- Email: [email protected]
Derivation of the Boltzmann equation from the Liouville equation
Physica
59
(1972) 206-227 o North-Holland
DERIVATION
OF THE
FROM
THE
Publishing
Co.
BOLTZMANN
LIOUVILLE
EQUATION
EQUATION
*
B. LEAF State University
of New
York,
College at Cortland,
Received
12 July
Cortland,
New
York r3045,
USA
1971
synopsis The
derivation
previously
given
central-force
interaction
body
equation
is obtained,
distribution
function.
kinetic
one-particle mation
to this
density,
but without
for the kinetic
is extended
equation
with
to include and from
The Boltzmann of previous
equation
for a classical
of external
this the exact
the assumptions
the limitation
equation
the effects
kinetic
The N-
equation
for the
is derived
of short-range
gas with
forces.
as an approxi-
interactions
and low
work to the case of weak interactions.
1. Introduction. In an earlier paper 1)) which will be cited in this article as paper I, an exact kinetic equation was derived for the one-particle distribution of a classical gas with central-force interactions. The starting point for the derivation was the Liouville equation; the method employed was the technique of projection operatorss). In the present paper the treatment is extended to the case in which external forces act on the gas. The assumptions of short-range interaction forces and low density are introduced, but not the limitation to the case of weak interactions as in paper I. If certain additional approximations are made, precise agreement is obtained with the full Boltzmann equation. The additional approximations are 1) Neglect of the elision by the projector 1[00] in the ordered exponential operator. This has the effect of making this operator into a displacement operator along a trajectory governed by a hamiltonian function [eq. (5.8)]. 2) Neglect of the spatial derivative part of the Liouville operator appearing in the ordered exponential operator. This may be considered as an extension of the assumption of molecular chaos [eq. (5.10)]. 3) Neglect of a derivative of the force of the collision interaction with respect to the relative velocity of the colliding particles [eq. (5.13)].
* This work was supported University of New York.
by a grant from the Research
206
Foundation
of the State
BOLTZMANN
EQUATION
FROM
LIOUVILLE
It must also be assumed that the two-particle the collision term can be factored into a product
EQUATION
207
distribution function in of one-particle functions
[eq. (5.21)]. The procedure followed in deriving the exact one-particle kinetic equation closely parallels that of paper I. In section 2, the Liouville operator is given. In section 3, the N-body kinetic equation is derived for the gas with external forces. The exact one-particle kinetic equation is given in section 4. It has the flow terms of the Boltzmann equation, including the contribution of external forces. In section 5, approximate one-particle kinetic equations are obtained. With the assumptions of short-range forces and dilute gas, which were considered in paper I, we obtain the approximate equation (5.1) with collision term (5.5). From this equation, on neglect of the external forces in the collision term and with the additional approximations listed above, the Boltzmann equation is derived. It is shown in section 6 that the momenta pi and p’ appearing in the Boltzmann equation (5.22) and defined by (5.19) and (5.20) in terms of the time-ordered exponential operator are the initial momenta in the inverse collision between two molecules which results in the final momenta p1 and p. In appendix A it is shown that an orthogonal set of projection operators can be constructed in terms of which a complete partition of the unit operator in the Hilbert space can be achieved. With these operators the hamiltonian and, correspondingly, the Liouville operator can be expanded in a systematic way, as required for the present formalism. In appendix B, properties of the time-ordered exponential operator W(T) of eq. (3.26) are considered. The time-dependent hamiltonian formalism, which results when the elision produced by the projector l[kl . . . kN] is ignored, is described. The derivation of the Boltzmann equation shows in detail the assumptions and approximations introduced in starting from the exact dynamical (hamiltonian) equations of motion, in the form of Liouville’s equation, and obtaining the dynamically irreversible Boltzmann equation. The exact oneparticle kinetic equation (4.5) contains none of these assumptions or approximations. Additional 2. Liouville hamiltonian
references operators.
to the literature Consider
are given in paper I.
a gas of N particles
described
by the
(2.1) where U(r,) is the potential energy of external forces acting on the ith particle at position ri, and 4(1 tti - rJ]) is the two-body, central-force interaction between the particles i and j. Expressed as Fourier integrals, U(r,) =
J ds ezxircms u(s), LOI
d(r) = ,J ds ezxires v(s).
(2.2)
B.LEAF
208 In these integrations
the point at s = 0 is omitted
since the contribution
to the integral for s = 0 is a constant, which we equate to zero in the hamiltonian without loss of generality. Equivalently we can require that U(O) = j dszl(s) 6(s) = 0,
~(0) = j ds ~(4 4s) = 0,
without exclusion of the point s = 0 from the integrations
in (2.2). Eq. (2.2)
implies that J d@ 1X LW)
+ X Z d(lrj -
i
i
In phase-space two parts, L = L1-t
representation
rd)l = 0. the Liouville
operator
is separated
L2,
into
(24
and L1 is separated further, L1 = Li + L;,
(24
where
L’; = i x XJ(rk)/i3rkai3/apk
= -
k
L2
=
iC
I: J ds e2rris’rkU(S) 2&?/apk,
(2.5)
k
a4(ir5-rki)
2
ar5
j
aPk
s
= _ F
-~a
.
&
e2nis*(rr-rj) p(s) zxs.
Liouville
-_?
_!? .
_
aPk
aP5
operators are defined by
9=
jdrNIrl...rN>L
p=
Ylf22,
(2.6)
with
so
(2.7)
S?l=9i+L?~,
that
9'; =
dk’N
-
s
x c
i
s
dkN[ki...
kk)
s
ds I-I 6(k;, - k, - SC?& u(s) 2xs.-
m
a
aP5
BOLTZMANN
9'2 =
-
EQUATION
J J dk’N
dkN ]ki
FROM
LIOUVILLE
EQUATION
201
...k;)
x T
?n
a
~ ap,
-
a aP, >
The ket ]rr . . . r~) is the eigenvector
(f-1 . . . rN/kl...
kN> = exp(2zi
(2.8;
of the N-particle
position operator,
2 rm-km). m
ant (2.9;
The Liouville operator _Yi was designed by Yr in paper I, where, in the absence of external forces, the operator 29 did not appear. The structure of these operators is made evident in eq. (A. 17) in terms oj a set of projection operators, which form a partition of the unit operator, as described in appendix A. It follows from the orthogonality of these projectors that P,uil[P]
= 1[8] SF@
(2.10;
= 0,
and
where the projector are defined as
g(kl... 1[8]
B = B(kl . . . kN) and its complement
1[P] = I[kl
kN) = jkl... kN>
a,,jat =
(2.121
Ikl... kN>(kl . . . kN[.
3. The N-body kinetic equation. from the Liouville equation
. . . kN:
To obtain
the N-body
kinetic
equatior (3.1
-i_Yp0
we proceed as in the derivation of eq. (4.9) of paper I. With the use of the orthogonality relations (2.10) and (2.1 l), corresponding to (3.19) and (3.20 of paper I, we obtain apo(t)iat i- iZlpo(t) = =
-iidP2o0(t) -
-i=Y2p0(t)
rd7
J dkNBYs
e-iP1[81r PPpo(t
-
T).
0
In this equation
_%‘2co(t)is given, as in eq. (4.13) of paper I, by
(3.2
B. LEAF
210
_Ys~e(t) = LYso’“(t) -
i rd7
j dkN 992
e-iS1lLPIT .Ps1[B]
oo(t -
T), (3.3)
0
where 1[8] e_(t) = e- iz[B’zlt 1[SP] @CO).
(3.4)
Using the formula for the exponential
A and B
of the sum of two operators
given in eq. (5.1) of paper I, eA+B = eA T exp ;dt e--t4 B et-4,
(3.5)
0
where T is the Dyson ordering operator, e-i~~~l[Bl
e -i7(ZLpl’+9~“)
=
e-iT2Wl
j-h . ..pN. =
=
e-i7_5fl’ =
e -i7(5?~‘+8~“+~2)1[~1
=
in terms of the operators wl(t,
I[?]
h
T exp{-_i
Wi(t)
we obtain Wl(d,
e-i7-LP1’
(3.6)
W(T)>
and W(t) defined as
. . . h) i &’
I[g]
eit’zl’
2:
e-it’zl’
w%
(3.7)
0 w(t,
PI
. ..PN.
h
=
T exp{-i
. . . kN)
s”dt’ 1[B] e’““” (9;
+ 92) e-it’Sl’
1[P]}.
(3.8)
0
Accordingly,
from (3.2), (3.3),
+0(t)/% + iPipo(t) 900(l)
(3.4) and (2. lo),
+ iY~p0(t) == -iY2ao(t)
OrdT j dkN BS?s e-iTzl’ W(T) (2; = 92@(t)
-
rds
S dkNPg2
+
92)
Ppo(t
-
(3.9)
4,
e-iTYLpI’Wi(T) 6psZ[g]
ao(t
-
T),
0
(3.10) 1[8] o:(t)
(3.11)
= ehitzl’ WI(t) 1[9] c&O).
From these Hilbert-space equations we obtain the phase-space equations with the help of eq. (3.7) of paper I. From (3.9) and (2.5), the N-body kinetic equation governing the time evolution of the probability density in r phase space, p(t, p1 . . . pN, t-1 . . . rN) is +(t,
pl
rl
. . . pN>
at
+
x
Pk
k
awk) -%G--•
. . . rN)
a&, k
pl
.
+(t,
pl
ark
P
...pN,
aPk
. . . PN,
rl
. . . rN)
rl
. . . IN)
BOLTZMANN
EQUATION
““‘ly
= -g
Ql) 3
FROM
LIOUVILLE
EQUATION
211
.(&-$--)o(t,Pl...PNA..YN)
m
-
PN, kl . . . kN)
drdkNg(T, PI...
s
0 X
n
J
dr’N e2rri?“*(‘r-rm’) ~(i! -
7, p1 . . . PN, ri , . . &),
(3.12)
where
g(T,PI . . . PN, kl .. . kN) =
“91’~(+?;
+ 92) Ikl... kN)-
(3.13)
As shown in eqs. (5.18) and (5.19) of paper I,
,-2~~~km~(~m+pm~id
x
...
