On the exact and phenomenological Langevin equations for a harmonic oscillator in a fluid

Physica II5A (1982) 1-20 North-Holland Publishing Co.

ON THE EXACT AND PHENOMENOLOGICAL

FOR A H A R M O N I C

OSCILLATOR

LANGEVIN

EQUATIONS

IN A FLUID

U. MOHANTY and K.E. SHULER*

Department of Chemistry, University of California, San Diego, La Jolla, CA 92093, USA and I. OPPENHEIMt

Department of Chemistry, Massachussetts Institute of Technology, Cambridge, MA 02139, USA Received 3 March 1982

We have derived, starting from the Liouville equation and using projection operators, the three-dimensional Langevin and Fokker-Planck equations for the time dependence of the momenta and position of two Brownian particles of mass M interacting with a harmonic potential (i.e., a harmonic oscillator) in a fluid of particles with mass m, with M ~>m. The resulting Langevin equation has a very complicated structure in that the friction coefficients, X, due to the hydrodynamic interaction between the oscillator particles, are functions of R(t), the timedependent separation of the oscillator particles, and the noise terms, though Gaussian, are non-stationary and of a generalized form, i.e., neither "additive" nor "multiplicative". In addition, there is a term involving the mean force exerted by the fluid on the oscillating Brownian particle. We then investigated the various approximations which must be made to reduce this "exact" Langevin equation to the frequently used one-dimensional, phenomenological Langevin equations (and corresponding Fokker-Planck equations) with purely "additive" and "multiplicative" noise. These approximations are of three types: (a)one must neglect the term arising from the rotational motion of the oscillator in the fluid; (b) one must neglect the R(t) dependence of ~[R(t)], leading to purely "additive" noise, or approximate x[R(t)] by a Taylor series expansion to quadratic order in R(t) - Ro (where R0 is the equilibrium separation of the oscillator particles), leading to "additive" and "multiplicative" noise; and (c) one must either neglect the mean force term or approximate it by a term linear in R(t)-Ro. We conclude from these results that one-dimensional phenomenological Langevin equations with simple noise structure are of doubtful validity for the dynamical description of the relative momenta (or vibrational energy) of oscillating molecules in a fluid.

1. Introduction The

problem

addressed

in t h i s

paper

is t h e

derivation

via projection

o p e r a t o r t e c h n i q u e s 1) o f t h e e x a c t L a n g e v i n e q u a t i o n f o r a h a r m o n i c o s c i l l a t o r w i t h h e a v y p a r t i c l e s in a h e a t b a t h o f light, s t r u c t u r e l e s s p a r t i c l e s s t a r t i n g w i t h the Hamiltonian

equations

of motion

for the oscillator-bath

system.

The

* Supported in part by the National Science Foundation under Grant CHE78-21361 and by a grant from Charles and Ren6e Taubman. t Supported in part by the National Science Foundation under Grant CHE79-23235.

0378-4371]8210000--0000]$02.75

( ~ 1982 N o r t h - H o l l a n d

2

U. MOHANTY et al.

principal objectives of this study are: ( a ) t h e comparison of this exact Langevin equation with phenomenological Langevin equations for the oscillator-heat bath system which have been employed by various investigators; (b) an analysis of the structure and statistics of the noise in the exact Langevin equation and the comparison with the simple Gaussian, deltacorrelated noise usually assumed for the phenomenological Langevin equation; and (c) a clarification of the approximations, and their physical implications, required to reduce the exact Langevin equation to the phenomenological ones. Previous work along these lines for a single heavy, structureless particle and for a collection of identical heavy particles in a heat bath of light particles can be found in the papers of Mazur and Oppenheim 2) and Deutch and Oppenheim3). The present work is based on the techniques developed in these papers. The description of the equation of motion (Brownian motion) of a harmonic oscillator in a heat bath via a momentum Langevin equation has been presented on a variety of levels4). Thus, Wang and Uhlenbeck 5) write, for the relative displacement x(t) and the relative momentum p(t), for a one-dimensional oscillator,

fc(t)

=

p(t)/l~,

f)(t) + (Jp(t) + ~w~x(t) = F(t),

(1.1)

where /3 is the damping constant, /~ = M/2 is the reduced mass of the oscillator, and the noise F(t) is assumed to be delta-correlated and Gaussian. They state, "This describes clearly the Brownian motion of a simple harmonic oscillator..." This Langevin equation (L.E.), which is the most common form of a "phenomenological" L.E., has been formulated with the simple prescription of adding a Gaussian, delta-correlated, stationary noise which is independent of the system variables x and p to the deterministic equation of motion. On another level of description, one writes yc(t) = p ( t ) / ~ ,

f)(t) + lip(t) + I~wz(t) x(t) + G(x, p) = F(t),

(1.2)

where to(t) is a fluctuating frequency centered around too and where G(x, p) is a nonlinear function of x and p. This type of Langevin equation has been used and discussed by a number of investigators6'7). Since the fluctuating frequency to(t) multiplies the system variable x(t), this type of noise is usually denoted by multiplicative noise to distinguish it from the additive noise F(t). We believe that this is an imprecise and misleading terminology. A more accurate terminology will be defined in the body of this paper. Eq. (1.2) has been derived for a model Hamiltonian4'~) which does not apply to a Brownian

