Cluster expansion for transport coefficients for dense simple fluids

(1)

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Cluster expansion for transport coeﬀicients for dense simple fluids

Byung Chan Eu

To cite this version:

Byung Chan Eu. Cluster expansion for transport coeﬀicients for dense simple fluids. Journal de

Physique I, EDP Sciences, 1991, 1 (11), pp.1557-1582. �10.1051/jp1:1991225�. �jpa-00246436�

(2)

J.

Phys.

I France 1

(1991)

1557-1582 NOVEMBRE1991, PAGE 1557

Classification Physics Abstracts

05.20 05.60 51.00

Cluster expansion for transport coefficients for dense simple

fluids

Byung

Chan Eu

(*)

Department

of

Chemistry,

McGill

University,

801 Sherbrooke Street West, Montreal,

Quebec,

Canada H3A 2K6

(Received

29 March 1991,

accepted

26

July

1991)

Abstract. Cluster

expansions

are

presented

for the collision bracket

integrals

for transport coefficients of _a dense

simple

fluid which appear in the kinetic

theory

of dense fluid

reported previously.

The

density

series calculated with the cluster

expansions

_are

given

in

exponential

forms and their

leading

terms consist of _a connected

three-particle

collision operator. ^The ^three-

particle

contributions _are discussed in terms of mass-normalized coordinates which make it

possible

to put them in forms

structurally

rather similar to the

two-particle

collision bracket

integrals

in the

Chapman-Enskog theory.

The three

particle

contributions _are thus cast into forms suitable for numerical

computation.

1. Introduction.

A

theory ill

Of transport prOceSSeS in _a dense fluid _was

developed

in

conjunction

with _a

theory [2]

of irreversible

thermodynamics

_some _years _ago

by

the

present

^author. ^Since ^the evolution

equations

for various

macroscopic

fluxes

appearing

in the

theory

_are amenable to

phenomenological

treatments

[3]

in which the

transport

coefficients _are

regarded

_as

phenomenological

parameters, ^the ^evolution

equations

have been

applied

_as if

they

_are

phenomenological,

and

consequently

the

transport

coefficients therein have not as

yet

been

seriously

studied from the

viewpoint

of molecular

theory. However,

it is essential _to calculate them from the molecular

viewpoint

if _one wishes to relate fluid flow

properties, namely,

fluid

dynamic variables,

to molecular structures and interaction forces between molecules. As

large

as our desire to

accomplish

such _an aim is the

challenge posed by

the

problem

since _even _an

approximate

but

sufficiently

accurate solution of the

many-particle problem

involved is not easy to obtain. Such _a

challenge

_appears to be

rarely

taken up these

days

since

nonequilibrium

molecular

dynamics

methods [4]

actively pursued by

_many in recent years appear ^to offer _a

relatively painless

alternative.

However,

since such numerical methods _are _a kind of

experiment

and not without

limitations,

theoretical endeavors of the kind

pursued

here _are still worthwhile and necessary. It

is,

of _course,

hopeless

to look for

analytic

methods to handle

satisfactorily many-particle problems involved,

but _one may still look for _some ways ^to

(*)

Also _at the

Physics Department

and the Centre for the

Physics

of Materials, McGill

University.

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1558 JOURNAL DE

PHYSIQUE

I lit 11

implement

numerical methods in _a _more economic _manner than the usual

nonequilibrium

molecular

dynamics

methods that

require

solution of

many-particle dynamics

for _a

large

number of

particles.

Studies

[5]

in the past ^{of the} ^virial

equation

of state, for

example,

indicate that _one can understand the

thermodynamic

behavior of

fairly

dense gases in terms of lower

order cluster

integrals

_or virial coefficients. _One _can then ask _a

question

_: whether _a similar

approach

may be taken in the _case of

transport

coefficients. The

present

^work ^is ^motivated

by

this

question.

This

question,

of _course, is _not _new.

Enskog [6]

^and ^others

[7]

devoted their effort to it. Here _we do not aim to calculate

many-particle quantities by analytic

_means, but to cast them in forms convenient to evaluate them

by

suitable numerical methods for _a small number of

particles

which is

usuafly

less

than,

_say, ten. With this

goal clearly

set in mind _we would like to

develop

cluster

expansions

for

transport

coefficients which

provide

well-defined

density

series for the latter and cast the

three-particle

terms therein into forms suitable for numerical

computation.

The cluster

expansions

will be

developed

such that

they

reduce to the

well-known

Chapman-Enskog

results

[8]

for the transport coefficients in

question.

This paper is

organized

_as follows _: in section 2 _a brief review will be

given

of the

dynamic

cluster

expansion

for the collision operator

appearing

in the

theory.

This cluster

expansion

is then used in section 3 to generate ^cluster

expansions

for collision bracket

integrals

with which

various

transport

coefficients _can be

computed.

In section 4 the results obtained _are considered in the _case of hard

spheres

in terms of mass-normalized coordinates which

put

^the

three-particle integrals

in forms that _can be also used for their numerical evaluation in the future. Their structures are rather similar _to the collision bracket

integrals

in the

Chapman- Enskog theory.

Section 6 is for discussion and

concluding

remarks.

2.

Dynandc

cluster

expansion

for collision

operators

: a brief review.

Cluster

expansions

have been used in different versions

by

various authors

[7,9]

in

connection with

transport

coefficients. In the

approaches

taken

by

other authors

[7b]

the evolution

(time-displacement)

_operator is

expanded

into clusters which

requires

_a _resumrna-

tion of series that is

divergent

in the limit

e - 0.

