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

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7.1 The Martin–Siggia–Rose theory 321

where the functional integral with Dψ



is according to the deﬁnition of (6.1.12).The

functional delta function is deﬁned as

δ(ψ − ψ



) = lim

→0

δ(ψ(i) − ψ



(i)), (7.1.4)

where i ∈

d+1

belongs to a (d +1)-dimensional lattice (including d, the number of spatial

dimensions, and time). Since ψ is a solution of the equation of motion (7.1.1) we replace

the delta function on the RHS of eqn. (7.1.3) with a change of coordinates to

f [ψ]=



Dψ J[ψ, θ] f [ψ]δ



∂ψ(1)

∂t

+ W [ψ]−θ(1)



, (7.1.5)

where

J [ψ, θ]=det

δθ

δψ

(7.1.6)

is the Jacobian of the transformation due to the change of the argument of the delta

function. Using a causal connection in the time discreatization, the Jacobian

of this trans-

formation is treated as a constant C

(say). Finally, on replacing the delta function on the

RHS of (7.1.6) by its functional Fourier transform in terms of a conjugate ﬁeld, we obtain

f [ψ]=C



Dψ



ψ f [ψ]exp



−i



ψ(1)

∂ψ(1)

∂t

+ W [ψ]

4

. (7.1.7)

We can now take an average over the randomness and obtain the corresponding average

over the noise as

f [ψ] = C



Dψ





ψ f [ψ]

exp



−i



ψ(1)



∂ψ(1)

∂t

+ W [ψ]−θ(1)

.

. (7.1.8)

Further simpliﬁcation of the RHS will require details of the nature of the randomness

described by θ . This should lead to an action functional A for developing an appropriate

ﬁeld-theoretic model that takes into account the role of the nonlinearities in the equation of

motion. The renormalized theory is then constructed order by order in a standard diagram-

matic approach. An example is the case of supercooled liquids. Let us now assume that the

noise is white and Gaussian. The corresponding average is then taken over the distribution

P[θ]=I

exp[−A

(θ)], where A

is a quadratic function of θ,

(θ) =



d1 d2 θ(1)[L

]

−1

(1, 2)θ(2)

, (7.1.9)

and the constant I

is deﬁned such that 1=1. Equation (7.1.9) gives the proper

ﬂuctuation–dissipation relation of the average correlation of the noise in terms of the bare

damping matrix L

The evaluation of the Jacobian has some ambiguity. We discuss this in Appendix A7.1.

322 Renormalization of the dynamics

The result (7.1.2) follows directly from the identity



D(θ)

δθ(1)

[θ(2)e

−A

(θ)

]=0 (7.1.10)

and taking [L

]

−1

to be a symmetric matrix. Using eqn. (7.1.1), the exponential term within

angular brackets on the RHS of eqn. (7.1.8) for the average of f (ψ) is written as

exp



−i



(1)

∂ψ(1)

∂t

+ W [ψ]−θ

− A

(θ)



. (7.1.11)

The exponent in (7.1.11) is reduced to a perfect square by adding an extra term quadratic in

ψ and the corresponding Gaussian integral in θ is easily performed. Denoting this integral

as I

, eqn. (7.1.8) is expressed in the form

f [ψ] = I



Dψ



ψ f [ψ]exp



−A[ψ,

ψ]

, (7.1.12)

where the action functional A is obtained as

A[ψ,

ψ]=



ψ(1)β

−1

(12)

ψ(2) +i



ψ(1)

∂ψ(1)

∂t

+ W [ψ]

(7.1.13)

The quantity I

on the RHS of eqn. (7.1.12) gives a normalization constant, which is

ﬁxed to ensure that 1=1, such that

f [ψ] =

Dψ

ψ f [ψ]exp



−A[ψ,

ψ]

Dψ

ψ exp



−A[ψ,

ψ]

. (7.1.14)

The expression (7.1.14) for the average of the functional f (ψ ) is used to write the averages

of ﬁelds and higher-order correlation functions in terms of a generating functional Z

Assuming f (ψ) ≡ ψ, we write

ψ(1)=

δξ(1)

ln Z

, (7.1.15)

with



Dψ



ψ exp



−A

[ψ,

ψ]

. (7.1.16)

We have deﬁned the generating functional Z

by including a linear current term in the

corresponding action functional A

[ψ,

ψ]=A[ψ,

ψ]−



d1 ξ(1)ψ(1), (7.1.17)

where A is given by (7.1.13).