(3.14)
rNl,
n
kN> = i T
92 [kl...
J
drN (rl
. . . rN) e2xif?‘*km (3.15)
Also, from (2.8), 2;
au(f-3)
(kl ..I kN> = i z
3
Accordingly, &,Pl
s
3
a
(3.16)
ap3
from (3.13) .*-pN,
x
drN\rl . . . mN>e2rriErm’km F.-.
z
[ jf
k1.e.
au(+) ark
kN)
.++zz j’
&j(lrb - &I) ar,
j’
It was shown in section 5 of paper I that ei7z,’
92
e-““”
=
Is
r,
p,(T,
(3.18)
PtPj),
j
where =923(7, ptp3)
=
J drN If-1 **. fN> &(T,
p43,
w3)
(3.19)
B. LEAF
212
- am -
(I+ -
r2 +
a
T/PI)
Pd
(3.20)
‘ark 1 ’
apk
the Liouville operator in the phase space of particles dependent interaction potential +(/rj - rt + (pj - pa) eirS1’ 2;
eWiTzLPI = C uj(~, pj), I
where YJ(T,
i and j with timeSimilarly,
T/,uj).
pj)
=
J drN
(3.21)
jr1 . . . rN> &(T,
pj,
rj)
(3.22)
. . . rNI,
with
-b(T, pj, rj) = i
au@, +&T/P)
.p a aP,
arjr
au(ri + PF/~U)
apj
a
lar
j
1
, (3.23)
the Liouville operator in the phase space of particle j with time-dependent external potential U(rj + pjT/,~). In k representation, 5Zj and Tt:, in (3.22) and (3.19) become &f,(T,
p,)
=
-2x
j dkN f dk’N Iki . . , k$>
J ds I1 6(k& - km - ~&n,~)u(s)
x
LOI m s . alap, - kjf a e2~iS*pf+/apj] x [e2ni*epdp
(3.24)
and u,(T> pi&)
x
s
= -2X
J
c dkN
s
dk’N 1ki . . . k&>
ds II d[% - km - +bn,j - &a,i)] p(s) m
LOI
a e2xis*(Pj-P+~/cc -
.(kj -
1
kt)
aPJ
According to (3.18) and (3.21) the exponential fined by (3.8), is given by w(T, PI =
. ..pN.
T
exp{-i
(3.25) operator
W in (3.17),
de-
kl -.. kN)
5 dT‘ I[B] n
&(T',
PI..
. PN)
1[8]},
(3.26)
BOLTZMANN
EQUATION
FROM
LIOUVILLE
EQUATION
213
where
A(T) is the Hilbert-space
Liouville
A(T) = s drN 111 . . . t-N> k?(~)
operator (3.28)
. . . rN\,
representation
M(T) = x &(T, PJ, rj) + x x -&(T, p&j I ZCI &f(T) is the Liouville tonian function
operator
v(T) = c u(r, +&T/P) i
(3.29)
r&).
for the time-dependent,
+ T
-
yg + (Pj -Pi)
interaction-hamil(3.30)
T/PI).
yc
The operator vO(T)
=
dri” Irr
j
. . . fN)
v(T)
(3.31)
. . . rNI,
depends on time through the similarity
transformation
v~(~) = eiT-%’ v. e-i~z~‘,
(3.32)
the same transformation as in (3.18) and (3.21) ; this is the transformation to picture of the “interaction” (or “Dirac”) picture from the “Schr%linger” the original Liouville equation (3.1). According to (3.26), -W(7 ) is determined, not by the Liouville operator h%!(T) arising from the time-dependent interaction hamiltonian k’(7) of (3.30), but by the projected operator I[g] d(T) l[P] from which the contribution at the point kr . . . kN has been elided. But, 1[9]
A(T)
=
1[9]
Jzt(T)
since, according
= -
I[P]
-
B(kl . . . kN)]
B(kl . . . kN)
to (3.24),
B(kl . . . kN) A(T) Therefore,
[I
J(T)
-
sdif(T)[f ./if(T)
-
B(kl . . . kN)] (3.33)
B(kl . . . kN),
(3.25) and (3.27)
B(kl . . . kN) = 0.
from (3.22), d(T) -
1[9’] s
=
j
drN Irr . . . rN> M(T)
drNS duNIt.
+ Ur, . . . . rN+
ffN)
. . . rNl e2rri:U’“+“M(T)
--drN~dUNIrl...rN>M(T)e-2rr’~um’*m<~l+U1,
. . . fNl
. . . . rN+UN(,
(3.34) which can no longer be expressed as a Liouville operator obtained from a classical hamiltonian function in the way that J(T) was obtained from V(T).
IX LEAF
214
In the language of diagrammatic representations, elision of the point at 1[B] in (3.26) is equivalent to the ki . . . kN by means of the projector statement that g(7, pi . . . pi, kl . . . kN) in (3.13) is a “diagonal fragment” s), a diagram in which none of the intermediate states is identical with the initial and final states, B(kl . . . kN) = Ikl . . . kN)(kl The properties of the operator W(T) in (3.26) appendix
. . . kNl. are described
further
in
B.
4. The one-particle kinetic equation. From the N-particle (3.12) we obtain the one-particle kinetic equation
I P . VlVPd -p. aW9 i3r ar !J
a/l(W) ____at =
s dr,
-r, n
x
Wr-
r’l)
at-
kinetic equation
VdtPr) aP
aS2(tprr') ap
l
M
1
dpN d(p,
/ 7s d
-P)
...PN.
dr'[a&,Pl
O&h)
0
eznikn’(r-r’) p(t
-
pl
7,
. . . pN,
r’),
(4.1)
where, as in paper I, fi(W)
= s drN J dpN c d(P,. - P) 6(r, n
the particle
x
o(k
rl . . . rN), (4.2)
density in the one-particle
.%(tprr’)= J drN
r) p(& p1 ...pN,
1 dpN C C
d(pn
phase space (normalized -p)
6(r,
-
r) d(r,
-
to N);
r’)
Wtf7l
pl...pN,
(4.3)
rl... IN),
a two-particle
density [normalized to N(N - l)] ; and g(~, p1 . . . pN, OLnl, kn) from g(~, p1 . . . pN, kl . . . kN) by setting all vectors kl . . . kN equal to zero except k,. Let
is obtained
v&> where from (3.17), comes
Vij = - Vji. Then the one-particle
I P . afl(tPr) au(r) ar p-e ar FL
Vi (W
at
=
dr,
f
a+(lr- f-7) ar
aS2(tprr’)
*
ap
afdtP4 aP
Pi . ..pN. k,), kinetic
equation
(4.4) be-
BOLTZMANN
V&T, PI
X
EQUATION
FROM
LIOUVILLE
kn) ezxikn’(‘--‘)P(t -
**-pN,
7,
pl
EQUATION
215
(4.5)
f’),
. . . PN,
where f’jn(T,
. . . PN, kn) =
PI
s
drN
dtlN 3(lr3
W(T,
p1 . . . PN,
au(f;)
ww - 64
I:
3f-i
j’
(
T/li)
arj OLnl, kn)
a
x 1f-i . . . t-k> e2xikn*rn’Z -.., at+ [ 3 z
rn + (Pj -Pn)
s
X e-2xi*n+n+pnr’p)
+
-
w a
a
’ --aP3*
(4.6)
Yjg’ >I
with, from (3.26) and (2.12), ‘r(T, Pi a.- PN, OLlzls kn)
= T exp{-i
i d$ I[O,,,, knl A(T’,
PI ..-PN) 1[&,
&I}.
(4.7)
0
Eq. (4.5) is an exact equation. The left-hand side exhibits the flow terms of the Boltzmann equation, including the contribution of an external force. If the external force is absent, we recover eqs. (6.11) and (6.12) of paper I. 5. Approximate one-particle kinetic eqzcation: the Boltzmann equation. Following the steps of section 7 of the previous paper, we now assume that the range of the intermolecular forces is short so that (1) we can set the term in SZ equal to zero in (4.5), (2) we can set k, = 0 in Vjn in (4.6) (“molecular chaos”), and (3) we can set 7 = 0 in p(t -- 7, p1 . . . pi, r’) in (4.5). These approximations give
dfl(tPr) -~ at
I P
afdtpd
af-
/
where the collision
C.T. = -
--. aw
ar
ap
* ”
(5.1)
term is
dpN T+F T&-S s
s
afl(tPd =; c T
s(pn -p)
00
x dT V&a 0
Pl -. . PN)
/‘(t, PI
. . . PN,
r),
(5.4
216
B. LEAF
where
Vjn = x
drN
drlN +(lrj
s
-
s
*n + (Pi at-j
-Pn) T//4)
rNI~(T,pl...pN,01...ON)Ir;...r~>
with W(T, Pl ...pN. 01 ...oN) =
T
exp{-i
i dT1 I[0 i . . . ON] d(T’,
0
pi . . .PN) I[& 1..ON]).
(54
Further we assume that the system is a gas at low density, in which case the collision term becomes, as in eq. (7.5) of paper I,
C.T. =
-
L.
