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Molecular theory of Brownian motion
Physica 50 (1970) 241-258
8 North-Holland Publishing Co.,
MOLECULAR
THEORY
OF BROWNIAN
MOTION
P. MAZUR Instituut-Lore&z,
Universiteit van Leiden, Leiden, Nederland
and I. OPPENHEIM * Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass., USA
Received 15 May 1970
synopsis A molecular theory of Brownian motion is presented which starts from the Hamiltonian equations of motion for a system consisting of one heavy particle of mass M and N light particles of mass m (bath). A projection operator which averages over the bath variables is utilized. Expansions in powers of 1 = (m/M)+ are obtained for the equation of motion for the heavy particle. It is rigorously demonstrated that the Langevin equation is valid up to 0(ns) for all times. The magnitude of the momentum of the heavy particle is restricted to be of order 1-l. It is also shown using linear response theory that the friction constant appearing in the Langevin equation is identical to the friction coefficient which characterizes the drag on a macroscopic body moving with a prescribed velocity. 1. Introduction. A number of investigatorsi-5) have used a variety of techniques to obtain a description of Brownian motion from molecular considerations. Their aim has been to derive either the Langevin equation for the motion of the heavy particle or the Fokker-Planck equation for the distribution function of the heavy particle in the appropriate limits, i.e. for long times and for small values of the ratio of the masses of the bath particles, m, to the mass, M, of the Brownian particle. Expansions in powers of 1 = (m/M)+ are obtained and higher-order terms are neglected in order to reduce the exact equations to the equations describing Brownian motion. However, the time dependence of these higher-order terms has not been sufficiently investigated to justify their neglect particularly for long times. In this paper we derive a rigorous equation for the motion of a heavy particle in a bath of light particles using a projection operatore). We are able to study the time dependence of the pertinent quantities that appear * Supported in part by the U.S. National Science Foundation. 241
P. MAZUR AND I. OPPENHEIM
242
in the exact equation in sufficient detail to state that the Langevin equation is valid to @(As) for all times provided that the momentum of the heavy particle is of order il- 1. The analysis depends on the basic assumption that correlation functions of bath variables in the presence of the fixed heavy particle are short lived. In section 2, we obtain the exact equation of motion for the heavy particle. In section 3 we investigate the behavior of the correlation function which appears in the equation obtained in section 2. The behavior of this function is contrasted to that of . In section 4, the Langevin equation is obtained both in the weak coupling limit (As -+ 0, Z + co, 1st fixed) and also for small il. In section 5, we demonstrate that the friction constant occurring in the Langevin equation is identical to the friction coefficient which characterizes the drag on a macroscopic body moving with prescribed velocity through a fluid. Finally, in the appendix we discuss the behavior of the correlation function . A preliminary report of this work was presented at the International Conference on Statistical Mechanics, Kyoto 19687). 2. An exact equation of motion for a particle interacting with a many-body system. In this section, we use projection operator techniques to obtain an exact form of the equation of motion of a particle immersed in a heat bath which is particularly suited for our purpose. We consider a classical system containing N light point particles of mass m (bath) and one heavy point particle of mass M in a volume I’. The Hamiltonian of the system is given bY H=--
P2
(2.1)
2M
where P is the momentum and R the position of the heavy particle; pN and rN are 3N dimensional vectors which specify the momenta and positions of the light particles; U(rN) is the short-range potential energy of interaction between the light particles which is translationally invariant ; and $( jr(- R I) is the short-range potential of interaction between the heavy particle and the ith light particle. We shall be concerned only with large systems in which the effects of the walls can be neglected. We shall find it convenient to rewrite eq. (2.1) as H=
$
(2.2)
+ Ho,
where Ho =
pN’pN 2m
+
U(fN) + @(R rN)
(2.3)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
243
and N @
=
z i=l
4(ln
-
RI).
(2.4)
Here H0 can be looked on as the Hamiltonian
of the N light particles
in an
external potential @ created by the fixed heavy particle and depends parametrically on R. The Liouville operator for the system is given by
where
F = -P&lo
= -VII@
c_
is the force on the heavy particle iL0 = $*V,
-
(2.6)
and
F#.N(U + @)*Q
= &PHo* VP -
F&Lo
l
I$
(2.7)
determines the motion of the interacting light particles in the presence of the fixed heavy particle. We now define a projection operator 9 by the equation ggB
~
(B>
se-BHO B drNdPN
~
sepBHo drN
dpN
’
(2.8)
where B is an arbitrary dynamical variable, j3 = (kT)-1, and dr* and dpN are 3N-dimensional volume elements. We use the operator identity e(A+B)t = eAt + I eA(t-7) B e(A+B)T d7, where A = iL and B = -iPL,
(2.9)
to write
F(t) = P(t) = F+(t) +- i elLctmT) iCP_LF+(T) d7,
(2.10)
where
F(t) = eiLt F
(2.11)
is the force on the heavy particle at time t given that the force at time t i 0 is F, P(t) is the time rate of change of the momentum of the heavy particle at time t, and F+(t) G ei(r-g)Lt The quantity
iPL
iPL F+(T) =
=
F.
(2.12)
F+(T) can be written ((
;
(vp-
.I’= + F.c7p F+(T) > >
(2.13a)
-&)a
(2.13b)
P. MAZUR AND I. OPPENHEIM
244
where we have used eqs. (2.5) and (2.8) and the fact that i&Ha to obtain
= 0 eq. (2.13a).
To obtain
VR
eq. (2.13b) -
we also have used the fact that
B
(2.15)
since H depends on position only through the quantities r( - R,i = 1, . . . , N. Substitution of eq. (2.13b) into eq. (2.10) yields the exact equation of motion
P(t) = F+(t) +
J$L(t-T){~p - &)a d7.
(2.16)
0 Eq. (2.16) is the starting point for our future developments. Note that eq. (2.16) is valid whether or not the bath of light particles is in equilibrium. We shall find is particularly useful when the bath is described by a canonical ensemble with Hamiltonian Ho. In our subsequent development, it will be useful to rewrite eq. (2.16) in terms of the reduced momentum P* where (2.17)
P" =(m/M)*P ZAP. Multiplication
of eq. (2.16) by I yields t
Pa = IF+(t)
+ A2
.
d7,
(2.18)
0
where p*.pR+F*Vp. m
+X0.
(2.19)
In the next section, we shall consider the properties of the correlation function for il Q 1 and for long times. This is necessary for the discussion of the limiting behavior of the exact equation of motion, eq. (2.18). 3. The correlation function . In the next section, we shall show that eq. (2.16) reduces to the Langevin equation for small 1 and for long times under the assumption that correlation functions for bath dynamical variables, in the presence of a fixedheavy particle, are short lived. In particular, we shall study eq. (2.18) in the limit as L -+ 0, t -+ 00, il2t finite and P* finite. The condition on P* implies that the momentum P of the heavy particle is of order Ma. As a first step, in this section, we shall investigate the behavior of the kernel where F+(t) is defined by eq. (2.12), in the above limits. We use eq. (2.9) with A = iL0
(3.1)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
245
and B = i(l - 9) L - iLe = i(l -
(3.2) to obtain
where
Fe(t) = eiLotF
(3.4)
is the force on the heavy
particle
at time t, given that
the force
at time
t = 0 is F, when the heavy particle is held fixed. The P” dependence of eq. (3.3) is explicit. Because of symmetry and since the factors of P* and PP. can be moved outside of the average, the terms in eq. (3.3) containing odd numbers of factors ((P*/m) PR + F- pp.) must vanish. Thus only even powers of 1 appear in eq. (3.3). We introduce the basic assumption that correlation functions of bath variables have the property l
= for T larger than some characteristic
(3.5)
time 76. In eq. (3.5), o and 6’ stand for
a set of non-negative time dependences produced by factors of eiLoO. The quantities A and B may be functions or operators which depend on the bath dynamical variables and parametrically on R. The characteristic time Tb is determined by the Liouville operator iL0 and is therefore independent of il.