~The meaning

of

e will be

given

in

(2.3) below.)

In the present paper a cluster

expansion

is

performed

_on the

N-particle

collision

operator appearing

in the collision bracket

integrals [lc],

and the

resulting

cluster

expansion

does not

require

_a resumrnation since the

expansion

does not involve

divergent

terns.

It _was shown in _a

[9]

_on a cluster

expansion

for _a

velocity

autocorrelation function that

N-particle

collision

operator

T~'~~ _can be

given

in terms of connected collision operators ^for ^clusters formed out of the

N-particles

in

question.

Let _us denote the Liouville

operator

^of ^N

particles by

L~'~~

L ^~'~~

=

i I(Pilmi) (?/?ri) (?v/or~) (o/op ~)j

=

l'~~

+

Ll'~~ (2. ')

and

£(N)

_~ (N)

= I

LjN) ~jN)

The resolvent operators are defined

by jlo(z)

=

(£j~~ z)~

_~,

(z

=

is,

e _> 0

(2.3)

jl

(z)

=

(£~~~ z)~

^'

(2.4)

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lit I I CLUSTER EXPANSION FOR TRANSPORT COEFFICIENTS 1559

Then the

N-particle

collision

operator

^T~'~~

obeys

the

integral equation T(N)

~

~jN) ~jN) ~~~~~

~n(N)

(~

_~~

which may be

equivalently

written _as

T(N)

~

~jN) ~jN) ~~~~ ~jN)

~~_~~

Equation (2.5)

is the classical

Lippmann-Schwinger equation

for the operator ^T~'~~. ^The system of N

particles

_may be

decomposed

into _a set of clusters

(cj,c~,...,c~)

where cj, c~,..., c~ denote clusters of _one

particle,

two

particles,

,

m

particles, respectively,

and there _are

NI

fl (m!

_)~~

~

if there are a~ clusters of size _m. Then it is

possible

to show

[9]

that T~'~~ is

decomposable

_as follows _:

T~~~

=

I

_l~c,

ci

cm/m;

Cm)

N N N

i I ~J

~

i I I ~ijk

+

Z Z Z Z _l~ijkl

₊ ₊ l$

j2 ~

l <J i «J « k

i <j < k

«

I

N

~

Z Z Z Z _~'J,~~

~

(~'~)

i «j «k «1

This is the classical collision

operator analog

of the cluster

expansion

for

quantum

^mechanical collision operator ^first

developed by Weinberg [10].

Here various collision operators are

defined

by

the

following integral equations

_:

(j

₌

£( _£)j 3lo(z) (y

,

(2.8a) Tjk

₌

ilk _£ljk Slo(z) Tjk

,

(2.8b) Tj

_ki

=

£[

_ki £

lj

ki

Slo (z _T;j

_ki

,

(2. 8C)

lsuk

=

Tok Tu

T<k

(k, ^(2.9a)

ls~j ;~i =

Tq;~i

T~~

T~i,

etc.

(2.9b)

The interaction Liouville

operator £[

^is ^for

particle pair (I,j),

_£)j~ is for

triplet

(I, j,

^k

), £)j.~i

^is ^for disconnected

pairs (I, j)

and

(k, I)

which do not interact with each other.

Therifore,

_there _is

no

potential

_energy term between the two

pairs.

The collision

operator ljj

^is ^for the

pair (I, j)

that is imbedded in

(N 2)

free

spectators

^which ^neither interact among themselves _nor

participate

in the collision process for the

pair (I, _j ).

The

operators lS;j~, lS;j;~i,

etc. describe collision _processes in which there _are _no

spectators

involved in the set

(I, j,

k

), (I, j,

k

I ),

etc., and this effect is achieved

by

the substitution of

terms T~y, ^etc. which describe _a collision _event in which _one of the three

particles

I, j

and k is _a

spectator

ⁱⁿ ^addition to the

(N-2)

_spectators in which

particles

I,

j

and k _are imbedded.

Therefore,

if _a

diagrammatic language

is used _to describe such collision events,

operators

such _as lS~j~ insure connected

diagrams

and thus _a true three-

particle

collision event, and

similarly

for other

operators.

In

fact,

if such

operators

are

expanded

in terms of

binary

collision

operators (~

then the

expansion

becomes free from

singularities arising

from disconnectedness of

diagrams

and there _are _no _e ^'

singularities

that

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1560 JOURNAL DE

PHYSIQUE

I lit 11

occur in _a naive

binary

coflision

expansion

for T~'~~ when N _m 3. In this _sense the cluster

expansion (2.7)

is well behaved with respect to _e and the

corresponding expansion

integrals

calculated with

(2.7)

is

expected

to be also well behaved. We end this brief review with the remark that _one should not _use _a

binary

collision

expansion

for N » 3 before inevitable disconnectivities of clusters _are

properly

taken _care of. This

point

is

discussed in detail in reference

[10]

and also in reference

[la].

3. Cluster

expansions

integrals.

