Pram~qa, Vol. I, No. 3, 1973, pp 129-34. Printed in India. © 1973

The kinet ic equat ion and the equat ion o f m o m e n t u m transfer for a p l a s m a

SAROJ K M A J U M D A R Saha Institute of Nuclear Physics, Calcutta, 700009

MS received 7 March 1973; in final form 23 June 1973

Abstract. An attempt is made to derive a simple form of the collision integral of the kinetic equation for a plasma, by using Rostoker's equation which expresses the pair correlation function in terms of the distribution functions of the particles, and the con- ditional probability of one particle shielding the other. The conditional probability function is assumed to be giveu by its equilibrium value. By taking first order velocity- moment of the resulting kinetic equation, the equation of momentum transfer has been obtained.

Keywords. Plasma; momentum transfer; collision integral; pair correlation function.

1. In troduct ion

According to the first order B-B-G--K-Y theory, a single-component electron plasma, neutralised by a medium of uniformly distributed static ions can be described by the following equation:

O lX, S I + v . l at Or m "av m Orlr-r' I"

Here, X = ( r , v) stands for the position and velocity of the electron, E(r , t) is electric field produced by the electron inhomogeneity, over and above the background ion distribution, f (X, t) is the distribution function, and G(X, X', t) is the two-particle correlation fimction of the electrons.

By using the test particle method, Rostoker (1964) has derived a very elegant ex- pression for G in terms o f f :

a(x, x', t)=f(x, t)P(x/x', t)+f(x', t)P(X'lX, t)

+ f dX"f(X", t)P(X"iX, t)P(X"/X', t) (2)

P(X[X', t) is the conditional probability fianction, which describes the probability for particle X to be found at X' , within the shield cloud around the particle X. It is our aim in this paper to use equation (2) along with equation (1) to derive the com- plete kinetic equation, and then obtain the equation of motion by taking the velocity moment of the kinetic equation. Rostoker (1964) has also given an equation which determines the function P in terms of f ' s , and one must use that equation in order to derive a r~gorous kinetic equation, which is always very complicated. Some simplifica- tions in the mathematical procedure are made by using a certain approximation in the function P (X/X', t). Since P (X[X', t) is a conditional probability that particle at

129

130 Saroj If Majumdar

X' takes part in shielding the particle X, it follows that the quantity D = fP(x/x', t) dr' is the density of the shield cloud around the particle X, formed by other particles. With the assumption that D is given by its equilibrium value, which is really the Debye shielding density, one can use the following equilibrium expression for P, also derived by Rostoker (1964):

P(X[X') = f ~dk exp [ik. ( r ' - - r )] P(k ,v /v ' ) (3)

where t

P(k, ,,Iv') = 4~e~__ 1 k'fo(v') (4) mk 2 e(k~ --ik'v) k'(v'--v)~-iO

and t

-ioD=l- f dv' k f o(¢) (5) mk ~ 3 k.v--co--iO

is the complex dielectric constant for the real frequency oJ. fo(V) is the equilibrium Maxwellian distribution of velocities:

So( --no oxp (_ "°'/ \2w7/ 2 T/

Notice that ¢(k, --ko) is a plus function, being analytic only in the upper half of the to-plane. One can now use equation (2) together with equations (3) --(6) in equation (1) in order to eliminate the function G, and thus obtain a kinetic equation in terms

the distribution functions are so normalised that f f(Xt)dX=total o f f only. All

number of particles in the system.

2. Calculation of the right hand side of equation (1)

If one denotes the right hand side of equation (1) by the symbol R(X, t), then use of equation (2) gives us the following expression for R(X, t).

R(X, t) = e~ O f dr, O 1 [ f m~--;" OrL'--r't f(Xt) P(X/X')dv'+

+ ff(x't)P(X'/X)dv'-t- f dX"f(X"t)P(X"/X) f e(x"/X')d,,'] (7)

It follows from (3) and (4) that

[ P(X"/X')dv = [ dk eik" (r'--r") --~(r '--r") (8) J J (2rr)" E(k, -- ik.v")

Then, use of equations (3), (4) and (8) in (7), gives in a straightforward manner the following expression for R(Xt) :

4rre' 1 c3 [ f (Xt ) (dk ik R(Xt) = m - (2zr)30v " L -- k2E(k,--ik'v) 4-

14-~-eZX. , . ' [ dk ' t / ) k . jo(v ) 1 [ + dr" X

j j (2~r)3 ~7: ( -~ - - - -v~ k"e (k ' , - - ik ' , v")

f ] × dk k~e( k , - ik . v") × B (9)

