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Hard-sphere dynamics and binary-collision operators
Physica
45 (1969)
127-146
0 North-Holland
HARD-SPHERE
Publishing
DYNAMICS
BINARY-COLLISION M. H. Universeel
Laboratorium,
for Fluid Dynamics,
ERNST
Bureau
Universiteit,
DORFMAN
Nijmegen,
R.
Nederland
*
University of Maryland, W.
National
AND
OPERATORS
Katholieke J. R.
Institute
Co., Amsterdam
College Park,
USA,
HOEGY**
of Standards,
Washington,
DC,
USA
and J. M. J. VAN Physics
Department,
LEEUWEN***
Temple
University,
Received
10 March
Philadelphia,
Pa,
USA
1969
Synopsis The dynamics suffer
of a classical
instantaneous
are not
accessible
straightforward particle
expressions cability
particles.
application
of the
Depending areas binary
so far proposed
are determined.
system
and that certain
to the
trajectories.
to the overlapping
hard-sphere
collisions
has as peculiarities (“overlapping”)
It is mainly usual
binary
the
collision
on how the relevant collision
are critically
expansions examined
The binary-collision
latter
that the particles
areas in phase feature
that
expansion,
streaming
generating
operators
The different
and the restrictions for potentials
a the
are extended
can be developed.
expansion
space
prevents
to their appliconsisting
of a
hard core and a soft tail is also discussed.
In the last few years it has been possible to evaluate some of the formal results obtained in non-equilibrium statistical mechanics, e.g., some of the terms in the density expansion of the transport coefficients 1. Ihoduction.
for gases composed of hard sphere molecules, obeying classical mechanicsi-5). Such computations have been carried out for simple fluids of hard sphere moleculesl* 2, or f or g as mixtures of light and heavy spheres. An important example of the mixture is the Lorentz models? 4) and the Ehrenfest wind-tree model 5). In discussing problems in non-equilibrium statistical mechanics one starts * Supported
by U.S.
Army
Research
Office
(Durham)
Grant
* * NRC-NBS Postdoctoral Research Associate 1968-l 969. *** Partially supported by the office of Naval Research under 127
AROD31-124-G783. contract
520-985-00.
ERNST,
128
DORFMAN,
from either of two equivalent J drp(f;
0) S+&(l’)
HOEGY
expressions,
= 1 dP/?(f) S-&r;
AND
VAN
LEEUWEN
i.e. 0),
(1.1)
where the phase point r represents the positions r( and velocities ve (i = = 1, 2, . ..) N) of all N particles in the system; p(Z’; 0) is the initial value of the N-particle distribution function and S-t p(P; 0) = p(f; t) is its value at time t; similarly h(r) is the initial value of some dynamical variable depending on the phase point r and S+&(P) = h(Pt) is its value at time t. The dynamics of the N particles is discussed in terms of forward (+) or backward (-) streaming operators S&t (1, 2, . . ., IV), which generate, when acting on the initial phases (ri, vi) of the particles, their phases at time i t. These streaming operators generate canonical transformations in r space, so that (S*t)+ = s,,.
(1.2)
The dagger denotes the hermitian adjoint. In fact, (1.2) has already been used in writing (1.1). The right hand side of (1.1) is the starting point when dealing with (reduced) distribution functions; the left hand side may be more convenient when the initial distribution function is given (e.g. equilibrium time correlation functions). There is a basic difficulty in the case of hard core interactions, connected with the fact that, physically, hard-sphere molecules are not allowed to overlap, a condition wherein the relative separation between the centers of any pair of spheres is less than their diameter u. Consequently, the streaming operators are not defined for overlapping configurations and Liouville’s theorem holds only in the restricted phase space of non-overlapping configurations. However, the N-particle distribution function p(r, 0) vanishes for overlapping configurations; or, more formally, one can say that p(r, 0) contains a factor W( 1, 2, . . . , N), which is equal to 1 for non-overlapping configurations and vanishes for overlapping ones. As long as the streaming operator stands to the right of the overlap function W, as is the case on the left hand side of (l.l), the dynamics, given by the combination WS+t is well-defined in terms of the initial values of positions and velocities, since W gives a vanishing weight to overlapping configurations. The combination S-tW, as appears on the right-hand side of (1. l), has a priori no meaning since the dynamics for overlapping initial configurations is not defined. In order to save the property (l.l), it is meaningful to define S-tW as the hermitian adjoint of WS+t, S-tW
= (Ws+t)+.
(1.3)
Furthermore, one has the formal property which is used quite often in equilibrium time correlation functions exp( -_BH) S+t = S+t exp(-_BH). For hard-sphere systems the equilibrium distribution function can be written
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
as exp(-_BH) = exp(-@Ho) W, w h ere H is the total Hamiltonian is its kinetic part. Since the kinetic energy is separately conserved,
129
and Ho we have
also the property WS+t = S+tW.
(14
Now, for systems of hard-sphere molecules it has been found to be very convenient to represent the dynamics of the N-particle system by means of the binary-collision expansion (BCE), that is, the dynamical behaviour of such systems is represented in terms of sequences of successive binary collisions among the particles. Since for hard sphere systems all binary collisions are instantaneous, and the probability is zero for three or more molecules to collide simultaneously, such a BCE seems to be a rather natural representation of the dynamics. However, in such a decomposition of the N-particle streaming operator S+t one readily encounters operators which have different domains of definition. For example, the free streaming operator, St,, is defined in the entire configuration space and may stream the particles into overlapping configurations. It seems logical then to extend the definition of all streaming operators to the entire space. Depending on how the definition of the streaming operators is extended, one can obtain different forms for the binary collision operator such as are given by: Weinstocke) ; Oppenheim and Kawasaki7) ; Haines, Dorfman and Ernst 2); Van Leeuwen and Weylanda) ; Hoegy4) ; Hauge and Cohens) ; and Hopfield and Bastins). As it was our experience that the comparison of the various results obtained, was seriously hampered by the fact that different binary-collision operators were used, we felt it to be useful to devote a separate paper to the comparison of the various methods as well as to determine the restrictions on their applicability. We will show that all binary collision operators yield identical results provided they are used in the right combination with the overlap function W( 1, 2, . . . . N). The formalism is given for forward streaming, indicated by a subscript (+). All formulae can be applied directly to backward streaming by reversing the direction of the velocities of all particles involved. Results for backward streaming are indicated by a subscript (-) . We discuss here a fluid of identical spheres, but the results can be applied directly to mixtures of unlike spheres with different diameters and different masses. For the description of a collision between two spheres (diameters (ra and Q, masses ma and mb) one uses the relative motion with effective diameter u = &(o, + 00) and reduced mass ,u%’ = m;’ + m;i. The Lorentz gas is just a special case of a binary mixture, in which the mass of the heavy particles goes to infinity and the density of the light particles becomes vanishingly small. In section 2 the binary collision expansion is briefly reviewed and the
ERNST,
130
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
binary kernels are introduced. This is done in a formal way, i.e., we assume that the motion of the N particles is determined by a well defined Hamiltonian with well-defined intermolecular forces. In section 3 the binary kernels are evaluated using the different extensions into the overlap region, and the geometrical meaning of the different operators is discussed. Through the BCE we obtain from these binary-collision operators extended streaming operators defined for the whole configuration space. We investigate in section 4 whether these extended streaming operators reproduce correctly the products W(BCE) or (BCE) W for the 2-particle case. In section 5 we generalize these results to the N-particle case by giving a proof of the validity of the BCE for hard spheres. This proof demonstrates under what conditions a given binary kernel is applicable. A discussion of the results is given in section 6. 2. Binary-collision
exfiamion.
through the smooth pair potential, can be represented as
For a system of N particles interacting V(rii), the streaming operators, S+ = S*t,
S* = exp(itL),
(2.1)
where the Liouville Hamiltonian H L = (
operator
L is given by the Poisson
brackets
with the
> H} = Lo + C Ll(a),
Lo = ;
&It.-
i=l
a
O1
art ’
(2.4
Ll(ij)
= -
1 m
awd a,,*(&-&)~
where the sum runs over all pairs, a, contained in the set ( 1, 2, . . . , N). Exponential operators satisfy integral relations of the type 9) S+ = So, + X S+L1(4* oi
(2.3a)
so,,
(2.3b)
S+ = So, + X So, * Ll(ol) s+, 01 where St is the free streaming operator St = exp(+tLo), asterisk represents a convolution product defined as
f * g =; d+ft-&
and where the
(2.4)
= j Wf&-Tl.
The convolution product f * g is occasionally written as ft + gt. The integral equations (2.3) are based on a separation of L into LO and 2 Li(ol). Similarly by splitting L into L(a) = LO + Lo and Cs+o( Ll(ol) one obtains the integral relations
s+L1(4 = B+(a) +a~BS+L~(B)* B+(4> Ll(4
s+=
C+(a) +
c C+(4
B#a:
*
Ll(B) s+,
(2.5a) (25b)
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
131
where the binary kernels B+(a) = B+t(ol) and C+(a) = C+t(ol) are defined as
B+(a) = .%(4 Ll(4>
(2.6a)
C+(a) = Ll(4
(2.6b)
S+(a)*
with S+(a) = exp tL(ol). A binary-collision expansion
for S+ results from iterating
(2.5) and in-
serting the result in (2.3), yielding s+ = so, + c B+(a) * so, + 5
01
OF B+(a) * B+(B) * So+ + *a. =
a
s+ = so, + x so, * C+(a) + 5 &S;
(Y
S+[B+l, (2.7a)
* C+(a) * C+(B) + .** =
S+[C+l. (2.7b)
An alternative definition of the binary kernels, B+(a) and C+(a), avoiding the operator Li(ol), which is singular for hard spheres, can be obtained using (2.3) for pair q: + B+(a) * so, = S+[B+I,
(2.8a)
S+(a) = so, + so, * C+(a) z S+[C+].
(2.8b)
S+(a) = s;
Differentiating
B+(4 = C+(a) =
(2.8) with respect
-$C&(4 [
to time yields
$1 - [S+(a) - S”,] Lo,
$ - Lo1 [S+(a) -
the explicit
expressions (2.9a)
(2.9b)
S”,].
For decently defined potentials the streaming operators are well defined for all configurations and all the above relations (a) and (b) are equivalent, (b) following from (a) by taking the hermitian adjoint and simultaneously reversing all velocities (B*(4)+
(2.10)
= C&).
This is a consequence
of Liouville’s
theorem
(1.2).
