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

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5.3 Linear ﬂuctuating hydrodynamics 231

The dynamic part

The dynamic part represented in eqn. (5.3.32) is a complicated expression and further

analysis of this involves dealing with the many-body dynamics. First, we analyze the above

formal expression to prove that the contribution to K

(d)

comes only from the nonlinear

part of L

. To demonstrate this, we separate the linear and nonlinear parts of the latter as

follows:

= a

, (5.3.42)

with

denoting the contributions involving (a) nonlinear products of the

and (b) those

which cannot be expressed in terms of the

. On substituting (5.3.42) into the expression

(5.3.32) for the part L

we obtain

(d)

=−{L

}R(z){a

}

+

R(z){a

}C

−1

(z){L

}R(z)

. (5.3.43)

On collecting the coefﬁcients of a

on the RHS we obtain the term



{L

}R(z)

−

R(z)

C

−1

(z){L

}R(z)



. (5.3.44)

The coefﬁcient of a

given above vanishes. To prove this, we note that by deﬁnition

we have



R(z)

C

−1

(z) ≡ C

(z)C

−1

(z) = δ

. (5.3.45)

Using this relation, the two terms within the square brackets in (5.3.44) cancel out. Simi-

larly, on expressing L

in the rest of the RHS of (5.3.43) with the expansion (5.3.42) it

follows in an exactly similar manner that the coefﬁcients of a

vanish. The dynamic part

of the kernel is therefore entirely controlled by the

(d)

=−

R(z)

+

R(z)

C

−1

(z)

R(z)

. (5.3.46)

The above analysis shows that the linear part of L

does not contribute to K

(d)

and the

latter is controlled by the nonlinearities in the dynamics as well as ﬂuctuations not included

in the

. This contribution to the kernel K

is termed the one-particle irreducible part. The

linear part of the dynamics, on the other hand, is included in the static contribution to K

Using this scheme of separating the kernel function into a static and dynamic part, the

Langevin equation (5.3.17) is now written as

∂φ

∂t

+i 

(t) +



t K

(d)

(t −

t)φ

(

t) = f

(t). (5.3.47)

From eqn. (5.3.47) an important relation between the noise and the dynamic part of the

memory function follows:

 f

(t) f



)=K

(d)

(t − t



. (5.3.48)

232 Dynamics of collective modes

The above relation is termed the second ﬂuctuation–dissipation theorem (FDT) and follows

from the assumption (5.3.15) that the noise has no projection onto the initial value of

(t).

The proof is given in Appendix A5.2. In the linear ﬂuctuating hydrodynamic approach dis-

cussed in this chapter the relation (5.3.48) between the noise correlation and the dynamic

part of the memory function forms the basis for dealing with the latter. We assume that

the dynamic part of the memory function is determined by the noise and represents the

dynamics over a time scale that is assumed to be much shorter than that of slow modes.

(d)

is thus used to deﬁne short-time kinetic coefﬁcients for the system. The bare trans-

port coefﬁcients determine the strength of the noise correlation in the system through the

so-called second FDT and are used as an input in the theory.

In the above discussion we have actually bypassed evaluating the dynamic part of the

memory function and identiﬁed the latter with the physical (linear) transport matrix in the

dissipative equations for the slow modes. Direct evaluation of the dynamic part using what

is termed as the mode-coupling approximation, has been done by Résibois and de Leener

(1966), de Leener and Résibois (1966), and Kawasaki (1970) (see also Andersen (2002,

2003a, 2003b) and Forster and Martin (1970) for further discussion on the dynamic part).

The linear equation for the slow modes gives rise to a linear equation for the correla-

tion function with a frequency-dependent memory function that satisﬁes the Green–Kubo

relation (discussed in Chapter 1,seeeqn. (5.2.23)) and is related to the noise through the

second ﬂuctuation–dissipation relation. In Appendix A7.4 we present, using the standard

Mori–Zwanzig projection operator scheme, an alternative scheme for evaluating this mem-

ory function, in particular, for the dynamics of density correlation functions. This approach

has traditionally been adopted in the literature for obtaining the simple mode-coupling

model (Bengtzelius et al., 1984; Götze, 2009) for glassy dynamics. In the present book we

will obtain the mode-coupling model starting from a set of nonlinear stochastic equations.

