An estimate of the first logarithmic term in the density expansions of transport coefficients of moderately dense gases

Physica 93A (1978) 191-214 ~) North-Holland Publishing Company

AN ESTIMATE OF THE FIRST LOGARITHMIC

TRANSPORT

TERM IN T H E D E N S I T Y E X P A N S I O N S O F COEFFICIENTS OF MODERATELY DENSE GASES* Yuen-Han KAN

Department of Physics and Astronomy and Institute for Physical Science and Technology, University of Maryland, College Park Maryland, 20742, USA

Received 7 March 1978 In this paper, we present an estimate of the coefficient of the first logarithmic term in the density expansions of transport coefficients of moderately dense gases. We formulate the theory for a gas of hard sphere molecules. Then the essential approximation made in this work is to replace the Lorentz-Boltzmann and the linearized Boltzmann operators by B.G.K. model collision operators. This allows us to compute the coefficient of interest in a simple way. We compare the results of this approximation with previous results.

1. I n t r o d u c t i o n

In a p r e v i o u s p a p e r l) h e n c e f o r t h i n d i c a t e d b y I), w e h a v e g i v e n a r e v i e w of t h e o c c u r e n c e o f l o g a r i t h m s o f the d e n s i t y , n, in t h e d e n s i t y e x p a n s i o n s f o r the k i n e t i c - k i n e t i c p a r t o f the t r a n s p o r t coefficients, tt KK, f o r m o d e r a t e l y d e n s e g a s e s . W e h a v e s h o w n t h a t t h e e x p a n s i o n f o r / z KK, a f t e r a r e s u m m a t i o n of the ring t e r m s is m a d e , is o f the f o r m IxKK/p,O = 1 + a ~ , ~p + (a~.~ + '~ _(2)~^2 ~,2m l n p + . . . ,

(1.1)

w h e r e ~0 is its v a l u e at low d e n s i t y as given b y the B o l t z m a n n e q u a t i o n a n d p = na 3, with n, the n u m b e r d e n s i t y a n d a, the d i a m e t e r o f a m o l e c u l e . T h e (1) a r e c o n t r i b u t i o n s f r o m the " r e n o r m a l i z e d " t h r e e b o d y coefficient a,,.l a n d a,,.2 ring e v e n t s w h i l e the coefficient t.t/.L.2 _(2) is the c o n t r i b u t i o n f r o m the " r e n o r m a l i z e d " f o u r b o d y ring e v e n t s . F o r h a r d s p h e r e m o l e c u l e s , w e h a v e g i v e n in I an e s t i m a t e o f ,*~.2. - " ) H o w e v e r , d u e to the c o m p l e x i t y o f the q u a d r a t u r e s i n v o l v e d , it is difficult to o b t a i n an e s t i m a t e o f ,*~,.2 _(2) e x c e p t b y c a r r y i n g o u t a m u l t i p l e d i m e n s i o n a l n u m e r i c a l i n t e g r a t i o n as h a d b e e n d o n e b y P o m e a u et al.:). *This work was supported by the National Science Foundation under Grant No. CHE 77-16308. 191

192

Y. KAN

The purpose of this p a p e r is to present the B.G.K. approximation by means of which an estimate for the complete coefficient a~.~ + a..2 (2) of the p2 In p term given in eq. (1.1) can be obtained. Using this approximation, we are able to reduce the dimension of the multiple integration such that it can be carried out by a simple method, namely the Gaussian quadrature. Due to the fact that the B.G.K. approximation for the linearized Boltzmann collision operators p r e s e r v e s all the general properties of these collision operators, the estimates given in this paper may provide an important estimate on the order of magnitude of the coefficient a,.2")+ ,*.,2.~(2) In section 2, we will give a brief review of the formal expressions for the coefficient of the p21np term in eq. (1.1), and will list from I those formulas we need for this paper. In section 3, we will introduce the B.G.K. model and the B.G.K. operators. In section 4. we will give an estimate of a,,2 ~) + a,.2 e-) and will c o m p a r e our results with results for hard sphere molecules. We conclude in section 5 with a n u m b e r of remarks. To avoid confusion, all the notations used in this paper are the same as those in I and readers are referred to I for detailed definition of the notations used here. 2. The formal expressions of the coefficient of the p2 In p term

In this section, we consider the p 2 l n p term in the density expansion of tt , given in eq. (1.1) for the coefficient of shear viscosity rl, the coefficient of thermal conductivity A, and the coefficient of self-diffusion D 3). For this reason, we will limit our attention to the r e s u m m e d ring term, /x~ K, that give rise to the pZ In p dependence. We have established in I that for a system of hard sphere molecules of diameter a and number density n, tt~ K is given by KK

(2.1)

where /z~ ) =

•--,o ,im,b fd ,fd =f ~ d, x (1 +

g" (v,) T,( l,

2)P,(v,. v2) T-,(1.2)

(r,P12)gg(vO6o(VOqJo(V2),

tx~ ) = lim.~0nb.

f fdvlf d ,dv2

~g.(v07~k(1,2)Pk(v.

(2.2a)

v2)A..ffv, v2; k)

×Pk(Vl, v2)T-~(1,2)(1 + cr.Pi2)g.(vO~bo(Vt)~bo(V2).

(2.2b)

nb. f dvl f dv2 f d ~ g.(vt)~k(1.2)[pk(v,,v2)aul(Vl,

t ~ ~ = lim.~0

×P~(vl, v2)T-k(1.2)(1 + tr.P12)g.(vOq'o(VOq'o(V2),

(2.2c)

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

6~R=limnb~.fdv, f d v 2 f

dk g~(v,)~(1,2)[P~(vbv2)A.l(vm, v2;k)]3

AN(Vt, v2) -- A..i(v~, ~P,2)g.(vOqJo(VO~Oo(V2),

x{~ + ik • v~2x (1 +

193

v2; k)}-' T - d l . 2) (2.2d)

w h e r e v~ d e n o t e s the velocity of the ith particle, o'. = 0 for ~ = D, and o'. = 1 for ~ = 7, A. T h e c o n s t a n t b~ is given by

b~-nm2fllO '

b~--

nka 3 ,

bo=~,

(2.3a,b,c)

with m, the m a s s of a molecule; fl = 1/kaT, T being the t e m p e r a t u r e of the s y s t e m and ka being the B o l t z m a n n constant. T h e o p e r a t o r T~(1,2) is a fourier r e p r e s e n t a t i o n of the binary collision o p e r a t o r of particles 1 and 2 and is defined by T~(1,2) = a 2

f

dd',2[v,2

• o',2[[eiak "°'2b,r -

e-re'°'2],

(2.4)

V12 ~'12>0 -

w h e r e t~t2 is a unit v e c t o r in the direction of the apse line of the binary collision and b. is an o p e r a t o r which r e p l a c e s the velocities v, and v2 b y their restituting velocities v] and v~ given by v~ = v , - (0,5" d')d~,

(2.5a)

v~ = v2+ (v,2- ~)d'.