The kinetic equation and the equation of momentum transfir for a plasma 131

where B stands for

=fdr'f(r", vL t) exp [ i (k+k' ) • ( r - - r") ] . B

writing r " - - r = R , we obtain

: f d R f ( r + R , v", t) exp [ - - i (k+k ' ) .R] B

Since correlation between two particles decreases as their distance of separation in- creases, one can assume the quantity R to be small, expand f (r +R) in Taylor series in powers of R around the point r , and keep terms only up to the first power of 11. In this way, one obtains from equation (9)

m (2,r) s 0v d k s e(k, - - ik - v)

- (j deYIr, ¢' , t) ×

xfdk k'f0 (~) it, ] k. (v -v") - i0 k~l <k, --ik. v")l a +

(4~s) s l . [fdv"fdk k ' f ' o~ v) l 0f k'/--2kk] + - (2-)'o-~" k.~--%'5-iok*l~l.o-~" k, (4~e 2)' 1 0 [fd. f dk k.f:(v) Of k

- ~ (2,,)s0v" k. (v--v')--i00r" k41, i ----~ x

× In e*(l~--ik" v") [ (I0) Ok J

In the third term of (10), the quantity I is the unit dyadic and e* is the complex con- jugate of ¢.

It is now easy to see that only the real parts of the four terms on the r.h.s, of (10) contribute to R(Xt). This is demonstrated only in case of the first term of equation (10). First of all, one sees from equation (5) that ¢(--ir~ + i k . v) is just the complex conjugate function of e(lr~ - - ik . v). Consider now the k-integrai of the first term of (10). If one changes the sign ofk in this integral, one can write

f 1 _ f - i k 1 k s e(lr~ ---ik "v) dk ~¥ E*(k, -- ik" v)

i.e. 2 (dkik [Real part °f I] =0 d k2

i.e. ,m[fdk~ !l:o Therefore, one picks out only the real parts of the terms on the r.h.s, of (10). For

this, one uses Plemelj formula

1 P 4- i*r 8(k" v - -k" v"),

k. (v-v")~=iO k- (v-v') where P denotes the Cauchy principal value, axtd write

e(l~ --ik" v") ----Re e+i Im e.

Thus the following expression for R(Xt) is obtained:

132 Saroj If Majumdar

R(Xt)=-- 4zte~ l --. [f(xt) f dk~Im!] m (2~r)3 Ov

(~-~1 [ f f (v) *rk8 (k" v-k" v'') 2 1 0 dr" f(r, v", t) dklt-.f~ k' I, 12 + (2~r)-------8 O v •

J

+ ~ J 12 k. v--k. v" t

_;dv, ,;dkkS.(v) P 0i 2 ,k ) 2 k41 el 2 k ' v - - k ' v " 0 r " k 2

_fdv,,fak Yo( P ,,of k_Olnl l el,l'k'v-k' Ok k ' - - f d v " r k "/°(v) v")0 f k O Ira,!] ) d k4" [ el 2 rr 8 ( k ' v - k " 0r--" o k t a n - t Re ,J (ll)

Here, • stands for e(k, - - ik . v) in the first term, whereas in all other terms • stands for e(k, --ik" v").

The expression on the r.h.s, of equation (11) is still very complicated. Some sim- plification can be made by treating the dielectric function e somewhat approximately.

k ' v k ' v " In the expression for e, assuming that the velocity ~ or - - is less than the

k k average thermal velocity v t = V'-T]m of the particles, one can write

, , ,

e ~ l + kD k ' v ~~ + i V 2 k2 -~

with IRe • I > [ Im • [, and then one can easily show that the fourth and fifth terms of equation (11)approximately cancel each other, and that the sixth term (last term) of equation (11) is much smaller compared to the second term. Therefore, one is left with the following expression for R(Xt) :

R(Xt) 4 " r r e 2 1 0 [ f(xt) f dk k !] = - - m" i2rr)3 0---v " kq Im

(_~_~) I f f • r r k ~ ( k ' v - - k " v")] 2 1 0 dv"f( r , v", t) dk k "fo(v)

t

' ( 4 ~ r e 2 ) 2 1 0 [ f d v " O f r d k k ' f°(°) P ] (12) ,--m--, (2~r)30v 0-r2 i i [ ~ [ g k ' v - - k ' v "

The integrals over k can be evaluated by using the same method as demonstrated by Montgomery and Tidman (1964), assuming k" v"[k to be less than vt, the average thermal speed of the particles. A lengthy calculation finally leads to the following expression for the collision term of the kinetic equation:

• 0 ~ R(Xt) = rre4 (In A) 0 .fdv"[fo(o")f(Xt) --fo(v)f(r, v", t)]v 0v0v Iv--v" [ mT Ov

+ (--~)s 0-v" L d Or O k ' l , l ' k . , - k . v "

The kinetiv equation and the equation of momentum transfer for a IJlasma 133

T 3 Here A -- and ~ = ~ (k , - - i k ' v " ) .