In the next chapter we demonstrate how to obtain a set of nonlinear equations of motion for

a chosen set of slow modes. The renormalization due to the nonlinearities in the equations

of motion is obtained using ﬁeld-theoretic techniques (Amit, 1999). The corresponding

renormalized theory due to the nonlinear dynamics of the collective modes gives rise in a

natural way to the self-consistent mode-coupling model. As we will see, the ﬁeld-theoretic

approach proves useful in understanding the full implications of the nonlinearities in the

many-body dynamics.

The Markov approximation

In dealing with the physical problem of the dynamics of the liquids we assume a complete

separation of time scale between the collective modes

and the f

. The fast modes or

the noise f

are assumed to be correlated over much shorter time scales than the

.This

implies that  f

(t) f

 decays to zero much faster than does 

(t)

. Thus  f

(t) f

 is

very sharply peaked near t = 0. The Markov approximation refers to the Markov process

(Markov, 1907) in which the dynamic evolution of the system in every step is determined

by the previous step only. In the present context, applying this approximation to the mem-

ory function amounts to assuming the absence of memory in the long-time limit. Thus in

5.3 Linear ﬂuctuating hydrodynamics 233

the Markov approximation the noise is in fact assumed to be delta-function correlated and

deﬁned in terms of a matrix L

which consists of the bare transport coefﬁcients,

 f

(t) f



)=2β

−1

δ(t − t



). (5.3.49)

is the white noise and its correlation is proportional to the temperature. From (5.3.48)

the dynamic part K

(d)

of the memory function is identiﬁed as

(d)

(t − t



) = 2β

−1

δ(t − t



). (5.3.50)

Using (5.3.41) and (5.3.50) in (5.3.47) we obtain the linear Langevin equation for the slow

mode φ

including the white noise f

∂φ

∂t

+ β

−1

−Q

+ L

−1

(t) = f

(t). (5.3.51)

The correlation of the white noise f

is given in (5.3.49) by the matrix L

. We can identify

the ﬁrst and second terms on the RHS of (5.3.51) with the time-reversal behavior of the

dynamics.

(i) From the deﬁnition of the Poisson bracket it follows that Q

(−t) =−



(t),

where 

denotes the time-reversal property of the slow mode

, i.e.,

(−t) =



(t). Therefore the second term in (5.3.51) remains invariant under time reversal

and represents the reversible part of the dynamics. In the linearized Langevin approach

the average of Q

gives the static part of the memory function.

(ii) Assuming that the matrix L

satisﬁes under time reversal the property L

(−t) =



(t), the third term on the LHS of (5.3.51) breaks the time-reversal symme-

try of the equation of motion and hence represents dissipation. This dissipative term

can be expressed in the form of a divergence ∇ · J

, with the current J

being propor-

tional to the gradient of the slow mode

through a linear constitutive relation. The

corresponding constants of proportionality are the linear transport coefﬁcients for the

system and are linked to the matrix L

. In the linearized Langevin equations the L

represent the physical transport coefﬁcients. We clarify this further below in discussing

the Langevin dynamics for the hydrodynamic modes of the liquid.

