Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond

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5.2 Hydrodynamic correlation functions 221

the tagged-particle velocity correlation obtained from the simulations of a Lennard-Jones

ﬂuid characterized by the appropriate interaction potential (see eqn. (1.2.117)). For this

system the time t, temperature T , and density ρ are expressed as dimensionless quantities

using the following deﬁnitions:

∗

with τ



mσ





1/2

, (5.2.28)

∗



and n

∗

= nσ

. (5.2.29)

Finally, for the inter-diffusion constant D

of a two-component system we can obtain a

similar Green–Kubo relation starting from eqn. (5.2.17). In this case, using the continuity

equation (5.1.58) for the concentration variable, we obtain the Green–Kubo relation

3χ



+∞

dt ϕ

(t), (5.2.30)

where ϕ

is the correlation of the current j

deﬁned microscopically in eqn. (5.1.59).

The two-component mixture reduces to a one-component system if the two species of

the mixture are identical. In this case the above result for the inter-diffusion constant

links to tagged-particle diffusion on taking N

=1 and N

= N −1. In the thermodynamic

limit x

→1 and x

→1. In this case, χ

→1 and the Green–Kubo relation (5.2.31)

for D

becomes identical to the relation (5.2.27) for the self-diffusion constant D

of a

one-component system.

In the above treatment we have expressed the various transport coefﬁcients as time inte-

grals of currents in the hydrodynamic correlation functions. Now, using the hypothesis

that these correlation functions are identical to their microscopic counterparts, the differ-

ent currents involved in the Green–Kubo relations are obtained from the corresponding

microscopic expressions. In this way the components of the stress correlation functions in

eqns. (5.2.23) and (5.2.25) are obtained from the Fourier transform of the expression in

eqn. (5.1.12). Similarly, the correlation of the current j

in the expression (5.2.31) for the

inter-diffusion constant is obtained from the microscopic expression (5.1.59). This results

in expressions for the transport coefﬁcients in terms of the microscopic interaction param-

eters of the system and provides a very useful tool for computing theoretically the transport

coefﬁcients in computer MD simulation.

An immediate implication of the Green–Kubo relations is the validity of linear constitu-

tive relations and the nature of the corresponding transport coefﬁcients in certain extreme

cases. For exponentially decaying current–current correlation functions the transport coef-

ﬁcient is simply proportional to the corresponding relaxation time and is well behaved.

However, it has been observed from computer MD simulations (Alder and Wainwright,

1967, 1970) that the velocity correlation function decays in a power-law form. In gen-

eral let us consider the velocity correlation function ϕ

(t) ∼ A

−d/2

in a d-dimensional

222 Dynamics of collective modes

system. This has been termed the long-time tail in the literature. That it has important

implications regarding the transport coefﬁcients of the ﬂuid can be seen directly from the

Green–Kubo relations. The transport coefﬁcient λ

is obtained as



(

t)d

t =

1 − d/2



1−d/2

. (5.2.31)

For d ≤ 2 the contribution from the upper limit of the integration becomes diverging. This

implies that conventional hydrodynamics will be invalid in this situation. The origin of

such effects, namely the power-law decay of correlation, comes from the collective effects

in the ﬂuid over semi-hydrodynamic length scales. This will be discussed in Chapter 6.

Bare transport coefﬁcients

In the classical statistical-mechanical approach, a microscopic description of the dynam-

ics is obtained from eqn. (1.1.39) for the time evolution of the one-particle distribution

function f (x, v, t). The effects of the interaction between the particles on the dynamics

are accounted for here through the collision term in this equation. In order to obtain the

transport coefﬁcient for a given ﬂuid in terms of its microscopic properties, i.e., the inter-

action potential between the particles, the Boltzmann equation is a suitable starting point.

These so-called bare transport coefﬁcients correspond to the short-time dynamics due to

uncorrelated binary collisions. Much progress on this has been achieved for the hard-sphere

interaction potential given by

(r) =

⎧

⎨

⎩

∞ for r ≤ σ,

0forr >σ,

(5.2.32)

where σ is the hard-sphere diameter. In a hard-sphere ﬂuid the local conserved densities of

mass, momentum, and energy {n(r, t), g(r, t), e(r, t)} are obtained from the moments of

the one-particle distribution function f (v, x, t):

n(r, t) =



dp f

(1)

(r, p, t),

g(r, t) =



dppf

(1)

(r, p, t),

e(r, t) =



(1)

(r, p, t). (5.2.33)

