Kinetic equation for the dilute Lorentz gas with attractive forces

Physica 84A (1976) 316-335 © North-Holland Publishing Co.

KINETIC E Q U A T I O N F O R THE DILUTE L O R E N T Z GAS WITH ATTRACTIVE FORCES R. WOJNAR* and J. STECKI

Institute of Fundamental Technological Research and Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland Received 10 February 1976

The kinetic equation for the dilute Lorentz gas of particles with repulsive-attractive potential, is derived. For that purpose the distribution function is decomposed into two parts: the one corresponding to the bounded motion of the marked particle and the second corresponding to its unbounded motion; only the second part, as representing the diffusion, is considered. After the appropriate modification of the projection operator, a general kinetic equation for the diffusion part of the distribution function is obtained. The low density limit of the scattering operator of this equation is found. The regions of bounded motion are excluded in the integration over position space. For the square-well potential, the Laplace-transformed kinetic equation is also given in full detail.

1. Introduction The kinetic theory of the Lorentz gas has been studied in the past extensively, a m o n g others by H a u g e l ) , b u t only for purely repulsive potentials. We have studied2), the kinetic e q u a t i o n for the m o t i o n of a m a r k e d particle in a classical fluid, following the earlier work by Lebowitz a n d Resibois 3) a n d we have f o u n d that those formal a n d general results were n o t directly applicable to systems with interaction potentials c o n t a i n i n g an attractive part. Therefore the kinetic equations have to be derived for the case of such interaction potentials in a separate effort. As a first step towards a general formalism, we have studied in detail the case of the dilute L o r e n t z gas for which the subdivision of the phase space is particularly simple. We have also specified our final e q u a t i o n s for the case of the square-well attrac* Partly based on a thesis submitted to the Institute of Physical Chemistry of the Polish Academy of Sciences, Warsaw, Poland. 316

KINETIC EQUATION FOR DILUTE LORENTZ GAS

317

tive potential which has been used extensively in molecular dynamics studies4), as well as in the approximate theories of a dense fluidS). Another general approach has been described by Hamer and Oppenheim6). A binary collision operator has also been derived by AltenbergerS). It is generally accepted that the Fourier transform of the one-particle distribution functionf(r~, vl, t), denoted b y f ( k , v~, t), satisfies the following linear kinetic equation 2, 3)

( - 2 + ikvl)f(k, vl,

t)=

oi G (k,~)f(k, v~,t .-r)dr,

(1.1)

where

f(k,

Vl, t) = j" dr t e -ik'' I dvN-1

(1.2)

drN-IFN (t),

has been so constructed as to yield the intermediate self-scattering function Is (k, t) and

f'x(t) = e-trNFN (0),

e-aV FN(0) = eikr'~M @1)"'" ~bM(VN)- -

Q

(1.3)

We denote here: v~, ri, the velocity and position vector of particle no. i; (~M(Ui), the Maxwell distribution function 1 md 3

CM(v,) =

exp

2

(-m,flv2/2),

where m~ is the mass of particle no. i, fi--1 = T is temperature,

kBT, kB is

(1.4)

the Boltzmann constant,

N

U = ~ u (Iri - ~J) = ~ uij, i
(1.5)

i
where ui~ is interaction potential between particles no. i and no. j, Q = ~dr Ne-ov,

(1.6)

.O

,(2 is the volume of the gas, N

8

i=~

8ri

8U

1

8

~
is the N-particle Liouville operator.

0.7)

318

R. WOJNAR AND J. STECKI

A l s o f ( k , vl, t) satisfies the initial condition f ( k , vl, O) = q~M(U1),

f ( r t , vl, O) = qbM(V,) 6(r,)

(1.8)

and at all times yields Is (k, t) = ( e x p { - i k r l

(t)} exp {ikrl (0)}> = S d v i f ( k , i)1, t).

(1.9)

An alternative form of the K.E. (1.1) is (-iz

+ ikvl)f(k, vl,z) -f(k,

vl,t

=0)

= G(k,z)f(k,

vl,z),

(1.10)

w h e r e f ( k , vl, z) is the Laplace transform o f f ( k , vl, t) as follows I m z > 0.

f ( z ) = S dt ei~t f ( t ) , o

(1.11)

F o r the dilute Lorentz gas G (k, z) takes the following form 2) G (k, z) = ~ -

dr ( - i z

+ ikVl) e -~u
e -ikr

oO

× ( dt e Izt (e -tK2 d o

-

e-tK2°)eikr

- - i Z + ikvl qbM(V) ,

r = r, -- r 2. (1.12)

Here K2 is the two-particle Liouville operator, see eq. (2.4) for N = 2. It turns out that this scattering o p e r a t o r is valid only for repulsive potentials as will be seen from the sequel.