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

3

oscillator in a fluid. An analysis of van Kampen s) has shown that one cannot give a stochastic description of a physical system simply by declaring some of the coefficients of the deterministic description to be random. Langevin equations can also be obtained by starting from a master equationg). If the master equation is based on the phenomenology of the dynamics, rather than on a derivation from the Hamiltonian equation of motion, the Langevin equations obtained in this manner have no more basis in microscopic dynamics than the parent master equation. From the research carried out over the last decade2'3'4'l°'ll), it has become clear that exact Langevin equations, i.e., Langevin equations derived from the Hamiltonian equation of motion, bear little relation to the phenomenological L.E. equations of the type of eqs. (1.1) and (1.2). Thus, for instance, the noise in the exact L.E. is of a much more complicated form than in the phenomenological L.E. In general it is variable dependent, i.e., of the form F[x(t), t], and not separable, except in very special cases7"~°), into "multiplicative" and "additive" parts. Furthermore, not surprisingly, the noise obtained in these derivations is very sensitive to the form of the interaction between the Brownian particle(s) and the heat bath and also to the averaging procedure used to separate the "systematic" part of the L.E. from the noise. Also, as it turns out, the derivation from the microscopic dynamics introduces nonlinear terms in the positions and momenta in the "systematic part" which are crucial in assuring the stability of the moments of the L.E. and which could not be intuited in a phenomenological approach. The phenomenological Langevin equations, owing to their deliberately chosen simplicity, can readily be used, via conversion to Fokker-Planck equations or through direct integration, for the calculation of various physical properties of interest. It is doubtful, however, whether such Langevin equations represent an accurate description of, or even a good approximation to a real physical s y s t e m - e x c e p t for very simple linear systems with weak interactionsS). It is therefore of interest and importance to understand the approximations which need to be imposed upon the mathematically much less tractable exact Langevin equations, both as regards the "systematic" part and the noise, to reduce them to the more tractable phenomenological L.E. of the types (1.1) and (1.2). Such knowledge will be very useful in assessing the ranges of validity of the phenomenological L.E. (1.1) and (1.2). As indicated above, this is one of the primary objectives of this work, The organization of this paper is as follows. In section 2 we derive, via projection operator techniques, the exact L.E. for a harmonic oscillator in three dimensions in a heat bath of light, structureless particles for an arbitrary general interaction between the oscillator and the heat bath. In section 2 we also derive the Fokker-Planck (F-P) equation corresponding to this L.E. We

4

u. MOHANTY et al.

then restrict these equations to the case where the harmonic oscillator is not free to rotate in order to compare the results with eqs. (1.1) and (1.2). In section 3 we define additive, multiplicative, and general noise. We then utilize the F - P equation obtained in section 2 to investigate the approximations required to reduce the exact L.E. to the W a n g - U h l e n b e c k eq. (1.1) and also discuss the properties of the noise of the exact L.E. and its relation to the Gaussian, delta-correlated, stationary noise of eq. (1.1). In section 4 we compare our results with those obtained for the Z w a n z i g - L i n d e n b e r g Seshadri model Hamiltonian 7'~°) and investigate the approximations required to reduce our exact results to the L.E. and F - P equations found by them. Finally, in section 5 we summarize our findings and discuss their implications. 2. The L a n g e v i n and F o k k e r - P i a n c k equations

Consider a classical system of N identical light, structureless bath particles each of mass m and an oscillator of two identical heavy particles each of mass M enclosed in a volume V. The heavy particles interact via a harmonic potential K. The Hamiltonian for such a system can be written as H = HB + Ho,

with

_Pt, Ha - ~

2

2

P2 . 1 . . . . ± ~ -r ~ r~ tat2 - R0) 2,

(2.1)

where Ri2 is the instantaneous displacement of the oscillator and R0 is the equilibrium displacement. The subscript B denotes the Brownian oscillator and the subscript O denotes the heat bath. Ho is the Hamiltonian of the bath particles in a field • created by fixed, heavy particles. Ho = P0NPN + 2m

U(r N) + ~(r N, Rt) + ~ ( r N, R2).

(2.2)

The positions and momenta of the Brownian particles are denoted by R ~2~and pC2~, respectively, and those of the bath particles by r N and pN. In (2.2), U(r N) is a short-range potential energy of interaction among the bath particles, and qb is the potential energy of interaction between the oscillator particles and all the bath particles, i.e., N

• (r N, Rv) - ~ qb(lRv - r,I);

v = 1, 2.

/=1

The Liouville operator for the system is given by iL = iLa + iLo,

(2.3)

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

5

where (2.4)

iLo = Vpt~Ho • Vr ~ -- VrNHo • VpN and

Pl

iLB = ~ "

P2

VR, + ~ "

VR2 + F~ • Vp. + F2" Vp2;

(2.5)

F~ is the f o r c e e x e r t e d on the vth B r o w n i a n particle Fv = - VR~[~ + ~ K ( R t 2 - R0)2],

v = 1, 2.