Eq. (3.5) implies that the bath is ergodic in the presence of the fixed heavy particle. We shall use eq. (3.5) to find an estimate for each term of @(An), m even, in eq. (3.3). We shall first consider the term of order 1s in eq. (3.3). This term is FerLO(r~-r~)(1
~s=Pik,ljdi.( 0 x
eiLo(~l-d
-
9)
(z__.
Pa
+
j’.
PP.
)
0 (1
_
q
_!?.pR m
+
F.Fp,
(3.6)
P. MAZUR AND I. OPPENHEIM
246
For small TO, TOI (7.21
<
76, the magnitude
of Tz is bounded by
(3.7)
127382,
value of the absolute, value of the integrand
where a2 is the maximum
in
eq. (3.6). To proceed further, we must make use of eq. (3.5). lf TO - 71 ) Tb the integrand factors and is zero both because = 0 and because of the first factor of (1 - 9’) from the left of the integrand. lf ~1 - 72 > Tb, the integrand is zero because of the second factor of (1 integrand is zero because = 0. Thus for Tb 5 TO I of T2 is bounded by 17-21 <
For 72
TO <
>
Tb
2
2
9). 3Tb,
lf 72 > Tb the the magnitude
2 Tba2.
(3.8)
376, at kaSt OIlI? Of the inequalities 70 71 > Tb, 71 must hold for every pair of values of 71 and 72. Thus,
T2 = 0,
70 >
-
72
>
Tb,
3Tb.
Or
(3.9)
The arguments presented above ;12n in eq. (3.3). This term is
are easily
extended
to the term of order
(3.10) We now have, for lT2nl I
where
Tb
(3.11)
the maximum For Tb 2 TO 5
a2n
value of the absolute
is
eq. (3.10). IT24
Finally,
TO 5
A2n 702n U2n>
5
for
Azn
(%Z
+
value of the integrand
(3.12)
7322s.
(2% +
TO >
1)
Tb,
(3.13)
Tzn = 0 Substitution
of eqs. (3.10)-(3.13)
i@F+(~o)>
-
=
(nTb)2
a2R2(j1Tb)
and 1l < (h)2” 3
in
1) Tb,
(1Tb)2n
@2nR2&Tb)
I
into eq. (3.3) leads to the conclusions m U2(n+1) (hb)2 U2 2 (i1Tb)2np n=O U2 (3.14)
Tb>TO>o
co U2~ax i=o
(nTb)2i
U2(n+i)
-___
U2n (3.15)
MOLECULAR
OF BROWNIAN
THEORY
MOTION
247
for (2?b+
l)Tb>Ta>(2%-
l)Tb
We make the reasonable
assumption
n2
1.
that the series in eqs. (3.14) and (3.15)
converge. function In the appendix we discuss the behavior of the correlation @F(t)) with F(t) = eiLtF. We show there (eq. (A.21)) that (FF(To)) is given bY
=
-
(m/b)
Y212
1 exp{-yT0~2}
+
o(14
(n2T0)‘),
(3.16) where
I is the unit tensor and y is defined by
y = (B/3m) ~(~&(T)>
(3.17)
d7.
It is obvious that there is a significant difference in behaviour between 0 2
g
dTo
j j
-
jhO(To)> 0
(TO)>
71) (ATb)2
-
a2R2
dTo1
@Fo
(ATb)
(TO)>1 dTo
+
2Tb
;
(lTb)2n
a2nR2n(ATb),
(3.18)
n=l
where (2% + 1) eq. (3.18) yields lim lim i t-m 1-o 0
Tb
>
t 2
(2%’
-
1) Tb.
In the limit as 1 -+ 0 and then t --f co,
(3.19)
dTa = (m/p) y I,
(FF+(TO)>
where we have used eq. (3.17). On the other hand, it also follows from eq. (3.18) that in the limit as t --f co and then il. -+ 0 that lim lim i
(TO)>
(3.20)
dTa = (ml/?) y I,
where we have used the fact that %(n+i) (nTb)2 R2(,+1,
_
1 _
aen&
1
<
1
(3.21)
R2n
for all n so that the series in eq. (3.18) converge as n’ --f CO. Finally, in the weak coupling limit as Ls --f 0, t --f CO, A2t fixed, we also obtain j
(m/b) Y 1.
(3.22)
P. MAZUR AND I. OPPENHEIM
248
Next we consider the integral from 0 to t of eq. (3.16) : i @F(To)) -
dTo = k dTo
(m/p) y212 1~eFyToA2dT0 + ;o(n4
(A2 70)~)
dT0.
(3.23)
0
In the first limit (A -+ 0, then t --f co) we again obtain lim lim s” t+w 1-o 0 However, i 0
dTe = (m//I) y I.
in the weak coupling
-+
(3.24)
limit, eq. (3.23) becomes
(m/B)y I - (m/p) y I{ 1 - e-“}
= (m/b) y 1 eC”, (3.25)
where s = A2t. The limit t --f co, then il --f 0, cannot be performed here because of lack of knowledge the term of @(A4 (n2T0)“). The analysis above demonstrates the power of the projection operator formalism when. detailed estimates of its effects can be carried through. In most previous treatments involving projection operators, it has been tacitly assumed, without investigation, that kernels of the form @F+(t)> were properly shortlived. In these treatments only the first term on the right-handside of eq. (3.3) was retained and the remainder was considered to be of lower order in the weak-coupling parameter without investigating its long-time behavior. On the basis of such an argument, however, one would obtain the same behavior for have the properties usually assumed depends critically on the physical system, the function F, and the projection operator 8. That the naive approach may lead to incorrect results has been discussed elsewheres). 4. The Lange&n equation. In this section we use the results of the last section to obtain the Langevin equation for the motion of the heavy particle. We first rewrite eq. (2.18) in the form t i’*(t)
-
IF+(t) + Pg.
s
d7
0
.{
--f&-
t -12i
7
)> * s
-;i*(t
0
+ mkT
7’)
-
) dT
d+ = A + B,
(4.1)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
249
where we have used the identity P*(t
-
T) = P*(t)
+ fi*
(t + T’) d#.
(4.2)
0
We can now present estimates of the orders of magnitude of the terms on the right-hand side of eq. (4.1). The first term on the right-hand side can be
written
cc--&
-
*vlf > Fo(4 >I dT-
F*Vp*
The difference of correlation functions in eq. (4.3) behaves in the same way as the difference in eq. (3.14) and (3.15). Thus the magnitude of A is bounded by an expression similar to that in eq. (3.18) where we must assume that P*(T), 0 < T I t, is bounded. The second term on the right-hand side of eq. (4.1) is then of order j1s since @Fo(T)> is zero for T > Tb. Thus we can write t
P*(t)
-
IF+(t)
+
p* mkT’
12
(4
d7 = @(13)
(4.4)
I 0
for all t. Eq. (4.4) is valid independent of the state of the bath. The state of the bath is relevant only when averages over the bath variables are taken. We introduce the time scale s = L2t to rewrite eq. (4.4) as dZI*(s) ___ ds
S
n*(s)
-E(s)+-&j?
s
do r
(4.5)
12
0
where n*(S)
3 P*(s/P),
E(s) =
(4.6)
F+(s/P) ;z ,
(4.7)
and o = 12T. In the weak coupling (4.5) b ecomes dlI* (s) ds
-
E(s) + yn*(S)
= 0,
limit
(A
zzc
s fixed),
eq.