2. ]['he K.E. for the Lorentz gas with square-well attraction

N o w we derive K.E. for repulsive-attractive type of potential of particles in Lorentz gas. Specifically, we assume

u(r) =

foo V -- < 0, k0,

r < RI R1 < r < R2 Re < r.

(2.1)

Fix a certain configuration of the scattering centers r2, ..., rN and the velocity vl. Then either for vl < vo = ~ / 2 V / m l

(2.2)

KINETIC EQUATION FOR DILUTE LORENTZ GAS

319

blj > bo = R2 x/1 - (Vo/VD2,

(2.3)

or for vi > Vo

and

where b~j is the impact parameter, lh~ = rl~ sin (vl, rlj), for any j for which also R1 < rlj < R2

(2.4)

the marked particle no. 1 is trapped in the potential well. In the dilute gas limit we will treat the attraction spheres of the scattering centres as non overlapping. Clearly the dense fluid case is just the opposite. We divide the accessible phase space of particle no. 1 into the following regions

l =(1 - H ) + H , using the definition N

H = H (rl, vl) = ~1 + ~ r/(R2

--

r~j) ~7(bo

-

blj)

~ (vi

-

Vo)

(2.5)

j=2

or what is equivalent N

H=~,

+ ~ ~/(R2-r,j)~/@l-

v,j)

J=2

with i'Jij = v0 [1 - (blj/R2)Z] -~ and N

"7, = 17[ ~ (r,j - R2)

(2.6)

j=2

where ,q=~(x)=

I0, 1,

x~<0 x>0.

The function (2.5) is equal to zero when particle no. 1 is in a bound state and equal to one when it is unbounded.

320

R. W O J N A R A N D J. STECKI

F o r the case of Lorentz gas it is sufficient to omit the velocity integrations in eq. (1.2). Then F(0)

. = e i k r ,qOM (V,) e - a V / Q

(2.7)

and K,

=

v, .

d

. . t~r 1

1

dU

. . . t;~ll t~r I

(~

v,.--

Cvj

8

1

~r 1

v

y,

D11 j=2

(~ulj.

8

~r I

~V 1

(2.s)

We representf(k, vl, t) as a sum f ( k , v , , t) = f A ( k , v~, t) + f ~ ( k , v, , t),

where

JA(0 = S d r N - ' S d r ~ H F ( t ) ,

.fB(t) = ~ dr N-' ~ dr, (1 - H ) F ( t ) .

(2.9)

The part fA corresponds to initial states in which particle no. 1 is unbounded. These definitions loose their physical meaning in a dense medium. Introduce now the projection operator, independent from time, PA("') = eikq

e-,,c f QA

dr N e i k q H ( . . . ) ,

(2.10)

where QA = .[ drN e - i k r l H eikr' e-/~v = .f drNH e-aU.

(2.11)

We will use the following notation PA(''') = ) ( ( ' ' ' ) ,

(2.12)

where

) =

e ikr~

e -'qU

QA

,

( = ~drNe

_ikrl

..

t,,

( ) = 1.

(2.13)

The following properties are readily proved : - P A P A ( ' " ) = PA("'),

• e-BU ;

P A F ([) = e 'k . . . .

f A ( k , v , , t),

QA (2.14) PAF(0)

" e pv 4,.(v0 = F(0), = e ikr, e -~v f d r N e-~kr~H QA J QA

(1 -- P A ) F ( 0 )

= 0.

KINETIC EQUATION FOR DILUTE LORENTZ GAS

321

With the aid of the operator PA we obtain the following K.E.

- - PAF (t) = -- PAKNPAF (t) ~t t

+ PAKN S dr exp {--r (1 -- PA) Ku} (1 -- PA) KNPAF(t -- r). o

(2.15)

Acting from the left with < we obtain

-

fA(k, vl, t) + ikv fA(k, vl, t) ~t ~ -

= l i d r K N e x p { - - r ( 1 - - P A ) KN}(1--PA)KNIfA(k,v,,t--7:).