The collision bracket

integrals required

for

calculating transport

coefficients for _a

single- component, simple

dense fluid _are

collectively given by

the formula

[16, lc]

J' ,J'

R(a)

=

f p

²_~

~ ~ ^~~(~v)

~ ^(«) ~ ~~n ~~~ ^~ ^(«)

~z(x)

_~~

_j~

~

j i k

~ ~'~~ ~~ ~ ~~

where

p

=

I/kBT;

_« _runs from I to

3; f~ =1/5

for _«

=

I,

I for _« =2,

1/3

for

«

=3;

g ₌

(mr/2 k~ T)"2/(nd)2 _(3.2)

with m~

denoting

the reduced _mass, _n the number

density

and d the size parameter of the molecule _; and the collision

operator

T~~~~

~~(z)

^is ^for a

system consisting

of N

subsystems

of

s

particles

which make _up the whole system of

A'particles

where A'= sN. The molecular

expressions h)"~

^for ^various

macroscopic

moments _are

~y j

h(1) ~_j~ ~_j(2)

_~

~ j~

_~

j(2) (~

~~

J J J J Jk jk _'

k»j

« =

2 hj~~~ ⁼

mj Cl

₊

Z _Fj~ _rj~ _mj pip

,

(3.4)

~j

~ ~~'

~/~~~ l("~j~~~~~ ~kj ~j+(~~ ~k~jk°~j~'llj(~' (~'~)

#j #J

where

ej

is the

peculiar velocity

of

particle _j

defined

by ej

₌ _v~ u

(u

= fluid

velocity), (3.6)

F~~ =

(3V/3q~)

_(r~~ ₌ _r~

r~), (3.7)

and

(

_denotes _the

enthalpy

_per unit _mass, _p the pressure, p the _mass

density,

and

[A

_]~~~ _means the traceless

symmetric _part

of second rank tensor A. The molecular moments,

h)'~, h)~~

^and h)~~, when

averaged

over the

phase

_space with the distribution function _as the statistical

weight, give

rise to the traceless

symmetric

part ^of ^the stress tensor, its _excess trace

part

and the heat flux of the

fluid, respectively. Therefore, R~'~

is related to the

viscosity,

R~~~ to the bulk

viscosity,

and R~~~ to the thermal

conductivity

of the fluid. We remark that

they

have been used _as

phenomenological parameters

ⁱⁿ ^the

generalized hydrodynamics study

up ^to now, but the

density dependence

of

R~"~

has not been calculated _as yet.

If the

similarity principle [lc, ld]

is used and it is the

principle

_on ^which the

present

kinetic

theory

is based then the collision bracket

integrals

_may be

computed

with collision operator T~'~~ defined

by (2.5)

instead of

(,___(z)

^which ^is a kind of _« distorted _wave _» form for T~~~'~

(6)

lit 11 CLUSTER EXPANSION FOR TRANSPORT COEFFICIENTS 1561

Therefore,

_we consider the

following

bracket

integral

in this work :

N N

R(a)= f p2~ ~ ~ ~~(N)~(«)~ ~~n(N)(~j~(«)~(N) (~_8)

~

j i k =1

~ ~ ~

Here N _now denotes the number of

particles

in the system. Ît îs ûseful to note that

T~'~~ in fact may be

regarded

_as the collision

operator

for the cluster set

(sj,

_s~,.., _s

N)

~

(l, I,...,

I

),

that

is,

the

decomposition

of N

particles

into N one-member

clusters,

and the

similarity principle

asserts that this cluster _set is

statistically

and

kinetically

siJnilar _to the cluster set

(sj,

_s~,

,

s

N)

where _s _» I.

It is convenient _to

separate h)~~

^into ^the ^kinetic ^and

potential

_energy

part

as follows _:

N

h~~~~ ⁼ I~ ⁺

jj _lI~~, (3.9a)

j~k

("~

N

= h~~~~ 8

(rj

r =

(

₊

jj ll~~, (3.9b)

j ~k

where

Iy

^and

(

_are the

single-particle

contributions that

depend

_on

only

the

momcptum

^of

particle j,

^and

l§j~

^and

$lj~

^are ^the interaction

energy-dependent

contributions that

generally depend

_on both

position

vectors and _momenta _see

(3.3)-(3.5).

Since the

potential

energy is

assumed to be

pairwise additive, Hj~

^and _$i~~ are also

pairwise

additive. It is also convenient to abbreviate

integrals

in the

following

_manner

(A T~'~~(B)

_m dx~'~~

FI'~~A

3

T~'~~B, _(3.10)

(A (C(B)~

=

V~(A (C(B) (3.ll)

In this notation _we _can write the bracket

integral

_as follows _:

N

N N N N N

R~~~

"

ifa P

^~_g

£ Jj

+

z £ tl~l

^T~'~>

£ Ik

+

z z wkl

J J t»j k k t»k

m

if~ p~gtN _(3.12)

where tN can be

decomposed

into the

following

four

components

tN = tKK+ t~p + tpK + tpp,

(3.13)

N

tKK ₌

jj (

T~'~~

jj ~)

,

(3.14)

j k

N

N N

tKp=

jj ( Tl'~~ jj jj ~i),

(3.15)

J k t»k

N

^N N

tpK "

Z Z

lI~-1 T~'~~

ilk

,

(3.16)

j f»j _k

N

^N N N

tPP _"

Z Z f-I

^T~'~~

jj jj

_Wk1

(3.17)

J f»J

k

f»k

Here tKK ^is ^the ^kinetic part

(I,e.,

the

single~partide contribution)

while tK~ tpK ^and

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1562 JOURNAL DE PHYSIQUE I M 11

tpp are either mixed contributions of the kinetic and

potential

_parts _or the

potential (many- particle)

contributions. These four contributions will be considered

separately.