(2.5b)

T h e function

g~.(vO is

the solution of the integral e q u a t i o n

-nX~(Vl)~bo(v,)g~,(v,) = ilto(vl)j~}(v,),

(2.6)

w h e r e qJ0(v) is the B o l t z m a n n f a c t o r which is given by

(tim ,¢12e-~'~212.

qJ0(v) = \~--~-]

(2.7)

T h e o p e r a t o r A,(Vl) is the linearized B o l t z m a n n collision o p e r a t o r for tz = rt, A and is the L o r e n t z - B o l t z m a n n o p e r a t o r for t~ = D. T h a t is

A.(vO[(vO~Oo(V,)= a2f

dr2

f

dd',2Oo(V,)~Oo(V2)Iv,2" o'12[

V12 - ~ 1 2 > 0

x [f(vD + f(v'z) - f(vO -

f(vz)],

(2.8a)

f o r / x = r/, A ; and

f dr2 f

da12qJo(V,)q~o(v2)lv,2,a~21ff(v~) - / ( v 0 ] ,

V12 • ~-12>0

(2.8b)

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Y. KAN

Here v', and v~ are the restituting velocities of the (12)-collision as given by eqs. (2.5). The current j~(vt) is given by j~(vt)

= vlvl

I

2

(2.9a)

- ~vll,

j~u(vO= ( - ~ v~- Ov,,

(2.9b)

j ~ ( v 0 = vl,

(2.9c)

here I stands for a unit tensor. The operatorsAN(Vl, v2) and A~.~(v~, v2, k) are the non-interacting and interacting part of the collision operator n{To(1,3)+ To(2, 3 ) + t r o t k(1,3)PI3 + Tk(2, 3)P23} and are given by

AN(Vl, V2) = --na2rr[t'(vt) + v(v2)],

(2.10a)

A~,j(vl, v2; k) dd-]vl3 • d'J[b~ + ~r,(b~ e

i~,

.~ _ eia, .a)Pl3]~b0(v3)

V l ~ * ~r~- 0

v2~ - 6->0

with

~'( vO

=

f dv~lv~ - v314,0(v3).

Here P# is a permutation operator which interchanges particle indices i and j. Lastly, Pk(v~, v2) is defined by

Pk(vl, V2) = {E + ik " Vl2 - AN(V,, V2)} 1.

(2.1 I)

It is important to recall that the term AN(V,, v 2 ) = - - n a 2 7 r [ v ( v O + v(v2)] has its origin from the non-interacting part of the binary collision operator and that the quantity na2rr~,(v) is the equilibrium collision f r e q u e n c y for a particle with speed v. This term, after the r e s u m m a t i o n is carried out, provides the exponential damping which takes into account the fact that it is not possible for a particle to travel large distances in a gas without suffering collisions with other particles. We also recall from I that t,~ ~ vanishes and that (1) 2 , In p / z ~ ) / g 0 = oei,,l p + ae~,,2p

-I- O ' ( p 2 ) ,

(2.12a)

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

/ ~ ) / ~ 0 = ,-~,,2p _<2~ 2.m p + ~?(p2),

195

(2.12b)

where a~,.lp is identical to the contribution t o ju, KK from the three body ring term before the resummation and ta" (1)~,2T±=-<2)~.,2jis identical to the contribution t o /.t KK from the four body ring term before the resummation with the logarithmic d i v e r g e n c e detected. T h e term 8ttR contains contribution to /~R~K f r o m the five and higher b o d y " r e n o r m a l i z e d " ring e v e n t s and does not a p p e a r to contain t e r m s of order p: In p 4). W e note f r o m I that: (i) the p2 In p term c o m e s only f r o m the part w h e r e the time interval b e t w e e n any two collisions is m u c h larger than the time td, w h e r e td is the time n e e d e d for a molecule to travel a distance of a few m o l e c u l a r d i a m e t e r s ; (ii) for k ~ 0, the o p e r a t o r Tk(i, J) introduces t e r m s of the f o r m e -+i~'o which h a v e no effect on the pZln p term. Thus, we can s e p a r a t e f r o m /~(~) and /z~ ) parts which do not c o n t r i b u t e to the p21n p term by (i) writing eqs. (2.2b,c) in time language, and taking only that part of the integral w h e r e t > To with To >> t-d, t, being the time interval b e t w e e n which the e v e n t h a p p e n s , td being the a v e r a g e time n e e d e d for the m o l e c u l e to travel a f e w m o l e c u l a r d i a m e t e r s ; (ii) replacing ]P~(i, j) in eq. (2.2b,c) by T0(i, j). T h u s we have

i

'=limnb fdte ,*o

-" dt, fdv, fd 2f ~dk

ro

gA,,0 ?o(1, 2)

o

x e x p { - i k • v12(t - tO + AN(v1, v2)(t -- tO}A ~j(vl, v2; 0) x e x p { - i k • thztl + AN(V1, v2)h}To(1, 2)(1 + o'~Pl2)g~(vO6o(VO6o(V2) •

(i)

+ ~.R,

/~)=limnb. ,-.o

(2.14a)

i TO

t

tI

dte-'tfd,,yd.fd,,,fd,,=f 0

~

dk

g~,(v,) To(l, 2)

0

x e x p { - i k • vn(t - tO + AN(V1, u2)(t -- tl)}A g,l(vl, v2; 0) x e x p { - i k • vj2(tl - t2) + AN(v1, v2)(tl - t2)}A .,l(vl, v2; 0) x e x p { - i k • v12t: + AN(Vb rE)rE}T0(1, 2)(1 + cr.P12)g.(vl)d/o(Vl)~bo(V2) •

(2)

+ ~ RR,

(2.14b)

w h e r e /~(~ and /~(~ d e n o t e the remaining part which does not contribute to the coefficient of the p2 In p term. T h e r e a d e r s are r e f e r r e d to I for a detailed discussion• W e w a n t to point out that the o p e r a t o r s AN(vl, v2) and A~•i(v~, v2; 0) can n o w be r e g a r d e d as the non-interacting and interacting part of the o p e r a t o r n{(1 - tr~),~D(VD + tr,,X~(vl)+ ,~n(vz)}. W e h a v e e s t i m a t e d in I the numerical value of ~,,,2 - " ) by making app r o x i m a t i o n on the f u n c t i o n v(v). H o w e v e r , such an a p p r o x i m a t i o n gives no

196

Y. KAN

simplification in the calculation of a,,.2 (2) and the multi-dimensional integration must be done by means of numerical integration. In the section 3, we will introduce the B.G.K. approximation. We will show that by means of this approximation, we can reduce the dimension of the multiple integration such that it can be done by a simple numerical method.

