Three-particle collisions in a gas of hard spheres

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PREPRINT
NASA TM X? (p 3 q ^.3
THREE-^ARTICLE COLLISIONS IN A GA,S
OF HARD SPHERES
W. R. HOEGY
3
J. ` . SENGERS
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TIIftI^;I'a-PARTICLE COLLISIONS IN A GAS OF HARD SPFIERES
VV, R, kIoegy
Laboratory for Planetary Atmospheres
Goddar, d Space Flight Center, Greenbelt, M^.,
,^ ;
and
^;
J, V, Stingers

^^ Institute for Molecular Physics
University of r^aryland, College Park, Md,
May 1970
^^^ .
GODDARD SF>ACE FLIGHT CENTER
Greenbelt, Maryland
r
a
....
PRE,.CEQII'VG PAGE BLANK NOT FILMED,
THREE-PARTICLE COLLISIONS IN A. GA.S OT FIA SPHERES
1^>;'1
^^^
,^.,
'vV. R. Hoegy
^.
Laboratory for Planetary Atmospl^e;^es
.`
^, i
and
i1
J. V. Stingers
1
Institute for Molecular Physics
j
3
1
'; ^'
r
^'' t
ABSTRACT
1
The triple collision. integrals which determine the first density correction
i
to the transport coefficients, are derived for a gas of haa^d spheres using the
binary collision expansion. This expansion provides a convenient tEChnique for
classifying the contributions in torms of sequences of successive binary col-
.
lisions between three molecules. Such sequences contain both interacting and ^
non-interacting collisions. It is demonstrated that for three hard spheres all.
sequences terminate after four successive collisions independent of th9 inter-
acting or non-interacting nature of the collisions. As a consequence the colli-
lion. integrals are related to a limited number of sequences with three and four
collisions only. It is shown that equivalent results are obtained from. the surface
integral forrr^ of the triple collision operator, derived earlier by Green and
Sengexs.
iii
'
r
„^-.^.^w^.-.
^.
^.
PREC^OIKIG PAGE aLANK NOT FILMED.
CONTENTS
Page
r^
ABS
'T'RACT ^ • • r • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • • • • r • ^1^,
^^ I. INTRODUCTION ...•......•.....•.••..•.•p•.•...•... I
i
IZ• TIIE TRIPLE COLLISION OPERATOR , . • • .. • ............ • . 3 ';
III. BINAN,"Y' COLLISION EXPANSION ....... • ..... • , .. • .. • . 9
I'V• REDUCTION OT TRIPLE COLLISION OPERATOR . • ...... , .. • • 17
a
V• COMPARISON WITH SURI'ACE INTEGRAL METHOD .. • ...... • . 22
VT. SUMMARX ... • • ........ • ... • . o • ................... 30
i
ACKNOWLEDGMENT... . • ........................•..... 3]. i
REFERENCES•....•
...................^.......•....... 31
I
APPEXDIX ..••.....•...^•..,.....•...........•....... 35
i
:^
t
f
f
i
^^^_
^' ^
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i ^
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V
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THREE-PARTICLE COL]'i^ISIONS IN A GAS OF HARD SPHERES
I. INTRODUCTION
Non-equilibrium statistical mechanics predicts that the first density cor-
section. to the transport coefficients of a gas can be represented by a ter, m linear
in the density n. That is, the thermal conductivity ^ and the shear viscosity ^
can be written as1,2
.. O ^1 n
^ + .^. ...