Let _us _now observe that the

following

identities hold for collision

operators

^which are

differential operators ⁱⁿ momenta

dx~'~~

T~y

F(x~'~~)

=

0

,

(3.18a)

dx~'~~

Y;~~

^F

^(x~'~~)

= 0

,

etc

(3.18b)

These _are easy to

verify by using

the definitions of collision operator or the classical

Lippmann-Schwinger integral equations (2.8a-c)

for them.

Therefore, by using (3.18a)

for

example,

_we _can show that

(()T~y[Ij)

₌ 0

(3,19)

identically,

_{if k #} I or

I

_# _I, _and

k, I

_#

j.

3.I COLLISION BRACKET INTEGRAL tKK. Substitution of the cluster

expansion (2.7)

for

T~'~~

yields

the collision bracket

integral

_tKK in the fornl

tKK =

z z Ii _j _Ill _Tki _llji

+

z z z Ill

_l~~im

_iji

+ +

~lJ~l ~ _«~«m

+

z z~z _z Ill _l~ki

;mp

ilj i

₊

(3.20)

k«I«m«p

We first consider the

binary

collision _ternl:

N N N

ti~i

»

z z z If _Tki _Ij1 _(3.21)

1=ij=ik«f

Since the

particles

_are identical and

n =

NIV (3.22)

in the

thernlodynamic limit,

the _sum _over _I _can be

replaced by

nV and thus

ti~i

_can be written

as

N N

t(~i

= n

£ £ £

V

(Ii

_T~y

(Ij) (3.23)

J=i k«I

Because of the

identity (3.19)

^there are

only (N

I

binary

collision operators ^that

give

rise to _a

nonvanishing integral,

and hence there follows from

(3.23)

ti~i

= n

(N

I V

(ii _Ti~ _fi

₊

_I~) _(3.24)

Note that the _sum _over

j

in

(3.23)

has

only

two tennis

Ii

and

I~

that

yield

_a

nonvanishing integral.

Since

Ti~

is

symmetric

with respect ^to the

particle indices,

_we

finally

obtain from

(3.24)

ti~i

= n~

(f~~~ Ti~

^1~~~)

~

(3.25)

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M 11 CLUSTER EXPANSION FOR TRANSPORT COEFFICIENTS 1563

where

I~~~

=

Ii

₊

I~

+ + I

(3.26)

and

similarly

for i~~~ See

(3.I I)

for the abbreviation

agreed

_upon for the bracket

integrals.

The

integral

on the

right

hand side of

(3.25)

still _can

depend

on

density

since the

pair

correlation function

appearing

in it

depends

_on

density.

^We will _pay attention to this aspect later. For _now _we tum to the

three~particle contribution,

which is

given by

N N N

ti~i

m

z z z z z If

_l~kim

_Ij1 _(3.2?)

1=ij=i k«I«m

Since the

particles

_are identical the _sum _over I _can be

replaced by

N times _a ternl and _we thus find

N N

tit

" n

£ £ £ £

V

(Ii _lskim IJI (3.28)

J=i k«f<m

Since for k _~ l the collision

operator ls~y~ gives

_a

vanishing

contributions

by

virtue of the

identity (3.18b)

there _are

(N

I

)(N 2)/2 equivalent

collision

operators,

^and

consequently (3.28)

_may be written _as

ti~i

=

n~ V~(ii

lSj~3

(I~~~)

=

I

n3

y(3)

_~~~~ _~

(3)j

~~_~~~

3 3

where the second line is obtained _on

symmetrization

of the

ii

factor. One _can _use _a similar

procedure

for other tennis in the cluster

expansion (3.20)

for tKK ^to obtain the

expansion

~ ~

t it>,

tKK ₌

z

n z _(f~~~ _lsc,

c~.., c

~

I~~~l_t

(3 .30)

f 2 <cmi

where the _sum _over

(c~)

is _an ordered _sequence

(fl' off particles

with the indices

increasing along

the _sequence,

namely,

_lSi~3 _lSi~34,

lsi~

34, etc. Because the _sequence is

ordered,

the ml factor that appears ⁱⁿ

(2.7)

is _not

necessity

in

(3.30). Equation (3.30)

is _a

dynamical analog of

the cluster

expansion for

the

configuration integral

in

equilibrium

statistical

mechanics

[I ii.

As mentioned

earlier,

since the

integrals

in

(3.30) depend

_on

density by

virtue of the distribution function

F~'~~ being dependent

_on

density, (3.30)

is _not _a

complete density

expansion

for

tK~

^In

^fact,

^since ^the

^I-th

^order ^ternl ⁱⁿ

^(3.30)

^involves ^the ^reduced distribution function

F~~ (x~

_~) defined

by

F)()(x~~))

₌ ^V ^-'~ ⁺

ldxt

~ i dx

~

F~'~) (3.31)

the

density dependence

of the _ternl will be that of

F)(),

which is

generally

weak in the _sense that reduced distribution functions

depend

_on

density,

but not so

strongly.

We

expand

F~)

in

density

series _:

~z(f)

~

=

z ^~z(f)

~m

(~ ~~)

~~

m o

~

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15M JOURNAL DE

PHYSIQUE

I M II

where

Fj~~

₌

,

~

F~~ (3.33)

m.