The matrix S

represents the equal-time correlation of the ﬁelds 

. We compute

this static correlation function in terms of the correlation of the coarse-grained variables

{φ} averaged over the function space with a coarse-grained free-energy functional F[φ]

(Mazenko 2002a, 2002b). The equilibrium average is obtained in terms of the probability

distribution determined with F,

≡φ

=

Dφφ

exp{−β F[φ]}

Dφ exp{−β F[φ]}

. (5.3.52)

234 Dynamics of collective modes

Using the functional identity



Dφ

δφ



−β F

= 0 (5.3.53)

we obtain that

δF

δφ

= δ

. (5.3.54)

Assuming F to be a quadratic functional of the ﬁelds φ

, the above equation leads to the

following expression:

φ]=



dx φ

−1

. (5.3.55)

In terms of F the linear Langevin equations (5.3.51) for the ﬁelds φ

reduce to the form

∂φ

∂t

+ β

−1

−Q

+ L

δF

δφ

= f

(t). (5.3.56)

The above equation gives the linear Langevin equations for the dynamics of a chosen set of

variables φ

with a part being controlled by the noise denoted as f

. The functional F[φ] is

the free-energy functional which determines the static correlation functions in equilibrium.

Indeed, as we will see in the next chapter, the stationary distribution corresponding to the

Langevin dynamics described by eqn. (5.3.56) is given by exp(−F[φ]). For constructing

the Langevin equations one needs this free-energy functional expressed in terms of the

coarse-grained densities φ. The construction of F is done mainly by maintaining consis-

tency with the static description of the system. We now examine the relevant Langevin

equations for the set of collective modes {φ

} for the two typical cases of the isotropic ﬂuid

and the crystalline solid.

Identiﬁcation of slow modes

Our discussion of the dynamics of ﬂuids so far has been based on the identiﬁcation of

a set of modes {

} in the many-particle system. The characteristic time scales for these

modes are much longer than those for the rest of the many other dynamic variables for

the system. The latter are identiﬁed as noise {f

} and assumed orthogonal to the set of

slow modes {

}, which is ensured with the orthogonality condition, see eqn. (5.3.15).We

will now examine the physical situations in which such slow modes occur. The existence

of slow modes in the ﬂuid can be inferred from the nature of the eigenmodes of decay.

Equivalently, the corresponding dispersion relations obtained from the pole structure of

the dynamic correlation functions in the frequency space are useful for understanding the

nature of the asymptotic decay. To explain this, let us consider here a single collective mode

denoted by

φ. The Laplace transform of the normalized equilibrium correlation function

5.3 Linear ﬂuctuating hydrodynamics 235

C(q, t) of

φ at two different times corresponding to wave vector q is obtained from eqn.

(5.3.23) as

C(q, z) =

z + M(q, z)S

−1

, (5.3.57)

where the memory function M(q, z) is obtained in the following form using eqn. (5.3.30):

M(q, z) =

φL

φ+{L

φ}R(z){L

φ}

−{L

φ}R(z)

φC

−1

(z)

φ R(z){L

φ}. (5.3.58)

is the equal time correlation function of the ﬁeld

φ. The corresponding pole of C(q, z)

is given by

z =−M(q, z)S

−1

(5.3.59)

and represents a slow mode if the RHS → 0inthesmall-q limit. This ensures that the

correlation of

φ over long distances decays at long times. Such a situation can result in one

of the following two ways.

(a) Conservation laws.Let

φ be a microscopically conserved density evolving with Liou-

ville operator L. It must satisfy the microscopic-balance equation

∂

∂t

+ L

φ =

∂

∂t

+∇· j

= 0, (5.3.60)

where j

is a current corresponding to

φ. Therefore the Fourier transform L

φ is of

O(q).Fromeqn. (5.3.58) it follows that the ﬁrst term on the RHS is of O(q) while

the second and third terms are of O(q

). Thus M(q, z) → 0asq → 0 while S

remains ﬁnite in the same limit. Thus the dispersion relation in the spectrum for the

correlation function is z → 0asq → 0, implying that the ﬂuctuations over long

distances decay in the long-time limit. The density of the microscopically conserved

property is a slow mode. For example, in an isotropic ﬂuid the conservation laws of

mass, momentum, and energy give rise to a set of hydrodynamic modes involving the

corresponding microscopic densities. Similarly, spin diffusion in a paramagnet with

conserved magnetization is a slow mode, and so on.

(b) Long-range order.