(2.16)

To obtain the second component in the left-hand side we used also the definition of QA, (12.11), and the vanishing of the integral (2.17)

f drsHK~ {~A-(~)fA(k,v~,t)=0. This may be seen after writing the integral in the form

dr u-~

drlH g e-Or ~rl

+

~t" fA/QA

ml13 0

and application of Gauss' theorem together with exploiting the symmetry properties of the integrand.

3. Low-density K.E. The formal K.E. (2.16) will be specified now for the case of binary encounters at low density. By analogy with numerous already known kinetic equations we will extract the term proportional to (N - 1), the number of scattering centers, from the right-hand side of (2.16). First we expand formally the right-hand side in powers of (-T), obtaining

( OTt

ikvl ) fA(k, Vl , t)

=-" / f O

drKN~./>o ( - r[)(~l n" ,

--PA) KN]"+llfA(k'v''t--T)

(3.1)

322

R. W O J N A R A N D J. STECKI

or

(-~t + ikvt)fg(t)=~o~' (--z)n

(3.2)

o

where

J(") = K~,,[(1 - PA) KN]n+t>

(3.3)

and we have omitted for abbreviation the arguments k, vt offA. Using the definition (2.13)1 and the relation (2.17) we obtain PAKN>fA(Vl)

=

(3.4)

>ikVlfA(V0,

SO

(1 - PA)/N>UA(I)I)

ea'r'KN~AfA(Vl)

=

(3.5)

and (3.3) may be rewritten in the form gin) = KN [(KN __ PAKN)]n eikrlKN e-BY QA

(3.6)

After introducing the notation:

po(...) =

drNH

e_By

(...),

(3.7)

QA N

Z =Z(ru,v,)

=KN fA(V~)

zj=K2(1,j)

fA(Vl) Q q~M(/)l) QA(v~)

K2(1,j)=v,

Q

= ~ ZJ, q~M(Vl) QA(Vl) S=z 1 ~uij g fA(Va) Q ~rl OVl q~M(Vl) QA(Vl)'

(3.8)

(3.9)

ml

~

1

eu,j

e

~rl

ml

0rl

Or1

(3.10) '

and using the commutation rule e-BU

KN-Q

4,~(vO (-.') -

e-BO

Q

4'M(V~) K~,('")

(3.11)

i

KINETIC EQUATION FOR DILUTE LORENTZ GAS

323

~ve have •

e-Or

J'"~fA(Vl) = e'k"~bM (Vl) ~

(ikvl + KN) A,Z,

(3.12)

where we abbreviate A , = [ikv, + KN -- po (ikvl + KN)]".

(3.13)

Defining Gtn)fA = (Jtn)fA

(3.14)

we obtain from (3.11) and (3.7) G'"'fA = QA eoq~M (v~) (ikvt + KN) A,Z.

Q

(3.15)

Because according to (3.7) and (2.11) po. C = C

(3.16)

for any constant C, and, moreover, as one can prove making use of (2.17), po (ikvl + KN) (1 - po) (ikvl + KN) ("') = P°KN (ikv~ + KN) ('.')

(3.17)

therefore G~,:,fA -- QA pOq5M (/)1) I N (ikvl + Kr) A , - 1 Z . Q

(3.18)

The factor Z contains ( N - 1) terms, each of which gives the same contribution after averaging. So we observe that P° Z is at least of the order 9 = NIl2, and that 0¢,_2(vl ) = po (ikvl + KN) An-zZ

(3.19)

is also at least of the order 9. Therefore, if we want to neglect all higher order terms, i e. containing 92, 93, ... etc., we should retain in (3.18) terms containing only one po (at the left). In passing, one also verifies that the QA/Q factor is not changing anything since QA/Q = 1 - 9yl (v~) + ..., where

•1(VI) is an already known function.

(3.20)

324

R. W O J N A R A N D J. STECKI

Therefore we are left with N

G ( , ) fa _

QA POCMKN (ikvl + K,,)" ~ ZJQ j:2

(3.21)

Recalling the definition of Kx, eq. (2.8), we see that it contains a summation over all the scattering centers. The only term giving rise to a single factor N after averaging is the one in which none of these summations is allowed to introduce additional N factors. It reads N

ubi.j"~(")"~A -

QA pooM ~ K,_ (1,j) [ikv, + K,_ (1,j)]" Zj. Q j=2

(3.22)

Now, ~1~ appearing in H, eq. (2.5) may be represented as '~

=I~[1

-

(R2

-

r,fl]

=

J

1 -

~,~(R2

-

r,:)

J

+ E E 7] ( R 2 -- r l j ) )] ( R 2 -- r l k ) j k

j~k

-- E E E ~ 7 j k l

(R2 - r~fl,i(R2 - r,k),](R2 - - r , , ) + ...

jCk, k~l j~-I

and in the low density limit, when particle no. 1 cannot be simultaneously in the attraction spheres of different particles j ~ k -¢ l, we have ~ll = 1 - ~ ] .J

(Rz - rift.