~~ n

o

On substitution of

(3.32)

into the

integral

ⁱⁿ

(3.30)

there results _a

density

series for the

integral

m

(i~~~

lS~

~ ~

I~~))

₌

£ A)~~(ci

c~ c

~) ^nJ. (3.34)

~ _~

j o

Here the coefficient

A)~~

^is ^defined

by A(f)(~~

_{~~,_} _~

=

l °~

(f(f)jl~

~

jI(f)j

,

(3,35)

~ ~

jl

3fl~ ^~~^~~ ~

n 0

which is

directly

related to the series

(3.32)

for

Fj~~. Substituting (3.34)

into

(3.30)

and

rearranging

the tennis in powers of _n, _we obtain the series in _n for tKK.

cJ f

tKK "

fl~ B)~~ £ )

a~~~

(3.36)

f ₀

where

I j ^If ^+2- ^jl'

~~~~ _~~

l~o fi ^~ij ^~~~

^~^~ ^~~~~ ^~~

^~~~~ ^'~

The factor B)~~ ^is a

quantity consisting

of the

binary

collision

integral

which _we will

specify

later when other contributions tKp, ^etc, are

expanded

in

density

series similar to

(3.36).

It will tum out to be

basically

the

Chapman~Enskog

collision bracket

integral [8].

The

meaning

of the _sum _over

(c~)

is the _same _as in

(3.30).

It is useful to write out

explicitly

_a few

leading

terms in

(3.36)

~~°~

~

Aj(12)/Bj, (3.38a)

a~'~

= _2!

A)~~(12)

+ 3!

A)(123)j/B), ^(3.38b)

a~~~

=

A)~~(12)

₊ A

)~~(123)

₊

A)~~(1234)

₊

A)~~(12

34

) /B)~~, (3.38c)

21 31 41 41

etc.

We remark that A

)~~(12)

involves

basically

the third virial coefficient

although

the collision operator ^involved ^is

binary. Similarly, Aj~~(12)

and

A)~~(123)

involve

basically

the fourth virial coefficient while the relevant collision operator for the latter is _{the lS123}

operator. Thus,

a~°~ is _a

two-particle contribution,

a~~~ _a

three-particle contribution,

etc.

3.2 COLLISION _BRACKET _INTEGRALS tKp, tpK AND tpp. ^The mixed and

potential

contributions tKp, tpK and tpp can be

expanded

ⁱⁿ

density

series in the _same _manner _as for tKK. We present ^the ^results

only

:

w ~ if)-

~KP ~

i

f§ I (~

^~~~ _l~ci

c~ c~

W~~)

,

(3.39a)

2 cmj

(10)

w

~y

if)>

tPK _"

I

fi I

W~~~ l~cj_c2 _c~ (1~~~) t ,

(3'39b)

f "2 cmj

cc ~

f If),

~PP ~

i

f§ I (~~~

_l~ci

c2 cm @~~~) t ,

(3.39c)

~ Cml

where

t

lI'~~~

=

£

_H§y

(3.40)

1«J

and

similarl~

^for

^ii/~~

Substitution of the

density expansion (3.32)

for reduced distribution function

F~~

^into ^the

^integrals

ⁱⁿ

^(3.38a-c) ^yields

w ~

f t~p =

n2Bl~~ z

n g#I, (3.41a)

t o

w ~

f

~PK "

fl~~~~~ £

fi ~#, _(~.~l~)

f 0

CC I

tPP " n

~Bj~~ I ) _g$~

,

(3Alc)

1= ₀

where

~~'

"

'~~lo ~ _~ji~' _~' _~~~~~~

~2

~m~'~'' ~~~~

~'k

=

ii i~ ~ _~)(~~~' Gill

~

~~(Ci _C2 Cm

)/B'~~, (3.42b)

=

~~

_~~

J o

(

₊

j ), ~~~

^~~

Gj(±

²

-J)(~

~~l

~ ~~

~m~'~', (~~~

~k

~ ~~~~~~ ~~ ^~

~

)

"

j $ _lP~~

~C<C2 Cm ~~'~~~i

,

(3.43a)

n o

~'~'

~

~~~~i ~2 ~

m ^"

) ) _l

_fl~~~

l~c,_c~ _cm ^I~~~l_t

~

,

(3.43b)

We _now choose B)~~ as follows _:

~j2)

~

l

j j ^/(2)

_~

^q#2)

_{~~~ ~}⁽²⁾ _~

~(2)j

_j~

~ ~~

~~

2 2

which is the

two~particle

contribution to

R~"~ Now, by combining (3.36)

and

(3Ala-c)

_we

finally

obtain the

density expansion

for tN

cc I

~N " ~ ~~~~~ ^l

£ ( ^ij _(~.~~)

l

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1566 JOURNAL DE PHYSIQUE I M II

where

ai = ia~~~ +

gii

+

gik

₊

gil~i

,

(3.46)

and thus

w I

R~"~

=

iP

^~

gfa

_n^~

]~~(

l

z )

aij ^(3.47)

1=1

It is convenient to _resum this series into _an

exponential

fornl

by using

the method of cumulants

[12].

We thus find

w I _w I

exP

~- ⁱ iffy

= I

I )ai _(3.48)

1= t

where the cumulant

fly

^is

given by

the formula

Pi

₌

f' ^~ Z (m

i

)' Z fl ) _(aj/Jil', _(3.491

i kj) j i

the _surn _over

(k~) being subject

to the conditions

t i

z jkj

=

I

,

z

_k~

= m

(3.50)

J i J =1

With

(3.48)

_we

finally

obtain the collision bracket

integral

in the forrn

« I

R~"~

=

iP~gfa n~Bl~~exP (- ^z ) _Pij _(3.sll

i=1

It is

helpful

to remark that

fly

are the

dynamical analogs

of the irreducible cluster

integrals

in

equilibrium

statistical mechanics _:

pi

involves three

particles, p~

four

particles,

etc. It is also

significant

to note that the

correspondence

between the

dynamical

and

equilibrium

irreducible cluster

integrals

is

brought

about

by resumming

the series in

(3.47)

into _an

exponential

form

by

_means of cumulants.

