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PHYSICAL REVIEW E 83, 041201 (2011)

Zwanzig-Mori equation for the time-dependent pair distribution function

Song-Ho Chong, Chang-Yun Son, and Sangyoub Lee*

Department of Chemistry, Seoul National University, Seoul 151-747, Korea

(Received 7 January 2011; revised manuscript received 7 March 2011; published 26 April 2011)

We develop a microscopic theoretical framework for the time-dependent pair distribution function starting

from the Liouville equation. An exact Zwanzig-Mori equation of motion for the time-dependent pair distribution

function is derived based on the projection-operator formalism. It is demonstrated that, under the Markovian

approximation, our equation reduces to the so-called telegraph equation that includes the potential of mean force

acting between the pair particles. With the additional approximation neglecting the inertia term, our equation

takes the form of Smoluchowski’s equation, which has been previously introduced with intuitive arguments and

shown to satisfactorily reproduce the simulation results of the particle-pair dynamics.

DOI: 10.1103/PhysRevE.83.041201 PACS number(s): 61.20.Lc, 05.20.Jj, 82.20.−w

I. INTRODUCTION

The time-dependent pair distribution function g(r,r?,t),

describing the probability density of finding a pair of particles

separated by r at time t given that they were separated by r?at

time zero, is one of the fundamental quantities characterizing

liquid-state dynamics [1]. It is a central quantity determining

collision-induced absorption and depolarized Rayleigh and

Raman scattering spectra [2,3]. Its knowledge also enables

the calculation of the first encounter time distribution and

survival probability of reactive molecules [4,5], and it has

played an important role in the rigorous formulation of

the diffusion-influenced bimolecular reaction kinetics [6,7].

However,thetheoreticaldevelopmentforg(r,r?,t)isstillinthe

primitive stage compared to that for the van Hove correlation

functions [1].

The function g(r,r?,t) was first introduced by Oppenheim

and Bloom [8] in their study of nuclear magnetic relaxation in

fluids. However, their theory is valid only in the limit of free

particles and leads to unsatisfactory results even for the short-

time regime in the presence of interparticle interactions [9].

The exact short-time dynamics of g(r,r?,t) were subsequently

derived by Balucani and Vallauri [10], but calculating the

dynamics in the longer time regime was outside the scope

of their work. On the other hand, Haan [11] studied the

dynamic behavior of g(r,r?,t) from a different approach and

demonstrated that Smoluchowski’s equation with a potential

of mean force satisfactorily reproduces the simulation results

oftheparticle-pairdynamics.However,hisapproachresortsto

intuitive arguments and is not based on a first-principle theory.

The pair distribution function has also been investigated based

on the kinetic theory [12], but its applicability is limited to the

low-density regime.

In this paper, we develop a basic theoretical framework

for the time-dependent pair distribution function, starting

from the Liouville equation and using the projection-operator

technique. Such a rigorous framework has served as a basis

for developing successful liquid-state theories for the van

Hove correlation functions [1]. It is demonstrated that the

exact short-time behavior derived before and Smoluchowski’s

equation, which satisfactorily reproduced the simulation

*sangyoub@snu.ac.kr

results in the longer time regime, naturally follow from the

exact equation of motion for g(r,r?,t) that we will derive

in the present work. Our result will therefore provide a

rigorous basis for developing improved theories dealing with

the time-dependent pair distribution function.

The paper is organized as follows. In the next section, we

derive an exact Zwanzig-Mori equation of motion for the

time-dependent pair distribution function, starting from the

Liouville equation for the whole system comprising a central

particle pair and surrounding solvent particles. Section III

discusses the implications of the derived equation, and the

Appendix is devoted to a derivation of the initial value of the

memory function.