(4.8)
where we have used eq. (3.17). Eq. (4.8) is the Langevin equation provided that E(s) has the proper stochastic properties for a bath in equilibrium. The
250
P. MAZUR
AND
I. OPPENHEIM
stochastic properties of E(s) are easily F+(t). First we show that
determined
from the properties
= 0, and therefore (F(s)>
of
(4.9)
that
= 0.
(4.10)
Eq. (4.9) follows from the fact that
$
= 0 and that
9) LF+(t)) = 0.
Next we study. This quantity
(4.11)
is in the weak coupling limit
--f lim
12
Ai0
(4.12)
*
Eq. (4.12) is zero, except for s = 0, which follows from eq. (3.15). That the object in eq. (4.12) can be represented by a delta function follows from eqs. (3.18) and (3.22). Thus we can write (4.13)
-+ 2(m//3)yl d(s). of
The value
in the weak coupling
(F+(s#) ... E(s2n+1)>
= lim
follows from
F+(s2n+1/A2)>
;12n+l
L-t0 z
...
--f lim 1-O
limit
(4.14)
0,
where the first equality follows from considerations similar to those in section 3 and the second from the symmetry of the system. Furthermore, one obtains in the same way
-+ lim
lim
2n
A+0
icf 1
Fo(s2n/12)>
12n
A+0
=
..-
Fo(sG2) > 12
where the sum is taken over all the different the 2n times sr, . . . , szn into n pairs and lim P
A+0
Eqs.
(4.14),
(4.15)
and (4.16)
= 2F
(4.15)
)
ways in which one can divide
(4.16)
yl S(S$ -- Si).
determine
the stochastic
properties
of E(s)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
251
and are identical
to those of the stochastic force in the phenomenological in the phenomenological The stochastic averages Langevin equation. Langevin equation are determined by statistical averages over the bath variables. Eq. (4.8) implies that, for a heavy particle with momentum P and mass M (M > m) the equation of motion is, for sufficiently long times (t >
Tb),
dP(t) =
-
dt
+ F+(t),
(4.17)
dr.
(4.18)
-PyP(t)
where Py
= &
s
0
In the derivation of eq. (4.17) we have imposed the condition that P(t) is at most of order (M/m)*. It is also clear from eq. (4.4) that if P is too small i.e. of order 1 or less, the term on the right-hand side of the equation may dominate the last term on the left-hand side. This implies that the very long-time behavior of P(t), t > (A‘+)-1, is not necessarily properly described by eq. (4.17). The relevant results for Brownian motion, i.e. the behavior of averages of powers of P(t), which are usually obtained from the Langevin equation, eq. (4.8), and the properties of E(s), can also be obtained directly from eq. (4.4) without use of the weak coupling limit. This is possible because we have shown that the right-hand side of eq. (4.4) is of order ;1s for all times; this does not imply that this term is independent of time but merely that it is bounded for all t. We shall illustrate this point by computing, from eq. (4.4). First, we rewrite eq. (4.4) as P(t) -
F+(t) + Py(t)
(4.19)
P(t) = O(i12,P*),
where t (4.20)
and y(t) = y for t > Tb. We have written P* explicitly on the right-hand side of eq. (4.19) to indicate that the ordering in 1 is valid when P* is 0( 1). The solution of eq. (4.19) is P(t) = exp(--i2iy(r) X (F+
Again
(T’)
we note that
+
dT} P(0) + / eXP{-A2jty(T) o(A2,
P*)}
eq. (4.21)
dT}
d+. is valid
(4.21) no matter
whether
the bath
is in
P. MAZUR
252
AND
I. OPPENHEIM
equilibrium or not. The averages discussed below, however, are taken assuming the bath to be in equilibrium with respect to the fixed Brownian particle. Taking the average of eq. (4.21) yields
i?(7) d7) P(0)
0
(4.22)
+ 6(1, P*),
where P(0) is @(A-r). For t 2 76, eq. (4.22) reduces to = exp{-A2yt)
P(0) + U(1, P*).
(4.23)
For times t such that Tb I t ( Z(L2y)-1, where I is a small integer, eq. (4.23) reduces to the usual Brownian motion result
= exp{--A2yt}
(4.24)
P(0).
For very long times, t > Z(lsy)-1, however, the last term on the right-hand side of eq. (4.23) may dominate. This term, however, is small compared to (AM”)* = 0 (A-r) w h’ICh is a measure of the equilibrium fluctuations in P. The average of is obtained from eq. (4.21) as
= exp{-2A2 + idr’
{y (T) dT) P(0) P(0)
0
I’dT” exp(-As
iy(~)
x + [exp{-A2
dT -
1s iy(~)
d7)
i Y(T) d+ P (0) + 13 0( 1, P*), (4.25) 0
where we have used eq. (3.14). For t > Tb, eq. (4.25) becomes
= exp{-221s it) P(0) P(0) + MkT I{ 1 + {e+yt
e--2A8ytj (4.26)
P(0) + 1) O(1, P*),
where we have used eq. (3.17). For times t such that 76 < t < Z(A2y)-l eq. (4.26) reduces to the usual Brownian motion result
= exp{-21syt)
P(0)P(0)+ MkT I{1 - e-2A*yt),
where we have neglected terms of order 1-r compared For very long times, t > Z(Lsy)-1, eq. (4.26) becomes
(4.27)
to terms of order Ad2.
(4.28)
we have neglected terms of 0( 1) compared to terms of 0(A-2). Eq. gives the equilibrium value for. procedure above can be extended to compute the average values of powers of P(t).
MOLECULAR
5. Friction that
THEORY
OF BROWNIAN
constant for a macroscopic
expression
(4.18)
for the friction
MOTION
253
body. In this section we shall show constant
occurring
in the Langevin
equation describing the stochastic motion of a heavy particle is identical to the friction coefficient, which characterizes the drag on a macroscopic body of mass M moving through the fluid with prescribed slowly varying velocity Ve(t). Consider a system body with position this system is
of N point particles &(t)
and velocity
of mass m in interaction
Ve(t) = A&).
Ro@)) = Bo(rN, PN) + igm
Q’?PN, where fJo(rN,pN)
=
v
with a
The Hamiltonian
- Ro@)I),
for
(5.1)
+utrN)j
U(rN) is the short-range potential of interaction between the point particles which is translationally invariant; and 4(lrr - Ra(t) I) is the short-range potential of interaction between the body and the ith bath particle. We now define a generating function for a time-dependent canonical transformation by N G
=
Z i=l
(Q
-
Roti))
.Pt,
w
where PC is the transformed equations are
i3G
Rc= - api
= rz -
momentum
$
(5.4
vector of particle i. belonging to the new
= Ao(RN, PN) + 2 +(IRtj) i
The new Hamiltonian
i. The transformation
Ro(t),
with Rt the transformed position The transformed Hamiltonian momenta is H = A+
of particle
coordinates
2I Pi* Vo@). i
and
(5.6)
H is of the form
H = Ho(RN, PN) - z Pt. V,(t), i
(5.7)
where Ho(RN, PN) = Bo(RN,
PN) + z +(IRfl)
i
(5.8)
P. MAZUR
254
AND
I. OPPENHEIM
is the Hamiltonian of the N bath particles in an external potential by the fixed body. It is therefore identical with the Hamiltonian
created defined
in eq. (2.3). We can now compute the linear response of any bath dynamical function to the prescribed motion of the body. We shall assume that at t --f - 00, the body was at rest, and the bath in equilibrium distribution function of the bath is given by
f (RN,
PN) = e--BHo/f dRN dPN eePHo.