3. The B.G.K. model

The B.G.K. model was first introduced by Bhatnagar, Gross and Krook ~) and independently by Welander6). The idea behind this model is to simplify the Boltzmann collision operators and at the same time retain those properties which are independent of the force lawT). Since it is a general belief that the details of the two body interaction are not likely to influence significantly the qualitative description of a system, this model is widely used to avoid the complicated structure of the collision operators. Let us consider the L o r e n t z - B o l t z m a n n collision operator Ao(v) and the linearized Boltzmann collision operator A~(v), for /,t = rl, A, as defined in eqs. (2.8). The properties of A,(v) which are independent of the hard sphere potential are: A. The eigenvalues of A~(v) are all negative except with a non-degenerate zero eigenvalue for )to(v) and a five-fold degenerate zero eigenvalue for A,(v), for tt = "q, A. B. The zero eigenfunctions of A~(v) are f ( v ) = 1, f o r / z = D and J:(v) = 1, v, v 2, for tz = A, n. Now let us consider the operators A~(v) and Ae(v) defined by f

Y~,(,,,)4,0(~l)f(v,)

-3',~bo(VOf(vO + "Yl J dv:~o(vO~o(v:)f(v:),

~:(v,),l,o(vl)f(v,)

--'YS~0(vl)f(vl) + "y: f dvs~Jo(vl)~0(Vs)

(3. la)

3

v) (3.1b)

where 3,1, and 3'2 are constant parameters to be fixed. It is easy to discover for Y~l(v) and Yt:(v) that while the zero eigenvalue and the corresponding eigenfunctions are the same as that of Xo(v) and X,(v) for tz = n, A respectively, all the negative eigenvalues have been collapsed into a single eigenvalue of infinite degeneracy, - y . , for Yt~(v) and -3'2, for Yts(v). Thus, Ytl(v) and ~,:(v) are respectively the desired operators needed for

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

197

simplifying ;to(v) and A,(v), for /x = ~/,X, so as to obtain a qualitative description for a dynamical system, and we will refer to them as the B.G.K. operators. The constant p a r a m e t e r s 3'1 and 3"2 can be fixed by replacing Ao(v) and A,(v), for /~ = 7/, A, respectively by Al(V) and A2(v) in the calculation of the transport coefficient in their zero-th order approximation. Thus we have

1 770 = 8 3 ' 2 '

A0=-~kB

(3.2a)

1

raft3"2'

(3.2b)

1

D o - nflm 3"~'

(3.2c)

where 7/0, ;to, Do are respectively the zero-th order approximations for the coefficients of shear viscosity, thermal conductivity and self-diffusionS). For hard spheres of diameter a, the values of tz0 in their first Enskog approximation are 9)

5 X/m~,

'0o = l - - ~ a

75k~ A0 - 64a2 ~ , Do -

3

8na2X/ ~.mfl"

(3.3a)

(3.3b) (3.3c)

Thus, Tt can be fixed by eqs. (3.2c), (3.3c) while T2 can be fixed either by eqs. (3.2a), (3.3a), or by eqs. (3.2b), (3.3b). In this paper we will use eqs. (3.2a), (3.3a) to fix 3'2. It will be found that the order of magnitude of t,t/z,2 ~(1) ~T_ I-g ~(2) /L,2 is not effected by either method of fixing 3'2. We will return to this point in section 5. Thus, we have 3'1 = ~ a Z ~ / 3 ,

(3.4a) (3.4b)

Having introduced the B.G.K. operators Al(v) and A2(v) as the approximation to the operators A~(v), we are now ready to construct an approximation to the operators AN(vt, v2) and A,,.l(vl, v:; 0). Referring to eqs. (3.1), we can separate the Al(v) and A2(v) operators into an "interacting" part and a "non-interacting" part as we did with the A,(v). We shall call the first term as the "non-interacting" part and the second term as the "interacting"

198

Y. KAN

part. such that under the B . G . K . a p p r o x i m a t i o n . AN(v~, v2) and A..~(v~. v2;0) becomes AN(V~, v2) = --ny~ = --n [(Yl + Ye)+ ~r~ (~/. -- Yl)].

(3.5a)

fi...~(v,, v2) = n [(1 -- o'.)A ~(v~) + ~.A ~(v,) + A ~(v2)],

(3.5b)

;~ ~(v,)q,o(v,)f(v,)

'Yl

(3.6a)

X~(v,),~,,(v,)f(v,)

3'2 1 dv3q~o(v~hl'o(vO[1 + f l m v l " v~

with

f dv3~bo(vl)tbo(v3)f(v3),

+ 3

v'~--~]\---j-- va -

(3.6b)

f(v3).

In section 4, we will replace AN(V~, v2) and Auj(v~, v2;O) by AN(V~,v2) and d~.~(v~,ve) in eqs. (2.14a) and (2.14b). W e will s h o w that u n d e r such a r e p l a c e m e n t , the n u m e r i c a l integration n e e d e d for the e v a l u a t i o n of ~,.2 ~ and ~2~ au,: are r e d u c e d to o n e - d i m e n s i o n and f o u r - d i m e n s i o n s , respectively.

4. E s t i m a t e

of

~) ~,2 + ~ ~2~ ~,.2 i n t h e B . G . K .

approximation

In this section, we are going to calculate ~-m+~.2u-~2~.,2 in the B.G.K. app r o x i m a t i o n . We do so b y replacing A N ( v t . v2) and A . . ~ ( v l , v2; 0) by AN(V~. v2) and A u . t ( v l , v2) in eqs. (2.14a) and (2.14b). T h u s we h a v e

i i;sI

dte-" dt~ dt2 dVl dv2 (--~-~)_[j~'(v~)+~ruj~'(v2)] To 0 0 x T0(l. 2) e x p { - i k • v~2(t - t ~ ) - n y . ( t - t,)}[(1 - ~.)~.l(v~) + A ~(v2)]

12~'=limn2b.F~

× e x p { - i k • v~2t, - ny.t~}'r0(I. 2)(1 + ~r.P,2)tj~)(v,) + ~rd~)(v2)] .

£~

=limn-b~F'~.~o,

~ RR,

(4. I a)

dte-" To

dtx 0

dt2

dv~

dk d v . _ (-~-~w)_[J~(v~) +

loft ~J.~ "(I}tt~ol ~"1

0

x T0(1, 2) e x p { - i k • v~2(t - t,) - n y . ( t - t 0}[(1 - cr.)A](v,)+ ~.A~(v~) + X ~(vg] e x p { - i k • vj2(t~ - t2) - n y . ( t ~ - t:)}[(l - cr.)A ~(vr) + A ~(v:)] x e x p { - i k • v,2t2 - ny.t2}T0(l, 2)(1 + x 6o(v,)q,o(V:) + ~* ~2~.

.r.P,2)lj~'(v,) + ,rd~'(v:)] (4.1b)

H e r e t2~ ~ and /i~ ~ d e n o t e the c o n t r i b u t i o n f r o m the " r e n o r m a l i z e d " three and

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

199

four body events in the B.G.K. approximation. The terms t x ~ and t x ~ denote the remaining part of t2~ ~and t2~ ~ which does not contribute to the coefficient of the 0 2 In p term. We note that in getting eqs. (4.1a,b) we have replaced the quantities g~,(v~) in eqs. (2.14a,b) by their values in the first Enskog approximation which are given by

g~,(vj) = F~,j~,;")lvtu,~

(4.2)

where Fn = ~ "O0,

(4.3a)

F~ = ~2tim X0,

(4.3b)

Fo = flmDo,

(4.3c)

We have also used the symmetric property of /i}~~, for i = l, 2, between the particle index 1 and 2 and the fact that for the case of self-diffusion T0(1,2)j ~)(l~ 1) = -- T0( 1,2)] g)(v2).