Jdpl 1.dTJ’7(7,
PIP) fz(t, PIP, r),
aP where
fz is the
0’
two-particle
V(~,p~p) =
probability
Miri
density normalized to N(N -
[dri[drsSdr;[dri
x
+
(5.5)
- *;I) i3ri
l), and
29(‘r1-r2+~~1-p2)T’P’) kir6>
a . --~
(
ah
a
(5.6)
ap2)I ’
with the matrix element in the space of two particles, of the operator, Wis(7, ~1~2) = T exp{--i
[ &-’ 12[001 s drl s dr2 Inm
x [Ll(T', pl> fl) + Ll(T', p2, r2) + L12(7', plp2, x
rlf2)1
(5.7)
The Liouville operators Li and Liz are given in (3.23) and (3.20). The collision term (5.5) differs from that in paper I by the presence of terms arising from the external potential U(r). If we assume that over the short range of the collision the forces arising from U(r) are negligible compared to the intermolecular forces, then these extra terms can be dropped,
BOLTZMANN
so that the collision
EQUATION
FROM
term becomes
LIOUVILLE
217
EQUATION
the same as that obtained
in eqs. (7.5)
and (7.13) of paper I. In paper I we considered the weak-interaction case in which the ordered exponential operator wrs(~, prps) was replaced by the first term 1s in its power-series expansion. In this case the collision term agrees with the Boltzmann collision term for small-angle scattering. We shall now show that the approximate one-particle kinetic equation (5.1) with the collision term (5.5) gives, with further approximations, the full Boltzmann equation. As we have seen in (3.34), the time displacement effected by the operator process because of the elision of W12(7, P1P2) in (5.7) is not a hamiltonian the point (00) by the projector Zs[OO] = 1s - ]OO>
so
that
T
PlP2)
exp{-i
If;&
id+
Lis(~‘, ~1~2, r1r2)) d(rl - ri) d(rz -
0
(5.6) becomes,
(5.8)
rh),
when terms in U(t) are dropped, W(lr1
-
(P1-
r2 +
P2) d/4)
arl 7
x Texp
d+Lrs(T’,
PrPs,
V’s)
0
X
We further
%Wl arl
(
w2)
this expression =
i
%wl
(5.9)
aP2 )
aP1
approximate
L12(7> PlP2,
a
a f-21) .-----_-_
-
r2 +
by writing
a
(Pl - P2) hi) arl
5
)
(5.10) ignoring the part of the operator in (Z+b/i3pl).(a/arl - a/are). This approximation can be regarded as a further consequence of the assumption of molecular chaos in which spatial derivatives of the distribution functions are neglected. The collision term of (5.5) can now be written with these approximations as
B. LEAF
218 T
x Texp x
IS dT,
WS + (Pl -PI T’IPI)
ag
F*(&-$)f2& a4M
PIP,
(5.11)
f9>
where g = rr - rs, and we have used the theorem of eq. (8.19) in the previous paper to suppress the apparent divergence arising from the double integral j drr j drs of a function depending only on rr - rs. The theorem can be applied directly to (5.6) when Is[OO] is ignored. On adding the vanishing integral of a divergence, we obtain the collision term in the form
jdpljd++& _ +).[a""5 + (;;-p'r'p"
C.T.=
0 7
As a final approximation, we neglect the derivative of a$( ILj+ (pr 85 with respect to the relative momentum pr - p, writing
(&
WIS
=
+
(Pl -PI d/4)
-+J[
w!? + (Pl -PI d/4) ap .(&f). a!t
Since the derivative
can be written WI5
+
(++[ neglecting cordingly,
it is consistent
1 (5.13)
ag
the
]=(&
previous
$+[i$jp approximation
C.T. = [dp,[d5S;r$ 0 7 x
T exp
dT’ IS
0
W) x ;*(&-
am
f
T/P\)/
as
(Pl -PI d/4) with
- p)
(Pl-PP)
T'/P~)
aI
&)f.(t,PIP,
r)
(5.10).
Ac-
BOLTZMANN
EQUATION
FROM
LIOUVILLE
a+v2 + (2+ Prdd21i
ag
0
EQUATION
219
*(&A+] (5.15)
where b and z are the components ly to the direction of the relative
Pr=Pl--p,
z = 22,
t=b+z,
With z’ = z + prr/p C.T. =~d~~~
a9w2+
a+(b2 + 2’2)t agl
w
(5.16)
-co
P
s
dpl
2 = PrlPr.
dbJdz
ag
=
and parallel respective-
and 1’ = b + 2’2, we obtain
dz’ ___ PIP x
of g perpendicular momentum pr :
s
dbfi
P
00
-00
dz __ P,IP
a#p + .22)* 35
B. LEAF
220
jdp’sdb$[Texp{ ldTY.(&- 6)) - 1]
=
-- co x f2(4 PlP9
(5.17)
t),
where now, z = pg/,~. Accordingly, C.T. =
s
where
IPl -PI db I p s
dpi
we obtain
Vz(k
pi = ~exp{~d~~~o(-&
Pip’, r) L
r)l,
f2(& Pip,
(5.18)
$))p’,
(5.19)
$-)]p,
(5.20)
--m and
m [diF.(&
p’ = Texp{
-
-cc
Assuming that f2 can be factored f2(& pip,
we obtain
fl(tplr)
r) =
the collision
_afdw)
P
I
at
f-J
=
dpl s
6. Disczlssion
.
(5.21)
fl(tpr),
term of the Boltzmann
-m. au ar
afl(tpr)
au
db ‘p1;-p’ s of pi and p’.
afdtpr) ap
[fi(tPir) fl(tP’r) The interpretation
(5.19) and (5.20) can be understood operator,
W12(7-)= T exp{--i
equationd),
-
fl(tplr)
fl(tpr)].
(5.22)
of pi and p’ defined
from the properties
in
of the phase-space
d dT’ L12(7’,~1~2, rlr2)},
(6.1)
which appears in (5.8), where the projector 12[00] is ignored. For purposes of this discussion we can reinstate the negligible portion of the Liouville operator Liz which was omitted in (5.10). In the phase space of two particles, Wrs(~) has the same form as w(T) considered in appendix B, and defined in (B.11). The inverse operator, therefore, as in (B.20), is w;-,'(T) =
T exp{i i dT'L12[7', f'l(T')j'2(T'),rl(T') r2(7')1,
(6.2)
where, as in (B.26), L12[7,p1(7)
p2(7), n(T)
r2(7)1 =
&t(T)L12(7,
plp2,
w2)
w12(7).
(6.3)
BOLTZMANN
According
to (3.20),
dependent
EQUATION
L&T,
~1~2,
hamiltonian
Lrs[+r, PI(T)
p4~),
Ml(4
+(lrl -
FROM
LIOUVILLE
r1r2)is the Liouville r2 +
= &W
$(lrl--
12 +
221
operator for the time-
~/pi).According
(~1 - p4
rl(7)r2(7)]is the Liouville
- Q(7) + [Plb) -
EQUATION
to (B.15),
operator for the hamiltonian
P2(7)1 d/4) (pl -p2)
T/PI) W12(7).
(6.4
The operator W;~(T) effects a displacement along the phase-space trajectory from coordinates and momenta at the initial point to those at a point 7 later in time. The operator WIT effects the reverse displacement. If the initial time were taken at --oo instead of 0, so that initially Wrs(--00) = Wi_,‘(-co) = 1, instead of Wrs(0) = W;,‘(O) = 1 in (B.25), then we would have the displacement operators W;:(T)
=
T exp{i
i
d7’ L12[7’, PI(+) ~~47’)~ rl(T’) eMI},
--co
WE+) According 47, where A(T, trajectory.
= T exp{--i
(6.5)
i &’ L12(7’, ~1~2, w2)). -co
to (B. 19), for any dynamical PI(T) p2(7), u(7) r2(dl
= W&J
property, 4~
P1P2, w2L
(6.6)
prps, rlr2)is the value of the property at the initial point of the Accordingly, since Wrs(7) effects the reverse displacement,
W12(7) P,(T) = T expf-i In particular,
i dr’ L12(7’, prps, -co
rla))Pj(7)
for 7 -+ co, with the approximation
?.exp{ rdTt ~-(-&-
=Pj(-a).
(6.7)
(5.10) for Lrs,
- $)}p&) =pl(-m), (6.8)
Therefore in (5.19) and (5.20), if p1 = pi(m) and p = p(co) are the final momenta in a collision process governed by the hamiltonian + of (6.4), then pi = pr(-co) and p’ = p(-co) are the initial momenta: pi and p’ are initial momenta in the inverse collision5) between two molecules which results in the final momenta p1 and p.
B. LEAF
222
APPENDIX
In addition
Orthogonal set of firojection operators. fined in (2.12), construct the projectors Bi =
S dkiB(kr Wtl = j dki9(kl
where integration Pjm==
j dkj [&I
A
to the operator
9 de-
. . . kj’... klv) . . . k;...
kN)[l - S(k;-
is performed
omitting
kj)],
(A.])
the point kj = kj;
S dk&,(kI.,.kj...k&...kr&
(A4
Wml
of any pair of and so forth. Clearly 9’~~~... is symmetric to interchange subscripts. The unit operator in the Hilbert space of N particles can be partitioned in the following way: 1 = J dk’B(ki
. . . kN) + I[kl . . . klv]
. . . k;T) = B(kl
-~+;r:~~+C~~~m+~CC~~rnn+‘... I
(A.3
For a given set of values (kl . . . kN}, each projector in this partition projects onto a different part of the Hilbert space, so that the set of projectors is an orthogonal set; for example,
A hamiltonian to (A.0) as
operator
in the Hilbert
space can be expanded
according
Ho = S dk HoB(kl . . . kN) =Sdk[BHoB+~~~Hob+CC~5mHo~+...l, j
+ C BHoPj i representation
(A4
+ C x PHoPjm j
+ . ..I.
of this operator
is given [paper I, eq.
. ..pN. rr . . . TN) =
where 101 . . . ON> is the ket therefore
Ikl . . . kN) in which
H = j dkN
+ x gjHo@ I
(A4
(A-7) all kt = 0, i =
+ . ..) 101 . . . ON).