with it. Thus at t + -00,
the
(5.9)
In particular we are interested in the average value of the force exerted by the bath on the body to terms linear in Vo(t). The force on the body is given by (see eq. (2.6)) (5.10) By standard
methods
we obtain
~@‘Fo(T)>. Vo(t - 7) d7,
t = - ,8
(5.11)
0
where the symbol < >t denotes the average value at time t, the symbol < > is defined in eq. (2.8), and Pa(t) is defined by eq. (3.4). If Vo(t) varies sufficiently slowly (5.7) becomes 0%
= -
j3 r
d7. Vo(t)
0 =
-qlPo(t),
(5.12)
where Pa(t) = M Vo(t), and where Asy is given by eq. (4.16). Comparison of eq. (5.12) with eq. (4.15) demonstrates the equivalence of the friction constant occurring in the Langevin equation and the friction constant characterizing the drag on a macroscopic body. This equivalence has also been derived by Lebowitz and RCsiboisa) by a different method. 6. Remarks. The techniques utilized here have enabled us to derive a simple equation of motion for the heavy particle which is valid to order A2 for all times. For t > 70, this equation reduces to the LangeVin equation. We have restricted our considerations to situations in which the magnitude of the reduced momentum, P*, of the Brownian particle is of the same order in A, i.e. La, as the magnitude of the momentum, p, of a bath particle. The Langevin equation, however, is valid for a much larger range of values of P*. In fact, it follows from eqs. (4.1) and (3.3) that the left-hand side of eq. (4.1) is of lower order in I than the right-hand side as long as O(P*) - o(~&p), - 1 < e. If E I - 1, the expansion in eq. (3.3) is not useful.
MOLECULAR
THEORY
OF BROWNIAN
It is possible to extend our considerations
MOTION
255
to treat Brownian
motion in an
external field of force and Brownian motion of particles with internal structure. The equations obtained are the phenomenological equations which have been discussed e.g. by Chandrasekhara) and by Pragerra), respectively. The equations for nonspherical-Brownian particles exhibit a coupling between the various components of the center-of-mass momentum and the angular momentum. Acknowledgement. hospitality extended
The authors over the years.
thank
the reciprocal
institutions
for
APPENDIX
In this appendix, we study the where F(t) is defined by eq. presented here is considerably more section 3 for . We use eq. (2.9) with A = iL.0, eq. B = i(L eq. (2.19),
behavior of the correlation function (2.11). We shall see that the analysis complicated than that presented in (2.77, and
La) = n((P*/m) - FR + F-V,.>,
to obtain
@F(t)> = + 1 i dr,
(A. 1)
0
where Fe(t) is defined by eq. (3.4). Iteration of eq. (A.l) results in
; n=l
x
7”-1
1%TdTi jld?s . . . j dTn 0
0
0
x eiLO(k-l-d (01 + 02) eiLoTn F), where t has been replaced
(A4
by TO,
and 0s = F-V,,.
(A.9
The P* dependence of eq. (A.2) is explicit and resides completely in the operator 01 and 0s. Because of symmetry and since the factors of P* and Vp. can be moved outside of the average, the terms in eq. (A.2) containing
P. MAZUR
256
AND
I. OPPENHEIM
odd numbers of factors (01 + 0s) must vanish. It is also clear that only those terms with proper ordering of the operators Or and 0s can survive. In particular, each 0s must have at least one more 01 to its right than it has 02’s to its right. Thus, there can be at most rt/2 0s operators in each term of eq. (A.2) and the last factor of 01 + 0s on the right reduces to 01. It is clear that for short times can be approximated by
eih(TJ-l-TJ)
o1
eih(TJ-TJ+l)
the
B(a’)>. (-4.5)
Since we need consider only those correlations in eq. (A.2) which contain an even number of factors 02, i = 1 or 2, there are two cases of eq. (A.5) that will be of interest. In the first case, A contains an even number of factors of 02 and B an odd number of Og. In this case, application of (3.5) leads to ci =
(A (a)) (01 eiLa(rJ-‘j+l) B(d)> [(B(o’)>
= 0 = 0
73'>
7b
(A.6a)
Tj+l >
Tb.
(A.6b)
Tj-1 -
Tj -
The result for eq. (A.6a) is zero because A contains an odd number of factors of 0~ and F. The result for eq. (A.6b) is zero because
c
= 1
[
eiLo(7J-1-7f) Ol>(B(o’)>
Tj-1 = 0
ri -
Tj >
Tj+l
>
76
(A.7a)
76.
(A.i’b)
The result for eq. (A.7a) need not be zero since A is odd in 0s and contains one F. The result for eq. (A.7b) is still zero for the same reason that (A.6b) is zero and also because B is even in Og and contains one F. Next, we consider the correlation function C2 = (/I@)
eih(TJ-l-TJ)
o2
&-h(5J-7J+1)
B(a’)>. (A.4
With A even in 02 and B odd, application cs =
<02 eiLa(rJ-rJ+l) B(d)>
of eq. (3.5) leads to
= 0
[
Tj-1 -
75 >
Tj - Tj+l >
76 Tb.
(A.9a) (A.9b)
The result for eq. (A.9a) is zero for the same reason as eq. (A.6a). The result for eq. (A.9b) need not be zero. For A odd in Or and B even, C2 becomes cs =
<02 eiLoCTj-‘j+l)B(d)> [ = 0
TJ-1 - Tj >
rb
(A. 1Oa)
73 - Tj+l >
Tb,
(A. lob)
where (A. 1Oa) need not be zero and (A. lob) is zero since B contains an even of Oa and one F.
number of factors
MOLECULAR
To summarize
THEORY
the results
OF BROWNIAN
of eqs. (A.6)-(A.lO),
times TJ-1, ~5, T~+I mzlst be close together occurs with an even number of factors will be of lower order in TO for TO > Tb each n in eq. (A.2). Because of this and is limited the leading term of order PZin An
fdT1(dT2...y-i+,
o2
we note that
257
the three
for a nonzero result only when Or
of 02 to its left. Thus, these terms than the other cases considered for because the number of factors of 0s eq. (A.2) is of the form eiL~(~~-~a)
O2
O1
0
0
...
MOTION
eiLo(sn-t-~n)
o1
eua+nF),
(A. 11)
with alternating factors of Or and Oz. Here and below we always assume To > Tb. In order to obtain this result we start our considerations on the right. The first 01 must be Or since there are no factors of P* to its right. The second Oa must be 0s or else eqs. (A.6) would apply. The third 0~ must be Or because there are no free P* factors to its right. The argument proceeds in this fashion until the last Oa on the left. The leading contribution to eq. (A.1 1) is obtained by allowing the largest possible lattitude to the time variables consistent with retaining a nonzero integrand. For most of the range of integration, the integrand of eq. (A. 11) breaks up, according to eq. (3.5), into
eiL~(~e4-~~-~l
02)
03) ~01
. . .
03)
eiLOra F>.
(A. 12)
Substitution (2.6) yields
of eqs. (A.3) and (A.4) into eq. (A. 12) and use of eqs. (2.15) and
(-
- 73)) . . .
p/m)*n
-
Tn-1))’
(A.13)
.
At this point we use the fact that, for reasons of symmetry, (A. 14)
= & 1, where I is the unit tensor. eq. (A.2) now becomes
The dominant
contribution
to the nth term in
(n-W2
x{
,po.
723+1)>)
(%)>
1,
where we recall that n is even. The dominant contribution easily obtained using eq. (3.5) and the fact that To > Tb.
(A. 15) to eq. (A. 15) is We use the fact
258
MOLECULAR
THEORY
OF BROWNIAN
MOTION
that oj
=~(F.F,,(O))
do 3 01,
T 2 TZ,(see (3.5)). The time integrations
(A. 16) in (A. 15) result in (A. 17)
for n = 2 and (A.18) for n > 2. Substitution
of (A.17) and (A.18) into eq. (A.2) yields
I = (-yn)P)k IX k=l k!