7.1 The Martin–Siggia–Rose theory 323

The multi-point correlation functions of the variables ψ are obtained from the generating

functional,

ψ(1)...ψ(m)=

δξ(1)

...

δξ(m)

ξ=0

. (7.1.18)

Note that, from the expression for the MSR action (7.1.13) and the equation of motion

(7.1.11), it follows that with the linear part of the dynamics, i.e., terms in W [ψ] linear in

the ﬁelds, ψ produces an MSR action functional that is quadratic (Gaussian) in the ﬁelds.

The MSR response function

In the RHS of the equation of motion (7.1.1) the constant term U

does not involve the

ﬁeld variable ψ. Let us consider the change in the ﬁeld ψ corresponding to an increment

in U

→ U

with an external ﬁeld term in the MSR action:

δψ(1)=



Dψ D



−A+i

ψ(2)

(2)

− e

−A

ψ(1)

= i



ψ(1)

ψ(2)

(2)d2 + O

(

)

. (7.1.19)

Since according to causality the ﬁeld ψ(1) at time t

is dependent only on

and noise at

an earlier time, we must have t

< t

for nonzero values of ψ(1)

ψ(2). The correlation

between the hatted and unhatted ﬁelds is therefore like a response function and hence

time-ordered.

It is important to note that the above-mentioned response function is generally not the

response function associated with a physical ﬁeld h

which couples to the ﬁeld ψ in the

equivalent Hamiltonian, i.e.,

ext

∼



dx ψ(x)h

(x, t). (7.1.20)

is simply proportional to h

then the MSR response function becomes the same as the

physical linear response function. In the equilibrium statistical mechanics the correlation

and linear response functions are related through ﬂuctuation–dissipation relations. (See

eqn. (1.3.70) in Section 1.3.3.) Hence in such a situation the elements G

ψψ

and G

the matrix G are related in MSR theory. In order to check whether this is the case, we note

that

is present in the driving term in the equation of motion for ψ. In the ﬂuctuating-

hydrodynamics description, the equation of motion for the ﬁeld ψ as given in eqn. (6.1.76)

has driving terms like



− L

δF

δψ

. (7.1.21)

According to eqn. (7.1.20), the functional derivative (δ F/δψ

) is proportional to h

Hence

is proportional to h

only if the Poisson bracket Q

and the bare transport

coefﬁcients L

are constants, i.e., independent of the ﬁelds ψ. We have seen in Chapter 6

that (in general) both these conditions can hold if the dynamics is linear. In this case the

324 Renormalization of the dynamics

correlation G

between a hatted and an unhatted ﬁeld is related through a ﬂuctuation–

dissipation relation to the correlation G

ψψ

between two corresponding unhatted ﬁelds. In

the case of nonlinear dynamics, however, such linear FDTs are not always satisﬁed. In

particular, neither of the two models we consider in the present book, namely the FNH

model in Section 6.2 and the DDFT model in Section 8.1.6, belongs to this category. In

the former the Poisson bracket Q

depends on the ﬂuctuating ﬁelds, whereas in the latter

the transport coefﬁcient is dependent on the density. However, certain relations between

ψψ

and G

still hold as generalized ﬂuctuation–dissipation relations. We discuss this

in Section 7.3.1.

The Schwinger–Dyson equation

To facilitate the discussion of the renormalization due to the nonlinearities in the dynamics,

the action functional A

is written in a polynomial form. We consider here the form

[%]=



1,2

%(1)G

−1

(12)%(2) +



1,2,3

V (123)%(1)%(2)%(3)

−



%(1)ξ(1), (7.1.22)

with the corresponding partition function given by the compact form



D% e

−A

[%]

. (7.1.23)

We have adopted a compact notation above in which the spatial coordinate x

and time t

both for the hatted and for the unhatted ﬁelds are all incorporated into a single index 1 of

the vector-ﬁeld variable %(1). The vertex functions V (123) are deﬁned in such a way that

they are symmetric under the exchange of the indices. For simplicity we have included in

the above expression for the MSR action (7.1.22) terms of up to cubic order vertex in the

action. The more generalized treatment with quartic nonlinearity in the action is available

elsewhere (Das and Mazenko, 1986). The Gaussian terms in the action consist of two

types of terms: ﬁrst, with two

ψ ﬁelds as a result of averaging over the noise; and second,

a product of

ψ and ψ coming from the linear part of the equations of motions. If we make

a transformation

ψ →

ψ/

√

β and ψ → ψ

√

β, the Gaussian terms become of O(1) while

nonlinear terms of nth order in the ψs in the equation of motion are O[(k

T )

(n−1)/2

] in

the action. The renormalized theory which takes into account the role of the nonlinearities

(beyond Gaussian terms) is obtained using standard Feynman-graph (Amit, 1999) methods

of ﬁeld theory. In the functional-integral formulation (Bausch et al., 1976; De Dominicis

and Pelti, 1978; De Dominicis and Pelti, 1978; Janssen, 1979; Jensen, 1981)oftheMSR

theory, the renormalized expressions for the transport coefﬁcients appear in a fully self-

consistent form.