(2.6)

We wish to write the e x a c t equation of m o t i o n of an aribtrary dynamical variable, X [ P ~2),R~2)], in the form which is suitable for approximations. In order to accomplish this, we introduce the projection operator, P, which is defined by:

P A =- (A) = Z -1 f dr N dp N e-~U°A,

(2.7)

where

Z= f dr N dp N e -~n°, fl = l[kT

(2.8)

and w h e r e the b r a c k e t ( ) d e n o t e s the average o v e r the bath equilibrium distribution. We use the o p e r a t o r identity t

e iLt i L = etl-V~iLt(1 -- P ) i L + eiLtp i L + f eiL"-')P i L e"-P)iL'(1 - P ) i L d r 0

(2.9)

acting on X [ P ~2~,R ~2)] to obtain [

X'(t) = K+(t) + eiLt(iiX(O)) + I eiL(t-')(iLK+('r)) dr,

(2.10)

0

where

m ( t ) = m[pt2)(t), R(:)(t)], X(0) - X[p(2)(t = 0), R(Z)(t = 0)] and

K+(t) = e"-P)IL'[iLX(0) - (iLB)X(0)].

(2.11)

It is e a s y to show that i L X ( 0 ) - (iLB)X(0) = #~. Vp, X(0) + #2" Vp2X(0),

(2.12)

6

U. M O H A N T Y et al.

where /~v = F~ - (F.). The function

K+(t) can thus be written as

K+(t) = e"-e)iU(Fl • Vp, + 1~2" Ve2)X(0).

(2.13)

Note that the average of K+(t) is zero, (2.14)

( K + ( t ) ) = ( K + ( 0 ) ) = 0.

The term eiL'(iLX(0)) in eq. (2.10) can be rewritten as e

iLt I-P1

P2

LM" VRt + ~ " VR2+ (F1)" Vp, + (F2)" Vp2]X(0).

(2.15)

The term (iLK+(~-)) is given by (2.16) Using the identity

VR~(K+(T)) =/3(~.K+(~)) + (VR K+(r)) = 0,

(2.17)

one can rewrite eq. (2.16) as

(2.18)

,u.=l

From (2.13), (2.15), and (2.18) the exact equation for lb~ then becomes l~v(t) = e i L t ( F v ) + e(1-P)iLt]~ v t

~ f d~e,u'-*)[rr t"",, - ~

/z=l

P, ]" (#,(0) e"-P)'C" F")" -

(2.19)

0

Eq. (2.19) is another way of writing Newton's Law for the time dependence of Pv(t). We wish to reduce this exact equation of motion to a Langevin equation using the fact that the mass of a Brownian particle, M, is very large compared with the mass of a bath particle, m. We shall not be interested in the long-time tail phenomena associated with the perturbation of the bath motion by the Brownian oscillator and shall follow the analysis of Mazur and Oppenheim:). The reader is referred to that paper and to the paper by Deutch and Oppenheim 3) for details. The basic assumptions required to reduce eq. (2.19) to a Langevin equation are the following: a) The correlation functions of bath dependent dynamical variables, A and B,

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

7

have the p r o p e r t y (Ae ~c°'B)= (A)(B) for t > % where % is a typical relaxation time for the bath particles in the p r e s e n c e of the Brownian particles fixed in position; b) The change of the m o m e n t a of the Brownian particles during the time interval zo is small c o m p a r e d with their m o m e n t a , i.e., ]I0]To/IP] ~ 1; and c) The velocities of the Brownian particles are less than the velocities of the bath particles, i.e., [PI/M
u = 1, 2,

(2.20)

/.t

where Q~[/L2(t)] = eiLt(F~).

(2.21)

Rlz(t) is the interparticle distance of the oscillator at time t, and where the r a n d o m force is given b y K+~(t) =

e(l-/')iLtl'~.

(2.22)

The friction tensors g~ are defined by * It is, however, a valid approach in calculating the rate of change of vibrational energy [or a quantum mechanical H.O. in a heat bath [see e.g.E.W. Montroll and K.E. Shuler, J. Chem. Phys. 26, 545 (1957)] since in that case of the ~(105), collisions between bath particles and the oscillator are required for a transition between adjacent vibrational energy levels, i.e., the vibrational energy is indeed a slowly changing variable.

U. MOHANTY et al. t

---

e

'y,~r, RI2),

(2.23)

0

where 2/is defined through eq. (2.23) and where (2.24)

l ~ ° ( r ) = eiL°~,,.

For t > to, the u p p e r limit of the integral in eq. (2.23) can be set to oo. The corresponding friction tensors will be denoted by ~,~[R12(t)] and 3,,~[R,2(t)]. The friction tensors which describe the m o m e n t u m damping due to collisions are time dependent through the instantaneous interparticle separation R,2(t). The friction tensors can be shown to have a long-range character at large interparticle separations. The h y d r o d y n a m i c interaction between the two particles which constitute the harmonic oscillator gives rise to cross-friction tensors which are of order a/[R12(t)] and which modify the self-friction tensor of an isolated Brownian particle b y terms of order (a/[R~2(t)]) 2. H e r e a is the radius of an oscillator particle. Eq. (2.20) is the momentum Langevin equation for the Brownian oscillator

provided K+dt) has the correct statistical properties for a bath in equilibrium. We have already seen that (K+~(t)) = 0. U n d e r the assumptions (a), (b), and (c) above, one can also show that + +~(t)) --- (K.(O)K~ + o (t)) = -2-B ~[R,2(O)]8(t). (K.(O)K

(2.25a)

where

K°(t) = e~L°'Kv.