A general strategy for computation of the transport coefﬁcients is to obtain the hydro-

dynamic balance equations for the dynamics of the conserved densities. This involves

taking moments of eqn. (1.1.39). Alternatively, the Green–Kubo expression for the trans-

port coefﬁcient is obtained using the microscopic expressions for the stress tensor. The

kinetic theory of the N -particle system (controlled by the Liouville operator L) is approx-

imated here in terms of the two-body dynamics. For the hard-sphere system this involves

approximating L in terms of the two-body hard-sphere collision operator (Résibois and de

5.2 Hydrodynamic correlation functions 223

Leener, 1977). The Boltzmann transport coefﬁcients for the hard-sphere system, namely

the shear viscosity η

, the thermal conductivity λ

, and the self-diffusion coefﬁcient D

of a tagged particle are obtained as



√



σ t

, (5.2.34)



√



σ t

, (5.2.35)



√



σ t

, (5.2.36)

where t

= σ/v

is the time taken by the particle with thermal velocity v

= 1/

√

βm to

cover the length of the hard-sphere diameter σ and n

is the number of particles per unit

volume.

In the case of hard spheres the collisions are instantaneous and hence even at high

densities multi-particle collisions are unlikely. Transport coefﬁcients at high densities are

therefore computed by improving the collision term in the Boltzmann equation with the

Enskog approximation (Résibois and de Leener, 1977). The latter involves (a) account-

ing for the higher rate of collision in the dense system in terms of the pair correlation

function g(r) and (b) computing the transfer of ﬂux in the transport through collision

between the particles. In a dense system the collisional contribution to the transport process

is comparable to and even dominant over that from actual movement of the ﬂuid particles.

Using the Enskog model, the corresponding Boltzmann transport coefﬁcients are corrected

(Hirschfelder et al., 1954)to

= 4ϕ



+ 0.8 +0.761Y



, (5.2.37)

= 4ϕ



+ 1.20 +0.755Y



, (5.2.38)

4ϕ

, (5.2.39)

where the quantities with subscript E are the Enskog values, ϕ = πnσ

/6 is the pack-

ing fraction, and Y = 4ϕg(σ ), where g(σ ) is the hard-sphere pair correlation function

at contact. The different terms in the above expressions for the transport coefﬁcients can

be understood from the corresponding Green–Kubo expressions. Thus, for example, the

shear viscosity is obtained from the time correlation of the off-diagonal stress tensor σ

which has a kinetic and a potential contribution (see eqn. (5.1.12)). Hence the correlation

function involving products of two σ s has three parts: a purely kinetic contribution, cor-

responding to the transfer of momentum from free streaming of the particles; a potential

224 Dynamics of collective modes

term from the interaction of the particles, giving the purely collisional contribution to the

transport; and a cross term. At liquid densities the collisional contribution is most domi-

nant (the third term on the RHS of eqn. (5.2.37)). The bulk viscosity ζ

, on the other hand,

makes only a collisional contribution since the kinetic part is diagonal and is subtracted

out in the expression (5.2.25).

The Enskog approximation to the transport coefﬁcients works well for liquids up to mod-

erate densities n

≤ 0.3. The Enskog expression for the transport coefﬁcients does not

conform to the Stokes–Einstein relation. On approaching the freezing point the discrepancy

with computer-simulation results grows. The Enskog value of shear viscosity becomes less

than that from simulations almost by a factor of 2 at the freezing transition point. The

self-diffusion coefﬁcient, on the other hand, shows an opposite trend. The product D

calculated from simulations is almost constant for moderate densities n

≥ 0.3 and

the value of this constant is close to that predicted from the Stokes–Einstein (SE) relation

(within 10%). Thus, while at low densities the SE relation is violated, it holds for moderate

liquid densities. In the supercooled liquid the discrepancies with Enskog theory grow much

more rapidly, and the SE relation is again violated. At high density the disagreement of the

shear viscosity with simulations stems from the fact that the latter has large contributions

from slowly decaying long-time tails of the stress correlation function. This applies to the

viscosities. The agreement between theory and simulation is much better in the case of the

thermal conductivity. The origin of the long-time tails can be understood in terms of cor-

related dynamics in the ﬂuid, which can be understood in terms of mode-coupling theories

to be discussed later.