I1 + ‘($)}-jJ
(A. 19)
where we have used eq. (A.14) and y is defined by P y =3mOL. The terms of ~(Q/To) in eq. (A.19) are a proper estimate of all the approximations made in proceeding from eq. (A.2) to eq. (A.19). The summation of the leading terms in eq. (A. 19) can be easily performed to yield = Q
TO)k
I -
(ml/?) y2 I2 I exp{--yT0L2} (A.21)
A4 Tb),
where k is some arbitrary
integer.
REFERENCES
1) Lebowitz, J. and Rubin, E., Phys. Rev. 131 (1963) 2381.
2) 3) 4) 5) 6) 7) 8) 9) 10)
Resibois, P. and Davis, R., Physica 30 (1964) 1077. Lebowitz, J. and Resibois, P., Phys. Rev. 139A (1965) 1101. Mori, H., Progr. theor. Phys. 33 (1965) 423. Kubo, R., Repts. Progr. Phys, 24 (1966) part 1 255. Zwanzig, R., in Lectures in Theoretical Physics, edited by W. E. Brittin, B. W. Downs and J. Downs, Interscience Publ., Inc. (New York, 1961) vol. III. 106. Mazur, P. and Oppenheim, I., J. Phys. Sot. Japan 26 (1969) suppl. 35. Terwiel, R. and Mazur, P., Physica 32 (1966) 1813; 36 (1967) 289; Resibois, P., Brocas, J. and Decan, G., J. math. Phys. 10 (1969) 964. Chandrasekhar, S., Rev. mod. Phys. 15 (1943) 1. Prager, S., J. them. Phys. 23 (1955) 2404.
8 North-Holland Publishing Co.,
MOLECULAR
THEORY
OF BROWNIAN
MOTION
P. MAZUR Instituut-Lore&z,
Universiteit van Leiden, Leiden, Nederland
and I. OPPENHEIM * Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass., USA
Received 15 May 1970
synopsis A molecular theory of Brownian motion is presented which starts from the Hamiltonian equations of motion for a system consisting of one heavy particle of mass M and N light particles of mass m (bath). A projection operator which averages over the bath variables is utilized. Expansions in powers of 1 = (m/M)+ are obtained for the equation of motion for the heavy particle. It is rigorously demonstrated that the Langevin equation is valid up to 0(ns) for all times. The magnitude of the momentum of the heavy particle is restricted to be of order 1-l. It is also shown using linear response theory that the friction constant appearing in the Langevin equation is identical to the friction coefficient which characterizes the drag on a macroscopic body moving with a prescribed velocity. 1. Introduction. A number of investigatorsi-5) have used a variety of techniques to obtain a description of Brownian motion from molecular considerations. Their aim has been to derive either the Langevin equation for the motion of the heavy particle or the Fokker-Planck equation for the distribution function of the heavy particle in the appropriate limits, i.e. for long times and for small values of the ratio of the masses of the bath particles, m, to the mass, M, of the Brownian particle. Expansions in powers of 1 = (m/M)+ are obtained and higher-order terms are neglected in order to reduce the exact equations to the equations describing Brownian motion. However, the time dependence of these higher-order terms has not been sufficiently investigated to justify their neglect particularly for long times. In this paper we derive a rigorous equation for the motion of a heavy particle in a bath of light particles using a projection operatore). We are able to study the time dependence of the pertinent quantities that appear * Supported in part by the U.S. National Science Foundation. 241
P. MAZUR AND I. OPPENHEIM
242
in the exact equation in sufficient detail to state that the Langevin equation is valid to @(As) for all times provided that the momentum of the heavy particle is of order il- 1. The analysis depends on the basic assumption that correlation functions of bath variables in the presence of the fixed heavy particle are short lived. In section 2, we obtain the exact equation of motion for the heavy particle. In section 3 we investigate the behavior of the correlation function
P2
(2.1)
2M
where P is the momentum and R the position of the heavy particle; pN and rN are 3N dimensional vectors which specify the momenta and positions of the light particles; U(rN) is the short-range potential energy of interaction between the light particles which is translationally invariant ; and $( jr(- R I) is the short-range potential of interaction between the heavy particle and the ith light particle. We shall be concerned only with large systems in which the effects of the walls can be neglected. We shall find it convenient to rewrite eq. (2.1) as H=
$
(2.2)
+ Ho,
where Ho =
pN’pN 2m
+
U(fN) + @(R rN)
(2.3)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
243
and N @
=
z i=l
4(ln
-
RI).
(2.4)
Here H0 can be looked on as the Hamiltonian
of the N light particles
in an
external potential @ created by the fixed heavy particle and depends parametrically on R. The Liouville operator for the system is given by
where
F = -P&lo
= -VII@
c_
is the force on the heavy particle iL0 = $*V,
-
(2.6)
and
F#.N(U + @)*Q
= &PHo* VP -
F&Lo
l
I$
(2.7)
determines the motion of the interacting light particles in the presence of the fixed heavy particle. We now define a projection operator 9 by the equation ggB
~
(B>
se-BHO B drNdPN
~
sepBHo drN
dpN
’
(2.8)
where B is an arbitrary dynamical variable, j3 = (kT)-1, and dr* and dpN are 3N-dimensional volume elements. We use the operator identity e(A+B)t = eAt + I eA(t-7) B e(A+B)T d7, where A = iL and B = -iPL,
(2.9)
to write
F(t) = P(t) = F+(t) +- i elLctmT) iCP_LF+(T) d7,
(2.10)
where
F(t) = eiLt F
(2.11)
is the force on the heavy particle at time t given that the force at time t i 0 is F, P(t) is the time rate of change of the momentum of the heavy particle at time t, and F+(t) G ei(r-g)Lt The quantity
iPL
iPL F+(T) =
=
F.
(2.12)
F+(T) can be written ((
;
(vp-
.I’= + F.c7p F+(T) > >
(2.13a)
-&)a
(2.13b)
P. MAZUR AND I. OPPENHEIM
244
where we have used eqs. (2.5) and (2.8) and the fact that i&Ha to obtain
= 0 eq. (2.13a).
To obtain
VR
eq. (2.13b) -
we also have used the fact that
B
(2.15)
since H depends on position only through the quantities r( - R,i = 1, . . . , N. Substitution of eq. (2.13b) into eq. (2.10) yields the exact equation of motion
P(t) = F+(t) +
J$L(t-T){~p - &)a
(2.16)
0 Eq. (2.16) is the starting point for our future developments. Note that eq. (2.16) is valid whether or not the bath of light particles is in equilibrium. We shall find is particularly useful when the bath is described by a canonical ensemble with Hamiltonian Ho. In our subsequent development, it will be useful to rewrite eq. (2.16) in terms of the reduced momentum P* where (2.17)
P" =(m/M)*P ZAP. Multiplication
of eq. (2.16) by I yields t
Pa = IF+(t)
+ A2
.
d7,
(2.18)
0
where p*.pR+F*Vp. m
+X0.
(2.19)
In the next section, we shall consider the properties of the correlation function
(3.1)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
245
and B = i(l - 9) L - iLe = i(l -
(3.2) to obtain
where
Fe(t) = eiLotF
(3.4)
is the force on the heavy
particle
at time t, given that
the force
at time
t = 0 is F, when the heavy particle is held fixed. The P” dependence of eq. (3.3) is explicit. Because of symmetry and since the factors of P* and PP. can be moved outside of the average, the terms in eq. (3.3) containing odd numbers of factors ((P*/m) PR + F- pp.) must vanish. Thus only even powers of 1 appear in eq. (3.3). We introduce the basic assumption that correlation functions of bath variables have the property l
(3.5)
time 76. In eq. (3.5), o and 6’ stand for
a set of non-negative time dependences produced by factors of eiLoO. The quantities A and B may be functions or operators which depend on the bath dynamical variables and parametrically on R. The characteristic time Tb is determined by the Liouville operator iL0 and is therefore independent of il.