(2.25b)

The equality in (2.25a) is true on the m o m e n t u m relaxation time scale of the oscillators. Eq. (2.25a) is a generalization of the fluctuation dissipation theorem. We will show in section 3 that the autocorrelation function(K+~(t)K+~(t + r)) is not stationary in time. In the w e a k coupling limit A ( - m l M ) t / 2 ~ O , t ~ oo such that ;~2t = constant, it can readily be shown 2) that K+(t) is a delta-correlated Gaussian stochastic process with zero mean and second m o m e n t given by (2.25a). The coupled nonlinear Langevin equations (2.20) can be put in a more t r a n s p a r e n t f o r m by the introduction of relative and center of mass coordinates:

P~I=P

= ~P1-P2 ,

Pcm = P1 + P2,

Rrel= R = R1 - R:;

Rcm = R~ +2 R2

(2.26a) (2.26b)

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

9

With this transformation, the Langevin equations (2.20) become

~'(t) = q_(t)2 - (~ - F ) . P(t) + e_(t)2 '

(2.27a)

lbcm(t) = q+(t) -- (~ + F ) . Peru(t) + e+(t),

(2.27b)

where

e=(t) = K~(t) -7-K~(t)

(2.28)

q=(t) = eiL'[(F,) -7-(F2)]

(2.29)

and

and where "~- ~11 = ~22,

r~-~

~21 = ~12-

(2.30)

The ~ and F in (2.27) are the "self-" and "cross-frictional" tensors, respectively. As pointed out before, these frictional tensors are time dependent through the instantaneous interparticle separation Rlz(t). The explicit time dependence of the friction tensor and the complicated structure of the noise tensor e~_(t) lead to a highly nonlinear structure of the Langevin equations (2.27). The terms q_(t) and ( ~ - F ) . P(t) are, respectively, the systematic and the dissipative part o f the momentum Langevin equation in relative coordinates. One can easily show that q+(t) is zero while q_(t) can be rewritten as

q_(t)2 = - K [ g ( t ) - g0]/~(t) + ~1 eiLtVRlog Z(IRD,

(2.31)

w h e r e / ~ ( t ) is a unit vector in the direction of R(t) and Z has been defined in (2.8). The quantities e_(t) and e+(t) are, respectively, the generalized noise for the relative and the center of mass Langevin equations. The fluctuating forces e~(t) satisfy the following moment properties: (e_(t)) = O,

(e+(t)) = O,

(e_(O)e_(t))

= 4_~ (~ _ r)~(t),

(e+(O)e+(t))

=~

(~ + r)~(t),

(2.32)

(e_(O)e+(t)) = O, (e+(O)e_(t)) = O.

Before we can relate the exact L.E. (2.27) to the phenomenological L.E.

10

U. MOHANTY et al.

(1.1) and (1.2), as we will do in sections 3 and 4, it is necessary to derive the F - P equation corresponding to the L.E. (2.27). This will also yield some insight into the structure of the noise terms e~_(t). The Fokker-Planck equation for the distribution function for the harmonic oscillator, @[]~(2) p(2), t], in a heat bath follows directly from the calculation in ref. 3. If we assume that the initial distribution function for the system is of the form

f [ r N, pN; R~2), p(2), 0] = pot~ [R (2), p(2)],

(2.33)

where (2.34)

po = e-oH°/Z, then 0~b[R(2), p(2), t] M with initial condition +[R(2), p(2), 0] = cb[R (z), p(:)].

(2.36)

The distribution function of the relative coordinates can be obtained from the Fokker-Planck equation

O~b(R,otP' t ) = {_ [ _ ~ . V R _¢ [ ( F , ) ( R ) -P( F 2 ) 2( R ) ] . V ] + vp- [~(R) -

M Vp ]},(,.,, f + ~-~ r(l~)] • LP

t).

(2.37)

Note that the terms (F0 - (F2) and f - F are functions of R. Using symmetry arguments we can write (F~)-(F2)

2

=

3W(R) f~ OR - [ K ( R - Ro) - k T ~

log Z(R)]/~

(2.38)

and

- F = a ( R ) l + O(R)RR,

(2.39)

with the coefficients a(R), 0(R) defined by (2.39). The quantity W ( R ) is the potential of the mean force between the oscillator particles. When R is large compared with or, the radius of a bath particle, W(R) is the harmonic

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

11

potential. When R is large compared to a, the radius of the Brownian particles, g - F, reduces to the friction tensor of a single Brownian particle and becomes independent of R. We wish to obtain a Fokker-Planck equation for the distribution function of the scalar distance R and the component Pr of the relative momentum along R. In order to do this, we write eq. (2.37) in polar coordinates and integrate over the angles 0 and ~b and the momentum Po and P,. The result is 0¢(R, Prt)

at

[ Pr O OW O = _ -~ O-R }" ~ O-P, ~ [a(R) + O(R)R ~] 0 ( P r + - ~ r ) ] l l l ( R , Pr, t)

- fdO dP0 d~b d P , [ P " ~-~ P-Palj 00p r I/t(R, P, t).