3. Binary kernels. In the case of hard spheres the definitions (2.6) are not suitable for computing the binary kernels, as they contain the derivatives of the hard-sphere potential. Instead, we can either construct B+t(a) and C+t(a) such that (2.8) is fulfilled, or obtain them from (2.9). This is basically a two-particle problem, and we may furthermore restrict ourselves to the relative motion. The extension to the inclusion of the center of mass motion and the motion of the other particles is trivial, and is postponed to the end of this section. As mentioned in the introduction we have to extend S+$(m) to overlapping
ERNST,
132
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
Fig. 1. Collision cylinder.
areas. This will be done by considering 1 = (S+t -
S”,t, h(f, q>
(3.1)
where r and v refer to the relative position and velocity. First we note that I is different from zero only in the so called collision cylinder and eventually for overlapping configurations (see fig. 1). We choose a coordinate system with a; axis parallel to v, and decompose
r into, r = r,6
where b is a unit vector in the plane perpendicular
+ r/j?, (ci = a//al)
to v. The point of impact
0 will be decomposed similarly into, G = o,b^ - ~0. Since /u) = cr is the hard sphere diameter we have, u2 = 021 + ~2. The geometry tells furthermore that Y, = ol. We also define the collision time, tX = -(y/l + 7)/v (note that r/l is negative for points in the collision cylinder). Now for points in the collision cylinder (i.e. /~,j < cr, r/j < -y) we have the basic relations : S+tv = 0’ = v -
2(v. G) 6
.c +g -
v’(t -
ri = u +
S+tv = v s+tr = rt = r + vt 1
f) 1
for
t > t*, (3.2)
for
t < t”.
So far the definitions are unambiguous. Now Van Leeuwen and Weyland (LW) extend the definition of S +t for Irl < 0, by putting S+t = St, in overlapping regions. This corresponds to a model of a semi-transparent action sphere: a particle originally inside the action sphere can move out, but a particle originally outside can never penetrate inside. This model can not
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
133
describe the time reversed motion, of course. In Weinstock’s paper the same extension is implicit, and Hauge and Cohen obtain the same results. We have from this extension ILw = e(a -
r,){0(rj,
+ y + vt) -
B(rl/ + y)){Ql,
u’)--h(rt,
(3.3)
u)),
where 0(x) = 1 for x > 0 and e(x) = 0 for ;w< 0. It is not hard to write this result in a form like (2.8) so that the corresponding
binary
S$”
kernels
can be recovered,
namely
= .Stt + SO,,T+ x So,, = So,, + So,, x T+SO,,,
where T+ is defined as the sum of a real or interacting virtual or non-interacting part, TV,, T+ = T:
(3.4) part,
T’,,
and a
+ T;,
WJ)
witht) Y,) d(rll + 7) b, = ~8-1 j d;
T1; = &(a -
lu.Gl d@)(r -
a) b,,
i7.u < 0, T;
=
+A(0
-
(3.6)
j d;
IJ S(Y,, + 7) = -e-l
luo~\ W)(r -
a),
2J.a < 0. The operator
b, acts only on the relative
b,v = v’ = v -
2(v.;)
velocity
v and is given by
$.
(3.7)
The integrations over the solid angle 5 in (3.6) are restricted to the precollision hemisphere; s indicates the number of dimensions; and d@(r - a) is an s-dimensional Dirac a-function. The operators T’, and TV, have a clear geometrical meaning. They check whether at a certain time the conditions for a collision are satisfied, which read: r = u is on the surface of the action sphere and the particles are approaching, veu < 0. The expression for the binary B’:; = So+t T +
kernel
C$
and
then
follows directly
from
= TiSo tt.
(3.4) : (3.8)
By using the relation &(rt, v) = Lo(ri, v’) = v a/&,, one verifies easily that (2.9) leads to the same results. Hoegy considers the hard sphere interaction as a limiting case of an t The results for backward streaming are obtained by reversing the direction of the . relative velocity, i.e. the point of impact is given by u = o,b + 75 and the collision operators are T’_ = vO(a -
YJ 6(r,, -
TV_ =
-
--&(a
y) b, =
YJ I!+,/ -
y) =
us-1 J d6 ju.BI W(r -
e-1
J dB lu.61 W(r
-
u) b,,
(u-u < O),
- u), (usa > 0).
(3.6)
ERNST,
134
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
(infinitely high) potential barrier. Consequently, he defines S+t for Irl < G to be equal to S “,1for such (small) times that the particles remain inside each other, and to be equal to zero for such times that the particles move out of each other. In the latter case they would gain infinite velocities and their contributions would vanish due to the presence of normalized velocity distribution
functions.
In = ILW -
Thus we obtain from Hoegy’s
qo -
II) qr,,
= ILW + f(r) eh
-
+ y){e(Y,, - y + vt) -
y + it) hh
where f(r) is the Mayer f-function otherwise),
interpretation qr,, --y)}h(rt,
v) =
u),
(3.9)
(i.e., - 1 in the overlapping
area and 0
which reads in our notation
f(r) = e(c -
r,){e(rll
-
Y) -
(3.10)
e(rll + Y)}.
The result (3.9) can be cast in the form (2.8) by writing s$
= so+, + sO,,[T+ + e(2y -
(3.1 la)
d) CF;] ++so,, =
= SO,, + SO,,x [F+ + e(2y -
(3.1 lb)
2d) 2-y so,,,
where t) (3.12)
F+ = T’, + T’; and 5?‘; = --ve(o
-
I,_) d(Y//-
y) = -us-l
J d& lu.;l
W)(r
- 0); (3.13)
2J.U > 0.
Eq. (3.11) can be verified by carrying out the time integrals. The difference between pil; and TV, is a matter of registration of the virtual collision: TV, does it when the particle enters the $recollision hemisphere, while p”; registers the collision when the particle leaves the postcollision hemisphere. From (3.11) we have for the binary kernels in Hoegy’s interpretation
By; =
SO,,[T+ + e(ay -
cC2’ + = [F, + e(2y -
vt) q.],
(3.14)
vt) T’,] SO +t’
Again one can verify that application of (2.9) leads to the same results. The Laplace transform of BE) and Cc”) has actually been used by Hoegy. Haines, Dorfman and Ernst (HDE) take the viewpoint that any initial configuration, in which the particles are inside each other, leads immediately t In the case of backward streaming one has of course !i_ = p”_ =
--v19(a -
YJ S(r,, + y) =
-us--l
J dB Iu.BI S)(r
-
a);
T’_ + T’y with (u-0
< 0) (3.13’)
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
135
to infinite velocities, so that they put S +t equal to zero for ]rj < ot). This interpretation of the streaming operators for overlapping configurations has also been used by Choh and Uhlenbeckia), From HDE’s interpretation it follows that IHDE = ILW -
e(a -
Y,)(e(Y,,
+ y) -
Sengersi)
f3(r,, -
SYFE
Cohen ll).
y)} h(Q, V) = (3.15)
= ILW + f(r) h(rt, u). Again this can be expressed
and
in the form (2.8) as
= so,, + SO,,[T+ + 6(+)(t) f(r)] x so,, = = so,t + so,t * [F+ + s(+)(t)
f(r)1 so,,,
(3.16a) (3.16b)
where 6(+)(t) is a Dirac S-function on the interval (0, co). Thus from (2.8) and (3,16) one finds for the binary kernels in HDE’s interpretation
f(r)l,
B!$
= SO,,[T+ + 6(+)(t)
c(jj
= [G!+ + 6(+)(t) f(r)] so,,.
HDE have used the Fourier and Laplace From (3.15) one can obtain expressions (2.9)) yielding B(4) +t
=
So
T
C$
= T +SO +t*
+t
(3.17) transform of B’_3) and By’. for the binary kernels by applying (3.18a)
. f,
(3.18b)
Here we notice a discrepancy between on the one hand the formal properties (2.8) and (2.9), and on the other hand the results (3.17) and (3.18). It is seen that insertion of (3.18a, b) into (2.8a, b) respectively does rtot give back the extended streaming operator SyFE in (3.16), from which we started. Equations (a) yield actually SL+y, while equations (b) give a new extension of the streaming operator, as SO, * C$!) # By’ ;)(;St. The discrepancy between the binary kernels obtained from the formal properties (2.8) and (2.9) may be attributed to the fact that (formally) (3.1) is zero at t = 0. This property is satisfied by the extension, I” and I=, but is not satisfied by IHDE. However, what is important is whether the binary kernels and the extended streaming operators reproduce correctly the 2- and N-particle dynamics for the relevant configurations. This question will be considered in later sections. The operator Cfj or rather p- has been used by Hopfield and Bastin, as we will see in section 5. t One can actually show that this interpretation of the intermolecular potential hard-sphere potential.
V(r) = b(l -
can be obtained as a limiting case
V/G) for b -+ 03, which approaches the
ERNST,
136
Finally particle
DORFMAN,
we extend operators
T1;(12)
HOEGY
the expressions
by stating
= as-1 J d;
AND
VAN
LEEUWEN
as given in the relative
the general definition
~vis.~l W)(ris
-
space to N-
of the T-operators
u) b,,
Viz’0
< 0;
TV,(12) = -08-l
J d;
ivis.;l
W)(ris
-
a),
V12’U < 0;
5!-‘V,(12) = -OS-1
s d;
lvis.;l
W)(ri2
-
a),
Vl2’U > 0,
(3.19)
with vi2 = vi -
vs;
bovr = vi = vi -
ris = Ii -
rs,
(Vi2.G) 6,
(3.20)
bov2 = vi = vs + (Vi2.G) 6, bovi = v2
(i = 3, . . . . N).
Using these definitions in (3.4), (3.11) and (3.16) with I --f rr2, v --f v12, y --f 7~12= /tiis. u/, we get the corresponding expressions for Syy( 12)) .Syt( 12) and SAKE. 4. Two-particle dynamics and the BCE. In the previous section we have obtained several possible binary kernels, which give via the BCE’s, S+[By’] or S+[Cy’], possible extensions of the streaming operators to the whole configuration space. In this section we investigate whether the extended a-particle streaming operator, say for the pair 01, generates correctly the dynamics for the relevant configurations. The BCE’s S+[By)] and S+[Cy’] f or i = 1, 2, 3 are given by the right hand side of (3.4), (3.11) and (3.16), respectively, as follows from (2.8) and the explicit expressions for the binary kernels. They were constructed in such a manner as to coincide with the hard sphere streaming operator for non overlapping isitial configurations. Therefore, W(r,)
S+(D) = W(r,) = W(r,)
S+[Bci)]
(i=
1,2,3,4),
(4.1 a)
s+&
(i =
1, 2, 3),
(4. lb)
since W(r,) = 0 for overlapping initial configurations, and By’(a) = By’(a). However, S+[C($ streams non-overlapping initial configurations into overlapping final configurations, and hence W(rLX)S+(a) # W(r,)
S+Kyl.