(3.23)

Therefore (3.20) reads b,oJA G(") +" = ~I- f dri y.

1 --,=2Z '~ '](R2 - r j , )

-'[- 1=2~~] (R2 -- I'll) t] (b° - bit) t] (Ul -- u°) 1 N

x e-eUqSM
and again for low density we m a y retain only terms with j = l. N o w we have ( N - 1) identical terms and after integration over (N - 2) positions we obtain finally

G(") b l . ar A = ~) S d r ~ z H j z g ( r , z ) ~ M ( V l ) K z ( i k v l

q_ K2)nK2 -fA(/J') -

0M(Vl) '

(3.24)

KINETIC EQUATION FOR DILUTE LORENTZ GAS

325

where H~2 = H12 (r,2, v,) = r] ( q 2 - R2) + ~] (R2 - r12)r] (bo - b12)~7 (vl - Vo), (3.25)

g(rl2) --

N-lD2Qf e-avdr3drN'N

(3.26)

K, = /(2 (1, 2)

(3.27)

and Q/QA = 1 in virtue o f (3.20). Using the relation ,~- ikrl lun ,fikr~(

(ikVl + K2)" (...) . . . .

2 '-

~'")

we obtain G{,,) bi.Jr a = 0 5 dr12 H12g @12) q~M@I)K2 ,~-ik,,r,., ~ "'2

eikqK2

fA(Vl)

(3.28)

and resuming all /,-~7 (.n ) , s we have t

G ~ i . A = e I d r ~ H~2g ( q 2 ) ~ ( v 0

~ dr K2 e -ik~ e -~'~ eak'~K2f~(k' v,, t -o

CM(v,)

v)

(3.29) This is one possible f o r m o f the scattering operator; another is obtained by resuming (3.24) t

Cb,nfA = 0 ~ d"12/C12g (r12) CM(<) I d~/~2 e-~"*~+"~/~2 f~(k, ~ , t o

~)

4',~(<) (3.30)

F o r finite t, the po and time integration can be c o m m u t e d : however, for t --+ o% it can be shown 7) that different results are obtained according to the order o f these two operations. Here g(r12) is the radial distribution function (for the pair light particle-scatterer). It will be approximated by its low-density limit g (rl, r2) = e -pu'2 (1 + ...)

(3.31)

in detailed computations which will reduce the scattering operator to a practical form.

326

R. WOJNAR AND J, STECKI

A Laplace transformed f o r m of the kinetic equation proved most practical for the step-wise potential considered here. We define the Laplace transformation by (1.11) and obtain

G~i, (k, Z)fA (k, vl, z) = 9 5 drlzH12g (r12) OM(v,) x K2 e -ikq

1

--iz + K2

eikqK2 fA(k, Vl, Z) 4~M(vl)

or GbinfA = Q I drl2Hlxg (r12) q~M(V,) e -ik'' ( - i k v l + K2) oo

x 5 dr e i=r e -~K: ( - i k V l + K2) e ik'~ fA(V0

(3.32)

¢.(<)

o

Integrating by parts we obtain co

f elZr e-~Ka _

,

Imz

>

0,

-iz + K 2 0

f e t z ~ ( - - ~ z ) e -'r2 = - l i+m ( ie i ~ze - ',K 2_) + + ~1

(3.33) -iz + K2'

0

o(2

e iz~

e -~K= = - l i m (ei=~K2 e -~K:) + K2 ~T 2

z ~ co

0

-

iz lim (e ~= e - ~ 2 ) + iz + (iz) 2 - i z + K2

-¢~oo

If we assume that for ~ -+ oo and I m z > 0, then e izr e -rK2 --+ 0,

i.e. after action on a s m o o t h function ~ (rl, vl) and integration over positions a vanishing result is produced• So we see that C¢2