Eq. (3.5) implies that the bath is ergodic in the presence of the fixed heavy particle. We shall use eq. (3.5) to find an estimate for each term of @(An), m even, in eq. (3.3). We shall first consider the term of order 1s in eq. (3.3). This term is FerLO(r~-r~)(1
~s=Pik,ljdi.( 0 x
eiLo(~l-d
-
9)
(z__.
Pa
+
j’.
PP.
)
0 (1
_
q
_!?.pR m
+
F.Fp,
(3.6)
P. MAZUR AND I. OPPENHEIM
246
For small TO, TOI (7.21
<
76, the magnitude
of Tz is bounded by
(3.7)
127382,
value of the absolute, value of the integrand
where a2 is the maximum
in
eq. (3.6). To proceed further, we must make use of eq. (3.5). lf TO - 71 ) Tb the integrand factors and is zero both because
For 72
TO <
>
Tb
2
2
9). 3Tb,
lf 72 > Tb the the magnitude
2 Tba2.
(3.8)
376, at kaSt OIlI? Of the inequalities 70 71 > Tb, 71 must hold for every pair of values of 71 and 72. Thus,
T2 = 0,
70 >
-
72
>
Tb,
3Tb.
Or
(3.9)
The arguments presented above ;12n in eq. (3.3). This term is
are easily
extended
to the term of order
(3.10) We now have, for lT2nl I
where
Tb
(3.11)
the maximum For Tb 2 TO 5
a2n
value of the absolute
is
eq. (3.10). IT24
Finally,
TO 5
A2n 702n U2n>
5
for
Azn
(%Z
+
value of the integrand
(3.12)
7322s.
(2% +
TO >
1)
Tb,
(3.13)
Tzn = 0 Substitution
of eqs. (3.10)-(3.13)
i@F+(~o)>
-
=
(nTb)2
a2R2(j1Tb)
and 1
in
1) Tb,
(1Tb)2n
@2nR2&Tb)
I
into eq. (3.3) leads to the conclusions m U2(n+1) (hb)2 U2 2 (i1Tb)2np n=O U2 (3.14)
Tb>TO>o
co U2~ax i=o
(nTb)2i
U2(n+i)
-___
U2n (3.15)
MOLECULAR
OF BROWNIAN
THEORY
MOTION
247
for (2?b+
l)Tb>Ta>(2%-
l)Tb
We make the reasonable
assumption
n2
1.
that the series in eqs. (3.14) and (3.15)
converge. function In the appendix we discuss the behavior of the correlation @F(t)) with F(t) = eiLtF. We show there (eq. (A.21)) that (FF(To)) is given bY
=
-
(m/b)
Y212
1 exp{-yT0~2}
+
o(14
(n2T0)‘),
(3.16) where
I is the unit tensor and y is defined by
y = (B/3m) ~(~&(T)>
(3.17)
d7.
It is obvious that there is a significant difference in behaviour between
g
dTo
j j
-
jhO(To)> 0
(TO)>
71) (ATb)2
-
a2R2
dTo1
@Fo
(ATb)
(TO)>1 dTo
+
2Tb
;
(lTb)2n
a2nR2n(ATb),
(3.18)
n=l
where (2% + 1) eq. (3.18) yields lim lim i t-m 1-o 0
Tb
>
t 2
(2%’
-
1) Tb.
In the limit as 1 -+ 0 and then t --f co,
(3.19)
dTa = (m/p) y I,
(FF+(TO)>
where we have used eq. (3.17). On the other hand, it also follows from eq. (3.18) that in the limit as t --f co and then il. -+ 0 that lim lim i
(TO)>
(3.20)
dTa = (ml/?) y I,
where we have used the fact that %(n+i) (nTb)2 R2(,+1,
_
1 _
aen&
1
<
1
(3.21)
R2n
for all n so that the series in eq. (3.18) converge as n’ --f CO. Finally, in the weak coupling limit as Ls --f 0, t --f CO, A2t fixed, we also obtain j
(m/b) Y 1.
(3.22)
P. MAZUR AND I. OPPENHEIM
248
Next we consider the integral from 0 to t of eq. (3.16) : i @F(To)) -
dTo = k
(m/p) y212 1~eFyToA2dT0 + ;o(n4
(A2 70)~)
dT0.
(3.23)
0
In the first limit (A -+ 0, then t --f co) we again obtain lim lim s”
dTe = (m//I) y I.
in the weak coupling
-+
(3.24)
limit, eq. (3.23) becomes
(m/B)y I - (m/p) y I{ 1 - e-“}
= (m/b) y 1 eC”, (3.25)
where s = A2t. The limit t --f co, then il --f 0, cannot be performed here because of lack of knowledge the term of @(A4 (n2T0)“). The analysis above demonstrates the power of the projection operator formalism when. detailed estimates of its effects can be carried through. In most previous treatments involving projection operators, it has been tacitly assumed, without investigation, that kernels of the form @F+(t)> were properly shortlived. In these treatments only the first term on the right-handside of eq. (3.3) was retained and the remainder was considered to be of lower order in the weak-coupling parameter without investigating its long-time behavior. On the basis of such an argument, however, one would obtain the same behavior for
-
IF+(t) + Pg.
s
d7
0
.{
--f&-
t -12i
7
)> * s
-;i*(t
0
+ mkT
7’)
-
d+ = A + B,
(4.1)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
249
where we have used the identity P*(t
-
T) = P*(t)
+ fi*
(t + T’) d#.
(4.2)
0
We can now present estimates of the orders of magnitude of the terms on the right-hand side of eq. (4.1). The first term on the right-hand side can be
written
cc--&
-
*vlf > Fo(4 >I dT-
F*Vp*
The difference of correlation functions in eq. (4.3) behaves in the same way as the difference in eq. (3.14) and (3.15). Thus the magnitude of A is bounded by an expression similar to that in eq. (3.18) where we must assume that P*(T), 0 < T I t, is bounded. The second term on the right-hand side of eq. (4.1) is then of order j1s since @Fo(T)> is zero for T > Tb. Thus we can write t
P*(t)
-
IF+(t)
+
p* mkT’
12
(4
d7 = @(13)
(4.4)
I 0
for all t. Eq. (4.4) is valid independent of the state of the bath. The state of the bath is relevant only when averages over the bath variables are taken. We introduce the time scale s = L2t to rewrite eq. (4.4) as dZI*(s) ___ ds
S
n*(s)
-E(s)+-&j?
s
do r
(4.5)
12
0
where n*(S)
3 P*(s/P),
E(s) =
(4.6)
F+(s/P) ;z ,
(4.7)
and o = 12T. In the weak coupling (4.5) b ecomes dlI* (s) ds
-
E(s) + yn*(S)
= 0,
limit
(A
zzc
s fixed),
eq.
(4.8)
where we have used eq. (3.17). Eq. (4.8) is the Langevin equation provided that E(s) has the proper stochastic properties for a bath in equilibrium. The
250
P. MAZUR
AND
I. OPPENHEIM
stochastic properties of E(s) are easily F+(t). First we show that
determined
from the properties
of
(4.9)
that
= 0.
(4.10)
Eq. (4.9) follows from the fact that
$
9) LF+(t)) = 0.
Next we study
(4.11)
is in the weak coupling limit
12
Ai0
(4.12)
*
Eq. (4.12) is zero, except for s = 0, which follows from eq. (3.15). That the object in eq. (4.12) can be represented by a delta function follows from eqs. (3.18) and (3.22). Thus we can write (4.13)
The value
in the weak coupling
(F+(s#) ... E(s2n+1)>
= lim
follows from
F+(s2n+1/A2)>
;12n+l
L-t0 z
...