(4.2)
For backward streaming, indicated by the subscript (-), one simply reverses all velocities in the previous equations. As explained in the introduction, S+(a) W(r,) has no meaning for overlapping initial configurations. One can, however, define this quantity to be equal to the hermitian adjoint of W(r,) S-(a) S+(a) W(r,)
= (W(r,)
S-(Ol))t,
(4.3)
HARD-SPHERE
DYNAMICS
AND BINARY-COLLISION
OPERATORS
137
and use the result (4.1) to extend streaming operators, which precede distribution functions (in this case W(r,)), to the whole configuration space. From
(4.1)-(4.3)
one finds
S+(a) W(r,)
= s+[C:‘l
W(r,)
(i = 2, 3, 4),
(4.4a)
= S+[BY’]
W(r,)
(i = 2, 3),
(4.4b)
S+(a) W(r,)
#
S+[ql
JJqra)
(i =
S+(a) W(r,)
#
.wC!ylW(rcx)
1,4),
(4.4c)
(i= 1).
(4.4d)
In the derivation of (4.4) one needs the relations St x Cf) = By’ x St for i = 1, 2, 3 and the hermitian adjoints which are defined for an arbitrary operator
A by:
J dr dz@,
v) Ath(r,
v) = j dr dvh(r, v) A@,
v)
(4.5)
for arbitrary functions g(r, v) and h(r, u). From (4.5) and the relations, du = dv’ and v-0 = -v’ ‘0, the hermitian adjoints follow t (Bz’)t
(T*)T = !&;
(S”,)-i = so,;
= C$)
(i = 2, 3, 4), p:“)t
= qy-0,
= q;
(cy)t
= spi,.
(4.6)
Until now we have separately extended the definitions of streaming operators for the case WS and SW. However, we have seen in section 1 that on formal grounds, WS = SW, and, therefore, we have to verify that this equality also holds for the extended operators, i.e. that the right hand sides of (4.1 a, b) are equal to the right hand sides of (4.4a, b). As S+[By’] and S+[Cf’] for i = 2 or 3 do not stream overlapping configurations into nonoverlapping ones nor vice versa, we have W(7,)
S+[B!y]
=
S+[~~‘l
W(r,)
S+[CY’] = S+[CY’l W(r,).
In view of the equalities completeness W(7,)
W(7a), (i
=
2, 3)
in (4.1) and (4.4) this suffices
(4.7) for our proof. For
we note that
s+[By)]
=
S+[C:“‘]
W(7,).
(4-B)
In the arguments that lead to (4. l), a minor difficulty presents itself when the binary kernels appear in combination with functions discontinuous at those values where the argument of the delta function in the binary kernelvanishes, e.g., W(7,) B?‘(a) or f(ra) By’(a) for t = 0. To express the fact that t We see that (Bg))t # C($, which is in conflict with the formal property (2.10). This is a consequence of the lack of symmetry under time reversal for the transparent action sphere model.
ERNST,
138
DORFMAN,
HOEGY
AND VAN LEEUWEN
we want the extended streaming operators to coincide with the hard-sphere streaming operators for any non-overlapping initial configurations, even those for which there is only an infinitesimally small interval between initial time and time of the first collision, it is convenient to add a factor S& to the left of the binary kernels used in the combination IV+ (BCE), and let 7 -+ 0 at the end of the calculations. (Similarly one adds SO,, to the right of the binary kernels in the combination (BCE) . W.) This addition determines the examples given above uniquely for t --f 0, e.g., lim f(r,) 1)+0
So = +9P(a) +
0.
The introduction of the factor So,, allows us also to give a very elegant representation of the extended streaming operators for the case i = 1, and i = 4. Due to this addition, convolution products of two binary kernels referring to the same pair of particles are well defined, and vanish for the case i = 1,4, e.g. lim 7-0
lim SO,,B~‘(ol) x SO,,,B~‘(IU) = 0 ++o
(while the products do not vanish for the case i = 2,3). In the case i = 1,4 it is, therefore, convenient to absorb the factors SO,, in the operators Tk and F* whose refined definitions read t
(4.9) and F+(a)
.;I = lim as-1 1 d6 1~)~ Ia-
{W)(r,
-
a) b, -- W(r,
+ o)} So,,,,
u‘e
(4. IO)
and which satisfy the relations SO,T”,(cx) x S”,Tb,@)
= 0,
F’“+(a) so+ x P+(P%)so, = 0 (4.11)
(a, b = I, v).
Due to (4.11) the BCE’s for i = 1,4 become simply equal to the iterated identities (2.3) with Li(ol) replaced by T + (01 ) or T+(a) and with only one pair 01present, i.e. S+[BY’]
= S+[C:“]
S+[C:4)] = sY+?,
= S+[@)]
= sy,
(4.12) (4.13)
t The operators T_ and T- for backward streaming are obtained from (4.9) and (4.10) by reversing the directions of the velocities.
HARD-SPHERE
DYNAMICS
where the “pseudo’‘-streaming
AND BINARY-COLLISION
operators
OPERATORS
139
are defined as
Sp+s(cy)= so, + SO,T+(cll) * so, + so, T+(a) * SO,T+(ol) * so, + ... = =
exp[tlo
+
tT+(a)l
(t2 0)
(4.14)
and P+)
= so, + so, * T+(a) so, + so, * F+(a) so, * F+(a) so, + . . . = = exp[tle
+ tT+(a)]
(4.15)
(t 2 0).
The operators T+(a) and F+(a) in (4.14) and (4.15) play the same role as the intermolecular forces Li(ol) in (2.2). These pseudo streaming operators are defined for all configurations, and ST coincides with S+ for all non-overlapping initial configurations. The operator fly is directly connected with the hermitian adjoint of Sy, i.e. (sP,s(cu))t = Spp(o1), and satisfies W(r,) according
(4.16)
the relation (4.17)
Sp+B(cy) = BP,“(Ly)W(r,), to (4.8). TABLE
I
This table indicates which binary kernels can be used in a BCE of W&t or S*tW. Note that backward streaming (-) is obtained from forward streaming (+) by reversing the directions of all velocities, and not by changing the sign of t. BCE in terms of
for W&
for .S*tW TZO
yes Yes
no no yes yes yes pseudo streaming operators
.S$ =
exp[ft
%*t =
exp[ftL0
LO
+
2 lZ T+(a)1 01
+
t
xOLC!*(m)~
yes
no
no
yes
Since the results of this section remain valid for the BCE’s of the N-particle streaming operators (as will be shown in the next section) we have summarized in table I possible forms of the binary kernels for hard sphere interactions, and we have indicated which binary kernels can be used in a BCE of the
ERNST,
140
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
streaming operators in the combination WS and SW, i.e., when the Nparticle distribution function W(1, 2, . . . . N) is placed before or after the streaming operator. We have also indicated which combination is valid for the pseudo streaming operators. 5. N-body hard .s$here dynamics and BCE. As stated earlier it is sufficient to require that the BCE generates the correct N-particles dynamics in nonoverlapping situations, i.e., W(BCE) = WS. We consider therefore an arbitrary N-particle phase point r, corresponding to non-overlapping initial configurations (W(r) = l), starting from which the particles collide in the sequence olr at t;, or2at ti, . , . . From the geometrical meaning of the T’, operator it follows that the N-particle dynamics for such a collision sequence and for t”, < t < tz,, is generated by Sf
= S”,T’,(al)
x .SO,T+)
x . . . ++SO,T:(a,)
I So,.
(5.1)
The problem is then to show that for the phase point r and the time interval the BCE (2.7) with the binary kernels of section 3 reduces t; < t < t;,,, precisely to (5.1). For the proof we introduce the auxiliary operator R:‘(a) defined by R$‘(ol) = So+ + Bz By’(B) x So, + & cx
y~pB~)(,!) x B:‘(y)
x So, + . . . (5.2)
i.e., the sum of all pairs in the BCE, S+[B!$, except those for which the first binary kernel involves the pair (II. We first prove by complete induction that the BCE for the phase point r and times t > tz reduces to Sf[By’]
= SO,T’,(w)
x . . . x SO,T;(ol,)
* I?!$,).
(5.3)
Then we prove that SS[BT’] reduces to ST for t”, < t < tz,,. To achieve the first step in the complete induction proof we write the BCE > S+[Bci’] + ’ as S+[BT’] = R$xl)
+ Bci’( + a1 ) * R!$l)
= So, + C B:‘(a) oi
(5.4a)
=
(5.4b)
x R:‘(a).
We start with the case i = 1,4 and split the binary kernel given in (3.8) into its real and virtual part, which yields for (5.4a) S+[B?f’] = R!$I)
+ SO,T’,(q)
x R?‘(w)
+ So,T?+(w) x R?‘(w).
(5.5)
Now for the phase $oint r and times t > tT the first and second term on the right hand side of (5.5) cancel each other, as can be shown by using (5.4b). Thus we obtain the first step in the induction proof Sf [By’]
= SO,T; (al) x R?‘(q)
(t > t;,,
(5.6)
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
141
where immediately after the oli-collision no particles are overlapping. In the case i = 2 or 3 equation (5.5) contains an additional term proportional to which vanishes for the phase point r f(r,,) (compare (3.9) and (3.15)), (non-overlapping initial configurations), and again (5.6) follows. For the induction step we assume next that for the phase point r and times t > t”,_r the BCE, S+[By’], reduces to s+ [BY’] = sO,T;(cui)
* . . . * s”,iry, (G-1)
* R:l)(%-1)
(t > t;_,,,
(5.7)
where no particles are overlapping immediately after the ol,-i-collision. Then, we reorder the terms in Ry’(a,_i) with the aid of (5.4) as R:1’(+1)
= R:‘)(rX%) + B:‘)(+&) x RY’(LXn) -
+
B:l)(c+&_i) * RY’(ar,_i) = (5.8a)
S”,T?+(a,) * Ryya,) - .qT+(a,-1) * RLy(an-l).