II. EXACT EQUATION OF MOTION

A. Liouville equation

We consider a classical fluid of N spherical particles of

mass m at a temperature T confined in a volume V, in which

two tagged particles, A and B, are dissolved. For simplicity,

the tagged particles are assumed to be mechanically identical

to solvent particles. The Hamiltonian of the total system is

given by

H = K + U,

(1)

with the kinetic energy part

K =

p2

2m+p2

AB

2m+

N

?

i=1

p2

2m,

i

(2)

and the potential energy part

U = φ(rAB) +

N

?

i=1

[φ(rAi) + φ(rBi)] +1

2

N

?

i,j (i?=j)

φ(rij).

(3)

Here, piand ridenote the momentum and position vectors

of the particle i, respectively, and we have assumed that the

total potential is represented by a sum of radially symmetric

potential functions φ(rij) that depend only on the particle

separation rij= |ri− rj|.

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SONG-HO CHONG, CHANG-YUN SON, AND SANGYOUB LEEPHYSICAL REVIEW E 83, 041201 (2011)

According to classical mechanics, Newton’s equation of

motion of a dynamical variable—say, A(t)—can be written in

the form

d

dtA(t) = {H,A(t)} ≡ iLA(t),

which is called the Liouville equation [1]. Here the symbol

{ , } denotes a classical Poisson bracket, and the Liouville

operator iL of the total system is given by

?

(4)

iL =

i=A,B,1,...,N

?pi

m·

∂

∂ri

−∂U

∂ri

·

∂

∂ pi

?

.

(5)

B. Time-dependent pair distribution function

Of primary interest in the present work is the joint

probability distribution function

G(r,r?,t) ≡ V?δ(r − rAB(t))δ(r?− rAB(0))?.

Here, rAB(t) = rA(t) − rB(t) denotes the separation of the

particles A and B at time t, and ?···? represents the canonical

ensemble average

?

i=A,B,1,...,N

In this expression, β = 1/(kBT), with kBbeing Boltzmann’s

constant and Z denoting the partition function

?

i=A,B,1,...,N

The initial value of G(r,r?,t) is given by

(6)

?···? =1

Z

?

[d pidri]e−βH···.

(7)

Z =

?

[d pidri]e−βH.

(8)

G(r,r?,0) = Vδ(r − r?)?δ(r − rAB)? = δ(r − r?)g(r)

in terms of the radial distribution function g(r) [1]. This result

accounts for the name time-dependent pair distribution func-

tion given to G(r,r?,t). In the long-time limit, the average in

Eq. (6) can be factored, yielding

(9)

lim

t→∞G(r,r?,t) =1

Vg(r)g(r?).

(10)

Let us also introduce the conditional distribution function

g(r,r?,t) ≡ G(r,r?,t)/g(r?),

(11)

which is proportional to the probability of finding a pair of

particlesseparatedbyr attimet,giventhattheywereseparated

by r?at time zero. The initial value and the long-time limit are

given by

g(r,r?,0) = δ(r − r?) and lim

The time-dependent pair distribution function obeys the

Liouville equation

t→∞g(r,r?,t) =1

Vg(r). (12)

d

dtG(r,r?,t) = iLG(r,r?,t),

(13)

and so does the conditional distribution function g(r,r?,t). In

the following, we shall rewrite this equation of motion using

the projection-operator formalism.

C. Projection-operator formalism

Here,wesummarizetheprojection-operatorformalismthat

is to be used in deriving the exact equation of motion for

G(r,r?,t). Let us consider time-correlation functions formed

with a set of dynamical variables {Ai(r)}:

Cij(r,r?,t) ≡ (Ai(r,t),Aj(r?,0)) ≡ V?Ai(r,t)Aj(r?,0)?.

(14)

Hereafter,theabsenceoftheargumentt impliesthatassociated

quantities are evaluated at time t = 0. For example, we shall

denote the initial value of Cij(r,r?,t) as

Cij(r,r?) = (Ai(r),Aj(r?)) = V?Ai(r)Aj(r?)?.

Letusintroducetheprojectionoperatorontoasetofdynamical

variables {Ai(r)} via

?

(15)

PX(r) ≡

j,?