4~no ee In equation (13), the first term is the familiar collision integral, but a little modified and somewhat simpler, whereas the second term denotes the effects of the long-range collective phenomena brought in by the correlations among the particles. Thus equa- tion (1), with R(Xt) given by equation (13) gives the complete kinetic equation.

3. Equation of momentum transfer

To derive the equation of momentum transfer, it is easier to start with the r.h.s, of equation (1) as given in equation (12). First of all, one sees from equation (5), that

Re E ---- 1-- 4"n'e* P f d v ' k "f:(v') mkZ k" v ' - -k" v ~

4~e2 f Im • = - - ~ dv'k-fo(v')cr3(k" v ' - -k" v") (14) mk 9.

Rel----Re--•and Im 1 _ Ime

We then multiply equation (12) by v and integrate over v. After integrating by parts and then using equation (i4), it is easy to see that the first and second terms of (12) cancel each other, with the result

,_ fdk Ro,- m (27r)sd Or ~ [,(k,--ik'v")l"

V u The k-integral in equation (15) can be evaluated for k" less than the thermal speed

k vt=-X/iT[m ) using the following approximate expression for the dielectric function: W- [ (k v"/. + ="""1 • (k, - - ik . v") --1 -k ,op 1 -- i (16)

Using (!6) for ~, the integral over k in equation (15) can be evaluated without any difficulty. "Keeping only terms up to the second power of v"/vt,

4zte~_ 1 k~ct' F f R(Xt) v dv = m (2~r)s oJ, [ vt D f dv" f (r' v"' t)-- ~ ~rf dv'v"~]

(17) Note the definitions of the following quantities:

f f ( r , v, t) dv----n(r, t), particle density

1 [ f v d v = u ( r , t), local mean velocity

n . / (18)

½m Traceff("--")'r = dv=,r, pressure tensor J Multiplying equation (1) with the r.h.s, as given in equation (12) by v, and then inte- grating over v, one obtains from equations (1), (17) and (18) the following equation

134 Saroj K Majumdar

of momentum transfer:

Ou Ou 0 ~ eE n - - + n u . + - - . - - + n -

Ot Or Or m m 1 Tk~On . . . . 1 k~ 0 [ TrY]

=8r rm n00r 487r n o Or nu2+ ~ - (19)

In the simplest case of isotropic distribution of random velocities, W is diagonal and

, r - 0p (20) Or Or

where p~-nT is the scalar pressure. For the isothermal case, one can linearize equa- tion (19) about the equilibrium state characterised by n(r, t)=no, u(r, t)----0, E(r, t) =0, and T, by writing n----n0+N(r , t). Thus N(r, t), E(r, t) and u(r, t) denote the departure from the equifibrium state and are considered to be small quantities. Neg- lecting second order terms in N, 1~., and u, one obtains from (19) and (20), the follow- ing linearised equation of momentum transfer:

otOu T ( _e~kD ~ 031 T ] _~ eE no _--7+ 1 + n o - - =0. (21) m

Notice here that replacing the factor 1--e2kD/4 T in the second term of equation (21) by unity, leads to the ordinary linearised equation of momentum transfer for a plasma. Because this term arises directly from the pressure tensor, the effect of correlation between particles is equivalent to modifying the pressure tensor, so far as the equation of momentum transfer is concerned.

k " V M In evaluating the k-integral in equation (15), it is assumed that ~ is less than

k

the thermal speed Vt=~/-T-~ of the electron. Since k . v"~-¢o implies that each particle velocity contributing to the k integral in equation (15), follow a point

k o v tt

of constant phase, the assumption ~ < V t implies that the average thermal velo- k

city of the particles is always greater than the phase velocities of the individual waves of the particles. This is generally true only for small wavelengths, wbich are usually Landau damped, and do not contribute to the collective behaviour of the plasma. Therefore, the evaluation of the collision-integral (equation 15) only takes account of the micro-field, and thus neglects the collective behaviour of the particles.

References

Montgomery D O and Tidman D A 1964 Plasma kinetic theory (McGraw Hill, New York) chapter 7 Rostoker N 1964 Phys. Fluids 7 491