(5.8b)
For the phase point r and after the time ti_i the pair an. is aimed to collide at tz. So inserting (5.8b) into (5.7) yields that the first and second term in (5.8b) cancel each other for the phase point r and times t > ti, whereas the fourth term in (5.8b) vanishes because S+TT+(%-1)
* S+T+(%h-1)
=
0
due to (4.11). Thus only the third term of (5.8b) survives in (5.7), which completes the induction step. For i = 2 or 3 equation (5.8b) contains additional terms proportional to f(r*,) or f(ra,J. Since immediately after the ol,_i-collision no particles are overlapping, these additional terms vanish upon insertion of (5.8b) into (5.7), and for these cases the induction proof is valid also. Now that (5.3) has been proved, it is furthermore clear that for times t; < t < tX,+, no new collision will take place after the or,-collision and thus Ry)(ar,) is effectively equal to S “+. Therefore, (5.3) reduces to (5.1) for the phase point r and t: -=ct i tz,,. The proof shows that apart from some algebraic manipulations, the only point involved is that the combination So, + S”,TV,(a) x St vanishes for configurations in which the or-collision is possible and for times after the collision. As this is basically a matter of two-particle dynamics the problem of validity is reduced to the investigation carried out in section 3. Since we started from non-overlapping initial configurations we have proved (4.1) for the N-particle case. The inequality (4.2) is valid in the general case, too, as follows from the example for N = 2. The results (4.4) also hold in general, as can be shown in exactly the same way as in section 4 from the N-particle analogue of (4.3). Thus we have shown that all binary kernels discussed, describe correctly the N-particle dynamics, when used in the appropriate combination WS or SW, as listed in table I.
142
ERNST,
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
Using the refined definitions of the operators T+(a) and F+(a), given in (4.9) and (4.10) one has again that @(LX) x B:)(a) and Cci’( C?)(a) +oI*) vanish for i = 1,4. This allows us to drop the conditions on the summations in. the BCE that no successive pairs be equal, and the BCE’s (2.7) in this case become simply equal to the iterated identities (2.3) with Li(a) replaced by T+(a) or T+(a), which can be written as pseudo streaming operators ST = exp[tlo
+ t C T+(a)] ;
A?? = expjtL0
+ t i T+(a)] OL
(5.9)
and they satisfy the N-particle analogue of (4.16). The pseudo streaming operator ST is the adequate extension of the streaming operator when it stands to the right of the N-particle distribution function, W(l, 2, . . . . N), while s”,” is the adequate extension when the streaming operators stands to the left of the distribution function. Hopfield and Bastins) have in fact used the pseudo streaming operator sp_” which is identical to the BCE S_[@]. The extensions of WS and SW are tied together according to (1.4) and we have to show that the analogs of (4.7), (4.8) and (4.17) also hold in the N-particle case. It suffices to prove one of these relations for which we choose WS””+ = spew + .
(5.10)
NOW (5.10) will hold if for arbitrary
functions
g(f)
and h(F) (5.11)
J drgWSP,sh = J dQL?P,“WJz. Using the N-particle
analogue
of (4.16) this can be written
J W dQSP,“h = J W drh.SYg.
as (5.12)
But (5.12) is an integral over the restricted non-overlapping space, where Sy = S+ and Sp_S= S_, and where due to (1.2) (S+)t = S-. Hence, we can write (5.12) as J W drgS+h
= J W dZ’Jz(S+)tg,
(5.13)
which is nothing else than the definition of the hermitian adjoint in the restricted non-overlapping space, and (5.10) is proved. A summary of the results of this section is also contained in table I. 6. Lkcussion. In systems of hard spheres the streaming operators S&t which generate the N-particle dynamics are not defined for overlapping configurations. However, time dependent statistical averages, (1.1)) where the N-particle distribution function stands in front of the streaming operator, formally WS+t, are well defined, since the overlap function gives a vanishing
HARD-SPHERE
DYNAMICS
weight to overlapping stands
cal interpretation, pendent operator
AND
BINARY-COLLISION
initial configurations.
OPERATORS
When the distribution
143
function
I,4 in (3.8) and (3.18) have a simple geometrifree streaming followed by (or vice versa) a time-indedescribing the actual collision (T, or F*), while for i = 2 i.e.
and 3 in (3.14) and (3.17) the actual collision is described by a time-dependent operator. From the geometrical interpretation of the binary kernels for i = 1,4 it is also obvious that these operators are nilpotent by virtue of (4.11) Bf’(or) x B:‘(m) = C(j)(,) * C~)(U) = 0
(i = 1,4).
,This property does not hold for i = 2,3. Due to (6.1) the BCE’s for i = 1,4 can be written namely as pseudo streaming operators -%
=
t
form,
X T+(a)],
expkW0
+
expkl30
+ t Z T&)1,
LY
S$ =
in an elegant
(6.1)
-
(t 2 0)
(6.2)
a
where .Syt or s$ are appropriate when the streaming operator stands spectively behind or in front of the distribution function (see table I).
re-
ERNST,
144
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
These pseudo streaming operators coincide with the actual streaming operators for the relevant configurations and contain T,(a) and T*(a), which play the role of pseudo intermolecular forces, Li(~l) in (2.2). This extension is analogous to the method of pseudo potentials used for quantum mechanical hard-sphere systemsia). The idea of pseudo forces has a bearing on more general hard-core potentials V(r), which can be written as the sum of a hardcore part, V”.‘.(r), and a smooth tail, Vta”(r). In this case one can also introduce pseudo streaming operators .!?yt and ~?“,“t, which are defined analogous to (2.1) with Lp,” = f T*(a) + L4&“(ar); zy
= & F*(a) + Ltl”“(a).
They are appropriate when the streaming operators stand respectively behind or in front of the distribution function. The property (6.1) allows one to drop the restriction on successive identical pairs in the BCE’s for i = 1,4, which has certain advantages when diagrammatic methods are used to compute time-dependent statistical averages. In connection with diagrammatic methods there is a useful relation between the operators T+ and p+, obtained from (3.16a, b) SO,T&) x So, + f(r,) So, = So, x T&x) So, + So,f(rn). (6.4) By repeatedly applying this relation one can convert an expansion in terms of TA into one in terms of F, or vice versa, or one can analyse the Iz-tuple collision integrals for hard spheres in the kinetic equation derived by Zwanzig139 14) in terms of proper or self-energy diagrams, and apply known many-body techniques such as the summation of ring diagramsls) or the Bethe-Salpeter equation to renormalize the divergent collision integral 16t 17). The operator syi is very convenient when dealing with the Liouville equation and (reduced) distribution functions for hard-sphere systems, as has been done already by Hopfield and Bastin. The Liouville equation has then the form -
;T, p+Lop =
c T-(a) p, a
and from here the BBGKY-hierarchy for hard spheres can be obtained directly*). We notice that F-( 01 ) is closely related to the collision operator in the Enskog-Boltzmann equation, i.e. J dus drsT-(21)
f(ri, ~1) f(rz, ~2) =
= j dvsoz J d+&)
[f(rr, v;) f(rr + (T, u;)-f(~i,
021’Q > 0, where f(r, V) is the l-particle
~1) f(rr-Q,
Q)l> (6.6)
distribution
function.
HARD-SPHERE
DYNAMICS
Let us also consider generate relation
canonical
AND
BINARY-COLLISION
the question:
do the extended
transformations
holds for the binary
OPERATORS
in r-space,
streaming
or is (.!Y$)t
145
operators
= S$
? This
kernels B$’ and CT’ with i = 2, 3, since from
(4.6) follows (S*[BZ’])t
= S,[C$‘]
= S,[B$‘]
(i = 2, 3),
(6.7)
and consequently B’Sk[B$‘] = S*[Bf’] W. Similar results hold for Cy) (i = 2, 3). The remaining BCE’s, which can be written as .Syt or L!?Y~,do not generate canonical transformations, since (S$)t
= S$
# .s$,
(6.8)
and here one has WSyt = 8:D+stW# SytW. This discussion has not really answered the question: which binary kernel is the best or are there still better ones. But it seems to us that the simple structure of the binary kernels for i = 1 and 4 and the advantages of the pseudo streaming operators derived from them, make them preferable above the cumbersome expressions for i = 2 and 3. After this choice it is still possible to use either .Syt or I!??~when dealing with equilibrium time correlation functions; however, when dealing with non-equilibrium bution functions @” is the only appropriate choice. Acknowledgements. One of us (M.H.E.) wants to thank Beyeren and Dr. A. Weyland for many clarifying discussions.
distri-
Mr. H. van
REFERENCES 1) a. b. 2)
Sengers, Sengers,
J. V., Phys. of Fluids J. V., Boulder-Lectures
Breach,
Science
Publ.
Haines,
L. K., Dorfman,
Haines,
L. K., Thesis,
3)
Van
4) 5)
Hoegy, Hauge,
Leeuwen,
6) 7)
Weinstock, Oppenheim,
8)
Hopfield, Siegert,
(New
York,
J. R. and Ernst,
Univ.
of Maryland
J. M. J. and Weyland,
M. H., Phys. (1966)
(New York,
J. J. and Bastin,
Vol IXC,
Rev.
; Dorfman,
A., Physica
J., Phys. Rev. 126 (1962) 341. I. and Kawasaki, K., Statistical Inc.
1685; Physics,
36 (1967)
Mechanics,
J. F., Phys.
Rev.
E., Phys.
Rev.
168 (1968)
C. N., Phys. Rev. 105 (1957) 767; Mechanics, John Wiley and Sons,
Rev.
39 (1968)
35.
W.
A.
193.
110 (1958)
1232.
Huang, Huang,
Phys.
457;
ed. T. A. Bak,
12)
R.,
207;
preprint.
Choh, S. T. and Uhlenbeck, G. E., Thesis, Univ. Michigan Sengers, J. V. and Cohen, E. G. D., Physica 27 (1961) 230.
13)
144 (1966)
J. R., ref. lb.
10) 11)
Ch. 13. Zwanzig,
and
1967).
A. J. F. and Teramoto,
K. and Yang, K., Statistical
Gordon
1967).
W. R., Thesis, Univ. of Michigan (1967); E. H. and Cohen, E. G. D., preprint.
Benjamin, 9)
Inc.
9 (1966) 1333, in Theoretical
129 (1963)
486.
(1958).
Inc.