7.1 The Martin–Siggia–Rose theory 325

The one-point function G(1) =%(1) is obtained from the generating function Z

i in

terms of the derivative,

%(1)=

δξ(1)

ln Z

. (7.1.24)

We include the density variable in the set of slow variables % as δρ (1) = ρ(1) −ρ(1).

G(1) vanishes as ξ → 0 both for ρ and for g. The two-point function G(12) is given by

G(12) =

δξ(2)

G(1) =δ%(1)δ%(2), (7.1.25)

where δ%(1) = %(1) −%(1)≡%(1). The inverse of the two-point correlation matrix

G(12) is deﬁned through the relation



−1

(13)G(32) = δ(12). (7.1.26)

For a Gaussian action without any cubic or higher-order terms in the action, G(12) is equal

to G

(12). The role of the nonlinearities is expressed in terms of the so-called “self-energy”

matrix  through the Schwinger–Dyson equation,

−1

(12) = G

−1

(12) − (12). (7.1.27)

In Appendix A7.2 we show that, for the action functional (7.1.22) involving cubic ver-

tices V (123), the “self-energy” matrix  is self-consistently expressed in terms of the

correlation functions,

(12) =



3,4,5,6

V (134)G(35)G(46)R(526). (7.1.28)

The renormalized three-point vertex function denoted by R(123) in (7.1.28) is obtained

from the solution of a self-consistent integral equation,

R(123) = (123) +



4,5,6,7

T (1453)G(46)G(57)(726), (7.1.29)

where

(123) = 2V (123) +



4,5,6,7

U (1453)G(46)G(57)(726). (7.1.30)

The four-point kernel T (1234) is to be determined from the integral equation

T (1234) = U (1234) +



5,6,7,8

U (1564)G(57)G(68)T (7238). (7.1.31)

The four-point vertex function U (1234) is deﬁned as

U (1234) =

δ(12)

δG(34)

. (7.1.32)

326 Renormalization of the dynamics

Equations (7.1.28) and (7.1.29) are conveniently expressed in terms of the Feynman graphs

shown in Fig. A7.1 in Appendix A7.2. The zeroth-order matrix G

in the MSR action

(7.1.22) corresponds to a Gaussian form. The effects of nonlinearities in the equations

of motion are calculated by evaluating the different self-energies expressed in terms of

Feynman graphs. A graphical expansion for  in terms of the bare vertices V and the full

correlation functions is straightforward to obtain. In Fig. A7.2 the diagrammatic expansion

up to second order in k

T is shown. In practice the renormalization has generally been

considered only up to one-loop order. To lowest order we obtain the following one-loop

expression for the self-energy ,

(12) =



3,4,5,6

2V (134)G(35)G(46)V (526). (7.1.33)

The bare vertex function V (123) is determined from the nonlinearities in the equations of

motion for the collective modes. In the present case the approach outlined above provides

the renormalization of the bare transport coefﬁcients in a self-consistent form in terms of

full correlation functions.

We will apply the above-described general framework of the MSR ﬁeld theory to com-

pute systematically the corrections to the linear theory due to the nonlinear coupling of

the slow modes. In particular, our focus will be on the dynamics of a compressible liq-

uid for which the construction of the nonlinear Langevin equations was discussed in the

previous chapter. The MSR ﬁeld theory outlined above involves a matrix G of correla-

tion functions between the different ﬁelds. The elements of this matrix broadly belong to

two categories: the correlation functions G

ψψ

and the “response” functions G

. As has

already been pointed out (see eqns. (7.1.19)–(7.1.21)), the correlation between a hatted and

an unhatted ﬁeld is like a linear response function corresponding to an equivalent ﬁeld and

is time-retarded in order to maintain causality. The response functions satisfy the relation

(q,ω) =−G

∗

βα

(q,ω). (7.1.34)

From the Schwinger–Dyson equation (7.1.27) it then follows that the self-energy matrix

elements satisfy



(q,ω) =−

∗

βα

(q,ω). (7.1.35)

In the MSR theory the element of the correlation function matrix G corresponding to hatted

ﬁelds is always zero. This can be seen in the following way. From the construction of the

MSR action the Gaussian part [G

−1

]

αβ

is zero. Also, as a result of causality (Mazenko

and Yeo, 1994), the 

αβ

element of the self-energy matrix is also zero. Therefore elements

−1

αβ

between two unhatted ﬁelds are zero and hence the inverse G

ˆα

elements between

two hatted ﬁelds are zero.