1 . . . N,
(-4.8)
,..a, ~~~+sN>~(~~...PN,s~...sN)<~I...~N(, (A-9)
BOLTZMANN
where the Fourier
EQUATION
FROM
LIOUVILLE
223
EQUATION
of H in (A.7) is
transform
h(pr . ..PN. kr . . . kN) =,
(A. 10)
so that BHoP
= Ikl... kN>h(P1...&v,01...O~)
PjHoB
=
s ds Jkl....k,+
s, . . . kN>
to1 X h(pl
. ..pN.
01 . . . Sj . . .
(A. 1 1)
ON)(kl...knrj,
9J,mHoB =
j ds j ds’ Ikl . . . . kj+ s,....km+ s',...kN) LOI LOI X h(pl...pN,01...Sj...S7;2...ON)
Substitution function,
into (A.8) gives the expansion
H = h(& . ..pN. 01 . . . ON) + z
j
of the phase-space
hamiltonian
01...Sj
ds e2xis*rfk(pr . ..PN.
. . . ON)
3 ro1
+
Fez
IA
ds ,A ds’ e2rri(s’rr+S’*rm) (A. 12)
x h(pr . ..pN. 01 . . . sj . . . & . . . ON) + . . . . For the hamiltonian
given in (2.1) and (2.2)
k(pl
.-.pN,
01 ..a ON)
=
I; i
$,2/2/h
h(p1
. ..pN.
01 . . . Sj . . . ON) =
(A. 13)
for all j,
u(S),
01...sj...sh...ON)= 6(s+ d)v(s), for all j # m. h(Pl...PN. The corresponding
expansion
of the Liouville
u=SdkN[~~~+Z:83Y8+~:C~,dp~+...], 3
operator
Y is (A. 14)
j
or (A. 15) ~~=dkN[8~;48+~:88~~+C~C~~5,+...], I~k,.
ah(pl
i
9~99
. . . PN,
01 . . . ON)
aP3
ds)kl . . . . k,+s, ...kN)
= -2~ J lOI
x
h(pl
ah&
_ k 3’
a
. ..pN.
01 . . . sj . . . ON)s*---
. ..pN.
ah
01 . . . sj
aP,
. . . ON)
...&I, 1
(A.16)
B. LEAF
224
Pj,US
ds
= -2x s
lO1
LO1
x h(pl . ..pN. kj.--
X
a
ds’ Jkl . . . . k:, + s, . . . . km + s’, . . . kN) s
01 . . . sj . . . s& . . . ON) s*-
a aPm >
aPm
h(pl .., PN, 01 . . . Sj . . . Sh . . . ON)
For the hamiltonian of the present parison with (2.8), that Zi
aP5
+ s’.-
a
+ k,~-
aPI
a
problem,
using (A.13) we find on com-
= J dkN9’99, (A. 17)
APPENDIX
B
Properties of W( r ) in eq. (3.26). Time-dependent hamiltonian formalism. In (3.26), W(T) is the solution of the differential equation,
a7qT)/aT
=
-il[P]
An inverse operator w-+,
Jo
W(0) = 1.
1[9] W(T),
(B.1)
is defined by 6)
pi . . . PN, kl . . . kN) = T(‘) exp{i ; d7’ 1[9] .&(T’, pi
. . . pnr)
1[9’]),
(B.2)
0
where T(‘) is the reversed w(T)
“i+(T)
Introducing d@‘(T) we
see
=
time-ordering
=
1.
(W
w-l(T)
l[p]
A(T)
the differential
that
=
-iw(T)
PI
-..pN,
kl
T(‘) exp(-i
1[9]
(B.4)
w(T),
equation
(B.l)
A’(T),
which gives the alternative
=
so that from (3.26),
the operator
&v(T)/aT
yf(T,
operator,
can be written
W(0)
expression
for
W(T)
=
1,
as (W
as
..- kN)
i dT’ 0
J&(T’,
pl
. ..PN.
kl . . . kN)}.
W)
BOLTZMANN
EQUATION
FROM
LIOUVILLE
The inverse operator in (B.2) is the solution of the differential &Y--~(T)/% = iW-1(7) with W-i(O)
equation
1[8] &Y(T) 1[8] = i&!‘(7) W-~(T),
= 1, so that an alternative
225
EQUATION
(B.7)
to (B.2) is the expression
?cy--i(T, pi . ..pN. kl . . . kN) == T exp{i 1 dT’
J?(T’,
p1
kl . . . kN)}.
. . . PN,
(B-8)
0
The time displacement effected by the operator W(T) is a non-hamiltonian process according to (B. 1) and (3.34). But it may be remarked that if (B.l) is approximated by omitting the projector 1[P], ignoring elision of the point kl . . . kN, then ~W(T)/~T so
=
-id(T)
W(T),
(B.9)
that, from (3.28), w(T)
=
s
drN (,.I . . .
rN>
w(T)
(B.lO)
. . . rNl,
where
w(T) = T eXp(--i
id+
M(T’,
P1
. . . PN,
rl
. . . rN)},
(B.ll)
and W-~(T) = J drN
jr1
. . . rN>
wei(T)(rl
(B.12)
. . . rN1,
where = T eXp{i i dT’
w-l(7)
M’(T),
pl
. . . PN,
rl
. . . rN)},
(B. 13)
0
with M’(T)
=
w-l(T)
M(T)
(B.14)
w(T).
As noted in (3.29), &f(T) is the Liouville hamiltonian v(T) in (3.30). Likewise M’(T) ator for the hamiltonian UT, pi(~) ...PN(T). =
w-l(T)
v(T,
operator for the time-dependent in (B. 14) is the Liouville oper-
rl(7) . .. r&)1 pl
. . . pN,
rl
. . . rN)
(B. 15)
w(T).
It can be written
M’(T, pl
-9.
PN, rl . . . rN) =
M[T, pi(T) . . . PN(T), rl(T) . . . rN(T)]-
(B.16)
These results follow from general considerations for motion governed by a time-dependent hamiltonian. The equations of motion for the hamiltonian v[T, pi(T) ***PN(T), rl(T) -.-rN(T)] are
dP3(4 -----=-dT
i3V %(T)
drj(T)
aV
d7
2P, (4
-==. ’
(B.17)
B. LEAF
226
For any dynamical property A [T, PI(T) . . . IN, on the trajectory
Q(T) . . . t-N(T)] of the system
at time T,
dA j -= dT
= iM[T, PI(T) . . . PN(T), Q(T) . . . IN]
A + aA/a~.
(B. 18)
The solution of this equation is A [T, Pi(T) -. . &v(T), =
rl(T) . . . f-N(T)]
(B. 19)
A(T, p1 . . . pN, rl . . . rN),
w-l(T)
where w-l(T)
T eXp(i f ds’ M[T’, pl(T’)
=
0
. . . pN(T’),
rl(T’)
(B.20)
. . . rN(T’)].
To verify the expression (B. 11) for W(T), and hence (B. 13) for W-~(T), consider the Liouville equation for the probability density P(T, pl . . . pN, rl . . . rN) for an ensemble with time-dependent hamiltonian V(T) of (3.30), aP(T, pl . ..pN. rl . . . rN)/aT + ik!(T, p1 . . . pN,
rl
. . . rN)
P(T,
pl
. . . PN,
rl
. . . rN)
=
(B.21)
0.
The solution of this equation is rl-.. rN) =
P(T,Pl...j'N,
W(T)P(O,pl...pN,
where w(T) is given by (B.ll). p[T> h(T) =
Accordingly, dP[T,
The differential
. . . &V(T),
-iM(T,
-iw(T)
dW-l(T)/dT
JCT,
-
iw-l(T)
iM[T, hf(T,
with W(0) = W-r(O) = MIT,
P(o,pl...p~,
rl(T)
...
pl
. ..PN.
PI(T) PI(T) p1
and f’l
. . . PN(T),
ti
. . . PN(T), . . . pN,
rl
PI
governed by
7/(T),
(B.24) follow
w-l(T)
. . . rN)
w(T) (T)
. . . r%(T)], . . . IN]
W-~(T)
(B.25)
. . . rN),
1. Therefore, rl(T) ... ~N(T)I
=
pl...pN,
M(T,
(B.23)
t'N(T)]/dT = 0.
w(T)
rl... rN)
rl... rN).
PI(T) . . . PN(T), w-l(T)
W-l(T)P(T,pl...pN,
theorem results for the trajectory
equations for
dW(T)/dT =
=
w(T)
Liouville’s
PI(T)
=
Using (B.19) we obtain
'..$'N(T), 11(T) ... rN(T)] =
w-l(T)
(B.22)
rl...rN),
rl...
in agreement with (B.14) and (B.16).
IN)
w(T),
(B.26)
BOLTZMANN
EQUATION
FROM LIOUVILLE
EQUATION
227
According to (B.20), the operator W-~(T) effects a displacement of phasespace coordinates along the trajectory from the initial point to a point T later in time, the trajectory governed by V of (B.15). The operator W(T) effects the reverse displacement. This description in terms of a time-dependent hamiltonian holds only if the projector I[P] is ignored in (B.l). Relations analogous to those given here for W(T) and W-~(T), for systems with time-dependent hamiltonians, hold also in quantum mechanicss) .
REFERENCES 1) Leaf, B., Physica 58 (1972) 445. This reference is cited as paper I in the present work. 2) Leaf, B. and Schieve, W. C., Physica 36 (1967) 589; Leaf, B., J. math. Phys. 11 (1970) 1806. 3) Prigogine, I., Non-Equilibrium Statistical Mechanics, Interscience Publishers (New York, 1962). 4) Boltzmann, L., Wien. Ber. 66 (1872) 275. 5) Ter Haar, D., Elements of Statistical Mechanics, Holt, Rinehart and Winston, Inc. (New York, 1954). 6) Leaf, B., J. math. Phys. 9 (1968) 769.