(4.4)

Before carrying out any calculation, we would like to point out that in writing eqs. (4. la,b), we are applying the B.G.K. model to the gas molecules in an inconsistant way. In replacing AN(V~, V2) and A~,.i(v,, Vz; O) by A-N(V,, v2) and A~.~(v~, v2), the dynamics of the middle collisions in the collision sequence is being replaced by the B.G.K. model while collisions at the beginning and at the end of the collision sequence still obey hard sphere dynamics. However, since we are interested only in an estimate of the coefficient of the p2 In p term, we shall bear with this inconsistency. We also note that in using the B.G.K. approximation, the role of the collision frequency, na2"n'v(v) is now played by ny~, and n3'z which are both constants. From this point on, we shall limit ourselves to an outline on the calculation of the coefficient of the 0 2 In p term of the self-diffusion coefficient and the readers are referred to Appendix A for details. The first thing to do is to get the coefficient of p2 In p out f r o m / 2 ~ ~ and /2~ ~. This can be accomplished by taking the time integration after applying the T0(1,2) operator to the current jo and performing the k-integration. Noting that

I To

'

exp{-nyoTot + nvo Ei(-nyoTo),

(4.5a)

o0

f l e x p { - n y ° t } dt = -Ei(-nyoTo), ro

(4.5b)

200

Y.

KAN

w h e r e E i ( - x ) is the exponential integral f u n c t i o n which b e h a v e s as ~°) Ei(-x)

, In x,

(4.6)

for small

we get tl) 0/D,2 ~

nvDbDF~, 8~D0aZJ~

--

(

:

dz

0

f f ~2 , dx

0

v: _ dyexp{-k(9z2+2x'+2v

2

0

+ 2xy)}(x + y) I

x f du{( 1 - crt~)y~ + y2,~t~(x, Y, z, u )},

(4.7a)

I

n~bDF~ f _ , l ut f 64¢r V m[3 Doa ~ .t

(

dx f dy J dz

exp[-~(16t2 + 3x2 + 3y2 + 3 z 2 - 2x . y - 2x • z - 2y • z )]C~Dfft, x, y, z)

X

+ e x p [ - ~ ( 1 6 t 2+ 3 x : + 4y 2+ 3 z : - 4 x -4y

• y + 2x • z

• z)l~m,(t, x, y, z)}P(0/3/33, 0/4/34).

(4.7b)

We refer to A p p e n d i x A for the f u n c t i o n s ~D(x, v, z, u), c~D;(t, x,y, z), for i = 1,2 and P(0/3/33, 0/43J. N o w m o s t of the integrations in eqs. (4.7a) and (4.7b) can be carried out easily b y making simple t r a n s f o r m a t i o n s and we obtain ~/2

sin" 20(sin 0 + cos 0)(2 + sin 20) -~/2

0/D,. -- _ 1)

x I-5(1 - crD) + Y(D(0)], rr/2

0/D2-

•r/2

do

.

(4.8a)

~

fd fd0/ f d0/4

0

× sin 7 0 cos 0 sin3 0 c Os3 0 COS 0/4[,~D1(0, IJ), O/3, 0/4)

X ZIS(O,

~), 0/3, 0 / 4 ) -~-

,~D2(O,~,

0/3,

0/4)Z25(0,

(~), 0/3, 0 / 4 ) ] .

(4.8b)

T h e readers are referred to A p p e n d i x A for details and for the functions a~D(O), ~Di(O, C~, 0/~, 0/4) and Zi(O, 4a, 0/3, 0/4), for i = 1,2. Finally, the one and f o u r dimensional integration in eqs. (4.8a,b) can be d o n e by m e a n s of G a u s s i a n q u a d r a t u r e ~E) and we get 0/~1)

D,2 = 0.2162,

(21 O/D,: =-!.

1227.

(4.9a,b)

Thus, the coefficient of the p2 In p term for the serf-diffusion coefficient is

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

201

given by l) a ~,2 + a/~!2 = - 0.9065.

(4.10)

The coefficient of shear viscosity and the coefficient of thermal conductivity can be handled in the same way. A list of a ~,2 ") and a ~.2 ~2) obtained by using the B.G.K. approximation is given in table I. These estimates can be checked with the estimates obtained by making approximation on the damping factors v(vi) and with the exact results obtained by P o m e a u et al.2). As for comparison, a list of all the previous results are also given in table I. For this we would like to point out the following: 1. As mentioned in I, the factor 0.5 which appears in the results of 77 and A obtained by P o m e a u et al should not be there, and can be accounted for by a small error in the first few steps in their paper. 2. The estimated values /x2 obtained by using the B.G.K. approximation of the renormalized three body events agree with the exact values tz3 both in sign and in order of magnitude. H o w e v e r , these agreements do not apply to the renormalized four body events. We c o m m e n t on this further in the next section. TABLE I Coefficient of the p21np coefficients.

term in the density expansion of transport

~t

Self-diffusion D

a~!2 (2) ao2

~2

~3

0.2162 -- 1.1227 -0.9065

0.3696 --6.788 -6.418

-0.6014

-0.5461 1.187 0.641

-0.3108 x 0.5 4 2 x 0.5 42 × 0.5

0.6574

0.5130 - 1.544 -1.031

0.5596 × 0.5 422 × 0.5 421 ×0.5

0.3755

Oto.2 "~ + ao.2t2~ Shear viscosity ~

Heat conductivity A

a~ a~.2 .~ + O/n,2 ~2) O~n.2 a~,t.~ ~2) Otx,2 t -~ OfX, ~2)2 O~X,2

tt~: Values obtained by making approximation on the damping factor. p2: Values obtained by using B.G.K. approximation. tL3: Values obtained by Gervios, Alle and Pomeau.

5. Discussion

We conclude with a n u m b e r of remarks. 1) The main results of this work are that the coefficient of the p2 l n p term in the density expansion of the kinetic-kinetic parts of the transport

202

Y. K A N

coefficients for hard sphere molecules, as given in eq. (1.1), are, in the B.G.K. approximation given by a,~,2 '"- -1-a ';3.- = 0.641,

(5.1a)

o~¢I~ A,2 + a ¢2~, x.. = - 1 . 0 3 1

(5.1b)

O / (D.. I ) -I-

(5. I c)

a (2) ~.2 = -0.9065,

where o ~ ,. and a~,.2 ~2~ are the contributions f r o m the renormalized three body events and four b o d y events, respectively. 2) As mentioned in section 4, we are applying the B.G.K. model to the gas molecules in an inconsistent way. In replacing A N ( V ~ , v2) and A~,.l(V~, v2; 0) by AN(v~,v:) and X ~ , . , ( v l , v2), the dynamics of the middle collisions in the collision sequence is being replaced by the B.G.K. model while collisions at the beginning and at the end of the collision sequence still o b e y hard sphere dynamics. As a result, the approximation b e c o m e s rather uncontrollable. H o w e v e r , if we assume that the detail of the two body interaction is not likely to influence significantly the qualitative description of a system, this approximation can most likely give us an estimate of the order of magnitude of 0¢ ,~t.2~ _-~- O~ ~2~ kt,2. 3) As mentioned in section 3, we may use either eqs. (3.2a), (3.3a), or eqs. (3.2b), (3.3b) to fix the parameter 3,2. L e t us denote ~/2n and ~12~the values of ~/~ fixed by eqs. (3.2a), (3.3a) and eqs. (3.2b), (3.3b), respectively. We have

]/2~ = !~ a 2 V ' ~ [ 3 , 32

~

/

(5.2a)

t

(5.2b)

y2~ = T3 a "v Trl r n p .