(New York,
1963)
146 HARD-SPHERE
DYNAMICS
M. H., Haines,
AND
BINARY-COLLISION
14)
Ernst,
15)
Kawasaki,
16)
ref. lb. Weyland,
17) 18)
Ernst, M. H. and Van Beyeren, H., to be published. Chapman, S. and Cowling, T. G., The mathematical Cambridge
OPERATORS
L. K. and Dorfman, J. R., Rev. mod. Phys. 41 (1969) 296. J. R., I., Phys. Rev. 139 (1965) A1763; Dorfman,
K. and Oppenheim, A., preprint.
University
Press
(Cambridge,
1960).
theory
of nonuniform
gases,
45 (1969)
127-146
0 North-Holland
HARD-SPHERE
Publishing
DYNAMICS
BINARY-COLLISION M. H. Universeel
Laboratorium,
for Fluid Dynamics,
ERNST
Bureau
Universiteit,
DORFMAN
Nijmegen,
R.
Nederland
*
University of Maryland, W.
National
AND
OPERATORS
Katholieke J. R.
Institute
Co., Amsterdam
College Park,
USA,
HOEGY**
of Standards,
Washington,
DC,
USA
and J. M. J. VAN Physics
Department,
LEEUWEN***
Temple
University,
Received
10 March
Philadelphia,
Pa,
USA
1969
Synopsis The dynamics suffer
of a classical
instantaneous
are not
accessible
straightforward particle
expressions cability
particles.
application
of the
Depending areas binary
so far proposed
are determined.
system
and that certain
to the
trajectories.
to the overlapping
hard-sphere
collisions
has as peculiarities (“overlapping”)
It is mainly usual
binary
the
collision
on how the relevant collision
are critically
expansions examined
The binary-collision
latter
that the particles
areas in phase feature
that
expansion,
streaming
generating
operators
The different
and the restrictions for potentials
a the
are extended
can be developed.
expansion
space
prevents
to their appliconsisting
of a
hard core and a soft tail is also discussed.
In the last few years it has been possible to evaluate some of the formal results obtained in non-equilibrium statistical mechanics, e.g., some of the terms in the density expansion of the transport coefficients 1. Ihoduction.
for gases composed of hard sphere molecules, obeying classical mechanicsi-5). Such computations have been carried out for simple fluids of hard sphere moleculesl* 2, or f or g as mixtures of light and heavy spheres. An important example of the mixture is the Lorentz models? 4) and the Ehrenfest wind-tree model 5). In discussing problems in non-equilibrium statistical mechanics one starts * Supported
by U.S.
Army
Research
Office
(Durham)
Grant
* * NRC-NBS Postdoctoral Research Associate 1968-l 969. *** Partially supported by the office of Naval Research under 127
AROD31-124-G783. contract
520-985-00.
ERNST,
128
DORFMAN,
from either of two equivalent J drp(f;
0) S+&(l’)
HOEGY
expressions,
= 1 dP/?(f) S-&r;
AND
VAN
LEEUWEN
i.e. 0),
(1.1)
where the phase point r represents the positions r( and velocities ve (i = = 1, 2, . ..) N) of all N particles in the system; p(Z’; 0) is the initial value of the N-particle distribution function and S-t p(P; 0) = p(f; t) is its value at time t; similarly h(r) is the initial value of some dynamical variable depending on the phase point r and S+&(P) = h(Pt) is its value at time t. The dynamics of the N particles is discussed in terms of forward (+) or backward (-) streaming operators S&t (1, 2, . . ., IV), which generate, when acting on the initial phases (ri, vi) of the particles, their phases at time i t. These streaming operators generate canonical transformations in r space, so that (S*t)+ = s,,.
(1.2)
The dagger denotes the hermitian adjoint. In fact, (1.2) has already been used in writing (1.1). The right hand side of (1.1) is the starting point when dealing with (reduced) distribution functions; the left hand side may be more convenient when the initial distribution function is given (e.g. equilibrium time correlation functions). There is a basic difficulty in the case of hard core interactions, connected with the fact that, physically, hard-sphere molecules are not allowed to overlap, a condition wherein the relative separation between the centers of any pair of spheres is less than their diameter u. Consequently, the streaming operators are not defined for overlapping configurations and Liouville’s theorem holds only in the restricted phase space of non-overlapping configurations. However, the N-particle distribution function p(r, 0) vanishes for overlapping configurations; or, more formally, one can say that p(r, 0) contains a factor W( 1, 2, . . . , N), which is equal to 1 for non-overlapping configurations and vanishes for overlapping ones. As long as the streaming operator stands to the right of the overlap function W, as is the case on the left hand side of (l.l), the dynamics, given by the combination WS+t is well-defined in terms of the initial values of positions and velocities, since W gives a vanishing weight to overlapping configurations. The combination S-tW, as appears on the right-hand side of (1. l), has a priori no meaning since the dynamics for overlapping initial configurations is not defined. In order to save the property (l.l), it is meaningful to define S-tW as the hermitian adjoint of WS+t, S-tW
= (Ws+t)+.
(1.3)
Furthermore, one has the formal property which is used quite often in equilibrium time correlation functions exp( -_BH) S+t = S+t exp(-_BH). For hard-sphere systems the equilibrium distribution function can be written
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
as exp(-_BH) = exp(-@Ho) W, w h ere H is the total Hamiltonian is its kinetic part. Since the kinetic energy is separately conserved,
129
and Ho we have
also the property WS+t = S+tW.
(14
Now, for systems of hard-sphere molecules it has been found to be very convenient to represent the dynamics of the N-particle system by means of the binary-collision expansion (BCE), that is, the dynamical behaviour of such systems is represented in terms of sequences of successive binary collisions among the particles. Since for hard sphere systems all binary collisions are instantaneous, and the probability is zero for three or more molecules to collide simultaneously, such a BCE seems to be a rather natural representation of the dynamics. However, in such a decomposition of the N-particle streaming operator S+t one readily encounters operators which have different domains of definition. For example, the free streaming operator, St,, is defined in the entire configuration space and may stream the particles into overlapping configurations. It seems logical then to extend the definition of all streaming operators to the entire space. Depending on how the definition of the streaming operators is extended, one can obtain different forms for the binary collision operator such as are given by: Weinstocke) ; Oppenheim and Kawasaki7) ; Haines, Dorfman and Ernst 2); Van Leeuwen and Weylanda) ; Hoegy4) ; Hauge and Cohens) ; and Hopfield and Bastins). As it was our experience that the comparison of the various results obtained, was seriously hampered by the fact that different binary-collision operators were used, we felt it to be useful to devote a separate paper to the comparison of the various methods as well as to determine the restrictions on their applicability. We will show that all binary collision operators yield identical results provided they are used in the right combination with the overlap function W( 1, 2, . . . . N). The formalism is given for forward streaming, indicated by a subscript (+). All formulae can be applied directly to backward streaming by reversing the direction of the velocities of all particles involved. Results for backward streaming are indicated by a subscript (-) . We discuss here a fluid of identical spheres, but the results can be applied directly to mixtures of unlike spheres with different diameters and different masses. For the description of a collision between two spheres (diameters (ra and Q, masses ma and mb) one uses the relative motion with effective diameter u = &(o, + 00) and reduced mass ,u%’ = m;’ + m;i. The Lorentz gas is just a special case of a binary mixture, in which the mass of the heavy particles goes to infinity and the density of the light particles becomes vanishingly small. In section 2 the binary collision expansion is briefly reviewed and the
ERNST,
130
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
binary kernels are introduced. This is done in a formal way, i.e., we assume that the motion of the N particles is determined by a well defined Hamiltonian with well-defined intermolecular forces. In section 3 the binary kernels are evaluated using the different extensions into the overlap region, and the geometrical meaning of the different operators is discussed. Through the BCE we obtain from these binary-collision operators extended streaming operators defined for the whole configuration space. We investigate in section 4 whether these extended streaming operators reproduce correctly the products W(BCE) or (BCE) W for the 2-particle case. In section 5 we generalize these results to the N-particle case by giving a proof of the validity of the BCE for hard spheres. This proof demonstrates under what conditions a given binary kernel is applicable. A discussion of the results is given in section 6. 2. Binary-collision
exfiamion.
through the smooth pair potential, can be represented as
For a system of N particles interacting V(rii), the streaming operators, S+ = S*t,
S* = exp(itL),
(2.1)
where the Liouville Hamiltonian H L = (
operator
L is given by the Poisson
brackets
with the
> H} = Lo + C Ll(a),
Lo = ;
&It.-
i=l
a
O1
art ’
(2.4
Ll(ij)
= -
1 m
awd a,,*(&-&)~
where the sum runs over all pairs, a, contained in the set ( 1, 2, . . . , N). Exponential operators satisfy integral relations of the type 9) S+ = So, + X S+L1(4* oi
(2.3a)
so,,
(2.3b)
S+ = So, + X So, * Ll(ol) s+, 01 where St is the free streaming operator St = exp(+tLo), asterisk represents a convolution product defined as
f * g =; d+ft-&
and where the
(2.4)
= j Wf&-Tl.
The convolution product f * g is occasionally written as ft + gt. The integral equations (2.3) are based on a separation of L into LO and 2 Li(ol). Similarly by splitting L into L(a) = LO + Lo and Cs+o( Ll(ol) one obtains the integral relations
s+L1(4 = B+(a) +a~BS+L~(B)* B+(4> Ll(4
s+=
C+(a) +
c C+(4
B#a:
*
Ll(B) s+,
(2.5a) (25b)
HARD-SPHERE
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AND
BINARY-COLLISION
OPERATORS
131
where the binary kernels B+(a) = B+t(ol) and C+(a) = C+t(ol) are defined as
B+(a) = .%(4 Ll(4>
(2.6a)
C+(a) = Ll(4
(2.6b)
S+(a)*
with S+(a) = exp tL(ol). A binary-collision expansion
for S+ results from iterating
(2.5) and in-
serting the result in (2.3), yielding s+ = so, + c B+(a) * so, + 5
01
OF B+(a) * B+(B) * So+ + *a. =
a
s+ = so, + x so, * C+(a) + 5 &S;
(Y
S+[B+l, (2.7a)
* C+(a) * C+(B) + .** =
S+[C+l. (2.7b)
An alternative definition of the binary kernels, B+(a) and C+(a), avoiding the operator Li(ol), which is singular for hard spheres, can be obtained using (2.3) for pair q: + B+(a) * so, = S+[B+I,
(2.8a)
S+(a) = so, + so, * C+(a) z S+[C+].
(2.8b)
S+(a) = s;
Differentiating
B+(4 = C+(a) =
(2.8) with respect
-$C&(4 [
to time yields
$1 - [S+(a) - S”,] Lo,
$ - Lo1 [S+(a) -
the explicit
expressions (2.9a)
(2.9b)
S”,].