The renormalization scheme can be summarized as follows. The matrix G of full cor-

relation and response functions is obtained from the Schwinger–Dyson equation (7.1.27).

The zeroth-order matrix G

corresponds to the Gaussian part of the MSR action functional

7.2 The compressible liquid 327

and nonlinearities in the equations of motion give rise to nonlinear vertices. The renor-

malized theory is obtained by evaluating the different self-energies expressed in terms of

a corresponding set of Feynman graphs determined by the vertex structure of the action

functional or equivalently by the nonlinearities in the equations of motion for the slow

variables. A graphical expansion for  in terms of the bare vertices V and the full correla-

tion functions is straightforward to obtain. Equation (7.1.33) is the one-loop expression for

the self-energy in terms of the bare vertices V (123) in the MSR action functional. Speciﬁc

self-energy contributions provide the renormalization of the corresponding bare transport

coefﬁcients in a self-consistent form, i.e., in terms of full correlation functions G.This

renormalized theory involving self-consistent expressions for the transport coefﬁcients in

terms of correlation functions constitutes an important feedback mechanism. The latter

forms the basis of the self-consistent mode-coupling theory for the cooperative dynamics

of a dense liquid and is discussed in the next chapter.

7.2 The compressible liquid

Let us now focus on the case of compressible liquids and consider the structure of the

renormalized theory when the dynamics is described by the nonlinear hydrodynamic equa-

tions presented in Section 6.2.1. Since the self-consistent equations for the time correlation

functions are essential in the discussion of the mode-coupling models, we describe this

analysis in some detail. Before we turn to the full analysis of the equations for the com-

pressible liquid with the nonlinear constraint, let us brieﬂy take note of the case of the

incompressible ﬂuid, i.e., the case of constant density.

The incompressible case

In this case, since the density ρ is constant in time, ∇ · g = 0, i.e., k · g(k) = 0. So the

momentum ﬁeld is transverse to the direction of the wave vector k and is denoted by g

The equation for the momentum conservation reduces to the form

∂g

∂t

=−P



∇



+ η

∇

+ θ

, (7.2.1)

where θ

is the transverse component of the noise. The nonlinear term in (7.2.1) is due to

the convective nonlinearity, which in this case will be the transverse part of (g · ∇)g.The

transverse projection is done by the operator P

(x), whose Fourier transform is given by

(

k) = δ

−

. (7.2.2)

The effect of this nonlinearity on the bare viscosity η

was studied using standard theo-

retical techniques (Forster et al., 1977). The shear viscosity η develops a long-time tail of

power-law decay,

η(t) = A

−d/2

. (7.2.3)

328 Renormalization of the dynamics

The exponent of the power-law behavior, i.e., d/2, is the same as that obtained in computer

simulations as well as in kinetic-theory models. Moreover, the exponent A

was computed

to be

120π

3/2



7ν

−3/2

+ D

−3/2

, (7.2.4)

which is the same as the result obtained from a detailed microscopic kinetic-theory calcu-

lation (Pomeau and Résibois, 1975).

7.2.1 MSR theory for a compressible liquid

For the compressible case let us ﬁrst consider the explicit form the action (7.1.13) takes.

In this case, while g satisﬁes a Langevin equation with a noise term, the equation for

the density ρ is the continuity equation (6.2.16). Therefore the appearance of ˆρ in the

action is



d1 ˆρ(1)



∂ρ(1)

∂t

+∇

· g(1)



, (7.2.5)

so that the functional integral over ˆρ reduces to a delta functional enforcing the continuity

equation. Similarly, the nonlinear relation g = ρv is also taken in the form of a delta-

function constraint,

exp



ψ(1)W [ψ]



D(v)

δ[g − ρv]. (7.2.6)

This delta function is then represented in terms of another conjugate ﬁeld

v as



d1 ˆv

(1)

(1) − ρ(1)v

(1)