( - i k v l + K2) ~ dz e t'' e -'K2 ( - i k V l + K2) 0 oo

= iz -- ikVl + ( - - i z + ikvl) ~ dz e t= e -~r2 ( - i z + ikvl). 0

(3.34)

KINETIC EQUATION FOR DILUTE LORENTZ GAS

327

Further, we remark that if we define (3.35)

K2° = v ~ - - , grl the obvious equality holds oo

(-ikvl

0 + K2)

• ~ dr e 'z'

e - ~K~o( - - i k v l

o

+ K°2) e ikrl f A ( v l ) - 0 CM(v~)

(3.36)

and that eq. (3.34) is also satisfied if K2 is replaced by K 2. Therefore -iz

+ ikvl

(-iz

-

+ ikVl)~ dr e iZ: e -~r2° ( - i z

+ ikVl) -f ~ -( v , )

O

-

O.

CM(v,) (3.37)

Insertion of (3.34) and (3.37) into (3.32) gives GbinfA = 0 S dr12H12g

(r12)

•M(U1)

e ikrl (--iz

-4- ikvt)

zo

x S dr e Iz~ (e -~r2 - e -'r~°) ( - i z + i k v l ) e ikrl fA(Vx) o

(3.38)

CM(v3

The form of the kinetic equation is still given by (1.10), but for the initial condition of an unbounded particle 1 the appropriate and correct kinetic equation is given by (1.10) with (3.38). For the initial condition of a particle bounded to a given scattering center, no diffusion obviously occurs and a kinetic equation would be a purely formal device to describe the mechanical oscillations of bounded 1. The scattering operator in (3.38) differs from (1.12); formally it may be obtained from (1.12), if we modify the potential appearing in exp ( - f l u ) , by a further exclusion, expressed by (2.2-4) of regions of bounded motion, from the space of integration.

4. Motion of a light particle in the square-well potential In what follows we drop the particle indices I, 2 using simply r and vl. For the purpose of further transformations of Gbin, the action exp ( - r K ~ ) and exp (--TK2) on a function of r and v should be tion of the first one, being the operator of free motion, may be because e-~K:°r = r -- vr,

e-~Kz°v = V.

and v for r 12, ]'l of the operators known. The aceasily obtained,

(4.1)

328

R. WOJNAR AND J. STECKI

To find the action of the second operator, the motion o f a single particle in the field of a scatterer should be found = e

r,

v ( - r ) = e-~K:v.

(4.2)

Consider a cylindrical coordinate system l, b, e centered on the scatterer, with 1 axis parallel to v,. Then b is simply the impact parameter hi2. To find the motion of an u n b o u n d e d particle we divide the position space into following subregions (see fig. 1): (la)

R I > b > 0,

-

(lb) bo > b > R1,

" 2\,'R

b 2

> 1 >

-

\ ,/R z2-

b2

R 2 > v/l 2 + b 2 > R 1

(2)

R, > b > 0,

\ :/R 2,

(3)

b~ > b > 0,

oo > l >

(4)

R2 > b > bl,

_

b 2 > l>

\/R 2 - b 2

V'R~-b d

oo > 1 > ( R e - b 2

(4.3)

m>l>-oo

(5a) oo > b > R2, (5b) R 2 > b > O,

-

(R~-

b2 > l >

-oo

R 2 > 412 + b 2 > R 1

(6)

R2 > b > bo,

(7)

RI > \ ' 1 z + b 2 > O.

The special value o f the impact parameter bo is defined by (2.3), whereas bL is defined by b~ = R1 \/1 + (Vo/V)2,

(4.4)

with Vo given by (2.2). Furthermore, we denote by (1) the region c o m p o s e d from (la) and (lb), and by (5) the one c o m p o s e d f r o m (5a) and (5b).

5ct

F[7

f 6 ~

tb 0

\ R~

, R2

g

Fig. I. Division of the space l, b, e.