(2.40)

Note that the equation for ¢(R, Pr, t) is not closed due to the presence of the last term, which reflects the rotation of the Brownian oscillator. Finally, the Langevin equation for the component of momentum along R(t) can be obtained from eq. (2.27a). We use the facts that Pr(t) -= l~(t). P(t)

(2.41)

and -fir(t) = l~(t)" la(t) +

1

[P(t) • P(t)

-

P r2(t)],

(2.42)

to write Pr(t)

=

OW[R(t)] aR(t)

{a[R(t)+ O[R(t)]R2(t)}Pr(t)

e_(t) + ~ ( t ) [P(t) • P(t) - P r2(t)]. +/~(t) • T

(2.43)

Again, this is not a closed equation in PrO) and R(t) owing to the rotation of the oscillator. In order to make contact with the phenomenological equations, eqs. (1.1) and (1.2), which describe one-dimensional oscillators, we shall simplify the Fokker-Planck equation, eq. (2.40), and the Langevin equation, eq. (2.43), by fixing the orientation of the Brownian oscillator in space, i.e., we do not permit the oscillator to rotate. Under these conditions, eq. (2.40) becomes

OqJ(R, Pr, t ) = [ _ P r O aW 0 0 ( /~ O ) ] at I~ OR + OR OP~+ x(R) - ~ Pr + -~ ~ ql(R, Pr, t), (2.44)

12

U. MOHANTY et al.

where

x(R) - a(R) + O(R)R 2,

(2.45)

and the Langevin equation, eq. (2.43), b e c o m e s

OW[R(t)] OR(t)

fir(t) =

x[R(t)]Pr(t) + ½[1~ • e_(t)].

(2.46)

In eq. (2.46), the first term on the r.h.s, is the systematic part of the L.E., the second term is the dissipative part, and the last term is the generalized noise.

3. Some properties of the random force and relation of the exact to the phenomenological Langevin equation The quantities which are of physical relevance are the averages and/or the cumulants of the r a n d o m force K+(t) rather than the detailed structure of K+(t). With this in mind, it is possible to give a precise classification of stochastic properties of K+(t). If the averages of K÷(t) are state independent, i.e., are not a function of the dynamical variable of the L.E., then the noise can be denoted as additive as in eq. (1.1). If the averages of cumulants of K+(t) are state dependent, one can distinguish b e t w e e n two cases: (a) K+(t) = G(at)F+(t), with (F÷(t)) = 0; and (b) K+(t) = F(at, t), In case (a), the r a n d o m force can be explicitly factored into a product of the function G(ttf) of the dynamical variable(s) at and a stochastic force F*(t). It is reasonable to denote such random forces as multiplicative noise as in eq. (1.2). The m u c h more c o m m o n case (b), where no such factorization is possible, should then be denoted as generalized noise. We r e m a r k here that the o c c u r r e n c e of such a factorization in real physical s y s t e m s is the exception rather than the rule. We shall first discuss the properties of the quantities in the Langevin equation (2.27a) and investigate under what conditions this equation reduces to eq. (1.1). We rewrite eq. (2.27a) as

P(t) = - K [ R ( t ) - Ro] + k r 0 - - ~ log Z [ R ( t ) I lq(t) -

{a[R(t)ll + O[R(t)]R(t)R(t)}. P ( t ) + e_(t)/2,

(3.1)

where we h a v e used eqs. (2.31) and (2.39). Since we have already established the facts that [eqs. (2.32)] (e_(t)) = O ,

4M (e_(t)e_(O)) = ~ [~(R) - r(R)]~(t),

(3.2)

EXACT AND P H E N O M E N O L O G I C A L LANGEVIN EQUATIONS

13

it can readily be seen that the noise, e_(t), in the Langevin equation (3.1) should be described as generalized noise. Furthermore, we can show that the correlation functions of the noise K+(t) in the L.E. (2.20), (2.27) and (3.1) is not stationary in time. Let G~(t, ~) be the correlation function of the noise,

G.~(t, "r)=- (K,(t)K~(t + + + 7));

t~,v = 1, 2.

(3.3)

The differential equation satisfied by G~(t, ,r) is •

+

+

•

+

+

-0- G~(t, 7) =--(~L[K,(t)K~(t + 7)]) = OLB[K~(t)K,(t + ~)]) Ot

(3.4)

where we have used the fact that P iL0(...) = 0. Using the identity + + + T)) = ( V~o[K,( + t )K~(t + + 7)]) + 13(~'eK+~(t)K+~(t+ r)), 7R,(K,(t)K~(t

(3.5)

we find, substituting (3.5) into (3.4),

0=| P0

+

+

+ - ~ " VRo(K~(t)K~(t

+

~)) + VPo" (Fo)" (K~(t)K+~(t + 7))

0=1

The terms in the sum over 0 in the second equality in (3.6) are of higher power in )~ = (m/M) 11zthan the last term and can be neglected. The solution of (3.6) for small A thus becomes

Gg~(t, ~') =

citLs>tG,~,(O, "r) = eitL>tGt.,(O, 'r)

el'~-~ ~[R(0)]6(r).

(3.7)

The above result shows that the correlations function 6,~(t, z) of the random force is not stationary in time. The important point to observe in eq. (3.7) is that the time dependence of G~(t, 7) is computed by using the bath averaged Liouville operator (iL). Physically, the non-stationarity of the noise is due to the relative motion of the two Brownian particles. The fluctuation dissipation theorem for the oscillator with the above nonstationary noise looks rather complicated. From (3.7) one obtains c¢

M

f (K+~(t)K+~(t + ,r)) d,r = eitg~[R(0)]. 0

Since, by definition, the friction coefficient ~.~[R(t)] is given by

(3.8)