So far, we have used only 3'2~ in all the calculation. To show the way of fixing 3'2 has no effect on the order of magnitude of the coefficient of the p 2 In p term. We list in table II the values of ak'~. and otk2.~ obtained by using different

T A B L E I1 Coefficient of the p2 In p t e r m in the d e n s i t y e x p a n sion of the coefficient of h e a t c o n d u c t i v i t y .

Ill O~X,2 12~ x.2 (It

(2~ a~,2 + o~x2

7/'0: V a l u e s "0A: V a l u e s a n d 3"2~ in the hA: V a l u e s

r/rl

rlA

AA

0.5130 - 1.544 - 1.031

0.3420 - 1.029 -0.687

0.2280 -0.6862 -0.458

o b t a i n e d b y u s i n g 3"2, for b o t h 3'2. o b t a i n e d b y u s i n g 3"2. in o n e of the 3"2"s other. o b t a i n e d b y u s i n g 3"2A for b o t h 3"2.

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

203

-,,2 obtained in I. From this combination of "~2. and TEa and also the value of ,*"~ A.2 are not affected by table, we note that the order of magnitude of a ~',t and ~, -a~ the way of fixing 3~2. Moreover, we plot in fig. 1 the exact collision frequency

(5.3)

fo = n c r a 2 v ( v ) = n w a 2 f d r , I v - v,lq'0(v,),

along with the approximate collision frequencies

(5.4)

f, = n'),, = ~na2V'-~-ml3, h. = 'm.

.

(5.5)

~na2N/~[3,

(5.6)

= ~ .a2x/V~f3

f2x = nr2;~ =

and [3 = n T r a : v l ( V ) = nlra2V2/~'[Jm 2(0.124/3mv2 + 1.033).

(5.7)

Here, f3 is the approximate collision frequency used in I. We note from fig. 1 and table II that as the collision frequency is lowered, the values of OtX,2 "~ decrease in a consistent way. We also note from fig. 1 that 3'2, is more preferable than 3'2,.

2.20

i

1

i

I

2.00

I

/

1.80 1.60

fo.~///

/~--,

L~ ,.4o ,.2o

,.oo

Z"

0.80

i/f2x

0.60

0.50 1.00 1.50 2.00 2.50 3.00

I

I

I

1

I

d~,,

Fig. l. A plot of the collision frequencies. The solid line is the exact collision frequency as given in eq. (5.3). The dash lines are the approximate collision frequencies [0, .fl, f2n, f2, and [3 as indicated and which are defined by eqs. (5.4)-(5.7).

204

Y. KAN

4) As pointed out at the end of the last section, the estimated values /x2 obtained by using the B.G.K. approximation of the renormalized three b o d y events agree with the exact values Ix3 both in sign and in order of magnitude. H o w e v e r , these agreements do not apply to the renormalized four body events. As a result, it is hard to draw any conclusive remarks on both the sign and the order of magnitude of a~.~ and hence a~,2-~ ~i~_ a~.2. ~2~ Nevertheless, we would like to point out that, as shown in table I, the magnitude of a~.~ and a,.2 are of the same order in both the B.G.K. approximation, ~t2, and the approximation make on the damping factor,/x~. Also, as illustrated in table II, the order of magnitude of these numbers are insensitive to the way of fixing the p a r a m e t e r 3'2 in the B.G.K. approximation. 5) As mentioned in I, a n u m b e r of attempts have been made to experimentally detect the presence of terms of order iO2 in p in the density expansion [given in eq. (1.1)] of the transport coefficients. So far, there are no convincing e x p e r i m e n t verification of the presence of such terms. H o w e v e r , Kestin, Sengers, and P a y k o c t2) have given a lower bound for the coefficient of the 192 In p term in the density expansion of the viscosity, in order that it could be detected in their analysis. If r12 and r13, given in table 1 can be taken as giving the correct order of magnitude of the coefficient for more realistic potentials, then the 0 2 In p terms are just at the limit of detectability. The difficulty in detecting the p~" In p terms is connected with the facts that (a) these terms are second order corrections to the Boltzmann equation results and that p = na 3. thus the diameter of the molecule " a " as well as the linear term must be accurately determined first; (b) since the magnitude of the iO2 In p terms varies as one changes the argument of the logarithm, the theory must provide not only the coefficient of the p2 In 0 term but also the coefficient of the p2 t e r m in order to facilitate a c o m p a r i s o n between theory and experimental data; and (c) when the /9 2 In p and O2 terms are important, so probably are the 19 3 and p 4 and possibly higher order terms.

Appendix A In this appendix we give a detailed discussion on the calculation of the coefficient of the p2 In p term of the self-diffusion coefficient. The coefficient of shear viscosity and the coefficient of thermal conductivity can be handled in the same way. A . I . The " r e n o r m a l i z e d " three b o d y event

We begin by applying the T0(l, 2) operator to the current jo. We have f r o m

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

205

eq. (4.1a)

5(~ ) = bo(nrta2Fo) 2

x

dt~ N dvi~bo(t~i)

exp{--n~/ot} dt To

~)Co(l, f ( -dk

0

2)At-t,(1, 2)MD(I, 2, 3 ) + D g ~ ,

(A.1)

where Mr)O, 2, 3) = 7~Ar,(3, 2)~0(3, 2 ) + B(2, 3)At,(1, 3)~o(1,3),

(A.2)

o,~D(i, J) = 1UiflPij,

(A.3)

At(i, j) = e x p { - i k • v~it},

(A.4)

with

2

Integrating o v e r k, we get o¢

t

0 ~ ' = bn(n'rra:FD) 2 f exp{-nyDt} f dt, To o x

.=

dv~Oo(v~)¢D(1, 2)~e(1, 2, 3) + D ~ ,

(A.6)

where q~D(1, 2, 3) = 71(1 -- ~D)J~O(3, 2)8(Vi2(t -- t0 -- v23tl) + B(2, 3)~D(1, 3)8(e12(t -- tl) + vI3t0.

(A.7)

NOW, let us change the integration variables f r o m el, v2, v3 to x, y, z with

x = V ~ m ( v , - v2),

(A.8a)

y = V~m(v~-

(A.8b)

v3),

z = "V'~m(vl + v:+ v3)/3.