59
(1972) 206-227 o North-Holland
DERIVATION
OF THE
FROM
THE
Publishing
Co.
BOLTZMANN
LIOUVILLE
EQUATION
EQUATION
*
B. LEAF State University
of New
York,
College at Cortland,
Received
12 July
Cortland,
New
York r3045,
USA
1971
synopsis The
derivation
previously
given
central-force
interaction
body
equation
is obtained,
distribution
function.
kinetic
one-particle mation
to this
density,
but without
for the kinetic
is extended
equation
with
to include and from
The Boltzmann of previous
equation
for a classical
of external
this the exact
the assumptions
the limitation
equation
the effects
kinetic
The N-
equation
for the
is derived
of short-range
gas with
forces.
as an approxi-
interactions
and low
work to the case of weak interactions.
1. Introduction. In an earlier paper 1)) which will be cited in this article as paper I, an exact kinetic equation was derived for the one-particle distribution of a classical gas with central-force interactions. The starting point for the derivation was the Liouville equation; the method employed was the technique of projection operatorss). In the present paper the treatment is extended to the case in which external forces act on the gas. The assumptions of short-range interaction forces and low density are introduced, but not the limitation to the case of weak interactions as in paper I. If certain additional approximations are made, precise agreement is obtained with the full Boltzmann equation. The additional approximations are 1) Neglect of the elision by the projector 1[00] in the ordered exponential operator. This has the effect of making this operator into a displacement operator along a trajectory governed by a hamiltonian function [eq. (5.8)]. 2) Neglect of the spatial derivative part of the Liouville operator appearing in the ordered exponential operator. This may be considered as an extension of the assumption of molecular chaos [eq. (5.10)]. 3) Neglect of a derivative of the force of the collision interaction with respect to the relative velocity of the colliding particles [eq. (5.13)].
* This work was supported University of New York.
by a grant from the Research
206
Foundation
of the State
BOLTZMANN
EQUATION
FROM
LIOUVILLE
It must also be assumed that the two-particle the collision term can be factored into a product
EQUATION
207
distribution function in of one-particle functions
[eq. (5.21)]. The procedure followed in deriving the exact one-particle kinetic equation closely parallels that of paper I. In section 2, the Liouville operator is given. In section 3, the N-body kinetic equation is derived for the gas with external forces. The exact one-particle kinetic equation is given in section 4. It has the flow terms of the Boltzmann equation, including the contribution of external forces. In section 5, approximate one-particle kinetic equations are obtained. With the assumptions of short-range forces and dilute gas, which were considered in paper I, we obtain the approximate equation (5.1) with collision term (5.5). From this equation, on neglect of the external forces in the collision term and with the additional approximations listed above, the Boltzmann equation is derived. It is shown in section 6 that the momenta pi and p’ appearing in the Boltzmann equation (5.22) and defined by (5.19) and (5.20) in terms of the time-ordered exponential operator are the initial momenta in the inverse collision between two molecules which results in the final momenta p1 and p. In appendix A it is shown that an orthogonal set of projection operators can be constructed in terms of which a complete partition of the unit operator in the Hilbert space can be achieved. With these operators the hamiltonian and, correspondingly, the Liouville operator can be expanded in a systematic way, as required for the present formalism. In appendix B, properties of the time-ordered exponential operator W(T) of eq. (3.26) are considered. The time-dependent hamiltonian formalism, which results when the elision produced by the projector l[kl . . . kN] is ignored, is described. The derivation of the Boltzmann equation shows in detail the assumptions and approximations introduced in starting from the exact dynamical (hamiltonian) equations of motion, in the form of Liouville’s equation, and obtaining the dynamically irreversible Boltzmann equation. The exact oneparticle kinetic equation (4.5) contains none of these assumptions or approximations. Additional 2. Liouville hamiltonian
references operators.
to the literature Consider
are given in paper I.
a gas of N particles
described
by the
(2.1) where U(r,) is the potential energy of external forces acting on the ith particle at position ri, and 4(1 tti - rJ]) is the two-body, central-force interaction between the particles i and j. Expressed as Fourier integrals, U(r,) =
J ds ezxircms u(s), LOI
d(r) = ,J ds ezxires v(s).
(2.2)
B.LEAF
208 In these integrations
the point at s = 0 is omitted
since the contribution
to the integral for s = 0 is a constant, which we equate to zero in the hamiltonian without loss of generality. Equivalently we can require that U(O) = j dszl(s) 6(s) = 0,
~(0) = j ds ~(4 4s) = 0,
without exclusion of the point s = 0 from the integrations
in (2.2). Eq. (2.2)
implies that J d@ 1X LW)
+ X Z d(lrj -
i
i
In phase-space two parts, L = L1-t
representation
rd)l = 0. the Liouville
operator
is separated
L2,
into
(24
and L1 is separated further, L1 = Li + L;,
(24
where
L’; = i x XJ(rk)/i3rkai3/apk
= -
k
L2
=
iC
I: J ds e2rris’rkU(S) 2&?/apk,
(2.5)
k
a4(ir5-rki)
2
ar5
j
aPk
s
= _ F
-~a
.
&
e2nis*(rr-rj) p(s) zxs.
Liouville
-_?
_!? .
_
aPk
aP5
operators are defined by
9=
jdrNIrl...rN>L
p=
Ylf22,
(2.6)
with
so
(2.7)
S?l=9i+L?~,
that
9'; =
dk’N
-
s
x c
i
s
dkN[ki...
kk)
s
ds I-I 6(k;, - k, - SC?& u(s) 2xs.-
m
a
aP5
BOLTZMANN
9'2 =
-
EQUATION
J J dk’N
dkN ]ki
FROM
LIOUVILLE
EQUATION
201
...k;)
x T
?n
a
~ ap,
-
a aP, >
The ket ]rr . . . r~) is the eigenvector
(f-1 . . . rN/kl...
kN> = exp(2zi
(2.8;
of the N-particle
position operator,
2 rm-km). m
ant (2.9;
The Liouville operator _Yi was designed by Yr in paper I, where, in the absence of external forces, the operator 29 did not appear. The structure of these operators is made evident in eq. (A. 17) in terms oj a set of projection operators, which form a partition of the unit operator, as described in appendix A. It follows from the orthogonality of these projectors that P,uil[P]
= 1[8] SF@
(2.10;
= 0,
and
where the projector are defined as
g(kl... 1[8]
B = B(kl . . . kN) and its complement
1[P] = I[kl
kN) = jkl... kN>
a,,jat =
(2.121
Ikl... kN>(kl . . . kN[.
3. The N-body kinetic equation. from the Liouville equation
. . . kN:
To obtain
the N-body
kinetic
equatior (3.1
-i_Yp0
we proceed as in the derivation of eq. (4.9) of paper I. With the use of the orthogonality relations (2.10) and (2.1 l), corresponding to (3.19) and (3.20 of paper I, we obtain apo(t)iat i- iZlpo(t) = =
-iidP2o0(t) -
-i=Y2p0(t)
rd7
J dkNBYs
e-iP1[81r PPpo(t
-
T).
0
In this equation
_%‘2co(t)is given, as in eq. (4.13) of paper I, by
(3.2
B. LEAF
210
_Ys~e(t) = LYso’“(t) -
i rd7
j dkN 992
e-iS1lLPIT .Ps1[B]
oo(t -
T), (3.3)
0
where 1[8] e_(t) = e- iz[B’zlt 1[SP] @CO).
(3.4)
Using the formula for the exponential
A and B
of the sum of two operators
given in eq. (5.1) of paper I, eA+B = eA T exp ;dt e--t4 B et-4,
(3.5)
0
where T is the Dyson ordering operator, e-i~~~l[Bl
e -i7(ZLpl’+9~“)
=
e-iT2Wl
j-h . ..pN. =
=
e-i7_5fl’ =
e -i7(5?~‘+8~“+~2)1[~1
=
in terms of the operators wl(t,
I[?]
h
T exp{-_i
Wi(t)
we obtain Wl(d,
e-i7-LP1’
(3.6)
W(T)>
and W(t) defined as
. . . h) i &’
I[g]
eit’zl’
2:
e-it’zl’
w%
(3.7)
0 w(t,
PI
. ..PN.
h
=
T exp{-i
. . . kN)
s”dt’ 1[B] e’““” (9;
+ 92) e-it’Sl’
1[P]}.
(3.8)
0
Accordingly,
from (3.2), (3.3),
+0(t)/% + iPipo(t) 900(l)
(3.4) and (2. lo),
+ iY~p0(t) == -iY2ao(t)
OrdT j dkN BS?s e-iTzl’ W(T) (2; = 92@(t)
-
rds
S dkNPg2
+
92)
Ppo(t
-
(3.9)
4,
e-iTYLpI’Wi(T) 6psZ[g]
ao(t
-
T),
0
(3.10) 1[8] o:(t)
(3.11)
= ehitzl’ WI(t) 1[9] c&O).
From these Hilbert-space equations we obtain the phase-space equations with the help of eq. (3.7) of paper I. From (3.9) and (2.5), the N-body kinetic equation governing the time evolution of the probability density in r phase space, p(t, p1 . . . pN, t-1 . . . rN) is +(t,
pl
rl
. . . pN>
at
+
x
Pk
k
awk) -%G--•
. . . rN)
a&, k
pl
.
+(t,
pl
ark
P
...pN,
aPk
. . . PN,
rl
. . . rN)
rl
. . . IN)
BOLTZMANN
EQUATION
““‘ly
= -g
Ql) 3
FROM
LIOUVILLE
EQUATION
211
.(&-$--)o(t,Pl...PNA..YN)
m
-
PN, kl . . . kN)
drdkNg(T, PI...
s
0 X
n
J
dr’N e2rri?“*(‘r-rm’) ~(i! -
7, p1 . . . PN, ri , . . &),
(3.12)
where
g(T,PI . . . PN, kl .. . kN) =
“91’~(+?;
+ 92) Ikl... kN)-
(3.13)
As shown in eqs. (5.18) and (5.19) of paper I,
,-2~~~km~(~m+pm~id
x
...