--f lim 1-O
limit
(4.14)
0,
where the first equality follows from considerations similar to those in section 3 and the second from the symmetry of the system. Furthermore, one obtains in the same way
lim
2n
A+0
icf 1
Fo(s2n/12)>
12n
A+0
=
..-
Fo(sG2) > 12
where the sum is taken over all the different the 2n times sr, . . . , szn into n pairs and lim
A+0
Eqs.
(4.14),
(4.15)
and (4.16)
= 2F
(4.15)
)
ways in which one can divide
(4.16)
yl S(S$ -- Si).
determine
the stochastic
properties
of E(s)
MOLECULAR
THEORY
OF BROWNIAN
MOTION
251
and are identical
to those of the stochastic force in the phenomenological in the phenomenological The stochastic averages Langevin equation. Langevin equation are determined by statistical averages over the bath variables. Eq. (4.8) implies that, for a heavy particle with momentum P and mass M (M > m) the equation of motion is, for sufficiently long times (t >
Tb),
dP(t) =
-
dt
+ F+(t),
(4.17)
(4.18)
-PyP(t)
where Py
= &
s
0
In the derivation of eq. (4.17) we have imposed the condition that P(t) is at most of order (M/m)*. It is also clear from eq. (4.4) that if P is too small i.e. of order 1 or less, the term on the right-hand side of the equation may dominate the last term on the left-hand side. This implies that the very long-time behavior of P(t), t > (A‘+)-1, is not necessarily properly described by eq. (4.17). The relevant results for Brownian motion, i.e. the behavior of averages of powers of P(t), which are usually obtained from the Langevin equation, eq. (4.8), and the properties of E(s), can also be obtained directly from eq. (4.4) without use of the weak coupling limit. This is possible because we have shown that the right-hand side of eq. (4.4) is of order ;1s for all times; this does not imply that this term is independent of time but merely that it is bounded for all t. We shall illustrate this point by computing
F+(t) + Py(t)
(4.19)
P(t) = O(i12,P*),
where t (4.20)
and y(t) = y for t > Tb. We have written P* explicitly on the right-hand side of eq. (4.19) to indicate that the ordering in 1 is valid when P* is 0( 1). The solution of eq. (4.19) is P(t) = exp(--i2iy(r) X (F+
Again
(T’)
we note that
+
dT} P(0) + / eXP{-A2jty(T) o(A2,
P*)}
eq. (4.21)
dT}
d+. is valid
(4.21) no matter
whether
the bath
is in
P. MAZUR
252
AND
I. OPPENHEIM
equilibrium or not. The averages discussed below, however, are taken assuming the bath to be in equilibrium with respect to the fixed Brownian particle. Taking the average of eq. (4.21) yields
i?(7) d7) P(0)
0
(4.22)
+ 6(1, P*),
where P(0) is @(A-r). For t 2 76, eq. (4.22) reduces to
P(0) + U(1, P*).
(4.23)
For times t such that Tb I t ( Z(L2y)-1, where I is a small integer, eq. (4.23) reduces to the usual Brownian motion result
(4.24)
P(0).
For very long times, t > Z(lsy)-1, however, the last term on the right-hand side of eq. (4.23) may dominate. This term, however, is small compared to (AM”)* = 0 (A-r) w h’ICh is a measure of the equilibrium fluctuations in P. The average of
{y (T) dT) P(0) P(0)
0
I’dT” exp(-As
iy(~)
x
dT -
1s iy(~)
d7)
i Y(T) d+ P (0) + 13 0( 1, P*), (4.25) 0
where we have used eq. (3.14). For t > Tb, eq. (4.25) becomes
e--2A8ytj (4.26)
P(0) + 1) O(1, P*),
where we have used eq. (3.17). For times t such that 76 < t < Z(A2y)-l eq. (4.26) reduces to the usual Brownian motion result
P(0)P(0)+ MkT I{1 - e-2A*yt),
where we have neglected terms of order 1-r compared For very long times, t > Z(Lsy)-1, eq. (4.26) becomes
(4.27)
to terms of order Ad2.
(4.28)
we have neglected terms of 0( 1) compared to terms of 0(A-2). Eq. gives the equilibrium value for
MOLECULAR
5. Friction that
THEORY
OF BROWNIAN
constant for a macroscopic
expression
(4.18)
for the friction
MOTION
253
body. In this section we shall show constant
occurring
in the Langevin
equation describing the stochastic motion of a heavy particle is identical to the friction coefficient, which characterizes the drag on a macroscopic body of mass M moving through the fluid with prescribed slowly varying velocity Ve(t). Consider a system body with position this system is
of N point particles &(t)
and velocity
of mass m in interaction
Ve(t) = A&).
Ro@)) = Bo(rN, PN) + igm
Q’?PN, where fJo(rN,pN)
=
v
with a
The Hamiltonian
- Ro@)I),
for
(5.1)
+utrN)j
U(rN) is the short-range potential of interaction between the point particles which is translationally invariant; and 4(lrr - Ra(t) I) is the short-range potential of interaction between the body and the ith bath particle. We now define a generating function for a time-dependent canonical transformation by N G
=
Z i=l
(Q
-
Roti))
.Pt,
w
where PC is the transformed equations are
i3G
Rc= - api
= rz -
momentum
$
(5.4
vector of particle i. belonging to the new
= Ao(RN, PN) + 2 +(IRtj) i
The new Hamiltonian
i. The transformation
Ro(t),
with Rt the transformed position The transformed Hamiltonian momenta is H = A+
of particle
coordinates
2I Pi* Vo@). i
and
(5.6)
H is of the form
H = Ho(RN, PN) - z Pt. V,(t), i
(5.7)
where Ho(RN, PN) = Bo(RN,
PN) + z +(IRfl)
i
(5.8)
P. MAZUR
254
AND
I. OPPENHEIM
is the Hamiltonian of the N bath particles in an external potential by the fixed body. It is therefore identical with the Hamiltonian
created defined
in eq. (2.3). We can now compute the linear response of any bath dynamical function to the prescribed motion of the body. We shall assume that at t --f - 00, the body was at rest, and the bath in equilibrium distribution function of the bath is given by
f (RN,
PN) = e--BHo/f dRN dPN eePHo.
with it. Thus at t + -00,
the
(5.9)
In particular we are interested in the average value of the force exerted by the bath on the body to terms linear in Vo(t). The force on the body is given by (see eq. (2.6)) (5.10) By standard
methods
we obtain
~@‘Fo(T)>. Vo(t - 7) d7,
(5.11)
0
where the symbol < >t denotes the average value at time t, the symbol < > is defined in eq. (2.8), and Pa(t) is defined by eq. (3.4). If Vo(t) varies sufficiently slowly (5.7) becomes 0%
= -
j3 r
d7. Vo(t)
0 =
-qlPo(t),
(5.12)
where Pa(t) = M Vo(t), and where Asy is given by eq. (4.16). Comparison of eq. (5.12) with eq. (4.15) demonstrates the equivalence of the friction constant occurring in the Langevin equation and the friction constant characterizing the drag on a macroscopic body. This equivalence has also been derived by Lebowitz and RCsiboisa) by a different method. 6. Remarks. The techniques utilized here have enabled us to derive a simple equation of motion for the heavy particle which is valid to order A2 for all times. For t > 70, this equation reduces to the LangeVin equation. We have restricted our considerations to situations in which the magnitude of the reduced momentum, P*, of the Brownian particle is of the same order in A, i.e. La, as the magnitude of the momentum, p, of a bath particle. The Langevin equation, however, is valid for a much larger range of values of P*. In fact, it follows from eqs. (4.1) and (3.3) that the left-hand side of eq. (4.1) is of lower order in I than the right-hand side as long as O(P*) - o(~&p), - 1 < e. If E I - 1, the expansion in eq. (3.3) is not useful.