?

dr?

?

dr??(X(r),Aj(r?))C−1

j?(r?,r??)A?(r??).

(16)

Here, C−1

defined through

ij

denotes an element of the inverse matrix of Cij

?

?

?

dr??Ci?(r,r??)C−1

?j(r??,r?) = δijδ(r − r?).

(17)

The complementary operator is defined by Q ≡ I − P, with

I being the identity operator. One can easily show that the

operators P and Q are idempotent and Hermitian.

Once the projection operator satisfying the idempotency

and Hermitianity is introduced, it is straightforward to obtain

from the Liouville equation

d

dtCij(r,r?,t) = iLCij(r,r?,t)(18)

the following exact Zwanzig-Mori equation of motion [1]:

?

?

×C?j(r??,r?,τ).

Here, the frequency matrix is defined by

?

d

dtCij(r,r?,t) =

?

−

?

dr??i?i?(r,r??)C?j(r??,r?,t)

?

?

dr??

?t

0

dτ Ki?(r,r??,t − τ)

(19)

i?ij(r,r?) =

?

?

dr??(iLAi(r),A?(r??))C−1

?j(r??,r?), (20)

while the memory-function matrix reads

?

in terms of the fluctuating force given by

Kij(r,r?,t) =

?

?

dr??(Ri(r,t),R?(r??))C−1

?j(r??,r?),

(21)

Ri(r,t) = eiQLQtRi(r),

(22)

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PHYSICAL REVIEW E 83, 041201 (2011)

with

Ri(r) = iLAi(r) −

?

?

?

dr?i?i?(r,r?)A?(r?).

(23)

D. Exact equation of motion for the time-dependent pair

distribution function

Here we derive the exact equation of motion for G(r,r?,t)

using the result from the previous subsection. To this end, we

introduce dynamical variables

ρ(r,t) = δ(r − rAB(t)),

j(r,t) = vAB(t)δ(r − rAB(t)),

(24)

(25)

so that A0(r) = ρ(r), A1(r) = jx(r), A2(r) = jy(r), and

A3(r) = jz(r). Here, vAB= pA/m − pB/m denotes the rel-

ative velocity. Notice that the function G(r,r?,t) in which

we are interested is given by the (0,0) component of

Cij(r,r?,t),

C00(r,r?,t) = G(r,r?,t).

(26)

For later convenience, let us also introduce the following

notation:

Hx(r,r?,t) ≡ C10(r,r?,t),Hy(r,r?,t) ≡ C20(r,r?,t),

(27)

Hz(r,r?,t) ≡ C30(r,r?,t).

We first evaluate the elements Cij(r,r?) and C−1

staticensembleaveragesformedwiththevariablesinEqs.(24)

and (25), we obtain

?ρ(r)ρ(r?)? =1

?ρ(r)jβ(r?)? = ?jα(r)ρ(r?)? = 0,

?jα(r)jβ(r?)? =1

Here,α andβ refertox,y,orz,andv2≡ kBT/μisthethermal

velocity with the reduced mass μ = m/2. We therefore obtain

from the definition (15)

ij(r,r?). For

Vδ(r − r?)g(r?),

(28)

(29)

Vδαβv2δ(r − r?)g(r?).

(30)

C(r,r?) = δ(r − r?)g(r)

⎛

⎜

⎜

⎜

⎝

1

0

0

0

0

v2

0

0

0

0

v2

0

0

0

0

v2

⎞

⎟

⎟

⎟

⎠,

(31)

and from Eq. (17), the inverse is given by

C−1(r,r?) = δ(r − r?)

1

g(r)

⎛

⎜

⎜

⎜

⎝

1

0

0

0

00

0

0

0

0

1/v2

0

0

1/v2

01/v2

⎞

⎟

(32)

⎟

⎟

⎠.