For decently defined potentials the streaming operators are well defined for all configurations and all the above relations (a) and (b) are equivalent, (b) following from (a) by taking the hermitian adjoint and simultaneously reversing all velocities (B*(4)+
(2.10)
= C&).
This is a consequence
of Liouville’s
theorem
(1.2).
3. Binary kernels. In the case of hard spheres the definitions (2.6) are not suitable for computing the binary kernels, as they contain the derivatives of the hard-sphere potential. Instead, we can either construct B+t(a) and C+t(a) such that (2.8) is fulfilled, or obtain them from (2.9). This is basically a two-particle problem, and we may furthermore restrict ourselves to the relative motion. The extension to the inclusion of the center of mass motion and the motion of the other particles is trivial, and is postponed to the end of this section. As mentioned in the introduction we have to extend S+$(m) to overlapping
ERNST,
132
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
Fig. 1. Collision cylinder.
areas. This will be done by considering 1 = (S+t -
S”,t, h(f, q>
(3.1)
where r and v refer to the relative position and velocity. First we note that I is different from zero only in the so called collision cylinder and eventually for overlapping configurations (see fig. 1). We choose a coordinate system with a; axis parallel to v, and decompose
r into, r = r,6
where b is a unit vector in the plane perpendicular
+ r/j?, (ci = a//al)
to v. The point of impact
0 will be decomposed similarly into, G = o,b^ - ~0. Since /u) = cr is the hard sphere diameter we have, u2 = 021 + ~2. The geometry tells furthermore that Y, = ol. We also define the collision time, tX = -(y/l + 7)/v (note that r/l is negative for points in the collision cylinder). Now for points in the collision cylinder (i.e. /~,j < cr, r/j < -y) we have the basic relations : S+tv = 0’ = v -
2(v. G) 6
.c +g -
v’(t -
ri = u +
S+tv = v s+tr = rt = r + vt 1
f) 1
for
t > t*, (3.2)
for
t < t”.
So far the definitions are unambiguous. Now Van Leeuwen and Weyland (LW) extend the definition of S +t for Irl < 0, by putting S+t = St, in overlapping regions. This corresponds to a model of a semi-transparent action sphere: a particle originally inside the action sphere can move out, but a particle originally outside can never penetrate inside. This model can not
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
133
describe the time reversed motion, of course. In Weinstock’s paper the same extension is implicit, and Hauge and Cohen obtain the same results. We have from this extension ILw = e(a -
r,){0(rj,
+ y + vt) -
B(rl/ + y)){Ql,
u’)--h(rt,
(3.3)
u)),
where 0(x) = 1 for x > 0 and e(x) = 0 for ;w< 0. It is not hard to write this result in a form like (2.8) so that the corresponding
binary
S$”
kernels
can be recovered,
namely
= .Stt + SO,,T+ x So,, = So,, + So,, x T+SO,,,
where T+ is defined as the sum of a real or interacting virtual or non-interacting part, TV,, T+ = T:
(3.4) part,
T’,,
and a
+ T;,
WJ)
witht) Y,) d(rll + 7) b, = ~8-1 j d;
T1; = &(a -
lu.Gl d@)(r -
a) b,,
i7.u < 0, T;
=
+A(0
-
(3.6)
j d;
IJ S(Y,, + 7) = -e-l
luo~\ W)(r -
a),
2J.a < 0. The operator
b, acts only on the relative
b,v = v’ = v -
2(v.;)
velocity
v and is given by
$.
(3.7)
The integrations over the solid angle 5 in (3.6) are restricted to the precollision hemisphere; s indicates the number of dimensions; and d@(r - a) is an s-dimensional Dirac a-function. The operators T’, and TV, have a clear geometrical meaning. They check whether at a certain time the conditions for a collision are satisfied, which read: r = u is on the surface of the action sphere and the particles are approaching, veu < 0. The expression for the binary B’:; = So+t T +
kernel
C$
and
then
follows directly
from
= TiSo tt.
(3.4) : (3.8)
By using the relation &(rt, v) = Lo(ri, v’) = v a/&,, one verifies easily that (2.9) leads to the same results. Hoegy considers the hard sphere interaction as a limiting case of an t The results for backward streaming are obtained by reversing the direction of the . relative velocity, i.e. the point of impact is given by u = o,b + 75 and the collision operators are T’_ = vO(a -
YJ 6(r,, -
TV_ =
-
--&(a
y) b, =
YJ I!+,/ -
y) =
us-1 J d6 ju.BI W(r -
e-1
J dB lu.61 W(r
-
u) b,,
(u-u < O),
- u), (usa > 0).
(3.6)
ERNST,
134
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
(infinitely high) potential barrier. Consequently, he defines S+t for Irl < G to be equal to S “,1for such (small) times that the particles remain inside each other, and to be equal to zero for such times that the particles move out of each other. In the latter case they would gain infinite velocities and their contributions would vanish due to the presence of normalized velocity distribution
functions.
In = ILW -
Thus we obtain from Hoegy’s
qo -
II) qr,,
= ILW + f(r) eh
-
+ y){e(Y,, - y + vt) -
y + it) hh
where f(r) is the Mayer f-function otherwise),
interpretation qr,, --y)}h(rt,
v) =
u),
(3.9)
(i.e., - 1 in the overlapping
area and 0
which reads in our notation
f(r) = e(c -
r,){e(rll
-
Y) -
(3.10)
e(rll + Y)}.
The result (3.9) can be cast in the form (2.8) by writing s$
= so+, + sO,,[T+ + e(2y -
(3.1 la)
d) CF;] ++so,, =
= SO,, + SO,,x [F+ + e(2y -
(3.1 lb)
2d) 2-y so,,,
where t) (3.12)
F+ = T’, + T’; and 5?‘; = --ve(o
-
I,_) d(Y//-
y) = -us-l
J d& lu.;l
W)(r
- 0); (3.13)
2J.U > 0.
Eq. (3.11) can be verified by carrying out the time integrals. The difference between pil; and TV, is a matter of registration of the virtual collision: TV, does it when the particle enters the $recollision hemisphere, while p”; registers the collision when the particle leaves the postcollision hemisphere. From (3.11) we have for the binary kernels in Hoegy’s interpretation
By; =
SO,,[T+ + e(ay -
cC2’ + = [F, + e(2y -
vt) q.],
(3.14)
vt) T’,] SO +t’
Again one can verify that application of (2.9) leads to the same results. The Laplace transform of BE) and Cc”) has actually been used by Hoegy. Haines, Dorfman and Ernst (HDE) take the viewpoint that any initial configuration, in which the particles are inside each other, leads immediately t In the case of backward streaming one has of course !i_ = p”_ =
--v19(a -
YJ S(r,, + y) =
-us--l
J dB Iu.BI S)(r
-
a);
T’_ + T’y with (u-0
< 0) (3.13’)
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
135
to infinite velocities, so that they put S +t equal to zero for ]rj < ot). This interpretation of the streaming operators for overlapping configurations has also been used by Choh and Uhlenbeckia), From HDE’s interpretation it follows that IHDE = ILW -
e(a -
Y,)(e(Y,,
+ y) -
Sengersi)
f3(r,, -
SYFE
Cohen ll).
y)} h(Q, V) = (3.15)
= ILW + f(r) h(rt, u). Again this can be expressed
and
in the form (2.8) as
= so,, + SO,,[T+ + 6(+)(t) f(r)] x so,, = = so,t + so,t * [F+ + s(+)(t)
f(r)1 so,,,
(3.16a) (3.16b)
where 6(+)(t) is a Dirac S-function on the interval (0, co). Thus from (2.8) and (3,16) one finds for the binary kernels in HDE’s interpretation
f(r)l,
B!$
= SO,,[T+ + 6(+)(t)
c(jj
= [G!+ + 6(+)(t) f(r)] so,,.
HDE have used the Fourier and Laplace From (3.15) one can obtain expressions (2.9)) yielding B(4) +t
=
So
T
C$
= T +SO +t*
+t
(3.17) transform of B’_3) and By’. for the binary kernels by applying (3.18a)
. f,
(3.18b)
Here we notice a discrepancy between on the one hand the formal properties (2.8) and (2.9), and on the other hand the results (3.17) and (3.18). It is seen that insertion of (3.18a, b) into (2.8a, b) respectively does rtot give back the extended streaming operator SyFE in (3.16), from which we started. Equations (a) yield actually SL+y, while equations (b) give a new extension of the streaming operator, as SO, * C$!) # By’ ;)(;St. The discrepancy between the binary kernels obtained from the formal properties (2.8) and (2.9) may be attributed to the fact that (formally) (3.1) is zero at t = 0. This property is satisfied by the extension, I” and I=, but is not satisfied by IHDE. However, what is important is whether the binary kernels and the extended streaming operators reproduce correctly the 2- and N-particle dynamics for the relevant configurations. This question will be considered in later sections. The operator Cfj or rather p- has been used by Hopfield and Bastin, as we will see in section 5. t One can actually show that this interpretation of the intermolecular potential hard-sphere potential.
V(r) = b(l -
can be obtained as a limiting case
V/G) for b -+ 03, which approaches the
ERNST,
136
Finally particle
DORFMAN,
we extend operators
T1;(12)
HOEGY
the expressions
by stating
= as-1 J d;
AND
VAN
LEEUWEN
as given in the relative
the general definition
~vis.~l W)(ris
-
space to N-
of the T-operators
u) b,,
Viz’0
< 0;
TV,(12) = -08-l
J d;
ivis.;l
W)(ris
-
a),
V12’U < 0;
5!-‘V,(12) = -OS-1
s d;
lvis.;l
W)(ri2
-
a),
Vl2’U > 0,
(3.19)
with vi2 = vi -
vs;
bovr = vi = vi -
ris = Ii -
rs,
(Vi2.G) 6,
(3.20)
bov2 = vi = vs + (Vi2.G) 6, bovi = v2
(i = 3, . . . . N).