φ is a slow variable if it corresponds to a thermodynamic state with

long-range order, i.e., its static correlation S

diverges as q

−2

for small q. This applies

to a dynamic variable irrespective of whether or not it represents a conserved prop-

erty. In this case, since the continuity equation of (5.3.60) no longer holds in general,

we have

lim

q→0

M(q, z) = 0. (5.3.61)

However, since S

−1

∼ q

, in the hydrodynamic limit we still can satisfy similar dis-

persion relations as in case (a) above, i.e., z → 0asq → 0. Hence

φ represents

236 Dynamics of collective modes

a slow mode. Therefore diverging static correlation in the small-wave-number limit

results in new slow modes in addition to the usual conservation laws. This behavior of

the static correlation functions represents long-range correlations in the system and is

associated with the breaking of a continuous symmetry. A classic example of this is

the elastic-displacement degrees of freedom in a solid, which constitute the transverse

sound modes. These are absent in the isotropic liquid, which has only longitudinal

sound modes. This result is referred to as the Goldstone theorem (Goldstone, 1961;

Nambu, 1960a, 1960b) and will be discussed in the next section.

To summarize, in an isotropic liquid consisting of similar particles, the slow modes are

the conserved densities of mass, momentum, and energy. In the present section we will

consider these hydrodynamic modes for the liquid state and their plausible extensions to

short wavelengths. In the next section we consider the case of the solid, which in addition

to the conserved modes also has the Nambu–Goldstone modes due to long-range order in

the crystalline or even in the glassy state.

5.3.2 The liquid-state dynamics

The isotropic liquid provides the simplest set of slow modes in terms of the conserved

densities of mass, momentum, and density {ρ,g,}. In the present analysis we make the

further simpliﬁcation of dealing with only the momentum and mass density ﬂuctuations.

The dynamical equations for the ﬂuctuating variables for the isotropic ﬂuid are obtained

with the standard recipe outlined in Section 5.3.1. The description of the linearized dynam-

ics in terms of a set of Langevin equations (Kim and Mazenko, 1990) is given by eqn.

(5.3.56). The construction of these linearized equations requires the kernels Q

and L

respectively, for the reversible and dissipative parts of the equations of motion for the

momentum density.

The collective densities are deﬁned in terms of the phase-space variables {r

, p

} as

ˆρ(x, t) =



mδ(x − r

), (5.3.62)

(x, t) =



δ(x − r

). (5.3.63)

The Poisson brackets Q

are computed from the corresponding deﬁnitions (5.3.62) of the

respective slow modes in terms of the phase-space variables. Using the canonical Poisson-

bracket relation (Landau and Lifshitz, 1975a),

, p

= δ

αβ

, (5.3.64)

we obtain the following results for the Poisson brackets between hydrodynamic variables:

ρρ

≡{ˆρ(x), ˆρ(x



)}=0, (5.3.65)

≡{ˆg

(x), ˆg



)}

5.3 Linear ﬂuctuating hydrodynamics 237

=−∇

[δ(x − x



) ˆg

(x)]+∇



[δ(x − x



) ˆg



)], (5.3.66)

ρg

≡{ˆρ(x), ˆg



)}=−∇

[δ(x − x



) ˆρ(x)]. (5.3.67)

Using the above values of the Poisson brackets, we obtain for their equilibrium averages

ρρ

= 0, (5.3.68)

= 0, (5.3.69)

ρg

=−∇

[δ(x − x



)]ρ

. (5.3.70)

where the average values of ρ and g

are taken as ρ

and 0, respectively, for the isotropic

liquid.

The dissipative coefﬁcients L

are chosen so as to match the dissipative parts of the

equations of motion for the momentum densities g

to those in the corresponding equa-

tions of macroscopic hydrodynamics discussed in Chapter 1 (see eqn. (5.1.36)). Upon

identifying in the linear case the velocity ﬁeld as v

= g

/ρ

, we obtain

≡ L

=−η



∇

+ δ

∇



− ζ

∇

, (5.3.71)

where ζ

is the bare bulk viscosity and η

is the bare shear viscosity. The longitudinal

viscosity is deﬁned as 

= ζ

+ 4η

/3. The other elements of the dissipative matrix L

are chosen as

ρg

= L

ρρ

= 0, (5.3.72)

a choice that is necessary in order to keep the form of the continuity equation unchanged.