KINETIC EQUATION FOR DILUTE LORENTZ GAS

329

Now, the motion of the particle no. 1 may be found according to the values r and v, which determine the initial conditions for motion backwards in time, in conformity with the operators exp ( - z K 2 ) or exp ( - z K ° ) . For r, v belonging t o subregion (5), the action of the operator exp (--TK2) is identical with that of the operator exp (--TK2°) as the particle moves freely, and the contribution of this region to Gb~, is equal to zero. The contribution of subregions (6) and (7) is also equal to zero: region (6) is not accessible to unbounded motion according to the shape of the function H~ 2, eq. (3.25) and region (7) as the region of hard core, where exp ( - f l u ) = O. In the remaining regions, until the time T~ of the first acceleration of the motion, the term with exp ( - r K °) cancels the term with exp (--TK2). Therefore we may change the lower limit in the time integral to z~. For that reason, the following time integrals are to be calculated in G ~ n f ( V ) : T O = e -ikr i~ dr e i= e -~K~° e ik~ i (z - k v ) g,(v)

(4.5)

Tt

and T ---- e -ikr ~ dr. e iz~ e - ~ 2 e ~*~ i (z - k v ) W(v),

(4.6)

Tt

where ,; = v,(v) = ~ (k, v, z) = f ( k ,

(4.7)

v, z)/4)M(v).

The first one equals T O = -et(=-*")~l~o (v),

(4.8)

because of (4.1). The value of the second integral depends on the functions r ( - z ) , v ( - z ) , cf. eq. (4.2), in different regions of phase space. Namely, the particle starting from the region (1) undergoes in its motion one acceleration, that starting from the regions (2) and (4) undergoes two accelerations and that starting from the region (3) three accelerations. In the last case the equation of the motion is: [ r(-~)

I I I v(-'O

I

-- r - ~v~ (~1 - T) -

[(~ -

~)

T z ) v " + 0:2 - z , ) v '

v' + ~ v ] ,~ (~ -

~1) ~ (~z - ~)

-

[(T -

-

[(T - ~3) v" + (T3 - ~2) v" + (Tz - ~ ) v' + ~ v ] ~ (~ - z s ) ,

= v,~ (~, - ~) + v'~ (~ -

+ ~v] ~ (r - ~)~

~ , ) ~ (Tz -

(~s -

~)

~)

+ v"r/(~ - ~2) r/(za - z) + v"rj (~ - Ta),

(4.9)

330

R. WOJNAR

AND

J. S T E C K [

where T1, Tz, T3 are the t i m e s o f successive a c c e l e r a t i o n s a n d v', v", v'" are the velocities after T1, r 2 , r 3 , respectively. I n t r o d u c i n g (4.9) i n t o (4.6) we o b t a i n the v a l u e o f the i n t e g r a l T -- 7-3 (the s u b s c r i p t 3 i n d i c a t e s the t h r e e f o l d accelerated m o t i o n ) , Ta = e i ( - ' - k ' ) ~ {[1 - e i ( Z - k ' ' ) ~ 2 - ~ ) ] ~v(v') -[- [1 -- e i ( z - k e ' ' ) ( r 3 - r 2 ) ] +

ei(Z-kv')(r2-z')~

e i ~z k,,")C~3-T2) e i (z-/,-v')(r2-r,)~v

(V")

(V"')} .

(4.10)

F o r ~3 ~ wJ we o b t a i n the i n t e g r a l T = 7"2, (4.6) f o r t w o f o l d accelerated m o t i o n a n d for also "r2 ~ ~ we o b t a i n T I . W i t h o u t difficulties, a p p l y i n g the p r i n c i p l e o f e n e r g y a n d m o m e n t u m c o n s e r v a t i o n , we find velocities v', v", v'" after first, s e c o n d a n d t h i r d a c c e l e r a t i o n respectively, for all cases o f initial p o s i t i o n s in s u b r e g i o n s i = 1. . . . . 4. Denoting tTOl = \ / U 2 - - U2 ,

(4.11)

I.'02 = "N./U2 ~- 1)2 ,

(4.12)

with Vo given by (2.2), we have the f o l l o w i n g explicit f o r m u l a e : v = v [1, 0]

(4.13)

v,~ = Vo~ [cos 7, sin 7]

(4.14)

with 7 = /3 - ~,

~ = arc sin ( b / R 2 ) ,

/3 = arc sin (b'/R2),

b' = (V/Vol) b

(see fig. 2a); here vl~ d e n o t e s the velocity after o n e a c c e l e r a t i o n for a particle s t a r t i n g f r o m the s u b r e g i o n i = l ; v2 ~ = - v [cos 2~, sin 2 ~ ] ,

(4.15)

t'22 ~--- --l)Ol [COS (2~ + a), sin (2o~ + 6)],

(4.16)

with :, = arc sin (b/R,),

a = ~ -

7 = arc sin (b'/R2),

b' = (V/Vol) b

/3,

fl = arc sin (b/R2),

KINETIC EQUATION FOR DILUTE LORENTZ GAS

331

(see fig. 2b); here v2~ and v2z denote the velocities after first and second acceleration respectively, for a particle starting f r o m the subregion i = 2; r41 = Vo2 [cos V, sin 7],

(4.17)

v42 = v [cos 27, sin 27],

(4.18)

with = arc sin (b/R2),

7=~-fl,

fl = arc sin (b'/R2),

6" @

e)

b" = (V/Vo2)b

~

cl)

Fig. 2. Different cases of unbounded motion in the square-well potential.