We next calculate static ensemble averages involving time

derivatives to obtain the expression for i?(r,r?). Due to the

time-reversal symmetry, the following equations hold:

?[iLρ(r)]ρ(r?)? = 0,

?[iLjα(r)]jβ(r?)? = 0.

(33)

For the rest, we use the continuity equation

iLρ(r) = −vAB· ∇δ(r − rAB) = −∇ · j(r),

to obtain

?[iLρ(r)]jα(r?)? = −1

(34)

Vv2∇α[δ(r − r?)g(r)],

(35)

?[iLjα(r)]ρ(r?)? = −?jα(r)[iLρ(r?)]?

=

1

Vv2∇?

α[δ(r − r?)g(r)],

(36)

where we have used the Hermitian property of L, which can

easily be derived from the definition (5). Here and in the

following, ∇α and ∇?

∇ ≡ ∂/∂r and ∇?≡ ∂/∂r?. Let us notice

1

g(r?)∇α[δ(r − r?)g(r)] = ∇αδ(r − r?),

αrefer to the x, y, or z component of

(37)

whereas

1

g(r?)∇?

= ∇?

= −∇αδ(r − r?) − βδ(r − r?)∇αw(r),

where we have introduced the potential of mean force [1]

α[δ(r − r?)g(r)]

αδ(r − r?) + δ(r − r?)∇?

αlog[g(r?)]

(38)

w(r) ≡ −kBT logg(r).

(39)

Using the results so far, we obtain from Eq. (20)

i?(r,r?) =

⎛

⎜

⎜

⎜

⎝

0

−∇xδ(r − r?)

0

0

0

−∇yδ(r − r?)

0

0

0

−∇zδ(r − r?)

0

0

0

−v2∇xδ(r − r?) − βv2δ(r − r?)∇xw(r)

−v2∇yδ(r − r?) − βv2δ(r − r?)∇yw(r)

−v2∇zδ(r − r?) − βv2δ(r − r?)∇zw(r)

⎞

⎟

⎟

⎟

⎠.

(40)

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Now we can calculate R(r) from Eq. (23). Using the

continuity equation (34), we have

tR(r) = (0

Rx(r)

Ry(r)

Rz(r)),

(41)

where the superscript “t” denotes the transpose, and

Rα(r) = iLjα(r) + v2∇αρ(r) + βv2ρ(r)∇αw(r).

One therefore obtains from Eqs. (21) and (32)

(42)

K(r,r?,t)

=

⎛

⎜

⎜

⎜

⎝

0

0

0

0

000

Kxx(r,r?,t)

Kyx(r,r?,t)

Kzx(r,r?,t)

Kxy(r,r?,t)

Kyy(r,r?,t)

Kzy(r,r?,t)

Kxz(r,r?,t)

Kyz(r,r?,t)

Kzz(r,r?,t)

⎞

⎟

⎟

⎟

⎠,

(43)

with

Kαβ(r,r?,t) ≡V

v2?Rα(r,t)Rβ(r?)?/g(r?).

(44)

The initial value of Kαβ(r,r?,t) is evaluated in the Appendix,

and the result under Kirkwood’s superposition approximation

for a triple-density correlation function [see Eqs. (A36) and

(A37)] is given in Eq. (A38).

Summarizing the results so far, we obtain the following

set of exact equations of motion involving G(r,r?,t) and

Hα(r,r?,t):

d

dtG(r,r?,t) = −∇ · H(r,r?,t),

(45)

d

dtHα(r,r?,t)

= −v2∇αG(r,r?,t) − βv2G(r,r?,t)∇αw(r)

−

β

?

?

dr??

?t

0

dτ Kαβ(r,r??,t − τ)Hβ(r??,r?,τ). (46)

Combining these two equations, one obtains

d2

dt2G(r,r?,t) = v2∇2G(r,r?,t) + βv2∇ · {G(r,r?,t)∇w(r)}

?

×Hβ(r??,r?,τ).

The equation just derived is still not in a useful form, and

further manipulation shall therefore be performed in the

following subsection.

+

α,β

?

dr??