14

U. MOHANTY et al. g,~[R(t)] = ciLia/R(0)/,

(3.9)

it is clear that the r.h.s, of eq. (3.8) is not equal to the friction coefficient as defined by (3.9). We now consider what approximations have to be made to the form of eq. (3.1) and to the properties of the fluctuating force, eqs. (3.2) and (3.7), in order to obtain the W a n g - U h l e n b e c k equation, eq. (1.1), with Gaussian, stationary white noise. The significant differences between eqs. (3.1) and (1.1) are: (1) the existence of the mean force term (O log Z/OR(t)); (2) the dependence of the friction tensors on R(t); and (3) the related properties of the noise correlation functions. We recall that the mean force term varies over distances comparable to the bath particle radii, or, and d e c a y s to zero when R - 2a is larger than several tr; the friction tensors and the moments of the fluctuating force in eq. (3.1) vary over distances comparable to the Brownian particle radii a and become independent of R ( t ) only for 3) R ( t ) ,> a. Finally, we must remember the restrictions under which eq. (3.1) has been derived. The Langevin equation (3.1) for Pr(t), the c o m p o n e n t of the relative momentum along/~(t), can be written as [see eqs. (2.38) and (2.46)] Pr(t) = ( - K [ R ( t ) - Ro] + k T

8

log Z [ R ( t ) ] ) - x [ R ( t ) ] P r ( t ) + er(t),

(3. lO) where (3.11)

er(t) = R • e_(t)/2

and the orientation of the Brownian oscillator is fixed in space. Since the dependence of x [ R ( t ) ] on R ( t ) is weak, it is useful to expand 2¢ in a Taylor series around R0, the equilibrium separation of the Brownian oscillator particles. Thus, x [ R ( t ) ] = ~¢o+ x l [ R ( t ) - R0] + X2

[R(t)

- Ro] 2

2

~-. • •,

(3.12)

where ~(o~ x ( R 0 ) ,

dx. (R0) , etc. Xt -= dRo

(3.13)

If the extension of the oscillator b e y o n d its equilibrium separation is sufficiently small, i.e., when R(t)-

Ro ,g, a,

(3.14)

we can assume that x[R(t)] = X0 = constant. Note that due to the restrictions under which eq. (3.1) has been derived, the force constant of the oscillator is

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

15

restricted in magnitude since (Kltx)J12"ro~ 1, and thus the radius a of the Brownian particle has to be large in order for eq. (3.14) to hold. If x[R(t)] -~ Xo, the moments of the random forces e_(t) in eq. (3.1) and er(t) in eq. (3.10) b e c o m e state independent, the noise becomes purely additive and can be described by a stationary Gaussian white process, i.e., (e~(t)er) = (M/[3)XoS(t), (er(t + ~')er(~')) = (er(t)er),

(3.15)

etc. In order to further reduce eq. (3.10) to eq. (1.1), we must, in addition, make some assumptions concerning the term involving log Z[R(t)]. The properties of this term are the following: (1) it goes to zero for R ( t ) - 2 a > > ~ r ; (2) it varies rapidly in space and therefore in time for R ( t ) - 2 a = or; and (3) its magnitude for R ( t ) - 2 a - - ~ cr is proportional to the area of the Brownian particles. In many cases, over the time scales of interest for the calculation of the time dependence of Pr(t), this term can be neglected. In some cases, however, it may give rise to a fluctuating f r e q u e n c y of the oscillator. It would be of interest to carry out some detailed analysis of this case. Thus, only when the log Z term can be neglected and when x[R(t)] can be approximated by a constant Xo will eq. (3.10) reduce to the Wang-Uhlenbeck equation, eq. (1.1). The path from the exact three-dimensional Langevin equation (2.43) to the phenomenological, one-dimensional W a n g - U h l e n b e c k equation involves a large number of restrictions and approximations. Before using the Langevin equation (1.1) (or the corresponding F - P equation) for the description of the dynamics of the Brownian motion of a harmonic oscillator in a physical/chemical system, one would thus be well advised to verify its applicability.

4. Comparison with the Zwanzig-Lindenberg-Seshadri model Hamiitonian In 1973 Zwanzig introduced a model Hamiltonian for a one-dimensional system (wisely left unspecified as to its physical nature) interacting with a heat bath for which an exact nonlinear Langevin equation could be derived by direct integration J°). One of the attractive features of this model is the ability to establish explicitly the connection between the fluctuating force in the L.E. and the form of the interaction in the Hamiltonian. Zwanzig's model Hamiltonian involved specific restrictive assumptions about the nature of the heat bath, i.e., the bath Hamiltonian was assumed to have a quadratic form, but left the form in the system Hamiltonian arbitrary. Lindenberg and Seshadri 7)

16

U. MOHANTY et al.

have recently studied in detail a special case of the Zwanzig model in which the system was specified to be a harmonic oscillator whose interaction with the heat bath oscillators was assumed to be linear in the bath coordinates but arbitrary in the system coordinates. Since this is a specialized case of the Zwanzig model, it was still possible to carry out integrations over the bath coordinates analytically and thus to obtain an explicit Langevin equation. Lindenberg and Seshadri (L-S) showed that for a system-heat bath interaction which is nonlinear in the oscillator coordinates and linear in the bath coordinates one obtains a L.E. with multiplicative noise (as defined in section 3 above) and nonlinear dissipative terms. For the special case of an interaction which is linear plus quadratic in the oscillator coordinates, they obtain the momentum L.E. = pl~,

16 = -/x[to2 + ~/(t)]x - )top - 2),lxp - )t2x2p + f(t)lx,

(4.1)

where we have followed the notation of ref. 7. It should be noted that this L.E. is a specific example of the L.E. (1.2). In eq. (4.1), too is the fundamental frequency of the system oscillator, the fluctuations v(t) are related to a time-dependent frequency oJ(t) by to2(t) = to2q_ "y(t), with (to=(t))= to2, x and p are, respectively, the system oscillator displacement and momentum, the h~ are model dependent friction constants, and f(t) is an additive (variable independent) fluctuation. Specifically, Lindenberg and Seshadri show that

(f(t)f(t')) = 2D0~(t - t'),

(4.2a)

(V(t)~/(t')) = 2D28(t - t'),

(4.2b)

(f(t)3,(t')) = 2D~8(t - t'),

(4.2c)

Di = )tltx//3.