(see fig:. 2c).; here v41 and v42 denote the velocities after first and second acceleration respectively, for a particle starting f r o m the subregion i = 4; v31 = Vo2 [cos 7, sin 7],

(4.19)

v32 = -Vo2 [cos (7 + 28), sin (7 + 28)],

(4.20)

v3a = - v [cos 2e, sin 2e],

(4.21)

with 7 = ~ - fl,

o~ = arc sin (b/R2),

8 = arc sin (b'/R~),

e = ~ + Y,

fl = arc sin (b~/R2), b' = (V/Vo2) b

(see fig. 2d); here v31, va2, va3 denote the velocities after first, second and third acceleration respectively, for a particle starting f r o m the subregion i = 3.

332

R. WOJNAR

AND

J. S T E C K [

The formulae for the times of accelerations are also easily f o u n d :

/11

=

_i (/ + \/'R~- - b2), ,

(4.22)

l;

l.:l

=

1 --(]

-

\.,/ R 2 -

b2),

(4.23)

U _ l.,1

1

'

-v

/

2

tee

=

/'4-1

= - - (,l - \ /' R22 - b2),

+

(\. R2

__ 6 2 __ \ . /R

(4.24)

21 - - b Z ) ,

1

(4.25)

u

t+, = t4~ + - - 12 x / R

2 - (b') 2 ,

b' =

U02 t3~

=

1

--(1

!

-

\/R2

t32 : t31 + - -

1

v

b,

(4.26)

U02 2

-

(4.27)

b2),

[~/R 2

(b') 2 -- x/R1a -- (b')2],

b'=

U02

v

b,

(4.28)

U02

(4.29)

t33 = t32 -t- (t32 - - t31 ) .

Here ti~ are the times o f the first acceleration for a particle starting from the subregion i ( = 1 , . . . , 4 ) ; (t22 - t 2 ~ ) and (t42 - t 4 , ) are the time intervals between accelerations for a particle starting f r o m the subregions 2 and 4, respectively; (t32 - t 3 a ) is the time interval between the second and the first acceleration and (t33 -- t32 ) between the third and the second acceleration for a particle starting from the subregion 3.

5. The final expression for Gbi. for a square-well potential After performing the time integrations, the space integration in (3.38) remains yet to be done. D e c o m p o s i n g the position space into subregions described above we have Gbinf(V)

": ~ qSM(V) i (2" - - k v )

× [ S e " V ( r , - r °)+ Se"V(r2-r °)+ S ( r 2 - r °) t(l)

+

(2)

I (r3 -- TO)] H i 2 b db de dl. (3)

(4)

(5.1)

KINETIC EQUATION FOR DILUTE LORENTZ GAS

333

In the a b o v e integrands, the d e p e n d e n c e on l is f o u n d o n l y in its linear f u n c t i o n za o c c u r r i n g in the c o m m o n factor o f all T, n a m e l y exp i (z - kv) "cl. After that integration we have finally 2~ R2 Gblnf(v) = OVUM (v) ~ de ~ b db 0

0

x {eOVr/(v -- Vo) r/(bo - b) [All ~ (1~11) -{- A21"~ (v21) "3WA22~)(v22)] -1- A42~0 (vxl.2) Jr (A41 -[- A31)~)(~)31)

+ A3zV, (v32) + A33~ (v~) - Aoo~O(v)},

(5.2)

where A l l = ~ (R,

-

b) [1

-

exp {i (z

-

k v ) (t22

-

t21)}]

A2~ = ~ (R1 - b) [1 - exp { i ( z - kv) (t22 - t z 0 } ]

x [1 - exp {i (z - k v 2 0 (t22 - t 2 0 } ] , A22 = ~] (R1 - b) [1 - exp {i (z - kv) (t22 - tzl)}l x exp {i (z - kv21) (/22