4.

Transport

coefficients.

The collision bracket

integrals

in

(3.51)

_now _can be used to calculate various transport coefficients for _a dense

simple

fluid. We consider

only viscosity

and thernlal

conductivity, deferring

the consideration of bulk

viscosity

to _a separate treatment since it does not fit in the

general

result

(3.51)

obtained here.

4.I VIscosITY. Since in the modified moment method the

viscosity

of _a

simple

fluid is

defined

by [16, lc]

'l =

2P~gfl/R~~~ (4.1)

its

density expansion

is

easily

found from

(3.51)

_:

CC I

'l #

[10 kB ~Pilfl~Bj~~j

_eXp

i ) _fly _(4.2)

1=1

(12)

It _can be shown with the hard

sphere

^model that the front factor in the _square bracket in

(4.2)

is indeed the

Chapman-Enskog viscosity [8]. Therefore,

we will define it

by

the fornlula

~o = 10

kB Tp~lin~B)~~ (4.3)

The _excess

viscosity

_over and above the

Chapman~Enskog viscosity

at low

density

is then

given by

A~

_=~-~o

w

~i

= 110 exp

z

~ fly

-1

(4.4)

i=1

We _now show that ~o is indeed the

Chapman~Enskog viscosity.

For this _purpose _we

consider the hard

sphere

model for interaction. In that _case

B)~~

^is

given by

Bj~~

=

(i~~~( Ti~( I~~~)

~]~ o

(4.5)

Since

according

to the

intertwining

relation

[13, 14]

the

Ti~ operator

may be written _as

Ti~

= D

(12 )

£)~~ £)~~ ^D

(12 ) (4.6)

where

D(12)

is the Mo4ler _wave

operator [13, _14]

for Liouville

operator

£~~~ Since

I~~~

depends

_on momenta

only

and £)~~ is a derivative operator ⁱⁿ

position tj2) ~(2)

~ ~

and therefore

(4.5)

_may be written as

Bj~~

=

(i~~

£)~~ ^D

(12)

_{(1~~~) ~]~}

o

(4.7)

When the center of _mass and relative

position

vectors _are introduced and the Liouville

operator

£j~~ ^is

expressed therewith,

the

integral

in

(4.7)

_can be written in _a familiar form _as follows _:

l~)~~

"

(~ kB ~)~ dPl dP2

^~ ^~~

~~121~,

l~~~

fl(~l) f2(~2) (~.8)

where

lWW,Wwlmlwi*W?+W?W?-Wiwi-W2W21~~~.

~wf Wf

₊

Wf Wf

W

i

Wi W~

W~]~~~

,

(4.9) f;(W;)

=

(2 «mkB T)~~/~

_exp

(- lI~?), (4.10)

W;

=

(p;

mu

)1(2 mkB T)~'~ (4,

I

I)

The details of the derivation of

(4.8)

from

(4.7)

is

given

in

Appendix

A. The notation used here is the standard _one

commonly

found in the Boltzmann kinetic

theory.

The

integral

in

(4.9)

is in fact _one of the

D~integrals [8]

well known in the

Chapman~Enskog theory

of solution for the Boltzmann

equation

_:

j ² w co

4 fl ~~~(2) _w

~ dpi dp~ d~ dbbg j~lww,

ww

fi (wi f~(w~) (4.12)

o o

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1568 JOURNAL DE

PHYSIQUE

I M I1

which in the _case of _a hard

sphere

of radius

«/2

is calculated _to be

[8]

D ~~~(2) ₌ ² arm

~(k~ Tlarm)~/~ (4.13)

Therefore,

_we find

iB)~~

=

16

(k~ T)~

D ~~~(2)

(4.14)

and the

limiting viscosity

_{t~o in} the fornl

'lo "

kB T/8 n~~~(2)

=

~

(mk~ Tin )'/~ (4.15)

16 _tr

for hard

spheres.

This is

exactly

the

viscosity

fornlula

[8]

for hard

spheres

in the

Chapman~

Enskog theory.

It is

independent

of the

density.

In summary, for hard

spheres

the _excess

viscosity

is

given by

the fornlula

A~

=

~

(mk~ Tlar)'/~ exp ( _fly

_I

_(4.16)

16 _« ^~

i

where the

leading

term consists of the

three-particle

contribution. We will discuss this contribution in section 5.

4.2 THERMAL CONDUCTIVITY. In the modified moment method the lowest order solution

of the constitutive

equation

for heat flux

yields

the thermal

conductivity

of _a

simple

dense fluid defined

by

the formula

[16, lc]

"

(t~p TP)~flg/R~~~ (4.17)

where

©~

^is ^the

specific

heat per unit _mass _at constant pressure. Substitution of

(3.51) yields

A in the form

"

1?5(kB T)~/4 miB)~~l

_eXP

_iii I ) _Pi _(4.18)

where

B)~~

is the

two-particle contribution,

and

pi

the

(I

₊ 2

)-particle contribution,

to the thermal

conductivity.