(3.14)
rNl,
n
kN> = i T
92 [kl...
J
drN (rl
. . . rN) e2xif?‘*km (3.15)
Also, from (2.8), 2;
au(f-3)
(kl ..I kN> = i z
3
Accordingly, &,Pl
s
3
a
(3.16)
ap3
from (3.13) .*-pN,
x
drN\rl . . . mN>e2rriErm’km F.-.
z
[ jf
k1.e.
au(+) ark
kN)
.++zz j’
&j(lrb - &I) ar,
j’
It was shown in section 5 of paper I that ei7z,’
92
e-““”
=
Is
r,
p,(T,
(3.18)
PtPj),
j
where =923(7, ptp3)
=
J drN If-1 **. fN> &(T,
p43,
w3)
(3.19)
B. LEAF
212
- am -
(I+ -
r2 +
a
T/PI)
Pd
(3.20)
‘ark 1 ’
apk
the Liouville operator in the phase space of particles dependent interaction potential +(/rj - rt + (pj - pa) eirS1’ 2;
eWiTzLPI = C uj(~, pj), I
where YJ(T,
i and j with timeSimilarly,
T/,uj).
pj)
=
J drN
(3.21)
jr1 . . . rN> &(T,
pj,
rj)
(3.22)
. . . rNI,
with
-b(T, pj, rj) = i
au@, +&T/P)
.p a aP,
arjr
au(ri + PF/~U)
apj
a
lar
j
1
, (3.23)
the Liouville operator in the phase space of particle j with time-dependent external potential U(rj + pjT/,~). In k representation, 5Zj and Tt:, in (3.22) and (3.19) become &f,(T,
p,)
=
-2x
j dkN f dk’N Iki . . , k$>
J ds I1 6(k& - km - ~&n,~)u(s)
x
LOI m s . alap, - kjf a e2~iS*pf+/apj] x [e2ni*epdp
(3.24)
and u,(T> pi&)
x
s
= -2X
J
c dkN
s
dk’N 1ki . . . k&>
ds II d[% - km - +bn,j - &a,i)] p(s) m
LOI
a e2xis*(Pj-P+~/cc -
.(kj -
1
kt)
aPJ
According to (3.18) and (3.21) the exponential fined by (3.8), is given by w(T, PI =
. ..pN.
T
exp{-i
(3.25) operator
W in (3.17),
de-
kl -.. kN)
5 dT‘ I[B] n
&(T',
PI..
. PN)
1[8]},
(3.26)
BOLTZMANN
EQUATION
FROM
LIOUVILLE
EQUATION
213
where
A(T) is the Hilbert-space
Liouville
A(T) = s drN 111 . . . t-N> k?(~)
operator (3.28)
. . . rN\,
representation
M(T) = x &(T, PJ, rj) + x x -&(T, p&j I ZCI &f(T) is the Liouville tonian function
operator
v(T) = c u(r, +&T/P) i
(3.29)
r&).
for the time-dependent,
+ T
-
yg + (Pj -Pi)
interaction-hamil(3.30)
T/PI).
yc
The operator vO(T)
=
dri” Irr
j
. . . fN)
v(T)
(3.31)
. . . rNI,
depends on time through the similarity
transformation
v~(~) = eiT-%’ v. e-i~z~‘,
(3.32)
the same transformation as in (3.18) and (3.21) ; this is the transformation to picture of the “interaction” (or “Dirac”) picture from the “Schr%linger” the original Liouville equation (3.1). According to (3.26), -W(7 ) is determined, not by the Liouville operator h%!(T) arising from the time-dependent interaction hamiltonian k’(7) of (3.30), but by the projected operator I[g] d(T) l[P] from which the contribution at the point kr . . . kN has been elided. But, 1[9]
A(T)
=
1[9]
Jzt(T)
since, according
= -
I[P]
-
B(kl . . . kN)]
B(kl . . . kN)
to (3.24),
B(kl . . . kN) A(T) Therefore,
[I
J(T)
-
sdif(T)[f ./if(T)
-
B(kl . . . kN)] (3.33)
B(kl . . . kN),
(3.25) and (3.27)
B(kl . . . kN) = 0.
from (3.22), d(T) -
1[9’] s
=
j
drN Irr . . . rN> M(T)
drNS duNIt.
+ Ur, . . . . rN+
ffN)
. . . rNl e2rri:U’“+“M(T)
--drN~dUNIrl...rN>M(T)e-2rr’~um’*m<~l+U1,
. . . fNl
. . . . rN+UN(,
(3.34) which can no longer be expressed as a Liouville operator obtained from a classical hamiltonian function in the way that J(T) was obtained from V(T).
IX LEAF
214
In the language of diagrammatic representations, elision of the point at 1[B] in (3.26) is equivalent to the ki . . . kN by means of the projector statement that g(7, pi . . . pi, kl . . . kN) in (3.13) is a “diagonal fragment” s), a diagram in which none of the intermediate states is identical with the initial and final states, B(kl . . . kN) = Ikl . . . kN)(kl The properties of the operator W(T) in (3.26) appendix
. . . kNl. are described
further
in
B.
4. The one-particle kinetic equation. From the N-particle (3.12) we obtain the one-particle kinetic equation
I P . VlVPd -p. aW9 i3r ar !J
a/l(W) ____at =
s dr,
-r, n
x
Wr-
r’l)
at-
kinetic equation
VdtPr) aP
aS2(tprr') ap
l
M
1
dpN d(p,
/ 7s d
-P)
...PN.
dr'[a&,Pl
O&h)
0
eznikn’(r-r’) p(t
-
pl
7,
. . . pN,
r’),
(4.1)
where, as in paper I, fi(W)
= s drN J dpN c d(P,. - P) 6(r, n
the particle
x
o(k
rl . . . rN), (4.2)
density in the one-particle
.%(tprr’)= J drN
r) p(& p1 ...pN,
1 dpN C C
d(pn
phase space (normalized -p)
6(r,
-
r) d(r,
-
to N);
r’)
Wtf7l
pl...pN,
(4.3)
rl... IN),
a two-particle
density [normalized to N(N - l)] ; and g(~, p1 . . . pN, OLnl, kn) from g(~, p1 . . . pN, kl . . . kN) by setting all vectors kl . . . kN equal to zero except k,. Let
is obtained
v&> where from (3.17), comes
Vij = - Vji. Then the one-particle
I P . afl(tPr) au(r) ar p-e ar FL
Vi (W
at
=
dr,
f
a+(lr- f-7) ar
aS2(tprr’)
*
ap
afdtP4 aP
Pi . ..pN. k,), kinetic
equation
(4.4) be-
BOLTZMANN
V&T, PI
X
EQUATION
FROM
LIOUVILLE
kn) ezxikn’(‘--‘)P(t -
**-pN,
7,
pl
EQUATION
215
(4.5)
f’),
. . . PN,
where f’jn(T,
. . . PN, kn) =
PI
s
drN
dtlN 3(lr3
W(T,
p1 . . . PN,
au(f;)
ww - 64
I:
3f-i
j’
(
T/li)
arj OLnl, kn)
a
x 1f-i . . . t-k> e2xikn*rn’Z -.., at+ [ 3 z
rn + (Pj -Pn)
s
X e-2xi*n+n+pnr’p)
+
-
w a
a
’ --aP3*
(4.6)
Yjg’ >I
with, from (3.26) and (2.12), ‘r(T, Pi a.- PN, OLlzls kn)
= T exp{-i
i d$ I[O,,,, knl A(T’,
PI ..-PN) 1[&,
&I}.
(4.7)
0
Eq. (4.5) is an exact equation. The left-hand side exhibits the flow terms of the Boltzmann equation, including the contribution of an external force. If the external force is absent, we recover eqs. (6.11) and (6.12) of paper I. 5. Approximate one-particle kinetic eqzcation: the Boltzmann equation. Following the steps of section 7 of the previous paper, we now assume that the range of the intermolecular forces is short so that (1) we can set the term in SZ equal to zero in (4.5), (2) we can set k, = 0 in Vjn in (4.6) (“molecular chaos”), and (3) we can set 7 = 0 in p(t -- 7, p1 . . . pi, r’) in (4.5). These approximations give
dfl(tPr) -~ at
I P
afdtpd
af-
/
where the collision
C.T. = -
--. aw
ar
ap
* ”
(5.1)
term is
dpN T+F T&-S s
s
afl(tPd =; c T
s(pn -p)
00
x dT V&a 0
Pl -. . PN)
/‘(t, PI
. . . PN,
r),
(5.4
216
B. LEAF
where
Vjn = x
drN
drlN +(lrj
s
-
s
*n + (Pi at-j
-Pn) T//4)
rNI~(T,pl...pN,01...ON)Ir;...r~>
with W(T, Pl ...pN. 01 ...oN) =
T
exp{-i
i dT1 I[0 i . . . ON] d(T’,
0
pi . . .PN) I[& 1..ON]).
(54
Further we assume that the system is a gas at low density, in which case the collision term becomes, as in eq. (7.5) of paper I,
C.T. =
-
L.