. (7.2.7)

Thus, with the set of ﬁelds consisting of the slow modes {ρ,g, v} and the corresponding

hatted ﬁelds {ˆρ,

v}, the MSR action is given by

A =



⎡

⎣



i, j

ˆg

−1

ˆg

+i ˆρ



∂ρ

∂t

+∇· g





ˆg

⎛

⎝

∂g

∂t

+ ρ∇

δF

δρ



∇

(ρv

) + L

⎞

⎠



ˆv

(

− ρv

)



, (7.2.8)

7.2 The compressible liquid 329

where we have set the linear (in the ﬁelds) term involving the currents ξ explicitly to

zero. The nonlinearities in the generalized Langevin equation for g are determined from

the functional F

.IfF

is chosen to be a local quadratic functional of density with a

wave-vector-independent structure factor, the function f in (6.2.23) is given by

f (x) =

−1

δρ

(x). (7.2.9)

The corresponding equal-time density correlation function is then obtained as

δρ (q)δρ(−q)=

, for q ≤ ,

0, for q >,

(7.2.10)

where  is a large-wave-number cutoff corresponding to the shortest length scale for the

model. With the above choice (7.2.9) of the free energy F

, we obtain that the pressure

functional is given by

P[δρ]=ρ

−1

δρ +

−1

δρ

. (7.2.11)

Hence for the choice (7.2.9) of the free energy the Gaussian part of the MSR action is now

obtained in the form



⎡

⎣



i, j

ˆg

−1

ˆg

+i ˆρ



∂ρ

∂t

+∇· g





ˆg

∂g

∂t

+ c

∇

ρ + L

+ i



ˆv

(

− ρ

)



(7.2.12)

where c

= ρ

−1

is the hydrodynamic speed of sound. Note that the Gaussian free energy

gives rise to a nonlinear dynamics as a result of the Poisson-bracket structure for the

slow modes in the liquid. In this case we have a cubic term in the non-Gaussian part A

(which contributes beyond the Gaussian order) of the action involving the coupling of the

density ﬂuctuations,

= i





ˆg

(x, t)

−1

∇

δρ

(x, t). (7.2.13)

This cubic nonlinearity in the action is a consequence of the structure of the Poisson brack-

ets (6.2.5) between the slow variables. The nonlinearity in eqn. (7.2.13) is related purely to

the reversible part of the dynamics.

330 Renormalization of the dynamics

7.2.2 Correlation and response functions

The renormalized theory for the dynamics of the compressible liquids is developed in terms

of the correlation functions,

αβ

(12) =ψ

(2)ψ

(1), (7.2.14)

and the response functions,

(12) =

(2)ψ

(1). (7.2.15)

The averages denoted here by the angular brackets are functional integrals over all the ﬁelds

weighted by e

−A

. The nonlinearities in the equations of motion (6.2.20) and (6.2.22) give

rise to non-Gaussian terms in the action (7.2.8) involving products of three or more ﬁeld

variables.

Linear dynamics

The inverse of the matrix G

in the Gaussian part of the MSR action (7.2.12) is obtained

from the part which is of quadratic order in the ﬁelds. This part corresponds to the lin-

earized equations for the dynamics. The corresponding inverse of the matrix G

for the

model equations of the compressible liquid is shown in Table 7.1. For an isotropic liquid

we can treat the transverse and longitudinal parts of the vector ﬁelds, i.e., g, v, and their hat-

ted counterparts, separately. The transverse components correspond to q · g

= 0, and they

do not couple into the density or its hatted conjugate ˆρ. The longitudinal and transverse

components of the correlation functions in the linearized dynamics, denoted with sufﬁx L

and sufﬁx T, respectively, are obtained by inverting the corresponding components of G

−1

shown in Table 7.1. The longitudinal part of the Gaussian matrix G

is obtained in the

block-matrix form involving 3 × 3 matrices C and R. The matrix of correlation functions

described below involves the longitudinal and transverse parts of the dissipation matrix

. These are denoted by 

and η

, respectively (see eqn. (5.3.71) for their deﬁnition),



†





, (7.2.16)

where  denotes the 3 × 3 null matrix with all the elements equal to zero. C and R

denote the matrices of the correlation and response functions, respectively, and R

†

is the

Hermitian conjugate of R. The correlation-function matrix C is expressed up to a common

factor in the following form:

C = 2β

−1



∗

⎡

⎢

⎣

ρ g

qωρ

⎤

⎥

⎦

, (7.2.17)