Using these definitions in (3.4), (3.11) and (3.16) with I --f rr2, v --f v12, y --f 7~12= /tiis. u/, we get the corresponding expressions for Syy( 12)) .Syt( 12) and SAKE. 4. Two-particle dynamics and the BCE. In the previous section we have obtained several possible binary kernels, which give via the BCE’s, S+[By’] or S+[Cy’], possible extensions of the streaming operators to the whole configuration space. In this section we investigate whether the extended a-particle streaming operator, say for the pair 01, generates correctly the dynamics for the relevant configurations. The BCE’s S+[By)] and S+[Cy’] f or i = 1, 2, 3 are given by the right hand side of (3.4), (3.11) and (3.16), respectively, as follows from (2.8) and the explicit expressions for the binary kernels. They were constructed in such a manner as to coincide with the hard sphere streaming operator for non overlapping isitial configurations. Therefore, W(r,)
S+(D) = W(r,) = W(r,)
S+[Bci)]
(i=
1,2,3,4),
(4.1 a)
s+&
(i =
1, 2, 3),
(4. lb)
since W(r,) = 0 for overlapping initial configurations, and By’(a) = By’(a). However, S+[C($ streams non-overlapping initial configurations into overlapping final configurations, and hence W(rLX)S+(a) # W(r,)
S+Kyl.
(4.2)
For backward streaming, indicated by the subscript (-), one simply reverses all velocities in the previous equations. As explained in the introduction, S+(a) W(r,) has no meaning for overlapping initial configurations. One can, however, define this quantity to be equal to the hermitian adjoint of W(r,) S-(a) S+(a) W(r,)
= (W(r,)
S-(Ol))t,
(4.3)
HARD-SPHERE
DYNAMICS
AND BINARY-COLLISION
OPERATORS
137
and use the result (4.1) to extend streaming operators, which precede distribution functions (in this case W(r,)), to the whole configuration space. From
(4.1)-(4.3)
one finds
S+(a) W(r,)
= s+[C:‘l
W(r,)
(i = 2, 3, 4),
(4.4a)
= S+[BY’]
W(r,)
(i = 2, 3),
(4.4b)
S+(a) W(r,)
#
S+[ql
JJqra)
(i =
S+(a) W(r,)
#
.wC!ylW(rcx)
1,4),
(4.4c)
(i= 1).
(4.4d)
In the derivation of (4.4) one needs the relations St x Cf) = By’ x St for i = 1, 2, 3 and the hermitian adjoints which are defined for an arbitrary operator
A by:
J dr dz@,
v) Ath(r,
v) = j dr dvh(r, v) A@,
v)
(4.5)
for arbitrary functions g(r, v) and h(r, u). From (4.5) and the relations, du = dv’ and v-0 = -v’ ‘0, the hermitian adjoints follow t (Bz’)t
(T*)T = !&;
(S”,)-i = so,;
= C$)
(i = 2, 3, 4), p:“)t
= qy-0,
= q;
(cy)t
= spi,.
(4.6)
Until now we have separately extended the definitions of streaming operators for the case WS and SW. However, we have seen in section 1 that on formal grounds, WS = SW, and, therefore, we have to verify that this equality also holds for the extended operators, i.e. that the right hand sides of (4.1 a, b) are equal to the right hand sides of (4.4a, b). As S+[By’] and S+[Cf’] for i = 2 or 3 do not stream overlapping configurations into nonoverlapping ones nor vice versa, we have W(7,)
S+[B!y]
=
S+[~~‘l
W(r,)
S+[CY’] = S+[CY’l W(r,).
In view of the equalities completeness W(7,)
W(7a), (i
=
2, 3)
in (4.1) and (4.4) this suffices
(4.7) for our proof. For
we note that
s+[By)]
=
S+[C:“‘]
W(7,).
(4-B)
In the arguments that lead to (4. l), a minor difficulty presents itself when the binary kernels appear in combination with functions discontinuous at those values where the argument of the delta function in the binary kernelvanishes, e.g., W(7,) B?‘(a) or f(ra) By’(a) for t = 0. To express the fact that t We see that (Bg))t # C($, which is in conflict with the formal property (2.10). This is a consequence of the lack of symmetry under time reversal for the transparent action sphere model.
ERNST,
138
DORFMAN,
HOEGY
AND VAN LEEUWEN
we want the extended streaming operators to coincide with the hard-sphere streaming operators for any non-overlapping initial configurations, even those for which there is only an infinitesimally small interval between initial time and time of the first collision, it is convenient to add a factor S& to the left of the binary kernels used in the combination IV+ (BCE), and let 7 -+ 0 at the end of the calculations. (Similarly one adds SO,, to the right of the binary kernels in the combination (BCE) . W.) This addition determines the examples given above uniquely for t --f 0, e.g., lim f(r,) 1)+0
So = +9P(a) +
0.
The introduction of the factor So,, allows us also to give a very elegant representation of the extended streaming operators for the case i = 1, and i = 4. Due to this addition, convolution products of two binary kernels referring to the same pair of particles are well defined, and vanish for the case i = 1,4, e.g. lim 7-0
lim SO,,B~‘(ol) x SO,,,B~‘(IU) = 0 ++o
(while the products do not vanish for the case i = 2,3). In the case i = 1,4 it is, therefore, convenient to absorb the factors SO,, in the operators Tk and F* whose refined definitions read t
(4.9) and F+(a)
.;I = lim as-1 1 d6 1~)~ Ia-
{W)(r,
-
a) b, -- W(r,
+ o)} So,,,,
u‘e
(4. IO)
and which satisfy the relations SO,T”,(cx) x S”,Tb,@)
= 0,
F’“+(a) so+ x P+(P%)so, = 0 (4.11)
(a, b = I, v).
Due to (4.11) the BCE’s for i = 1,4 become simply equal to the iterated identities (2.3) with Li(ol) replaced by T + (01 ) or T+(a) and with only one pair 01present, i.e. S+[BY’]
= S+[C:“]
S+[C:4)] = sY+?,
= S+[@)]
= sy,
(4.12) (4.13)
t The operators T_ and T- for backward streaming are obtained from (4.9) and (4.10) by reversing the directions of the velocities.
HARD-SPHERE
DYNAMICS
where the “pseudo’‘-streaming
AND BINARY-COLLISION
operators
OPERATORS
139
are defined as
Sp+s(cy)= so, + SO,T+(cll) * so, + so, T+(a) * SO,T+(ol) * so, + ... = =
exp[tlo
+
tT+(a)l
(t2 0)
(4.14)
and P+)
= so, + so, * T+(a) so, + so, * F+(a) so, * F+(a) so, + . . . = = exp[tle
+ tT+(a)]
(4.15)
(t 2 0).
The operators T+(a) and F+(a) in (4.14) and (4.15) play the same role as the intermolecular forces Li(ol) in (2.2). These pseudo streaming operators are defined for all configurations, and ST coincides with S+ for all non-overlapping initial configurations. The operator fly is directly connected with the hermitian adjoint of Sy, i.e. (sP,s(cu))t = Spp(o1), and satisfies W(r,) according
(4.16)
the relation (4.17)
Sp+B(cy) = BP,“(Ly)W(r,), to (4.8). TABLE
I
This table indicates which binary kernels can be used in a BCE of W&t or S*tW. Note that backward streaming (-) is obtained from forward streaming (+) by reversing the directions of all velocities, and not by changing the sign of t. BCE in terms of
for W&
for .S*tW TZO
yes Yes
no no yes yes yes pseudo streaming operators
.S$ =
exp[ft
%*t =
exp[ftL0
LO
+
2 lZ T+(a)1 01
+
t
xOLC!*(m)~
yes
no
no
yes
Since the results of this section remain valid for the BCE’s of the N-particle streaming operators (as will be shown in the next section) we have summarized in table I possible forms of the binary kernels for hard sphere interactions, and we have indicated which binary kernels can be used in a BCE of the
ERNST,
140
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
streaming operators in the combination WS and SW, i.e., when the Nparticle distribution function W(1, 2, . . . . N) is placed before or after the streaming operator. We have also indicated which combination is valid for the pseudo streaming operators. 5. N-body hard .s$here dynamics and BCE. As stated earlier it is sufficient to require that the BCE generates the correct N-particles dynamics in nonoverlapping situations, i.e., W(BCE) = WS. We consider therefore an arbitrary N-particle phase point r, corresponding to non-overlapping initial configurations (W(r) = l), starting from which the particles collide in the sequence olr at t;, or2at ti, . , . . From the geometrical meaning of the T’, operator it follows that the N-particle dynamics for such a collision sequence and for t”, < t < tz,, is generated by Sf
= S”,T’,(al)
x .SO,T+)
x . . . ++SO,T:(a,)
I So,.
(5.1)
The problem is then to show that for the phase point r and the time interval the BCE (2.7) with the binary kernels of section 3 reduces t; < t < t;,,, precisely to (5.1). For the proof we introduce the auxiliary operator R:‘(a) defined by R$‘(ol) = So+ + Bz By’(B) x So, + & cx
y~pB~)(,!) x B:‘(y)
x So, + . . . (5.2)
i.e., the sum of all pairs in the BCE, S+[B!$, except those for which the first binary kernel involves the pair (II. We first prove by complete induction that the BCE for the phase point r and times t > tz reduces to Sf[By’]
= SO,T’,(w)
x . . . x SO,T;(ol,)
* I?!$,).
(5.3)
Then we prove that SS[BT’] reduces to ST for t”, < t < tz,,. To achieve the first step in the complete induction proof we write the BCE > S+[Bci’] + ’ as S+[BT’] = R$xl)
+ Bci’( + a1 ) * R!$l)
= So, + C B:‘(a) oi
(5.4a)
=
(5.4b)
x R:‘(a).
We start with the case i = 1,4 and split the binary kernel given in (3.8) into its real and virtual part, which yields for (5.4a) S+[B?f’] = R!$I)
+ SO,T’,(q)
x R?‘(w)
+ So,T?+(w) x R?‘(w).
(5.5)
Now for the phase $oint r and times t > tT the first and second term on the right hand side of (5.5) cancel each other, as can be shown by using (5.4b). Thus we obtain the first step in the induction proof Sf [By’]
= SO,T; (al) x R?‘(q)
(t > t;,,
(5.6)
HARD-SPHERE
DYNAMICS
AND
BINARY-COLLISION
OPERATORS
141
where immediately after the oli-collision no particles are overlapping. In the case i = 2 or 3 equation (5.5) contains an additional term proportional to which vanishes for the phase point r f(r,,) (compare (3.9) and (3.15)), (non-overlapping initial configurations), and again (5.6) follows. For the induction step we assume next that for the phase point r and times t > t”,_r the BCE, S+[By’], reduces to s+ [BY’] = sO,T;(cui)
* . . . * s”,iry, (G-1)
* R:l)(%-1)
(t > t;_,,,
(5.7)
where no particles are overlapping immediately after the ol,-i-collision. Then, we reorder the terms in Ry’(a,_i) with the aid of (5.4) as R:1’(+1)
= R:‘)(rX%) + B:‘)(+&) x RY’(LXn) -
+
B:l)(c+&_i) * RY’(ar,_i) = (5.8a)
S”,T?+(a,) * Ryya,) - .qT+(a,-1) * RLy(an-l).