For a general interaction potential multi-particle collisions are likely to contribute sub-

stantially to the transport process at high densities. Calculation of the transport coefﬁcients

in such cases is more involved and has been done along phenomenological lines with

ad-hoc extension of the Enskog model in terms of the so-called modiﬁed Enskog the-

ory or using numerical methods for computing the collision integrals (Fitts, 1966; Curtiss,

1967). Corrections to the short-time transport coefﬁcients at higher densities have also been

obtained by improving the Boltzmann equation in the corresponding situations. The trans-

port coefﬁcient is obtained in terms of a density expansion similar to the virial expansion

for the equation of state of the ﬂuid. The calculation involves treating the pair distribution

function in the collision integral as a functional of the one-particle distribution function.

The result of this rather complicated technical development is that we obtain the second-

order term ρ

ln ρ for the transport coefﬁcient λ(ρ) as nonanalytic (Ernst et al., 1969;

Curtiss, 1967),

λ(ρ

) = λ



1 + a

+ a

ln ρ

+···

, (5.2.40)

where ρ

is the equilibrium density and the a

are functions of temperature only. The

origin of such nonanalytic behavior is a consequence of collective effects coming from

semi-microscopic length and time scales. This will be discussed later, in Chapters 6–8.

5.3 Linear ﬂuctuating hydrodynamics 225

5.3 Linear ﬂuctuating hydrodynamics

The mechanical description of the liquid consisting of N particles involves many degrees of

freedom in the thermodynamic limit. The general strategy adopted in developing statistical-

mechanical models for the dynamics is to focus on a small set of dynamic variables with

time scales of variation much longer than those of the vast number of other degrees of

freedom for the system. These slow modes are the hydrodynamic modes. The rest of the

degrees of freedom are treated as noise. The simplest example of such a description of the

dynamics is the case of Brownian motion discussed in Chapter 1. In that case the relevant

variable, namely the velocity of the heavy particle, is slow due to its large inertia relative

to the surrounding ﬂuid particles. The origin of the slow mode can be different in different

physical systems. We discuss in the following a general scheme for obtaining the dynamics

for these slow modes.

The equation of motion for a set of variables {

} is usually obtained through the stan-

dard projection-operator scheme of Zwanzig (1961) and Mori (1965a, 1965b). This involves

a projection operator P, which is deﬁned so as to “project” any dynamic variable along the

chosen set in terms of a suitably deﬁned scalar-product operation. The effects of the rest of

the dynamic variables are treated as noise f

and spanned by the operator Q = 1 −P.The

deduction of the equations of motion for the variables

with the help of P is described

in the appendix of this chapter. We present below a somewhat different technique (Ma and

Mazenko, 1975; Mazenko, 2006a) for obtaining the equation of motion. This formalism is

referred to as the technique of ﬂuctuating hydrodynamics at the level of the linear dynam-

ics of slow modes. This is particularly useful in ﬁnally obtaining the nonlinear Langevin

equations to be discussed in the next chapter. These nonlinear equations form the basis

for constructing relevant theoretical models for the slow dynamics in the dense ﬂuid near

freezing and in the supercooled state.

5.3.1 The generalized Langevin equation

Let us consider a classical system of N particles. Let {

} for i = 1,...,n denote the

set of n collective densities that are functions of the phase-space variables {r

, r

,...,r

;

, p

,...,p

}. By suitably deﬁning the

we maintain the condition that



(t)=0. (5.3.1)

The time rates of variation of the collective variables are given by

∂

(t)

∂t



j,α



∂

∂ r

˙r

∂

∂p

˙p



, (5.3.2)

where p

and r

, respectively, denote the jth component of the momentum and the position

coordinates of the αth particle in the system. Using Hamilton’s equations of motion for the

particles in terms of the derivatives of the Hamiltonian H ,

226 Dynamics of collective modes

˙r

(t) =

∂ H

∂p

, (5.3.3)

˙p

(t) =−

∂ H

∂r

, (5.3.4)

the equation of motion of

is expressed in terms of Poisson’s bracket (Dzyloshinski and

Volvick, 1980)as

∂

(t)

∂t

, H}. (5.3.5)

The Poisson bracket {A, B} between two dynamic variables A and B is deﬁned as

{A, B}=



i,α



∂ A

∂r

∂ B

∂p

−

∂ A

∂p

∂ B

∂r



. (5.3.6)

Thus the equation of motion for the collective variable

φ(t) is written in terms of the

Liouville operator,

∂

(t)

∂t

=−{H,

}≡iL

(t), (5.3.7)

which can be formally integrated to obtain

(t) = exp(iLt)

, where

(t = 0).