(A.8c)

We also note that by solving the argument of the 8-function in eq. (A.6) most of the angular variable and the tt variable can easily be integrated. Defining u = .~ • ~

(A.10)

206

Y. KAN

we obtain n-a boFb

8~/~-~

/5(~) -

x2

z 2 dz 0

dx

0

I t2

) ,2 dy 0

TO

x exp{--n'yDt} dt e x p { - ~(9z z + 2x 2 + 2y 2 + 2xy)}(x + y) I

×

f du{(l --OVD)'Yl "~- T2~ZD(X, y , Z, /d)}"~- lr~(1) LU' RR,

(A.11)

I

where ~ o ( x , y, z, u) = 1 + ~[9z 2 - 2x 2 - 2y 2 - 5xy - 3(x - y ) z u ] + ~2{ ~1( 9 Z 2 + 4 X 2 + y2 + 4 x y _ 2 7 ) ( 9 Z 2 + X z + 4 y 2 + 4 X y - - 2 7 ) + ~[(9Z 2 + 7X 2 -- 2V 2 + 4xy -- 27)yz -- (9Z 2 -- 2X 2 + 7y 2 + 4xy -- 27)XZ]U -- ~(X + 2y)(2X + y )z2/d 2}. (A.12) N o w the coefficient o f the t e r m t921nt9 in /3~)/D0 c a n e a s i l y be i s o l a t e d . U p o n p e r f o r m i n g the t - i n t e g r a t i o n , we h a v e l e x p { - n y o t } dt = Too e x p { - n y o T o } + n'yD E i ( - n y D T o ) ,

(A. 13a)

TO

w h e r e E i ( - x ) is the e x p o n e n t i a l integral f u n c t i o n w h i c h b e h a v e s as m) Ei(-x)

for small x

, In x.

(A.13b)

T h u s , f o r small n, the coefficient a~!: is g i v e n b y

8V,~CtmDoa 2

z 2dz 0

x2dx

y2dy

0

0

X e x p { - 1(9z 2 + 2x: + 2y 2 + 2xy)}(x + y) I

x f du{(l - oro)y~

+

y2,~D(X,

y,

2,

U)}.

(A.14)

I

A t this p o i n t , the i n t e g r a t i o n o v e r u a n d z c a n e a s i l y b e d o n e a n d the r e s u l t , after the transformation x=rsin0,

y=rcos0

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

207

is given by *r/2

et D,2 l' =

--

~ X / - ~ f dr f dOr6sin2Ocos20(sinO+cosO) 0

0

x exp - -~ (1 + sin 0 cos 0) {~(1 - o'0) + ~o(r, 0)},

(A.15)

where

~o(r,O)

=f25_ r2 r4 t9 81(37 + 79 sin 0 c°s 0) + 4--8-6( l + 3 c ° s 2 0 + 4 s i n 0 c ° s 0 ) x(1 + 3 sin 2 0 + 4 sin 0 cos 0)}.

(A.17)

Finally, after performing the r-integration, we get ~r/2

a") D,2-- ~,"~

f dO sin 2 20(sin 0 + cos 0)(2 + sin 20) -'/2 0

x {-~(1 - cro) + ~(o(0)},

(A.18)

where ~o(0) = 11736 + 1576 sin 20 + 37 sin 2 201.

(A.19)

The one-dimensional integration can easily be done by means of Gaussian quadrature") and we have a(1) D , 2 - -_ 0.2162.

(A.20)

A.2. The "renormalized" four body event Applying the 2Po(1,2) operator to the current j~ we have from eq. (4.1b) /)~)=

bDn3('a'a2eo)2

dt exp{-nyot} To

x I-I,=, dv,~,0(v,)

dt~ 0

~o(1,

dt2 0

2)A,_t,(1, 2)Mo(1, 2, 3, 4) + D~)R, (A.21)

where MD(1, 2, 3, 4) = y~2At,_,2(3,2)A,2(4, 2),,~o(4, 2) + Y1Y2A,,-,2(3, 2)At2(3,4)B(2, 4)~D(3, 4) + "yt~t2atl_t2(1, 3)A,2(4, 3)B(2, 3)~o(4, 3) + y2A,,_,2(1, 3)at2(1, 4)B(2, 3)B(3, 4)~o(1, 4)

(A.22)

208

Y. KAN

I n t e g r a t i n g o v e r k, w e get

D~=

bDn3(Ira2FD)2y~ [ exp{-n-/Dt} dt To

f o

dt~ o

dvi~bo(Vi)~D(l, 2 ) ~ D ( I , 2, 3, 4) + ni2~ URn,

X

(A.23)

where ~ D ( 1 , 2 , 3, 4) = ~ F ( 1 3 , 1 4 ) ~ o ( 1 , 4 ) + 5 F ( 3 2 , 34)B(2, 4 ) ~ ( 3 , 4) + ~ F ( 3 2 , 34)B(1, 3)~0(3, 4) + F ( 1 3 , 14)B(2, 3)B(3, 4)~D(1,4),

(A.24)

with

F ( kl, ran) = ~ ( v l 2 ( t

-

tO +

Vkl(

t l -

t 2 ) + V,nnt2).

(A.25)

In o b t a i n i n g ~ D ( 1 , 2 , 3, 4), we h a v e i n t e r c h a n g e d the d u m m y i n d e x 1,2 in the first a n d t h e third t e r m . N o w let us c h a n g e the i n t e g r a t i o n v a r i a b l e s f r o m vl, v2, v3, v4 to

x = ~ / f l m ( v l - v2), ~/flm(vl-

(A.26a)

v3),

(A.26b)

z = X/tim (vl - v4),

(A.26c)

t = V~ 4

(A.26d)

y

=

(v~ + v2+ v3+ v4),

f o r t e r m s w i t h c u r r e n t ~ D ( 1 , 4 ) , and to

x = X / - f l m ( v l - v2),

(A.27a)

y = "X/flm(v2 - v3),

(A.27b)

z = N / ~ m ( v 3 - v4),

(A.27c)

t = V'~ 4

(v~ + v2+ v~+ v4),

(A.27d)

for t e r m s with c u r r e n t ~ o ( 3 , 4). W e also n o t e that b y s o l v i n g t h e a r g u m e n t of the & f u n c t i o n in eq. (A.23), the i n t e g r a t i o n o v e r the t ' s can e a s i l y be integrated. We then have

ff)~)--

bDn3F2Da4yz~ j d ,

f Oxf dy f dz

x { e x p [ - ~ ( 1 6 t 2 + 3x 2 + 3y 2 + 3z 2 - 2x • y - 2x • z - 2y • Z)]C~Dl(t, X, y, Z) + e x p [ - - ~ ( 1 6 t 2+ 3x 2+ 4y 2+ 3z 2 - 4 x • y + 2x • z - - 4y - z ) ] ~ m ( t , x , y , z)} r~2) (A.28) X Ei(-nTnTo)F(ot3fl3, ot4f14) + IJRR,

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS w h e r e (a3,/~3) and the v e c t o r $, a n d

(Or4,

209

~4) a r e the a n g u l a r p o s i t i o n s o f ~ a n d ~ with r e s p e c t to

XZ ff'(Ot3B3' O/4~4) = y sin a3 sin a4

~(~4

--

Or + /33))~(a3 + a4 -- rr),

25

~m(t, X, y, 7.) = ~ COS 0t4{g + G3(t, x, y, z)G4(t, x, y,

z)},

c~m(t, x, y, z) = ~ c o s a4[Gl(t, x, y, z) + G2(t, x, y, z)].