(5.8b)
For the phase point r and after the time ti_i the pair an. is aimed to collide at tz. So inserting (5.8b) into (5.7) yields that the first and second term in (5.8b) cancel each other for the phase point r and times t > ti, whereas the fourth term in (5.8b) vanishes because S+TT+(%-1)
* S+T+(%h-1)
=
0
due to (4.11). Thus only the third term of (5.8b) survives in (5.7), which completes the induction step. For i = 2 or 3 equation (5.8b) contains additional terms proportional to f(r*,) or f(ra,J. Since immediately after the ol,_i-collision no particles are overlapping, these additional terms vanish upon insertion of (5.8b) into (5.7), and for these cases the induction proof is valid also. Now that (5.3) has been proved, it is furthermore clear that for times t; < t < tX,+, no new collision will take place after the or,-collision and thus Ry)(ar,) is effectively equal to S “+. Therefore, (5.3) reduces to (5.1) for the phase point r and t: -=ct i tz,,. The proof shows that apart from some algebraic manipulations, the only point involved is that the combination So, + S”,TV,(a) x St vanishes for configurations in which the or-collision is possible and for times after the collision. As this is basically a matter of two-particle dynamics the problem of validity is reduced to the investigation carried out in section 3. Since we started from non-overlapping initial configurations we have proved (4.1) for the N-particle case. The inequality (4.2) is valid in the general case, too, as follows from the example for N = 2. The results (4.4) also hold in general, as can be shown in exactly the same way as in section 4 from the N-particle analogue of (4.3). Thus we have shown that all binary kernels discussed, describe correctly the N-particle dynamics, when used in the appropriate combination WS or SW, as listed in table I.
142
ERNST,
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
Using the refined definitions of the operators T+(a) and F+(a), given in (4.9) and (4.10) one has again that @(LX) x B:)(a) and Cci’( C?)(a) +oI*) vanish for i = 1,4. This allows us to drop the conditions on the summations in. the BCE that no successive pairs be equal, and the BCE’s (2.7) in this case become simply equal to the iterated identities (2.3) with Li(a) replaced by T+(a) or T+(a), which can be written as pseudo streaming operators ST = exp[tlo
+ t C T+(a)] ;
A?? = expjtL0
+ t i T+(a)] OL
(5.9)
and they satisfy the N-particle analogue of (4.16). The pseudo streaming operator ST is the adequate extension of the streaming operator when it stands to the right of the N-particle distribution function, W(l, 2, . . . . N), while s”,” is the adequate extension when the streaming operators stands to the left of the distribution function. Hopfield and Bastins) have in fact used the pseudo streaming operator sp_” which is identical to the BCE S_[@]. The extensions of WS and SW are tied together according to (1.4) and we have to show that the analogs of (4.7), (4.8) and (4.17) also hold in the N-particle case. It suffices to prove one of these relations for which we choose WS””+ = spew + .
(5.10)
NOW (5.10) will hold if for arbitrary
functions
g(f)
and h(F) (5.11)
J drgWSP,sh = J dQL?P,“WJz. Using the N-particle
analogue
of (4.16) this can be written
J W dQSP,“h = J W drh.SYg.
as (5.12)
But (5.12) is an integral over the restricted non-overlapping space, where Sy = S+ and Sp_S= S_, and where due to (1.2) (S+)t = S-. Hence, we can write (5.12) as J W drgS+h
= J W dZ’Jz(S+)tg,
(5.13)
which is nothing else than the definition of the hermitian adjoint in the restricted non-overlapping space, and (5.10) is proved. A summary of the results of this section is also contained in table I. 6. Lkcussion. In systems of hard spheres the streaming operators S&t which generate the N-particle dynamics are not defined for overlapping configurations. However, time dependent statistical averages, (1.1)) where the N-particle distribution function stands in front of the streaming operator, formally WS+t, are well defined, since the overlap function gives a vanishing
HARD-SPHERE
DYNAMICS
weight to overlapping stands
cal interpretation, pendent operator
AND
BINARY-COLLISION
initial configurations.
OPERATORS
When the distribution
143
function
I,4 in (3.8) and (3.18) have a simple geometrifree streaming followed by (or vice versa) a time-indedescribing the actual collision (T, or F*), while for i = 2 i.e.
and 3 in (3.14) and (3.17) the actual collision is described by a time-dependent operator. From the geometrical interpretation of the binary kernels for i = 1,4 it is also obvious that these operators are nilpotent by virtue of (4.11) Bf’(or) x B:‘(m) = C(j)(,) * C~)(U) = 0
(i = 1,4).
,This property does not hold for i = 2,3. Due to (6.1) the BCE’s for i = 1,4 can be written namely as pseudo streaming operators -%
=
t
form,
X T+(a)],
expkW0
+
expkl30
+ t Z T&)1,
LY
S$ =
in an elegant
(6.1)
-
(t 2 0)
(6.2)
a
where .Syt or s$ are appropriate when the streaming operator stands spectively behind or in front of the distribution function (see table I).
re-
ERNST,
144
DORFMAN,
HOEGY
AND
VAN
LEEUWEN
These pseudo streaming operators coincide with the actual streaming operators for the relevant configurations and contain T,(a) and T*(a), which play the role of pseudo intermolecular forces, Li(~l) in (2.2). This extension is analogous to the method of pseudo potentials used for quantum mechanical hard-sphere systemsia). The idea of pseudo forces has a bearing on more general hard-core potentials V(r), which can be written as the sum of a hardcore part, V”.‘.(r), and a smooth tail, Vta”(r). In this case one can also introduce pseudo streaming operators .!?yt and ~?“,“t, which are defined analogous to (2.1) with Lp,” = f T*(a) + L4&“(ar); zy
= & F*(a) + Ltl”“(a).
They are appropriate when the streaming operators stand respectively behind or in front of the distribution function. The property (6.1) allows one to drop the restriction on successive identical pairs in the BCE’s for i = 1,4, which has certain advantages when diagrammatic methods are used to compute time-dependent statistical averages. In connection with diagrammatic methods there is a useful relation between the operators T+ and p+, obtained from (3.16a, b) SO,T&) x So, + f(r,) So, = So, x T&x) So, + So,f(rn). (6.4) By repeatedly applying this relation one can convert an expansion in terms of TA into one in terms of F, or vice versa, or one can analyse the Iz-tuple collision integrals for hard spheres in the kinetic equation derived by Zwanzig139 14) in terms of proper or self-energy diagrams, and apply known many-body techniques such as the summation of ring diagramsls) or the Bethe-Salpeter equation to renormalize the divergent collision integral 16t 17). The operator syi is very convenient when dealing with the Liouville equation and (reduced) distribution functions for hard-sphere systems, as has been done already by Hopfield and Bastin. The Liouville equation has then the form -
;T, p+Lop =
c T-(a) p, a
and from here the BBGKY-hierarchy for hard spheres can be obtained directly*). We notice that F-( 01 ) is closely related to the collision operator in the Enskog-Boltzmann equation, i.e. J dus drsT-(21)
f(ri, ~1) f(rz, ~2) =
= j dvsoz J d+&)
[f(rr, v;) f(rr + (T, u;)-f(~i,
021’Q > 0, where f(r, V) is the l-particle
~1) f(rr-Q,
Q)l> (6.6)
distribution
function.
HARD-SPHERE
DYNAMICS
Let us also consider generate relation
canonical
AND
BINARY-COLLISION
the question:
do the extended
transformations
holds for the binary
OPERATORS
in r-space,
streaming
or is (.!Y$)t
145
operators
= S$
? This
kernels B$’ and CT’ with i = 2, 3, since from
(4.6) follows (S*[BZ’])t
= S,[C$‘]
= S,[B$‘]
(i = 2, 3),
(6.7)
and consequently B’Sk[B$‘] = S*[Bf’] W. Similar results hold for Cy) (i = 2, 3). The remaining BCE’s, which can be written as .Syt or L!?Y~,do not generate canonical transformations, since (S$)t
= S$
# .s$,
(6.8)
and here one has WSyt = 8:D+stW# SytW. This discussion has not really answered the question: which binary kernel is the best or are there still better ones. But it seems to us that the simple structure of the binary kernels for i = 1 and 4 and the advantages of the pseudo streaming operators derived from them, make them preferable above the cumbersome expressions for i = 2 and 3. After this choice it is still possible to use either .Syt or I!??~when dealing with equilibrium time correlation functions; however, when dealing with non-equilibrium bution functions @” is the only appropriate choice. Acknowledgements. One of us (M.H.E.) wants to thank Beyeren and Dr. A. Weyland for many clarifying discussions.
distri-
Mr. H. van
REFERENCES 1) a. b. 2)
Sengers, Sengers,
J. V., Phys. of Fluids J. V., Boulder-Lectures
Breach,
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Publ.
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L. K., Dorfman,
Haines,
L. K., Thesis,
3)
Van
4) 5)
Hoegy, Hauge,
Leeuwen,
6) 7)
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8)
Hopfield, Siegert,
(New
York,
J. R. and Ernst,
Univ.
of Maryland
J. M. J. and Weyland,
M. H., Phys. (1966)
(New York,
J. J. and Bastin,
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; Dorfman,
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ed. T. A. Bak,
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Choh, S. T. and Uhlenbeck, G. E., Thesis, Univ. Michigan Sengers, J. V. and Cohen, E. G. D., Physica 27 (1961) 230.
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A. J. F. and Teramoto,
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Inc.
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DYNAMICS
M. H., Haines,
AND
BINARY-COLLISION
14)
Ernst,
15)
Kawasaki,
16)
ref. lb. Weyland,
17) 18)
Ernst, M. H. and Van Beyeren, H., to be published. Chapman, S. and Cowling, T. G., The mathematical Cambridge
OPERATORS
L. K. and Dorfman, J. R., Rev. mod. Phys. 41 (1969) 296. J. R., I., Phys. Rev. 139 (1965) A1763; Dorfman,
K. and Oppenheim, A., preprint.
University
Press
(Cambridge,
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theory
of nonuniform
gases,