MOLECULAR
THEORY
OF BROWNIAN
It is possible to extend our considerations
MOTION
255
to treat Brownian
motion in an
external field of force and Brownian motion of particles with internal structure. The equations obtained are the phenomenological equations which have been discussed e.g. by Chandrasekhara) and by Pragerra), respectively. The equations for nonspherical-Brownian particles exhibit a coupling between the various components of the center-of-mass momentum and the angular momentum. Acknowledgement. hospitality extended
The authors over the years.
thank
the reciprocal
institutions
for
APPENDIX
In this appendix, we study the
behavior of the correlation function (2.11). We shall see that the analysis complicated than that presented in (2.77, and
La) = n((P*/m) - FR + F-V,.>,
to obtain
@F(t)> =
(A. 1)
0
where Fe(t) is defined by eq. (3.4). Iteration of eq. (A.l) results in
; n=l
x
7”-1
1%TdTi jld?s . . . j dTn 0
0
0
x eiLO(k-l-d (01 + 02) eiLoTn F), where t has been replaced
(A4
by TO,
and 0s = F-V,,.
(A.9
The P* dependence of eq. (A.2) is explicit and resides completely in the operator 01 and 0s. Because of symmetry and since the factors of P* and Vp. can be moved outside of the average, the terms in eq. (A.2) containing
P. MAZUR
256
AND
I. OPPENHEIM
odd numbers of factors (01 + 0s) must vanish. It is also clear that only those terms with proper ordering of the operators Or and 0s can survive. In particular, each 0s must have at least one more 01 to its right than it has 02’s to its right. Thus, there can be at most rt/2 0s operators in each term of eq. (A.2) and the last factor of 01 + 0s on the right reduces to 01. It is clear that for short times
eih(TJ-l-TJ)
o1
eih(TJ-TJ+l)
the
B(a’)>. (-4.5)
Since we need consider only those correlations in eq. (A.2) which contain an even number of factors 02, i = 1 or 2, there are two cases of eq. (A.5) that will be of interest. In the first case, A contains an even number of factors of 02 and B an odd number of Og. In this case, application of (3.5) leads to ci =
(A (a)) (01 eiLa(rJ-‘j+l) B(d)> [
= 0 = 0
73'>
7b
(A.6a)
Tj+l >
Tb.
(A.6b)
Tj-1 -
Tj -
The result for eq. (A.6a) is zero because A contains an odd number of factors of 0~ and F. The result for eq. (A.6b) is zero because
c
= 1
eiLo(7J-1-7f) Ol>(B(o’)>
Tj-1 = 0
ri -
Tj >
Tj+l
>
76
(A.7a)
76.
(A.i’b)
The result for eq. (A.7a) need not be zero since A is odd in 0s and contains one F. The result for eq. (A.7b) is still zero for the same reason that (A.6b) is zero and also because B is even in Og and contains one F. Next, we consider the correlation function C2 = (/I@)
eih(TJ-l-TJ)
o2
&-h(5J-7J+1)
B(a’)>. (A.4
With A even in 02 and B odd, application cs =
of eq. (3.5) leads to
= 0
[
Tj-1 -
75 >
Tj - Tj+l >
76 Tb.
(A.9a) (A.9b)
The result for eq. (A.9a) is zero for the same reason as eq. (A.6a). The result for eq. (A.9b) need not be zero. For A odd in Or and B even, C2 becomes cs =
TJ-1 - Tj >
rb
(A. 1Oa)
73 - Tj+l >
Tb,
(A. lob)
where (A. 1Oa) need not be zero and (A. lob) is zero since B contains an even of Oa and one F.
number of factors
MOLECULAR
To summarize
THEORY
the results
OF BROWNIAN
of eqs. (A.6)-(A.lO),
times TJ-1, ~5, T~+I mzlst be close together occurs with an even number of factors will be of lower order in TO for TO > Tb each n in eq. (A.2). Because of this and is limited the leading term of order PZin An
fdT1(dT2...y-i+,
o2
we note that
257
the three
for a nonzero result only when Or
of 02 to its left. Thus, these terms than the other cases considered for because the number of factors of 0s eq. (A.2) is of the form eiL~(~~-~a)
O2
O1
0
0
...
MOTION
eiLo(sn-t-~n)
o1
eua+nF),
(A. 11)
with alternating factors of Or and Oz. Here and below we always assume To > Tb. In order to obtain this result we start our considerations on the right. The first 01 must be Or since there are no factors of P* to its right. The second Oa must be 0s or else eqs. (A.6) would apply. The third 0~ must be Or because there are no free P* factors to its right. The argument proceeds in this fashion until the last Oa on the left. The leading contribution to eq. (A.1 1) is obtained by allowing the largest possible lattitude to the time variables consistent with retaining a nonzero integrand. For most of the range of integration, the integrand of eq. (A. 11) breaks up, according to eq. (3.5), into
eiL~(~e4-~~-~l
02)
03) ~01
. . .
03)
eiLOra F>.
(A. 12)
Substitution (2.6) yields
of eqs. (A.3) and (A.4) into eq. (A. 12) and use of eqs. (2.15) and
(-
- 73)) . . .
p/m)*n
-
Tn-1))’
(A.13)
At this point we use the fact that, for reasons of symmetry, (A. 14)
The dominant
contribution
to the nth term in
(n-W2
x{
,po.
723+1)>)
(%)>
1,
where we recall that n is even. The dominant contribution easily obtained using eq. (3.5) and the fact that To > Tb.
(A. 15) to eq. (A. 15) is We use the fact
258
MOLECULAR
THEORY
OF BROWNIAN
MOTION
that oj
=~(F.F,,(O))
do 3 01,
T 2 TZ,(see (3.5)). The time integrations
(A. 16) in (A. 15) result in (A. 17)
for n = 2 and (A.18) for n > 2. Substitution
of (A.17) and (A.18) into eq. (A.2) yields
I = (-yn)P)k IX k=l k!
I1 + ‘($)}-jJ
(A. 19)
where we have used eq. (A.14) and y is defined by P y =3mOL. The terms of ~(Q/To) in eq. (A.19) are a proper estimate of all the approximations made in proceeding from eq. (A.2) to eq. (A.19). The summation of the leading terms in eq. (A. 19) can be easily performed to yield
TO)k
I -
(ml/?) y2 I2 I exp{--yT0L2} (A.21)
A4 Tb),
where k is some arbitrary
integer.
REFERENCES
1) Lebowitz, J. and Rubin, E., Phys. Rev. 131 (1963) 2381.
2) 3) 4) 5) 6) 7) 8) 9) 10)
Resibois, P. and Davis, R., Physica 30 (1964) 1077. Lebowitz, J. and Resibois, P., Phys. Rev. 139A (1965) 1101. Mori, H., Progr. theor. Phys. 33 (1965) 423. Kubo, R., Repts. Progr. Phys, 24 (1966) part 1 255. Zwanzig, R., in Lectures in Theoretical Physics, edited by W. E. Brittin, B. W. Downs and J. Downs, Interscience Publ., Inc. (New York, 1961) vol. III. 106. Mazur, P. and Oppenheim, I., J. Phys. Sot. Japan 26 (1969) suppl. 35. Terwiel, R. and Mazur, P., Physica 32 (1966) 1813; 36 (1967) 289; Resibois, P., Brocas, J. and Decan, G., J. math. Phys. 10 (1969) 964. Chandrasekhar, S., Rev. mod. Phys. 15 (1943) 1. Prager, S., J. them. Phys. 23 (1955) 2404.