [17 ­ 19]. Equation (28) achieves the consistency of the theory. Let us conclude with two final remarks. First, the 1 expression of Ja p1 , t given by Eq. (25) with Eq. (24), which is very useful to verify the conservation laws, can be expressed in a more compact form which can be used with benefit in other applications. Second, in many problems, nonlocality in space must be considered as well as non-Markovian character (or nonlocality in time); we hope to consider this topic in the near future. We are grateful to Professor R. Balescu for helpful discussion. This work is supported by INTAS under Contract No. 1010­CT93­0015.

The particle conservation is a trivial consequence of expression (1) while the momentum conservation follows directly from Eqs. (1) and (26). We know that kinetic 0 energy is conserved by Ja p1 , t , the Balescu-Lenard collision integral. Therefore, the variation of the kinetic energy is due to the non-Markovian part of the collision 1 integral Ja p1 , t : ! 2 X p1 1 XZ t 23 d k d p1 d p2 Fab k 2ma 8p ab a
1 3 k ? v1 2 k ? v2 gab p1 , p2 , k, t ,

(27) where we used Eqs. (1), (25), and (26). The three last terms of the rhs of Eq. (24) give no contribution to the rhs of Eq. (27) since they are proportional to R 0. The first term leads to d p p fa p , t Ü 2¿ X p1 1 1 XZ d k d p1 d p2 2t t 8p 3 2 ab a 2ma
0 3 Fab k gab p1 , p2 , k, t .

(28)

The rhs of Eq. (28) is precisely minus the variation of the potential energy at the order of approximation we consider. This potential energy can be written explicitly as XZ 1 0 d k d p1 d p2 Fab k gab t , k, p1 , p2 U 3 16p
1 16p 1 32p

XZ
3

ab

d k d p2 Fbb k fb p2 , t d k d p1 d p2 1
2 Fab k k?v1 2k?v2

n

12j´ k?v2 ,k j j´ k?v2 ,k j2

2

o

XZ
3

b

3 3

h

ab 1 j´ k?v2 ,kj
2

1 j´ k?v1 ,kj2 fb k ? p1

i

fa 2 fa k ?

p2

fb .

(29)

[1] S. Gasirovicz, N. Neuman, and R. J. Riddell, Jr., Phys. Rev. 101, 922 (1956). [2] R. Balescu, Fluids 3, 52 (1960). [3] I. Prigogine, Non-Equilibrium Statistical Mechanics ( Wiley, New York, 1963). [4] A. Lenard, Ann. Phys. ( N.Y.) 10, 390 (1960). [5] N. N. Bogoliubov, in Studies in Statistical Mechanics, edited by J. de Boer and G. E. Uhlenbeck ( North-Holland, Amsterdam, 1962). [6] O. V. Konstantinov and V. I. Perel, Sov. Phys. JETP 12, 597 (1961). [7] H. W. Wyeld, Jr. and D. Pines, Phys. Rev. 127, 1851 (1962). [8] L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics ( Benjamin, New York, 1964). [9] Yu. L. Klimontovich, Kinetic Theory of Nonideal Gases and Nonideal Plasmas (Academic Press, New York, 1975). [10] Yu. L. Klimontovich, Sov. Phys. JETP 35, 920 (1972). [11] P. RÈsibois, Phys. Fluids 6, 817 (1963). [12] J. R. Dorfman and E. G. D. Cohen, Int. J. Quantum Chem. 16, 63 (1982). [13] A. Z. Akcasu and J. J. Duderstadt, Phys. Rev. 188, 479 (1969). [14] D. Forster and P. C. Martin, Phys. Rev. A 2, 1575 (1970); D. Forster, Phys. Rev. A 9, 943 (1974); M. S. Jhon and D. Forster, Phys. Rev. A 12, 254 (1975). [15] K. Morawetz and G. Roepke, Phys. Rev. E 51, 4246 (1995). [16] K. Morawetz, R. Walke, and G. Roepke, Phys. Lett. A 190, 96 (1994); M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. KÆhler, in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, Binz (to be published). [17] R. Balescu, Statistical Mechanics of Charged Particles ( Wiley, New York, 1963). [18] S. Ichimaru, Basic Principles of Plasma Physics ( Benjamin, New York, 1973). [19] G. Kalman, in Plasma Physics ­ Les Houches 1972, edited by C. DeWitt (Gordon and Beach, New York, 1975).

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