(4.2d)

with

Since the fluctuations given in (4.2) are Gaussian and delta-correlated, it is possible to convert the L.E. (4.1) to a Fokker-Planck equation for the distribution function qJ(x, p, t I x0, P0) -- q',

O----t- - - ~ - ~ +

(V'(x) + ~t°P + 2"~txP + )t2x2p)

02 ] + (Do+ 2Dlx + D2x2) ~-~]qJ tL O ~ + ()to + 2)t,x + )t2x 2) -~pO(p + ~-~-~p)]qs, = [ _ p 0~_ -~- + V'(x) ol)

(4.3)

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

17

where V'(x) is the derivative with respect to x of the potential for the system oscillator, i.e., V'(x)= toExlx. For s y s t e m - b a t h interactions linear in the system coordinate, hi -- 0 for i = l, 2 . . . . and eq. (4.3) reduces to

(4.4) which is the Fokker-Planck equation corresponding to the phenomenological L.E. (1.1). We wish to compare the F o k k e r - P l a n c k equations, eqs. (4.3) and (4.4), with the F o k k e r - P l a n c k equation, eq. (2.44), which we obtained from the Langevin equation, eq. (2.46), for a space fixed, i.e., non-rotating, harmonic oscillator. We rewrite eq. (2.44) in the form

a~(R, Pr,

= t ~- ~ P, aa_R~- [ K ( R - R o ) - k T - ~ R l O g Z ( R ) ]

o

0 OP, ix 3

(4.5) where we have used eqs. (2.38) and (3.12). Eqs. (4.5) and (4.3) have the same form except for the term involving l o g Z ( R ) . This term arises in our formalism because we have chosen general interactions between the Brownian particles and the bath particles. For the specific choice made by Lindenberg and Seshadri of an interaction potential linear in the Brownian oscillator coordinate, R, i.e., the Zwanzig Hamiltonian, a log Z(R)/aR = O. While the F o k k e r - P l a n c k equations, eqs. (4.5) and (4.3), have the same form, the physical models from which they are obtained are quite different. This can be seen by comparing the Langevin equations, eqs. (4.1) and (2.46). The Lindenberg-Seshadri Langevin equation, eq. (4.1), in our notation becomes /5,(t) = -/x~oo2[R(t) - R0] - )toP~(t) - 2ht[R(t) - Ro]P,(t) - h2[R(t) - Ro]2p,(t) + Ix{f(t) + T(t)[R(t) - R0]},

(4.6)

whereas our Langevin equation is /~,(t) = -/xto0:[R(t) - R0] + k T a log Z[R(t)] 0R(t)

x[R(t)]Pr(t) + er(t).

(4.7)

The R dependence of the dissipative terms in eq. (4.6) arise from the specific form of the interaction between the bath oscillators and the Brownian oscillator, whereas the R dependence of the dissipative term in eq. (4.7) arises from the h y d r o d y n a m i c interaction between the two Brownian particles in the oscillator. The noise term in eq. (4.6) is multiplicative, whereas the noise term

18

U. M O H A N T Y et al.

in eq. (4.7) is generalized. Eq. (4.7) yields the same F o k k e r - P l a n c k equation as eq. (4.6) only when (a) the term involving log Z can be neglected and (b) more significantly only when the expansion of eq. (3.12) is valid.

5. Summary and conclusions In section 2 we have derived the Langevin equation, starting from the Liouville equation, for the time dependence of the momenta of two Brownian particles of mass M interacting with a harmonic potential in a fluid of particles of mass m. The conditions under which eqs. (2.27) have been derived are the following: ( a ) t h e momenta of the Brownian particles are large compared with the momenta of the bath particles; ( b ) t h e velocities of the Brownian particles are small compared with the velocities of the bath particles; and (c) the change in momenta of the Brownian particles in the time interval ~'o is small compared to the momenta of the Brownian particles, where ro is the relaxation time of the bath variables. These conditions are met if M -> m and if ~Oro< 1 where ~0 is the frequency of the Brownian oscillator. The L.E. (2.27) have a complicated structure. The friction coefficients are functions of R(t), where R(t) is the time dependent separation of the oscillator particles and the noise terms are generalized (i.e., not additive and not multiplicative), Gaussian and nonstationary. In addition, there is a term involving the mean force exerted by the fluid on the oscillating Brownian particles. We have also derived the F - P equations which arise from these L.E. We have then looked in particular at the L.E. for the c o m p o n e n t of the relative momentum of the Brownian particles along the line of centers, eq. (2.43), and again at the corresponding F - P equation, eq. (2.40). These equations have terms which describe the rotation of the oscillator in threedimensional space. Our principal aim in this paper has been to investigate the conditions under which the L.E. and the F - P equations derived from the Liouville equation reduce to the phenomenological L.E. (1.1) and (1.2) and the corresponding F - P equations. Eqs. (1.1) and (1.2) describe one-dimensional Brownian oscillators and therefore the first step in such a comparison involves the neglect of the terms due to rotation in eqs. (2.40) and (2.43). The resulting F - P equation is eq. (2.44) and the resulting L.E. is (2.46). In section 3 we study the properties of the noise terms, the dissipative terms and the mean force terms in the L.E. (2.27) in order to ascertain the approximations which must be made to reduce the L.E. (2.27) to the simple phenomenological L.E. (1.1). If we assume that R(t)-Ro=-AR(t) is small