The analysis which will follow is facilitated in terms of the Laplace transform deﬁned

(Abramowitz and Stegun, 1965)as

(z) =−i



∞

dt e

izt

(t) (5.3.8)

with Im(z)>0. On taking the Laplace transform of eqn. (5.3.7) we obtain the relation

(z) = R(z)

, (5.3.9)

where the resolvent operator R(z) is given by

R(z) =[z + L]

−1

. (5.3.10)

Equivalently, we obtain

(z) =

− L

(z). (5.3.11)

A key step in formulating the dynamics of the collective modes involves determining the

effect of the operator L on the Laplace-transformed

(z). In general the net result of

this operation can give rise to a complicated dynamics. The crucial approximation in this

analysis is approximating the quantity L

(z) as a sum of two parts.

(a) The ﬁrst part is linearly proportional to the ﬁelds

(b) The rest of the time evolution of ϕ

is treated as a noise f

(z).

5.3 Linear ﬂuctuating hydrodynamics 227

We write schematically

−L

(z) ≈ K

(z)

(z) +if

(z), (5.3.12)

where K

(z) is a kernel function that represents the projection of L

(z) on the subspace

of the collective modes

(z). Here and in the rest of this section we adopt the Einstein

summation convention that repeated indices are summed over. The leftover

contribution or the noise denoted by f

(z) in the second term on the RHS of eqn. (5.3.12)

behaves like a dynamic variable similar to

(z). The description of the dynamics of the

many-particle system in the above scheme involves two sets of variables, {φ

} and {f

which are independent at t = 0. At later times they mix through the nonlinearities in the

dynamical equations. Thus φ

(z) has components along both φ

and f

(z).

On taking the equilibrium average of (5.3.12) we obtain that the noise part f

(z) is zero

on average,

 f

(z)=0. (5.3.13)

The above result follows from two observations.

(a) From (5.3.1) it follows that 

(z)=0.

(b) We have, from the deﬁnition of the Liouville operator, the condition

L

(z)=−i



∞

dt e

izt

(−i)



(t)=0. (5.3.14)

Equation (5.3.14) is a consequence of the time translational invariance of the equilibrium

average. A second requirement on the noise is that f

(z) has no projection on the mode

(t) at t = 0,



(z)=0. (5.3.15)

The equation of motion for the collective modes maintains the two properties (5.3.13) and

(5.3.15). Using the ansatz (5.3.12) in (5.3.11), we obtain

zδ

− K

(z)

(z) = φ

+if

(z). (5.3.16)

Note that we have dropped the hat on the φ to imply that this approximate equation

(obtained with the use of eqn. (5.3.12)) is satisﬁed for a coarse-grained form of the col-

lective variable

φ. The inverse Laplace transform of (5.3.16) gives the equation of motion

for φ

(t):

∂φ

∂t



t K

(t −

t )φ

(

t ) = f

(t). (5.3.17)

The dynamics of the many-particle system is now described in terms of (a) a set of

slow modes denoted by {φ(t)}, which are coarse-grained functions with smooth space–

time variations and not dependent on phase-space variables; and (b) the noise f

(t), which

varies over time scales much shorter than that of φ

(t) and is treated as random with average

equal to zero. The dissipative parts in the generalized Langevin equation of motion for the

228 Dynamics of collective modes

(t) are related to the correlation of the noise at two different times. This is similar to the

Langevin equation introduced in Chapter 1 in reference to the discussion of the Brownian

motion of a heavy particle moving in a liquid. The approximation of separating the time

scales in the dynamics applies in that case because of the very large mass ratio between the

Brownian particles and the surrounding liquid particles.

The nature of the dependence of the dynamics on the φ

is obtained by focusing on the

kernel function K

. In general the kernel function K

can be divided into two parts, static

and dynamic. For the Laplace-transformed quantity we write

(z) = K

(s)

+ K

(d)

(z). (5.3.18)

(s)

is independent of z and is taken as a static contribution. The second part, with a

frequency dependence, is classiﬁed as the dynamic contribution K

(d)

. Using the above

separation, the Langevin equation reduces to

∂φ

∂t

+i K

(s)

(t −

t )φ



t K

(d)

(t −

t )φ

(

t ) = f

(t). (5.3.19)

We now analyze the kernel matrix K

further by expressing it in terms of the correlation

functions of the collective modes.