(A.29) (A.30a) (A.30b)

H e r e the f u n c t i o n _~ is defined b y {~ ~(x) =

f o r x ~>0, for x < 0,

(A.31)

and Gi(t, x, y, z) = 1 + l ( 4 t + b , ) ( 4 t + bi:)

+ ~[~(4t + bn) 2 - ~1[~2(4t + bn) 2 - -~],

(A.32)

with 3x - 2y + z,

bl! : b21 :

-

x -

b31=-3x+ b41 :

2y + z, y+z,

X-- 3y + Z,

b~2 = - x + 2y +

z,

b22 = - x + 2y - 3z, b32 =

x-3y+

z,

b42 =

x + y - 3z.

(A.33)

R e f e r r i n g to eq. (A.13b) f o r the a s y m p t o t i c b e h a v i o r o f t h e e x p o n e n t i a l i n t e g r a l f u n c t i o n , the coefficient o f t h e t e r m p21n p in D ~ I D o c a n e a s i l y b e isolated. We have

o.:=-641r4a2DoX/--~m~jdt__

dx

dy

dz

x { e x p [ - ~(16t 2 + 3x 2 + 3y 2 + 3z 2 - 2x • y - 2x • z - 2y • z ) ] ~ m ( t , x, y, z) + e x p [ - ~(16t 2 + 3x 2 + 4y 2 + 3z 2 - 4 x • y + 2 x • z - 4y • z)]~o2(t, x, y, z)} X /~(O/3~3, Ot4~4).

(A.34)

In A p p e n d i x B, w e s h o w s o m e i d e n t i t i e s f o r the a n g u l a r i n t e g r a t i o n of a unit v e c t o r [ = sin 0 c o s 4ff + sin 0 sin 4ff + c o s 0/~,

(A.35)

w h e r e f, j, /~ a r e t h r e e o r t h o g o n a l unit v e c t o r s . U s i n g t h e s e i d e n t i t i e s t h e i n t e g r a t i o n o v e r [ a n d m o s t o f the a n g u l a r v a r i a b l e s c a n be c a r r i e d out. W e

210

Y. KAN

then have ~

,,2, D,2

~r

V /,2of 3dxfydyf dzfd,, 0

0

0

0

f cosot4dot4 ~'-a

x { e x p [ - ~(3x ~-+ 4y 2 + 3 z 2 - 4 x • y + 2 x • z - 4 y • z ) ] ~ m ( x ,

y, z )

+ e x p [ - ~(3x 2 + 3y z + 3z 2 - 2 x • y - 2x • z - 2y • z)] ~ m ( x , y, z)}, (A.36) w h e r e t h e d o t p r o d u c t s b e t w e e n t h e v e c t o r s x, y, z a r e n o w

defined by

x • y = xy c o s a3,

(A.37a)

x • z = x z c o s a4,

(A.37b)

y • z = y z c o s ( a 3 + a4),

(A.37c)

and that ~ o , ( x , y, z) = ~6{85 - ~[9(b~, + b~2 + b 2, + b~2) - 28(b,, • b,2 + b2,. b22)1 1 t'K2

+ ~u

K2

2

"~

(A.38a)

11~"12 + b ~ l b ~ ) }

~D2(X, y, Z) = 61"25 g t ~ + g o t ( x , y, z)]. Here

is

~ol(x,y,z)

obtained

(A.38b)

from

~o~(x,y,z)

by

making

the

following

replacements: bll --) b31,

b12 -'-~b32,

b21 ---) b4~,

bt2--~b42.

N o w , let us m a k e t h e t r a n s f o r m a t i o n z = r sin 0 c o s 4~,

(A.39a)

z = r sin 0 sin 4~,

(A.39b)

y = r c o s 0,

(A.39c)

a n d i n t e g r a t e o v e r r, w e g e t ~r/2

o,2

~ ~/2

¢r/2

dO 0

7r

d~b 0

~r

dot3 0

f

da4

7r - a ~

x sin 7 0 c o s 0 sin 3 ~b c o s 3 ~b c o s a4[J;DffO, d~, a 3 , a , ) Z ~ ( O , + ,~oz(O, qb, a3, a 4 ) Z ~ 5 ( O , dp, a 3 , a4)],

da, a3, a4)

(A.40)

where ~ m ( 0 , ~b, a3, a4) = A[51 + ( - 3 6 Y1.1 + 56Yi,2)Z~l + 1 2 Y I , 4 Z ~ 2 ] ,

(A.41a)

DENSITY

EXPANSIONS

OF TRANSPORT

COEFFICIENTS

211

,~;D2(O, dp, aS, at4) = l - ~ [ 14429 + ( - - 1 7 8 8 0 Y2.1 + 2 8 8 8 0 Y2.2 + 1120 Y2.s)Z2 I + (6120 Y2,4 + 8 3 5 2 Y2,5 - 117121:2,6 + 9 6 0 Y2.7 + 18816 Y2.8 + 384 Y2.9)Z~ 2 + ( - 6 0 4 8 Y2.1o + 9408 I:2.11 + 1344 Y2.12)Z~ 3 + 5376 Y2:3Z~4], (A.41b) H e r e the Y's a n d Z ' s are d e f i n e d b y (A.42)

Zi = Bi,o

Yi,1 = Bi,5

Yi.2

-= Bi,6,

Yi,3 = Bi,7, Yi,4 -- Bi.tBi,: + Bi,3Bi.4, Yi.5 = Bi.sBi.9, Yi.6 = Bi.sBi.ll + Bi.9Bi.lO, El,7 = Bi.lBi.16 + Bi,2Bi,17 + Bi.3Bi,18 + Bi.4Bi,19,

(A.43)

Yi.8 = Bi, loBi, ll, Yi.9 = Bi, t2Bi,15 + Bi,13Bi,14, Yi, lO = Bi.sBi.3Bi.4 + Bi.9Bi.iBi.2, Yi, n = Bi, lBi,2Bi.ii + Bi,3Bi,4Bi,lO,

Yi,12 = Bi, IBi.3Bi, l~ + Bi.lBi,4Bi,14 + Bi,2Bi.aBi,13 + Bi.2Bi,4Bi.12, Yi.13 = Bi.lBi.zBi.3Bi,4, and that 6

B 0 =-- ( a b a2, a3, a4, as, a6) --- ~

I=1

atxl,

(A.44)

where XI~-- l~ X 2 = COS 2 0,

x3 = sin z 0 c o s z 4~, x4 = sin 0 c o s 0 sin 4, c o s a3, x5 = sin 2 0 sin ~b c o s $ c o s a4, x6 = sin 0 c o s 0 c o s $ c o s ( a 3 + a4),

(A.45)