It must be noted that

they

_are not the _same _as for the identical

symbols

used for the

viscosity

the former is defined

by (3.44)

while the latter is defined

by

the set of

definitions

(3.37), (3A2a-c), (3A3a~c), (3.46)

and

(3.49),

where

Ij

₌ _m~

Cl _e~

_m~

_e~ ⁽

,

(4.19)

lI§~ = V,~

e~

⁺ F;~ r;~

e~ (4.20)

In the _case of hard

spheres

of diameter _« it _can be shown that

iBj2)

=

18(k~ T)3jmi

n

(2)(2). _(4.21)

This derivation is

given

in

Appendix

A. We define the

pre-exponential'factor

in

(4.18)

_as the

limiting

thermal

conductivity lo

which with

(4.21)

takes the form

(14)

lo

₌

75(k~ T)~/4 miB)~~

25

dv _{kB T~}

(4.22)

"

16 fl ~~~(2)

where

dv

is the

specific

heat _per unit _mass at constant volume. This is

exactly

the

Chapman- Enskog

thermal

conductivity [8], and,

on substitution of D ~~~(2) from

(4.13),

becomes

25

dv

_T

_k~

_T _1/2

lo

~ ~

(4.23)

32 _« "m

for hard

spheres.

On

elimination,of D~~~(2)

with

(4.15)

for ~o we obtain the well-known relation

[8]

between thermal

conductivity

and

viscosity

lo

=

~

dv _T~o. _(4.24)

Note that there is _a factor of Tin the

present

formula because the thermal

conductivity

in the present

theory

is related to the

Chapman-Cowling

thermal

conductivity [8] by

the relation

lo

=

Tlo(Chapman-Cowling)

because

lo

is defined with respect to V In T instead of VT used for the

thermodynamic

force in the

Chapman-Cowling theory.

With the relation

(4.24)

the thermal

conductivity

formula

(4.18)

_may be written _as

A

=

Note that

pi

here _are _not the _same _as those for

viscosity

_; recall the remark made below

(4.18)

and the definitions

(4.20a, b).

We

explicitly

write out

pi

for thermal

conductivity

_:

Pi

= ai

"