Jdpl 1.dTJ’7(7,
PIP) fz(t, PIP, r),
aP where
fz is the
0’
two-particle
V(~,p~p) =
probability
Miri
density normalized to N(N -
[dri[drsSdr;[dri
x
+
(5.5)
- *;I) i3ri
l), and
29(‘r1-r2+~~1-p2)T’P’) kir6>
a . --~
(
ah
a
(5.6)
ap2)I ’
with the matrix element in the space of two particles, of the operator, Wis(7, ~1~2) = T exp{--i
[ &-’ 12[001 s drl s dr2 Inm
x [Ll(T', pl> fl) + Ll(T', p2, r2) + L12(7', plp2, x
rlf2)1
(5.7)
The Liouville operators Li and Liz are given in (3.23) and (3.20). The collision term (5.5) differs from that in paper I by the presence of terms arising from the external potential U(r). If we assume that over the short range of the collision the forces arising from U(r) are negligible compared to the intermolecular forces, then these extra terms can be dropped,
BOLTZMANN
so that the collision
EQUATION
FROM
term becomes
LIOUVILLE
217
EQUATION
the same as that obtained
in eqs. (7.5)
and (7.13) of paper I. In paper I we considered the weak-interaction case in which the ordered exponential operator wrs(~, prps) was replaced by the first term 1s in its power-series expansion. In this case the collision term agrees with the Boltzmann collision term for small-angle scattering. We shall now show that the approximate one-particle kinetic equation (5.1) with the collision term (5.5) gives, with further approximations, the full Boltzmann equation. As we have seen in (3.34), the time displacement effected by the operator process because of the elision of W12(7, P1P2) in (5.7) is not a hamiltonian the point (00) by the projector Zs[OO] = 1s - ]OO>
so
that
T
PlP2)
exp{-i
If;&
id+
Lis(~‘, ~1~2, r1r2)) d(rl - ri) d(rz -
0
(5.6) becomes,
(5.8)
rh),
when terms in U(t) are dropped, W(lr1
-
(P1-
r2 +
P2) d/4)
arl 7
x Texp
d+Lrs(T’,
PrPs,
V’s)
0
X
We further
%Wl arl
(
w2)
this expression =
i
%wl
(5.9)
aP2 )
aP1
approximate
L12(7> PlP2,
a
a f-21) .-----_-_
-
r2 +
by writing
a
(Pl - P2) hi) arl
5
)
(5.10) ignoring the part of the operator in (Z+b/i3pl).(a/arl - a/are). This approximation can be regarded as a further consequence of the assumption of molecular chaos in which spatial derivatives of the distribution functions are neglected. The collision term of (5.5) can now be written with these approximations as
B. LEAF
218 T
x Texp x
IS dT,
WS + (Pl -PI T’IPI)
ag
F*(&-$)f2& a4M
PIP,
(5.11)
f9>
where g = rr - rs, and we have used the theorem of eq. (8.19) in the previous paper to suppress the apparent divergence arising from the double integral j drr j drs of a function depending only on rr - rs. The theorem can be applied directly to (5.6) when Is[OO] is ignored. On adding the vanishing integral of a divergence, we obtain the collision term in the form
jdpljd++& _ +).[a""5 + (;;-p'r'p"
C.T.=
0 7
As a final approximation, we neglect the derivative of a$( ILj+ (pr 85 with respect to the relative momentum pr - p, writing
(&
WIS
=
+
(Pl -PI d/4)
-+J[
w!? + (Pl -PI d/4) ap .(&f). a!t
Since the derivative
can be written WI5
+
(++[ neglecting cordingly,
it is consistent
1 (5.13)
ag
the
]=(&
previous
$+[i$jp approximation
C.T. = [dp,[d5S;r$ 0 7 x
T exp
dT’ IS
0
W) x ;*(&-
am
f
T/P\)/
as
(Pl -PI d/4) with
- p)
(Pl-PP)
T'/P~)
aI
&)f.(t,PIP,
r)
(5.10).
Ac-
BOLTZMANN
EQUATION
FROM
LIOUVILLE
a+v2 + (2+ Prdd21i
ag
0
EQUATION
219
*(&A+] (5.15)
where b and z are the components ly to the direction of the relative
Pr=Pl--p,
z = 22,
t=b+z,
With z’ = z + prr/p C.T. =~d~~~
a9w2+
a+(b2 + 2’2)t agl
w
(5.16)
-co
P
s
dpl
2 = PrlPr.
dbJdz
ag
=
and parallel respective-
and 1’ = b + 2’2, we obtain
dz’ ___ PIP x
of g perpendicular momentum pr :
s
dbfi
P
00
-00
dz __ P,IP
a#p + .22)* 35
B. LEAF
220
jdp’sdb$[Texp{ ldTY.(&- 6)) - 1]
=
-- co x f2(4 PlP9
(5.17)
t),
where now, z = pg/,~. Accordingly, C.T. =
s
where
IPl -PI db I p s
dpi
we obtain
Vz(k
pi = ~exp{~d~~~o(-&
Pip’, r) L
r)l,
f2(& Pip,
(5.18)
$))p’,
(5.19)
$-)]p,
(5.20)
--m and
m [diF.(&
p’ = Texp{
-
-cc
Assuming that f2 can be factored f2(& pip,
we obtain
fl(tplr)
r) =
the collision
_afdw)
P
I
at
f-J
=
dpl s
6. Disczlssion
.
(5.21)
fl(tpr),
term of the Boltzmann
-m. au ar
afl(tpr)
au
db ‘p1;-p’ s of pi and p’.
afdtpr) ap
[fi(tPir) fl(tP’r) The interpretation
(5.19) and (5.20) can be understood operator,
W12(7-)= T exp{--i
equationd),
-
fl(tplr)
fl(tpr)].
(5.22)
of pi and p’ defined
from the properties
in
of the phase-space
d dT’ L12(7’,~1~2, rlr2)},
(6.1)
which appears in (5.8), where the projector 12[00] is ignored. For purposes of this discussion we can reinstate the negligible portion of the Liouville operator Liz which was omitted in (5.10). In the phase space of two particles, Wrs(~) has the same form as w(T) considered in appendix B, and defined in (B.11). The inverse operator, therefore, as in (B.20), is w;-,'(T) =
T exp{i i dT'L12[7', f'l(T')j'2(T'),rl(T') r2(7')1,
(6.2)
where, as in (B.26), L12[7,p1(7)
p2(7), n(T)
r2(7)1 =
&t(T)L12(7,
plp2,
w2)
w12(7).
(6.3)
BOLTZMANN
According
to (3.20),
dependent
EQUATION
L&T,
~1~2,
hamiltonian
Lrs[+r, PI(T)
p4~),
Ml(4
+(lrl -
FROM
LIOUVILLE
r1r2)is the Liouville r2 +
= &W
$(lrl--
12 +
221
operator for the time-
~/pi).According
(~1 - p4
rl(7)r2(7)]is the Liouville
- Q(7) + [Plb) -
EQUATION
to (B.15),
operator for the hamiltonian
P2(7)1 d/4) (pl -p2)
T/PI) W12(7).
(6.4
The operator W;~(T) effects a displacement along the phase-space trajectory from coordinates and momenta at the initial point to those at a point 7 later in time. The operator WIT effects the reverse displacement. If the initial time were taken at --oo instead of 0, so that initially Wrs(--00) = Wi_,‘(-co) = 1, instead of Wrs(0) = W;,‘(O) = 1 in (B.25), then we would have the displacement operators W;:(T)
=
T exp{i
i
d7’ L12[7’, PI(+) ~~47’)~ rl(T’) eMI},
--co
WE+) According 47, where A(T, trajectory.
= T exp{--i
(6.5)
i &’ L12(7’, ~1~2, w2)). -co
to (B. 19), for any dynamical PI(T) p2(7), u(7) r2(dl
= W&J
property, 4~
P1P2, w2L
(6.6)
prps, rlr2)is the value of the property at the initial point of the Accordingly, since Wrs(7) effects the reverse displacement,
W12(7) P,(T) = T expf-i In particular,
i dr’ L12(7’, prps, -co
rla))Pj(7)
for 7 -+ co, with the approximation
?.exp{ rdTt ~-(-&-
=Pj(-a).
(6.7)
(5.10) for Lrs,
- $)}p&) =pl(-m), (6.8)
Therefore in (5.19) and (5.20), if p1 = pi(m) and p = p(co) are the final momenta in a collision process governed by the hamiltonian + of (6.4), then pi = pr(-co) and p’ = p(-co) are the initial momenta: pi and p’ are initial momenta in the inverse collision5) between two molecules which results in the final momenta p1 and p.
B. LEAF
222
APPENDIX
In addition
Orthogonal set of firojection operators. fined in (2.12), construct the projectors Bi =
S dkiB(kr Wtl = j dki9(kl
where integration Pjm==
j dkj [&I
A
to the operator
9 de-
. . . kj’... klv) . . . k;...
kN)[l - S(k;-
is performed
omitting
kj)],
(A.])
the point kj = kj;
S dk&,(kI.,.kj...k&...kr&
(A4
Wml
of any pair of and so forth. Clearly 9’~~~... is symmetric to interchange subscripts. The unit operator in the Hilbert space of N particles can be partitioned in the following way: 1 = J dk’B(ki
. . . kN) + I[kl . . . klv]
. . . k;T) = B(kl
-~+;r:~~+C~~~m+~CC~~rnn+‘... I
(A.3
For a given set of values (kl . . . kN}, each projector in this partition projects onto a different part of the Hilbert space, so that the set of projectors is an orthogonal set; for example,
A hamiltonian to (A.0) as
operator
in the Hilbert
space can be expanded
according
Ho = S dk HoB(kl . . . kN) =Sdk[BHoB+~~~Hob+CC~5mHo~+...l, j
+ C BHoPj i representation
(A4
+ C x PHoPjm j
+ . ..I.
of this operator
is given [paper I, eq.
. ..pN. rr . . . TN) =
where 101 . . . ON> is the ket therefore
Ikl . . . kN) in which
H = j dkN
+ x gjHo@ I
(A4
(A-7) all kt = 0, i =
+ . ..) 101 . . . ON).