?t

0

dτ ∇αKαβ(r,r??,t − τ)

(47)

E. Further manipulation

The manipulation we will do here is to eliminate Hβ in

favor of G from the last term in Eq. (47). This is possible by

exploiting the isotropy of the system, according to which the

current density can be decomposed into the longitudinal and

transverse components

j(r,t) = jL(r,t) + jT(r,t),

(48)

satisfying

∇ × jL(r,t) = 0,

∇ · jT(r,t) = 0.

(49)

(50)

The longitudinal component jL,α(r,t) can be written in terms

of a scalar function ψ(r,t) as

jL,α(r,t) = ∇αψ(r,t).

(51)

Sincethedensityfluctuationcouplesonlywiththelongitudinal

current fluctuation in the isotropic system, there holds

Hα(r,r?,t) = (jα(r,t),ρ(r?,0)) = (jL,α(r,t),ρ(r?,0))

= ∇α?(r,r?,t),

where in the final equality we have introduced the time-

correlation function

(52)

?(r,r?,t) ≡ (ψ(r,t),ρ(r?,0)).

(53)

It then follows from Eq. (45) that

d

dtG(r,r?,t) = −∇2?(r,r?,t).

Thus, Hα and G are connected via the function ? through

Eqs. (52) and (54).

To see a more direct connection, it is more convenient

to work in the Fourier space. Let us introduce the Fourier

transform (FT) of G(r,r?,t) via

?

and its inverse relation by

?

TheFTsofotherfunctionsshallbedefined similarly.Onethen

obtains from Eqs. (52) and (54)

(54)

F(k,k?,t) =

dr eik·r

?

dr?e−ik?·r?G(r,r?,t),

(55)

G(r,r?,t) =

1

(2π)6

dke−ik·r

?

dk?eik?·r?F(k,k?,t).

(56)

Hα(k,k?,t) = −ikα?(k,k?,t),

d

dtF(k,k?,t) = k2?(k,k?,t),

(57)

and hence

Hα(k,k?,t) = −ikα

k2

d

dtF(k,k?,t).

(58)

Let us use this result to rewrite the last term in Eq. (47).

Using the product rule

?

=

(2π)3

the FT of the last term in Eq. (47) is found to be given by

??

?

dreik·r

?

dr?e−ik?·r???

?

dr??A(r,r??)B(r??,r?)

?

1

dk??A(k,k??)B(k??,k?),

(59)

FT of

α,β

?

dr??

?t

dτ KL(k,k??,t − τ)d

0

dτ ∇αKαβ(r,r??,t − τ)Hβ(r??,r?,τ)

?

= −

dk??

?t

0

dτF(k??,k?,τ),

(60)

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PHYSICAL REVIEW E 83, 041201 (2011)

where we have introduced

KL(k,k?,t) ≡

?

α,β

kαk?

(k?)2Kαβ(k,k?,t).

β

(61)

Equation (47) can then be written as

d2

dt2G(r,r?,t)

= v2∇2G(r,r?,t) + βv2∇ · {G(r,r?,t)∇w(r)}

−

0

with the memory kernel KL(r,r?,t), which is an inverse FT of

KL(k,k?,t).

Finally, we notice that all of the differential operators in

Eq. (62) do not involve r?, so this equation holds also for

the conditional distribution function g(r,r?,t) introduced in

Eq. (11).

?

dr??

?t

dτ KL(r,r??,t − τ)d

dτG(r??,r?,τ),

(62)

III. DISCUSSION

In this section, we discuss implications of the exact

Zwanzig-Mori equation (62) in connection with the previous

related work and with the corresponding equation for the van

Hove self-correlation function. We start from the short-time

expansion of G(r,r?,t),

G(r,r?,t) = G(r,r?,0) +t2

in which only even powers of time appear, due to the time-

reversal symmetry. The initial value is given in Eq. (9). For the

initial second time derivative, one obtains by setting t = 0 in

Eq. (62) and using the definition (39) for the potential of mean

force

¨G(r,r?,0) = v2∇2G(r,r?,0) − v2∇ · {G(r,r?,0)∇ log[g(r)]}

= v2∇2{δ(r − r?)g(r)} − v2∇ · {δ(r − r?)∇g(r)}.