The other necessary ingredient for the construction of the equations of motion is the

driving free-energy functional F expressed as a functional of the slow variables {ρ,g}.

Let F

be the part dependent on the momentum density g and let the potential part F

treated as a functional of the density ρ. We write the free energy in the following schematic

form:

F = F

+ F

−1



(x)

+ f [ρ(x)]

. (5.3.73)

The kinetic or the momentum-density-dependent part F

in the ﬁrst term on the RHS above

gives the equilibrium correlation functions for the momentum density as

g

(q)g

(q)=ρ

T δ

, (5.3.74)

so that the average kinetic energy per particle is

T in accordance with the equipartition

law (Huang, 1987). The density-dependent potential part F

is written here in terms of the

238 Dynamics of collective modes

functional f [ρ], which is the free-energy density. In the Gaussian approximation to the

free energy, the simplest choice is

f [ρ]≡

−1

δρ

(x) (5.3.75)

so as to correspond to a wave-vector-independent static structure factor or density correla-

tion function χ. We adopt this approximation in order to keep the treatment simple. This

is easily extended to the wave-vector-dependent structure factors by expressing the free

energy in terms of direct correlation functions (Hansen and McDonald, 1986).

Equations of linearized dynamics

Using the values of the kernels Q

and L

discussed above as well as the Gaussian

free-energy functional F[ρ, g], we obtain the linear equations of the ﬂuctuating hydro-

dynamics. The equation of motion for the density remains the continuity equation even for

the ﬂuctuating variables,

∂ρ

∂t

+∇· g = 0. (5.3.76)

The linear equations for the components of the momentum density g are obtained in a

generalized form of the Navier–Stokes equation:

∂g

∂t

=−ρ

∇

∂ f

∂ρ

−



+ θ

. (5.3.77)

TheﬁrsttermontheRHSofeqn. (5.3.77) takes the form ∇

P[ρ], which is similar to the

macro-hydrodynamic equation in terms of the pressure functional P with the identiﬁcation

P = ρ



∂ f

∂ρ



. (5.3.78)

For the choice (5.3.75) for the Gaussian free energy the pressure reduces to the form

P = ρ

−1

δρ. Hence the speed of sound c

in this case is obtained as c

= (∂ P/∂ρ) =

−1

. In the next chapter we will demonstrate that with the formulation of the nonlin-

ear ﬂuctuating hydrodynamics the relation (5.3.78) reduces to the usual thermodynamic

relation

P =−



∂ F

∂V



. (5.3.79)

The stochastic θ

term in the generalized Langevin equation (5.3.77) is assumed to repre-

sent Gaussian noise. The correlation of the noise is related to the bare damping matrix L

through the following ﬂuctuation–dissipation relation:

θ

(r, t)θ



, t



)=2k

δ(r − r



)δ(t − t



). (5.3.80)

Equations (5.3.76) and (5.3.77) form the basic set of linear ﬂuctuating hydrodynamic

equations for the set {ρ,g}.

5.3 Linear ﬂuctuating hydrodynamics 239

Correlation functions

We construct the equations of motion for the correlation functions for the slow variables

for the set

≡{ρ,g} using eqn. (5.3.23). The matrix M

is obtained in (5.3.18) as a

sum of the static and dynamic parts, denoted by K

(s)

and K

(d)

, respectively. These kernels

are obtained in terms of the average Poisson brackets Q

and the bare transport coefﬁcients

. The matrix of the equal-time or static correlation functions is diagonal with

ρρ

= β

−1

χ, (5.3.81)

= β

−1

. (5.3.82)

The static part is given by eqns. (5.3.33) and (5.3.41) and in the present case the only

nonzero components are

(s)