The kernel function

In evaluating the kernel function or the memory function K

we adopt the strategy of

expressing it in terms of the correlation function of the slow variables. The correlation of

and

(t) is deﬁned as

(t) =

(t). (5.3.20)

The Laplace transform of the correlation functions

(z) =−i



∞

dt e

izt

(t) (5.3.21)

is related to the resolvent R(z) deﬁned above in eqn. (5.3.10) through the relation

(z) =

(z)=

R(z)

. (5.3.22)

We now express the memory function K

in terms of the correlation functions C

.This

involves multiplying (5.3.16) by

and taking the equilibrium average:

(z) = S

+ K

(z), (5.3.23)

where S

=

 is the static or the equal-time correlation function among the slow

variables. Note that in reaching (5.3.23) we have used the ansatz (5.3.15) of zero projection

of the noise on the slow modes. On the other hand, on multiplying eqn. (5.3.11) by

and

taking the equilibrium average we obtain

(z) = S

−

R(z)L

. (5.3.24)

5.3 Linear ﬂuctuating hydrodynamics 229

In the second term on the RHS of (5.3.24) we have used the identity

LR(z) = L(z + L)

−1

= R(z)L. (5.3.25)

On comparing (5.3.23) and (5.3.24) we obtain

(z) =−

R(z)L

. (5.3.26)

On multiplying (5.3.26) by z and substituting (5.3.23) on the LHS for zC

(z) we obtain



−

R(z)L

=−

zR(z)L

=−

LR(z)L

, (5.3.27)

where we have used the identity zR(z) = 1 − LR(z). Since the Liouville operator L as

deﬁned in (5.3.3) is a derivative operator, the following property holds for the equilibrium

average for two variables A and B:

ALB=−{L A}B. (5.3.28)

The above relation is also a consequence of the time translational invariance of the equi-

librium distribution. The relation (5.3.27) now reduces to the following expression for the

memory function K in terms of the correlation function C:

=−

−

}R(z){L

}

− K

}R(z)

. (5.3.29)

The RHS of (5.3.29) still contains the memory function. This is eliminated formally by

inverting the relation (5.3.26) in terms of the inverse C

−1

of the correlation matrix C,

giving the relation

=−

−

}R(z){L

}

R(z)L

−1

(z)

}R(z)

. (5.3.30)

The ﬁrst term on the RHS of (5.3.30) is independent of z and is taken as a static contribution

(s)

to the memory function K as deﬁned in eqn. (5.3.18). The second and third terms with

frequency dependences are classiﬁed as the dynamic contribution K

(d)

in eqn. (5.3.18).We

therefore obtain the following deﬁnitions:

(s)

=−

, (5.3.31)

(d)

=−{L

}R(z){L

}

+{L

}R(z)

C

−1

(z)

R(z){L

}. (5.3.32)

We treat the static and the dynamic part of the kernel function K separately in the

following.

230 Dynamics of collective modes

The static part

Since the static part K

(s)

of K

(z) is a constant, we obtain by taking the inverse Laplace

transform

(s)

(t) = 2i

δ(t), (5.3.33)

where we denote 

≡

S

−1

. This part of the kernel can be expressed in terms

of static correlation functions. To compute this we express, using (5.3.7), the Liouville

operator L in terms of the Poisson bracket to obtain

−

=−i 

{H,

}, (5.3.34)

=−

Tr e

−β H

{H,

}, (5.3.35)

where the equilibrium average is being considered in the canonical ensemble with partition

function Z and the operation “Tr” denotes the classical trace (McQuarie, 2000), deﬁned as

Tr ≡

∞



N =0

N !



...



...



. (5.3.36)

Since the Poisson brackets involve derivative operators, we have the relation

−β H

{H,

}=−β

−1

−β H

}. (5.3.37)

Using this result, we obtain from eqn. (5.3.34) that

−

=−

iβ

−1

−β H

}

iβ

−1





∂

∂r

−β H



∂

∂p

−



∂

∂p

−β H



∂

∂r



. (5.3.38)

Now, partially integrating with the derivative of the Boltzmann factor e

−β H

simpliﬁes the

above expression to

−

=−

iβ

−1

Tr e

−β H



−

∂

∂r



∂

∂p



∂

∂p



∂

∂r



= iβ

−1

{

}. (5.3.39)

On denoting the Poisson bracket {

} between the slow variables as a function of

the

≡{

}, (5.3.40)

we obtain from the static part of the memory function the contribution



= iβ

−1

, (5.3.41)

where Q

=Q

 is the equilibrium average of the Poisson bracket Q