r1 = ^u + n + '' ' (1.1)
^1
As is well known, the transport coefficients ^ o and ^7o in the low density limit
^^.
are determined by the solution of the linearized Boltzmann equation. °The
Boltzmann equation takes into account only uncorrelated binary collisions.
The transport coefficients ^ a and ^ o can be expressed in terms of collision.
integrals which involve the velocities of two molecules before and after a binary
collision.
In the past decade many investigators have derived the integral equation.
for the coefficients ^ 1 and ^ i of the first density correction.. For a discussion.
and a bibliography we refer to a recent :review article of Ernst, Haines and
Dorfmana This integral equation leads to collision integrals :for the coefficients
^L 1 and ^ i that involve the. effects of collisions between three molecules. For a
gas of hard spheres it was shown by Green and Sengers nat the triple colli-
3 4 'g
sion integrals can be reduced to a surface integral form, analogous to the binary
1
^r
x
a
collision integrals determining and That is, the integrals could be ex-
%0 770.
pressed in terms of the initial and final velocities of three particles in specified
sequences of successive correlated collisions. Thus the problem was completely
reduced to a study of the dynamics of three spheres.
394
In the earlier formulation the dynamics of the molecules was described
,
with the aid of a time-displacement or streaming operator S_ t (1 • • ^,). The
i
operator is a substitution operator which replaces the phase vari-
S_ t (1 • • t)
ables of t particles at a given time by their values at a time t earlier. As an
i
j alternative approach several investigators have proposed to describe the dy-
namics using a binary collision expansion.5
Here we reconsider the reduction of the triple collision integrals for a gas
of hard spheres from the point of view of the binary collision expansion. This
expansion enables us to decompose the triple collision integrals into a series
of terms consisting of increasing numbers of successive binary collisions. The
decomposition, thus obtained, differs in appearance from the decomposition
i
derived earlier by the surface integral method .4 Nevertheless, we shall show
in Section V that the two decompositions are completely equivalent. One of the
1
advantages of the biwry collision expansion is that certain properties of the
1
triple collision operator, such as its symmetry, can be readily demonstrated.
More importantly, we shall use the binary collision expansion formalism to
eliminate those collision sequences that carrot occur according to the laws of
mechanics.
2
r
The generalization of the Boltzmann equation to include the effect of three-
particle collisions is obtained from a cluster expansion of the Liouville operator.
As a consequence the collision sequences involve not only interacting collisions,
but also non--interacting collisions. In the latter type of collision the particles
pass through one another, instead of exchanging their momenta. We shall prove
that all sequences of binary collisions between three hard spheres terminate
after four successive collisions. Several sequences of three and four successive
collisions can also be ruled out as being physically impossible. Thus the triple
collision integrals will be decomposed into a limited Number of sequences with
three and four collisions only.
H. THE TRIPLE COLLISION OPERATOR
A formal solution of the integral equation which describes the effect of triple
4 collisions, yields for the coefficients X and 771697
1
h
1
dpl
3kT2 A Pi^ I3 A^Pi^
Yr
i

TMY aPi BCPl) I3 B^pl) . (2.1)
f
w
ri
^r
C:
y 3
.0
V
#6
The functions A ( Pj and B('P,) which are functions of the momentumP ', only,
represent tho solutions of the linearized Boltzmann equations
0
5 PI
-y kT)
I2
A(pi) M
6
0
P1 P1
I
B p (2.2)
2* 01) M
The mass of a,molecule is denoted by m and 0(i) is the normalized Maxwell
distribution
function
mkT)-3/2
(21r exp(- p?/2m kT) (2.3)
The tensor is the traceless tensor associated with the dyam*c 'p, 'p,. The
'P,O 'P,
I
operators I and I are linearized binary collision and triple collision integral
2 3
operators. In the cluster expansion method developed by Green • and C6hen9
these operators can be related to the streaming operators S-
t
I
2
12 fdx2 012 j8(12) ;6(l);6(2) Pli (2.4)
L
jnj
1
3 f dx 2 dx 3 0 12 fh(123) — ,9(12) h(13) A(12) A(23) (2.5) 01
3 4
L
+ A(12)) 0(1) qb(2) 0(3) Pli
jul
4
r
V
with
It
l `im S-t ( 1 '' ,^) S+^ (i)
k (2.6)
f
We use the notation dx to indicate an integration over the momentum p and
i i
r
the position i of particle i. The permutation operator Pi interchanges the
i
indices 1 and i. The operator 6 i j is a differential operator
aui j avi j
eij _ OMr, • a^ap^ i * a-r4j • 4aa pj t2.7>

where t T , j is the pair potential of particles i and j .
EquattOn. (2.5) was the stutrting point of the previous reduction of the triple
collision operator.4p6 However, for the binary collision expansion it is more
convenient to use the c -method, first introduced by Zwanzig. 10 This method
uses resolvent operators G (1 • • t) which are the Isplace transforms of the
f-
r
streaming operators
m
G(1 • t) = dt a Et S-t (1 .. '0' f6+ ;l(1 • • t)]-1, (2.8)
o I.
c
where 9 (1 - • t) is the Uouville operator for the t particles
....t) (1 .. t) _ r T 6 i j
g(1 go
1<i< j (2.9)
<e
5