(~~~

+

~~~

(1~123 ~^~~~ ₊ ~~'~~~)2in 0

~

~~~~

~

~~~

_1~2 _~~~~

+ ifi~~~~)

j

~

j ~4~~ ~.~~)

We will examine this formula

together

with the

corresponding expression

for

viscosity

in section 5.

The bulk

viscosity requires

_a

separate

consideration in view of the fact that there is _no _two~

particle

contribution and thus the cluster

expansion

must be modified. It

will, therefore,

be examined elsewhere.

An

exponential density dependence

was used for

viscosity by

Diller

[15]

who

investigated

the

viscosity

of

parahydrogen

_over _a wide _range of

density

and

temperature.

^Similar

exponential

forms _are also used for the

viscosity

and the thermal

conductivity

of _argon

by

Ashurst and Hoover

[16]

who calculated the

transport

coefficients

by

_means of _a

nonequilib-

Rum molecular

dynamics

method. The cluster

expansions

obtained here for

viscosity

and thernlal

conductivity provide

_a kinetic

theory

foundation for the aforementioned

empirical

fornlulas.

5.

Three-particle

conkibufions.

The

density

correction for the

Chapman-Enskog transport

coefficients

requires

solution of

many-particle dynamical problems.

Since

they

_are not solvable in

analytical

fornl _as is well

(15)

1570 JOURNAL DE PHYSIQUE I M II

recognized,

it will be futile to try ^for ^such a

solution,

but it should be useful to cast

fit

in _a

forn1easily

amenable to numerical solution methods. Here _we pay close attention to

pi

in

particular,

which involves _a

three-particle problem.

In the conventional

approach

to a

three-particle problem

_one _uses relative and _center of

mass coordinates _so that the kinetic energy is written in _terms of the center of _mass and two relative kinetic

energies,

the latter

being given

in the two relative coordinates introduced. The relative kinetic

energies

_appear

asymmetrically

because of the difference in the reduced

masses : in _one the reduced _mass

of,

_say,

particles

I and

2,

and in the other the reduced _mass

of

particle

3 and the

particle pair (1, 2).

Such coordinates _are

generally

not so convenient for

treating three~partide dynamics,

and _we _use another set of coordinates _more suitable for

study.

This kind of coordinates _was

initially

introduced in nuclear

physics [17]

and used

by

some authors

[18, 19]

in connection with

many-particle dynamics.

Since

they

_appear to be little used these

days despite

their

potential usefulness,

_a brief review will be worthwhile and

given

below. It also defines the notation _necessary in the

subsequent

section.

S-I MAss-NORMALIzED COORDINATES. Let _us label three

particles by I,j

and

k,

and their

positions

and _momenta in _a fixed coordinate system _are denoted _r~ and

p~, a ₌

I, j, k, respectively.

Their _masses will be denoted _m~, _a

= I,

j,

^k. Define the

following symbols:

M=m,+m~+m~,

m;~ = m, + mj _, etc.

(5.I)

p ^~ = m, m~

m~/M,

d(

= m,~

Rim,

_m~

=

m~(M m~)/pM

₌ _m~

m,j/pM.

Here

d,

and

(

_may be defined

similarly

to

d~ by using I, j,

^and ^k

cyclically.

Note that

d~

^is dimensionless. Then _we introduce the

following

transformations of coordinates

(r,, _r~, r~)

to _a _new set of coordinates

((~i, (~~, (~3)

_:

kl

°

~k ~k

_~>

(~~ =

-d~ m,d~/m;~ m~d~/m;j

_r~

^(5.2)

(~3 m,/M m~/M m~/M

_r~

Here

(~j

is

essentially

the relative

position

vector of

particles

I and

j,

and

(~~

^the ^relative

position

vector between

particle

k and the center of _mass of the

pair (I,j),

while

(~~ ^is ^the center of _mass vector of

particles I, j

and k _as _a whole. The

subscript

k is attached to

(

to indicate that k is the

particle

whose relative distance from the center of

mass of the

pair (I, j )

is

(~~.

The

subscript

k _on

(

_may be used

cyclically

and two _more transfornlations

equivalent

to

(5.2)

_can be obtained

thereby.

The difference between the present transformations

(5.2)

and the conventional _one

already

mentioned is in the factor

d~.

^The ^usefulness ^of ^this ^factor ^will ^become ^clear as we

proceed.

Since the center of _mass

motion for the whole system ^is ^of ^little

interest,

_(~~ _may be set

equal

to _zero.

Let _us denote the vector from the center of _mass to

particle

k

by

_r~~.

~ck " ~k

(m;

_r; + mj

rj

⁺ mk

~k)/fit. (5.3)

Then there holds the

identity

jj

_{m~ r~~}

=

0

(5.4)

k

(16)

where the _sum is _over three

particle

indices. The

following identity

also holds _:

I _{~k ~kl}

" °

(~.5)

k

This is _easy to show vith the definition of

(~i, namely, (5.2).

Of considerable

physical

interest is the

length

of the six-dimensional vector

(

=

((~i, (~~ )

m

(£

j,

£~, £3, £4, £~, £~)

fornled with two relative

position

vectors

(~i

and

(~~.

6

P ^~ "

i ^it

=

fki £

_ki +

fk2 fk2 (5.6a)

t _i

This

quantity gives

_a _measure of

togethemess

of three

particles

and _can be written in the

follov4ng

four different modes _:

p~= p~' jj

3

m~rj~

k=1 3

"

I (1 "lk/~ll I(2

k =1 3

"

I (1 Dlk/~ll ill

k I

"

(H~f) ["lij

_~~ + '~lki ~(i +

'~ljk ~$i ^(5.~~)

which indicate that _p^~is related _to the moment of inertia for three

particles (I, j,

k

).

Since the

Lagrangian

L _can be written in the _new coordinate system as

L

=

§ _z (di;/dt)~ _v(1), _(5.7a)

the

generalized

momenta ar; are

given by

«, =

(aLjai,)

=

pi, _(5.7b)

where the overdot _means the time derivative. The

Hamiltonian, therefore,

is

given by

H

=

£ _«]3

+

£ _(«]1

+

«]2)

₊

v(f) (5.81

In accord vith the

ordering

convention introduced for the six-dimensional vector

(

we may write the momentum

similarly

_:

" "

(H~kl, H~k2)

"

("kl, "k2)

"

("1,

_"2, _"3, _"4, _"5,

"6)

Then in the center of _mass coordinate system the Hamiltonian _may be written in the fornJ

H

=

w2/2

_p ₊

v(1) (5.9)

where w~3 is set

equal

to zero and if

=

jar

₌

(w.

_w _)~/~. In this fornl the Hamiltonian looks like that for _a

two~partide

system except ^for

V(().

Since the _rate of

change

in p can be written _as

~i

=

P~ I I<

_",,

(5.io)

(17)

1572 JOURNAL DE PHYSIQUE I M

the

corresponding

momentum may be defined

by Pp

_# H

~

" P ^~~

~i i I,

_";

(5.ii)

TMs momentum will turn out to be

analogous

to the radial relative _momentum in

two-particle problems.

Smith

[19]

introduced

generalized angular

momenta

by

A,~ =

£;

_arj _£~ _ar,

,

(I, j

= 1,

2,..

,

6

(5.12)

or

A;jfl

=

i~;

_«p~

ip~

_«~,

,

(«, p

= 1, 2

) (5.12')

These

generalized angular

momenta have the

following properties

_among others _:

jj f,

_A,~ ₌ p^~ar~

£j pP~

,

(5.I3a)

1

jj

_A,~

_arj

₌ ²

pf;K-

_ar,

pP~, (5.I3b)

where K is the kinetic energy. With these

relations,

it is easy to show

A~m jj _(A;j)~

~

iJ

= 2 _pp ^~ K _p ^~

P( (5.13c)

Note that A is the

magnitude

of the

generalized angular

momentum tensor A

=

(A,j),

and the kinetic _energy in tennis of A takes _an

especially revealing

form _: K

=

P(/2

_p ₊

A~/2

_pp ^~

(5.15)

which is

isomorphic

to the relative kinetic energy of _two

particles

in

spherical

coordinates.

Hence,

the

ternlinology

radial momentum for

P~

we introduced earlier. Since the

centrifugal

term

A~/2 pp~

_in

_(5.15)

_vanishes _as

p - cc,

P~ asymptotically approaches

w ₌

jar

in the limit.

We _now introduce

hyperpolar

coordinates

[20]

ⁱⁿ ^the

following

manner :

f~

= p cos 0

~,

f5

₌ P sin 0 _cos

04,

~

=

~

~~ ^~jos

0~, ^'~

£~

= p sin

0~

^sin

04

^sin

03

^sin

0~

_cos

ii,

ii

= p sin 0

~ sin

04

sin

03

sin

0~

^sin _~§1

,

where