2

¨G(r,r?,0) + O(t4),

(63)

(64)

This expression can be rewritten in a more symmetrical form

with respect to r and r?as

¨G(r,r?,0) = −v2∇ · ∇?[δ(r − r?)g(r?)],

showing that¨G(r,r?,0) is negative definite when viewed as a

matrix with indices r and r?. This result agrees with the one

derived in the previous work [1,10]. The short-time expansion

for g(r,r?,t) can be obtained in a similar manner, with the

result

g(r,r?,t) = g(r,r?,0) +t2

in which g(r,r?,0) = δ(r − r?) and

¨ g(r,r?,0) = v2∇2δ(r − r?) + βv2∇ · [δ(r − r?)∇w(r)].

(65)

2¨ g(r,r?,0) + O(t4),

(66)

(67)

We next consider the long-time diffusive regime. Let us

introduce the following Markovian approximation for the

memory kernel

KL(r,r?,t) ≈v2

Dr

δ(r − r?)δ(t),

(68)

intermsoftherelativediffusionconstantDr.Onethenobtains

from Eq. (62) for g(r,r?,t) [see the comment below Eq. (62)]

d2

dt2g(r,r?,t) = v2∇2g(r,r?,t) + βv2∇ · [g(r,r?,t)∇w(r)]

−v2

Dr

d

dtg(r,r?,t).

(69)

By introducing

Dr=kBT

ζr

and ¯β =ζr

μ,

(70)

Eq. (69) can be rewritten as

¯β−1d2

dt2g(r,r?,t) +d

=kBT

ζr

dtg(r,r?,t)

∇ · [∇g(r,r?,t) + βg(r,r?,t)∇w(r)].

(71)

This equation is formally identical to the so-called telegraph

equation [13]. The telegraph equation is conventionally de-

rived startingfromtheFokker-Planck equation [13],butitalso

follows from the Liouville equation, as we have demonstrated

here.

If one further neglects the inertia term in Eq. (69), one

obtains the following equation, which takes the form of

Smoluchowski’s equation

d

dtg(r,r?,t) = Dr{∇2g(r,r?,t) + β∇ · [g(r,r?,t)∇w(r)]}.

(72)

This equation is the one proposed by Haan with intuitive

arguments, and has been shown to satisfactorily reproduce

the simulation result for g(r,r?,t) [11].

Finally,letuscomparetheequationsforthetime-dependent

pair distribution function g(r,r?,t) with the corresponding

equations for the more familiar van Hove self-correlation

function Gs(r,t) [1] to highlight the new feature in the former

equations. The van Hove self-correlation function, defined

by Gs(r,t) ≡ ?δ(r − [rA(t) − rA(0)])?, is associated with the

probability that a single tagged particle moves to a position

that is separated by r from its initial position during the

time t. The initial value and the long-time limit are given by

Gs(r,0) = δ(r) and limt→∞Gs(r,t) = 1/V, which are to be

compared with those for g(r,r?,t) given in Eq. (12). Using the

similar projection-operator technique presented in Sec. II, one

obtains the following Zwanzig-Mori equation for Gs(r,t) [1]:

d2

dt2Gs(r,t) = v2

s∇2Gs(r,t)

?

−

dr??

?t

0

dτ Ks,L(r−r??,t−τ)

d

dτGs(r??,τ),

(73)

where v2

ory kernel, which is to be compared with Eq. (62). From this

equation, the short-time expansion is given by

s= kBT/m and Ks,L(r,t) is the corresponding mem-

Gs(r,t) = Gs(r,0) +t2

2

¨Gs(r,0) + O(t4),

(74)

041201-5