--

t20},

A42 = ~ (b - 61) exp {i (z - kv41) (t4z - t,1)}, A41

=

~ (b

A31

=

~

--

bl) [1 - exp {i (z - kv31) (t,~2 - t4;)}],

(bl - b) [1 - exp {i (z - kv3,) (t32 - t31)}],

A32 = ~ ( b l

-b)[1

-exp{i(z-

kvs2) (t3z - t3,)}]

× exp (i (z -- k v 3 , ) (t32 - t3,)}, A33 = ~ (b~ - b) exp (i (z - kv32) (ta2 - t 3 0 ) exp {i (z - k v 3 0 (t32 - t3~),

Aoo = 1 + eBV~ (v -- Vo)~ (bo - b ) { 2 ~ (R~ -- b) x [1 - exp {i (z - kv) (t22 - t2,)}]

334

R. WOJNAR AND J. STECKI

a n d we have e x p l o i t e d the e q u a l i t y o f the expressions for v41 a n d v31, of. (4.17) a n d (4.19) a n d the e q u a l i t y o f the time intervals (t32 - t 3 0 a n d (t33 - t32), cf. (4.29). W e r e m e m b e r t h a t the tij a p p e a r i n g in Ak~ as well as the v;j- are functions o f b.

6. Conclusions W e iaave derived a new kinetic e q u a t i o n valid for a dilute L o r e n t z gas ~ i t h s h o r t - r a n g e a t t r a c t i o n a n d we have also given the explicit form o f the scattering o p e r a t o r for the case o f a square-well attractive potential. The m a t h e m a t i c a l form to which this scattering o p e r a t o r reduces for o n e - d i m e n s i o n a l collisions is also k n o w n 2"7) but has n o t been d e m o n s t r a t e d here. It is easy to see t h a t for k = 0 a n d z = 0 the scattering o p e r a t o r reduces to the B o l t z m a n n scattering o p e r a t o r . It also takes the f o r m o f the B o l t z m a n n o p e r a t o r for k = 0, any z, a n d sufficiently high velocity v. The first three terms in (5.2) c o n t a i n the restrictions on r-integration, whicleliminate the possibility o f b o u n d e d m o t i o n , a n d it is seen t h a t (5.2) differs from w h a t is (incorrectly) o b t a i n e d by specifying (1.12) for the p a r t i c u l a r case o f a square well potential. The solution o f this kinetic e q u a t i o n will be discussed later.

Aeknowledgement The a u t h o r s are i n d e b t e d to Dr. M. N a r b u t o w i c z for several discussions.

References 1) E. H. Hauge and E.G.D. Cohen, J. Mathem. Phys. 10, (1969) 397. E.H.Hauge, Phys. Fluids 13 (1970) 1201; Phys. Letters 39A (1972) 397; What can one learn from Lorentz models, in Transport Phenomena, Sitges Intern. School of Statistical Mechanics, June 1974, Sitges, Barcelona, G.Kirczenow and J. Marno, eds., Lecture Notes in Physics, Vol. 31 (Springer, Berlin, 1974). W.R.Hoegy, Thesis, Univ. of Michigan, Ann. Arbor, Mich., 1967. W. W. Wood and F. Lado, J. Comput. Phys. 7 (1971) 528. 2) R.Wojnar, Thesis, Instytut Chemii Fizycznej, Warszawa 1973. J.Stecki and R.Wojnar, Chem. Phys. Lett. 2 (1968) 343. 3) J.kebowitz and P.Resibois, Phys. Rev. 139 (1965) All01. P. Resibois and H. T. Davies, Physica 30 (1964) 1077. See also M. H. Ernst, Transport Coefficients from Time Correlation Functions, Theoretical Physics Institute, University of Colorado. Summer 1966. 4) B.J.Alder and T.E.Wainwright, J. Chem. Phys. 31 (1959)459. S.H.Sung, D.Chandler and B.J.Alder, J. Chem. Phys. 61 (1974) 932.

KINETIC E Q U A T I O N FOR D I L U T E LORENTZ GAS

335

5) H. Ted Davis, Kinetic theory of dense fluids and liquids revisited, in Advances in Chem. Physics, 1. Prigogine and Stuart A. Rice, eds., Vol. XXIV (Wiley, New York, 1973). 6) N. Hamer and I. Oppenheim, Physica 66 (1973) 217. 7) R.Wojnar, not published. 8) A. Altenberger, Physica 80A (1975) 46.