ρg

(q, z) =−q

, K

(s)

(q) =−q

, (5.3.83)

where the speed of sound is given by c

= ρ

−1

. The dynamic part K

(d)

is obtained by

comparing (5.3.48) with (5.3.80) and the nonzero components are given by

(d)

(q, z) =−iL

. (5.3.84)

Using the above expressions for K

, the correlation-function matrix C

is solved from

eqn. (5.3.23). In equilibrium time translational invariance holds and hence the correlation

(t, t



) of ﬂuctuations at two different times t and t



depends only on the difference

(t − t



) of the two times. Similarly, using the spatial translational invariance in equilib-

rium, the Fourier transform of the correlation functions depends on a single wave vector q.

The analysis of the correlation functions is particularly facilitated by using the isotropic

symmetry of the normal liquid state. For the isotropic system the Fourier transform of the

correlation function C

(q, t) is a function only of the magnitude of the wave vector q

and the vector ﬁeld g is split into components (a) g

parallel (longitudinal) and (b) g

perpendicular to the wave vector q. Functions involving the vector indices for Cartesian

components are expressed in terms of longitudinal and transverse parts deﬁned as

ρg

(q,ω) =ˆq

ρg

(q,ω), (5.3.85)

(q,ω) =ˆq

ˆq

(q,ω)+ (δ

−ˆq

ˆq

(q,ω). (5.3.86)

Thus the density ﬁeld couples to the longitudinal component g

. The matrix of the mem-

ory function K

(q) splits into a longitudinal and a transverse part decoupled from each

other. The computation is straightforward using the relations (5.3.41) and (5.3.50), respec-

tively, for the static and dynamic parts of K

. Equation (5.3.23) reduces to the following

explicit form:



(q) z + i 





ρρ

ρg

gρ



=−β

−1



χ 0

0 ρ



. (5.3.87)

240 Dynamics of collective modes

We obtain the Laplace transform of the density correlation function as

ρρ

(q, z) =−β

−1



z + i 

− c

+iz



(5.3.88)

with 

= D

/ρ

, where D

= ζ

+ 4η

/3 is the longitudinal viscosity. The frequency

(z)-dependent part within square brackets represents the decay behavior of the density

ﬂuctuations. This expression has two poles (up to leading order in q)at

z =±qc



, (5.3.89)

representing the two decay modes of the density ﬂuctuation. These correspond to the prop-

agating sound modes in the isotropic ﬂuid at speed c

in directions ±q. 

represents

the attenuation of the corresponding sound waves. For later reference we will write the

Laplace transform (for its deﬁnition, see eqn. (5.3.21)) of the density correlation function

ψ(q, z) normalized with respect to its equal-time value in the form of a continued fraction,

ψ(q, z) =



z −



z + iq





−1

, (5.3.90)

with 

= c

being the microscopic frequency of the liquid state. The transverse part

of the current (g

) correlation functions, on the other hand, is obtained as

(q, z) =

−1

z + iq

, (5.3.91)

where ν

= η/ρ

is the kinetic shear viscosity. The corresponding pole

z =−iq

(5.3.92)

represents a diffusion process for the transverse current ﬂuctuations. In three dimensions

there are two diffusive shearmodes for the two directionsperpendicular to q. The transverse-

current correlation φ

(q, z) normalized with respect to its equal-time value is expressed in

the form of a simple diffusive pole,

(q, z) =

z + iq

. (5.3.93)

These expressions (5.3.88) and (5.3.91) are respectively identical to (5.2.4) and (5.2.7)

for the density and current correlations obtained in Chapter 1, and both correspond to

exponential decay of correlations.

The Fourier transform

C(q,ω) of the density correlation function is proportional to

the intensity measured in scattering experiments. In Fig. 5.2 results from light-scattering

and neutron-scattering experiments on normal liquids are shown in a schematic form. Fig-

ure 5.2(a) displays schematically the nature of this dependence at a ﬁxed wave number q.