EXACT AND PHENOMENOLOGICAL LANGEVIN EQUATIONS

19

compared to a, where R0 is the equilibrium interparticle separation of the oscillator and a is the radius of the oscillator particle, we can neglect the R(t) dependence of the friction coefficient and the noise reduces to additive, Gaussian, delta-correlated, stationary noise. If, in addition, we neglect the mean force term, then the L.E. (2.46) reduces to the Wang-Uhlenbeck eq. (1.1). In section 4 we study the conditions under which the F - P eq. (2.44) reduces to the F - P equation of the L - S model, eq. (4.3). If we assume that the friction coefficient can be approximated by a Taylor series expansion to quadratic order in AR(t), eq. (3.12), and that the mean force term can be neglected, then eq. (2.44) reduces to eq. (4.3). Note that even with these approximations, our L.E. (2.46) or (4.7) still differs from the L - S Langevin equation (4.6) in that the noise term in our L.E. (2.46) is generalized noise as defined in section 3, whereas the noise term in (4.6) is multiplicative. The various approximations, detailed above, which must be made to reduce the L.E. derived from the Liouville equation to the simple, one-dimensional phenomenological L.E. (1.1) and (1.2) are of three types. First, the term arising from the rotation of the oscillator in the fluid must be neglected. This can only be justified for an oscillator in a crystal. Second, it is necessary to approximate the R(t) dependence of the friction coefficient x[R(t)] which arises from the hydrodynamic interactions between the Brownian particles. It can be shown 3) that ;~[R(t)] varies over distances which are of the order of the radius a of the oscillator particle. Thus, if AR(t)~a, x[R(t)] can be approximated by a constant. If AR(t)< a, one can employ a Taylor series expansion of x[R(t)] in powers of AR(t). This implies that V < ato, where V is the relative velocity of the oscillator particle in the absence of the fluid. Third, one must either neglect the mean force term or approximate it by a term linear in zaR(t). If R(t),> 2a +tr, where tr is the radius of the bath particles, the mean force term is zero. If the mean force is much less than KAR(t), it can be neglected. It is not clear under what conditions the mean force term can be approximated by terms linear in AR(t). In view of the many restrictions and approximations involved in reducing the " e x a c t " L.E., i.e., the L.E. derived from the Liouville equation, to the frequently used one-dimensional phenomenological L.E. of the type (1.1) and (1.2) for the Brownian motion of harmonic oscillators in a fluid, one would be well advised to check whether eqs. (1.1) and (1.2) are valid descriptions of the dynamics of the specific physical/chemical systems under study. As one of many examples, we mention the widespread use of simple L.E. of the form (1.1) for the study of the torsion dynamics of macromoleculest2). An analysis of the type developed above would be useful in establishing the range of validity of eq. (1.1) for this application.

20

U. MOHANTY et al.

References 1) R. Zwanzig, in Lectures in Theoretical Physics, Vol. III, W.E. Brittin, B.W. Downs and J. Downs, eds. (Interscience, New York, 1961), p. 106. 2) P. Mazur and I. Oppenheim, Physica 50 (1970) 241. 3) J.M. Deutch and I. Oppenheim, J. Chem. Phys. 54 (1971) 3547. 4) For an excellent discussion of this point, see R. Zwanzig, in Systems Far from Equilibrium, L. Garrido, ed. (Springer, Berlin, 1980), pp. 198-225. 5) M.C. Wang and G.E. Uhlenbeck, Rev. Mod. Phys. 17 (1945) 323. 6) For references and discussion, see e.g.K. Lindenberg, K.E. Shuler, V. Seshadri and B.J. West, in Probabilistic Analysis and Related Topics, Vol. 3, K.T. Bharucha-Reid, ed. (Academic Press, New York) (in press). 7) K. Lindenberg and V. Seshadri, Physica 109A (1981) 483. 8) N. van Kampen, Physics Reports 24 (1976) 171. 9) (a) D. Bedeaux, Phys. Lett. 62A (1977) 10; (b) P. Hanggi, Z. Phys. B31 (1978) 407; (c) P. Hanggi, K.E. Shuler and I. Oppenheim, Physica 107A (1981) 143. 10) R. Zwanzig, J. Stat. Phys. 9 (1973) 215. 11) H. Mori, Prog. Theoretical Physics (Japan) 33 (1965) 423. 12) See e.g.S.C. Lin and J.M. Schurr, Biopolymers 17 (1978) 425. S.A. Allison and J.M. Schurr, Chem. Phys. 41 (1979) 35. J.A. McCommon, P.G. Wolynes and M. Karplus, Biochem. 18 (1979) 927.