212

Y. KAN

with Bi.0 = (3, l, O, - 4 , 2, - 4 ) Bl.i = ( 9 , - - 5 , - 8 , - - 1 2 , 6 , - - 4 )

B2.o

B1•2 = ( 1 , 3 , 0 , - 4 , - 2 , 4 )

B2.2 = ( 1 , 8 , 0 , - 6 , 2 , - 6 )

B1,3

B2,3

=

(1,3, 0, 4, - 2 , - 4 )

= ( 3 , 0 , 0, --2, - - 2 , - - 2 ) B2.1 = ( 9 , - 8 , - 8 , - 6 , - 6 , 2 )

=

B2,2

Bi.4 = ( 1 , 3 , 8 , - 4 , 6 , - 1 2 )

B2.4 = ( 1 , 0 , 8 , 2 , - 6 , - 6 )

B1•5 = (3, 1,0, - 4 , 2, - 4 )

B2.5

=

(3, 2, 0, - 4 , - 2 , - 4 )

Bi,6 = ( - 1 , - 3 , 0 , 4 , 2 , 4 )

B2.6

=

(-I,-2,0,4,-2,4)

B2,7

=

( - 1,2, 0, 0, 2, 0)

B2.8

=

(5, 0, - 4 , - 6 , - 2 , - 2 )

B2.9

=

(1, 4, 4, - 2 , - 2 , - 6 )

BI,7

= (-

1, 1 , 0 ,

0, --2, 0)

B1,s = (5, - 1, - 4 , - 8 , 2, 0) B1.9

= (1,

3, 4, 0, 2, - 8 )

Bl,lo = ( - 3 , - 1,4, 8, 2, 0)

B2.1o = ( - 3 , 0, 4, 10, - 2 , - 2 )

Bl.ll = ( 1 , - - 5 , - - 4 , 0 , 2, 8)

B,~,ll = ( 1 , - - 4 , - - 4 , - - 2 , - - 2 , 10)

Bl.12 =

( - 3 , 7, 4, - 4 , 2, - 4 )

B2.12 = B2•,~

BH3 = ( - 3 , - 1,0, 8, - 10, 8)

Bz.13 = ( - 3 , 4, 0 , - 2 , 10, - 2 )

Bl.14 = (1, --5, 0, 0, --2, 0)

B2.14 = B2.2

Bl,15 = (1,3, - 4 , - 4 , 2, - 4 )

B2.15 = B2.11

BI,16 = (1, - 1, - 2 , - 2 , 0, 2)

B2,16 = ( 1,2, - 2 , - 4 , 0, 2)

BL17 = ( - 3 , 3 , 2 , 2 , - 4 , 2 )

B2,17 = ( - 3 , 2 , 2 , 4 , 4 , - 2 )

Bl,18 = ( - 1, I, - 2 , 2, - 4 , 2)

B2,18 = ( - 1,0, - 2 , - 2 , 4, 4)

BI,19 = ( - -

1, l, 2, --2, 0, --2)

(A.46)

B2,19 =

( - 1,4, 2, 2, 0, - 4 ) .

T h e four d i m e n s i o n a l integration can easily be done by m e a n s of G a u s s i a n q u a d r a t u r e ~l) and we get aCz) _ _ 1.1227. D•2--

(A.47)

T h u s for the coefficient of self-diffusion, the coefficient of the p2 In p term in eq. (1.1) is given by ol~12 ~.~2) • + OtD,2

-0.9065.

(A.48)

Appendix B In this a p p e n d i x we list the results of the angular integration of a unit vector t~= sin 0 cos ~bf+ sin 0 sin 4 , f + cos 0/~,

(B.I)

DENSITY EXPANSIONS OF TRANSPORT COEFFICIENTS

213

where the integrand involves only vector product of /', with some arbitrary vectors a~,az . . . . etc. and that f,f,/~ are three orthogonal unit vectors. Denoting It

A,=

f

2~r

sin0d0

I ,__n, d4~

(ai't),

n=0,1,2 .....

(B.2)

=

0

0

with ¢r

A0= f sin 0 dO f d4,, 0

(B.3)

0

we have A0 = 41r, 4~r

(B.4a)

Az = ~

al " a2,

(B.4b)

A4 = ~

[(al • az)(a3 " a4) + (al " a3)(a2 " a4) + (al • a4)(a2 • a3)],

(B.4c)

A6 = 1~5 {(al" a2)[(a3- a4)(as" a6)+ (a3" as)(a4 • a6)+ (a3" a6)(a4 • as)] + (al- a3)[(a2 • a4)(a5 • a6) + (az" as) (a4" a6) + (a2" a6)(a4 • as)] + (at" a4)[(a2" a3)(as" a6)+ (az" as)(a3, a6)+ (a2" a6)(a3" adl + (al" as)[(az • a3)(a4 • a6) + (a2" a4)(a3 • a6) + (a2" a6)(a3 • a4)] + ( z l " a6)[(az" a3)(a4" as) + (a2" a4)(a3 • as) + (a2" as)(a3 • a4)]},

(B.4d)

and that A, = 0

for n = odd integer.

(B.5)

Acknowledgments I would like to express my sincere appreciation to Dr. J.R. Dorfman for introducing me to this problem and for his continued interest, guidance, and kind assistance throughout the course of this work. I would also like to thank the Center for Statistical Mechanics and Thermodynamics, University of Texas at Austin, for its hospitality during the time in which part of this paper was prepared.

214

Y. KAN

References and footnotes 1) Y. Yan and J.R. Dorfman, Phys. Rev. A16 (1977) 2447. 2) Y. Pomeau and A. Gervios, Phys. Rev. A9 (1974) 2196; A. Gervois, C. Normand-Alle and Y. Pomeau, Phys. Rev. A12 (1975) 1570. 3) It appears that only the kinetic-kinetic part contribute to the coefficient of the 02 In p term in the density expansion of the transport coetficients. For additional comments, see ref. 1. 4) For a discussion on this point, see Appendix A in ref. I. 5) P.L. Bhatnagar, E.P. Gross and M. Krook, Phys. Rev. 94 (1954) 511. 6) P, Welander, Ark. Fysik 7 (6) 0954) 507. 7) A discussion on this model can also be found in C. Cercignani, Mathematical Method in Kinetic Theory (Plenum Press, New York, 1969). 8) Equations (3.2a) and (3.2b) indicate that there exists a discrepancy in the B.G,K. model when applied to monoatomic gases because they together give A0/rt0 = 5kB/2m, a value which does not agree with both the true Boltzmann equation for Maxwell model and the experimental d a t a - w h i c h agree in giving A0/r/0~ 5ka/3m. 9) See, S. Chapman and T.G. Cowling, The Mathematical Theory of Non-uniform Gases (Cambridge Univ. Press, London, 1970). 10) See, I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals Series and Products, A. Jeffery, ed. (Academic Press, New York and London, 1965) p. 925. ll) See, S.D. Conte, Elementary Numerical Analysis (McGraw-Hill, New York, 1965) p. 138. 12) J. Kestin, E. Paykoc and J.V. Sengers, Physica 54 (1971) 1.