1 . . . N,
(-4.8)
,..a, ~~~+sN>~(~~...PN,s~...sN)<~I...~N(, (A-9)
BOLTZMANN
where the Fourier
EQUATION
FROM
LIOUVILLE
223
EQUATION
of H in (A.7) is
transform
h(pr . ..PN. kr . . . kN) =
(A. 10)
so that BHoP
= Ikl... kN>h(P1...&v,01...O~)
PjHoB
=
s ds Jkl....k,+
s, . . . kN>
to1 X h(pl
. ..pN.
01 . . . Sj . . .
(A. 1 1)
ON)(kl...knrj,
9J,mHoB =
j ds j ds’ Ikl . . . . kj+ s,....km+ s',...kN) LOI LOI X h(pl...pN,01...Sj...S7;2...ON)
Substitution function,
into (A.8) gives the expansion
H = h(& . ..pN. 01 . . . ON) + z
j
of the phase-space
hamiltonian
01...Sj
ds e2xis*rfk(pr . ..PN.
. . . ON)
3 ro1
+
Fez
IA
ds ,A ds’ e2rri(s’rr+S’*rm) (A. 12)
x h(pr . ..pN. 01 . . . sj . . . & . . . ON) + . . . . For the hamiltonian
given in (2.1) and (2.2)
k(pl
.-.pN,
01 ..a ON)
=
I; i
$,2/2/h
h(p1
. ..pN.
01 . . . Sj . . . ON) =
(A. 13)
for all j,
u(S),
01...sj...sh...ON)= 6(s+ d)v(s), for all j # m. h(Pl...PN. The corresponding
expansion
of the Liouville
u=SdkN[~~~+Z:83Y8+~:C~,dp~+...], 3
operator
Y is (A. 14)
j
or (A. 15) ~~=dkN[8~;48+~:88~~+C~C~~5,+...], I
ah(pl
i
9~99
. . . PN,
01 . . . ON)
aP3
ds)kl . . . . k,+s, ...kN)
= -2~ J lOI
x
h(pl
ah&
_ k 3’
a
. ..pN.
01 . . . sj . . . ON)s*---
. ..pN.
ah
01 . . . sj
aP,
. . . ON)
...&I, 1
(A.16)
B. LEAF
224
Pj,US
ds
= -2x s
lO1
LO1
x h(pl . ..pN. kj.--
X
a
ds’ Jkl . . . . k:, + s, . . . . km + s’, . . . kN) s
01 . . . sj . . . s& . . . ON) s*-
a aPm >
aPm
h(pl .., PN, 01 . . . Sj . . . Sh . . . ON)
For the hamiltonian of the present parison with (2.8), that Zi
aP5
+ s’.-
a
+ k,~-
aPI
a
problem,
using (A.13) we find on com-
= J dkN9’99, (A. 17)
APPENDIX
B
Properties of W( r ) in eq. (3.26). Time-dependent hamiltonian formalism. In (3.26), W(T) is the solution of the differential equation,
a7qT)/aT
=
-il[P]
An inverse operator w-+,
Jo
W(0) = 1.
1[9] W(T),
(B.1)
is defined by 6)
pi . . . PN, kl . . . kN) = T(‘) exp{i ; d7’ 1[9] .&(T’, pi
. . . pnr)
1[9’]),
(B.2)
0
where T(‘) is the reversed w(T)
“i+(T)
Introducing d@‘(T) we
see
=
time-ordering
=
1.
(W
w-l(T)
l[p]
A(T)
the differential
that
=
-iw(T)
PI
-..pN,
kl
T(‘) exp(-i
1[9]
(B.4)
w(T),
equation
(B.l)
A’(T),
which gives the alternative
=
so that from (3.26),
the operator
&v(T)/aT
yf(T,
operator,
can be written
W(0)
expression
for
W(T)
=
1,
as (W
as
..- kN)
i dT’ 0
J&(T’,
pl
. ..PN.
kl . . . kN)}.
W)
BOLTZMANN
EQUATION
FROM
LIOUVILLE
The inverse operator in (B.2) is the solution of the differential &Y--~(T)/% = iW-1(7) with W-i(O)
equation
1[8] &Y(T) 1[8] = i&!‘(7) W-~(T),
= 1, so that an alternative
225
EQUATION
(B.7)
to (B.2) is the expression
?cy--i(T, pi . ..pN. kl . . . kN) == T exp{i 1 dT’
J?(T’,
p1
kl . . . kN)}.
. . . PN,
(B-8)
0
The time displacement effected by the operator W(T) is a non-hamiltonian process according to (B. 1) and (3.34). But it may be remarked that if (B.l) is approximated by omitting the projector 1[P], ignoring elision of the point kl . . . kN, then ~W(T)/~T so
=
-id(T)
W(T),
(B.9)
that, from (3.28), w(T)
=
s
drN (,.I . . .
rN>
w(T)
(B.lO)
. . . rNl,
where
w(T) = T eXp(--i
id+
M(T’,
P1
. . . PN,
rl
. . . rN)},
(B.ll)
and W-~(T) = J drN
jr1
. . . rN>
wei(T)(rl
(B.12)
. . . rN1,
where = T eXp{i i dT’
w-l(7)
M’(T),
pl
. . . PN,
rl
. . . rN)},
(B. 13)
0
with M’(T)
=
w-l(T)
M(T)
(B.14)
w(T).
As noted in (3.29), &f(T) is the Liouville hamiltonian v(T) in (3.30). Likewise M’(T) ator for the hamiltonian UT, pi(~) ...PN(T). =
w-l(T)
v(T,
operator for the time-dependent in (B. 14) is the Liouville oper-
rl(7) . .. r&)1 pl
. . . pN,
rl
. . . rN)
(B. 15)
w(T).
It can be written
M’(T, pl
-9.
PN, rl . . . rN) =
M[T, pi(T) . . . PN(T), rl(T) . . . rN(T)]-
(B.16)
These results follow from general considerations for motion governed by a time-dependent hamiltonian. The equations of motion for the hamiltonian v[T, pi(T) ***PN(T), rl(T) -.-rN(T)] are
dP3(4 -----=-dT
i3V %(T)
drj(T)
aV
d7
2P, (4
-==. ’
(B.17)
B. LEAF
226
For any dynamical property A [T, PI(T) . . . IN, on the trajectory
Q(T) . . . t-N(T)] of the system
at time T,
dA j -= dT
= iM[T, PI(T) . . . PN(T), Q(T) . . . IN]
A + aA/a~.
(B. 18)
The solution of this equation is A [T, Pi(T) -. . &v(T), =
rl(T) . . . f-N(T)]
(B. 19)
A(T, p1 . . . pN, rl . . . rN),
w-l(T)
where w-l(T)
T eXp(i f ds’ M[T’, pl(T’)
=
0
. . . pN(T’),
rl(T’)
(B.20)
. . . rN(T’)].
To verify the expression (B. 11) for W(T), and hence (B. 13) for W-~(T), consider the Liouville equation for the probability density P(T, pl . . . pN, rl . . . rN) for an ensemble with time-dependent hamiltonian V(T) of (3.30), aP(T, pl . ..pN. rl . . . rN)/aT + ik!(T, p1 . . . pN,
rl
. . . rN)
P(T,
pl
. . . PN,
rl
. . . rN)
=
(B.21)
0.
The solution of this equation is rl-.. rN) =
P(T,Pl...j'N,
W(T)P(O,pl...pN,
where w(T) is given by (B.ll). p[T> h(T) =
Accordingly, dP[T,
The differential
. . . &V(T),
-iM(T,
-iw(T)
dW-l(T)/dT
JCT,
-
iw-l(T)
iM[T, hf(T,
with W(0) = W-r(O) = MIT,
P(o,pl...p~,
rl(T)
...
pl
. ..PN.
PI(T) PI(T) p1
and f’l
. . . PN(T),
ti
. . . PN(T), . . . pN,
rl
PI
governed by
7/(T),
(B.24) follow
w-l(T)
. . . rN)
w(T) (T)
. . . r%(T)], . . . IN]
W-~(T)
(B.25)
. . . rN),
1. Therefore, rl(T) ... ~N(T)I
=
pl...pN,
M(T,
(B.23)
t'N(T)]/dT = 0.
w(T)
rl... rN)
rl... rN).
PI(T) . . . PN(T), w-l(T)
W-l(T)P(T,pl...pN,
theorem results for the trajectory
equations for
dW(T)/dT =
=
w(T)
Liouville’s
PI(T)
=
Using (B.19) we obtain
'..$'N(T), 11(T) ... rN(T)] =
w-l(T)
(B.22)
rl...rN),
rl...
in agreement with (B.14) and (B.16).
IN)
w(T),
(B.26)
BOLTZMANN
EQUATION
FROM LIOUVILLE
EQUATION
227
According to (B.20), the operator W-~(T) effects a displacement of phasespace coordinates along the trajectory from the initial point to a point T later in time, the trajectory governed by V of (B.15). The operator W(T) effects the reverse displacement. This description in terms of a time-dependent hamiltonian holds only if the projector I[P] is ignored in (B.l). Relations analogous to those given here for W(T) and W-~(T), for systems with time-dependent hamiltonians, hold also in quantum mechanicss) .
REFERENCES 1) Leaf, B., Physica 58 (1972) 445. This reference is cited as paper I in the present work. 2) Leaf, B. and Schieve, W. C., Physica 36 (1967) 589; Leaf, B., J. math. Phys. 11 (1970) 1806. 3) Prigogine, I., Non-Equilibrium Statistical Mechanics, Interscience Publishers (New York, 1962). 4) Boltzmann, L., Wien. Ber. 66 (1872) 275. 5) Ter Haar, D., Elements of Statistical Mechanics, Holt, Rinehart and Winston, Inc. (New York, 1954). 6) Leaf, B., J. math. Phys. 9 (1968) 769.