Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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6
Nonlinear fluctuating hydrodynamics
The fluctuating-hydrodynamics approach discussed earlier takes into account only the
transport properties at the level of completely uncorrelated motion of the fluid particles.
The corresponding dissipative processes are expressed in terms of bare transport coef-
ficients of the fluid. The strongly correlated motion of the fluid particles which occurs
at high density is not take into consideration here. This is reflected through the Markov
approximation of the transport coefficients and the short correlation of the corresponding
noise representing the fast degrees of freedom in the system. The Markovian equations for
the collective modes involving frequency-independent transport coefficients constitute a
model for the dynamics of fluids with exponential relaxation of the fluctuations. The cor-
responding equations of motion for the collective modes are linear. However, exceptions
occur in certain situations in which the description of the dynamics cannot be reduced to a
set of linearly coupled fluctuating equations with frequency-independent transport coeffi-
cients. In this chapter we will consider the nonlinear dynamics of the hydrodynamic modes
for studying the strongly correlated motion of the particles in a dense fluid.
6.1 Nonlinear Langevin equations
We present in this section the formulation of a set of nonlinear stochastic equations for the
dynamics of the many-particle system. We first discuss the physical motivation for exten-
sion of the fluctuating-hydrodynamics approach to include nonlinear coupling of the slow
modes. It is important to note that the fluctuating nonlinear equations for the slow modes
which we obtain below are treated as plausible generalizations of the macroscopic hydro-
dynamic laws and describe the actual motion of the hydrodynamic variables themselves,
rather than just that of their mean values.
6.1.1 Coupling of collective modes
The strong correlations that build up in a dense liquid show up in several ways. The striking
anomalies seen in the transport properties near critical points are an example. Inclusion in
the dynamical description of nonlinear coupling of the collective modes (Kawasaki, 1966a,
1966b) is required in order to explain the observed phenomena. Another example is the
271
272 Nonlinear fluctuating hydrodynamics
observation of long-time tails, which are related to power-law behavior in the asymptotic
decay of a correlation function in a fluid, e.g., the 1/t decay of the velocity correlation
of a tagged particle seen in computer simulations of hard disks (Alder and Wainwright,
1967, 1970). An implication of the power law in long-time tail decay is the breakdown
of conventional hydrodynamics in low-dimensional systems. This is directly seen from
the Green–Kubo-type relation in Section 5.1.2 in Chapter 5. If the integrand on the RHS
of this relation decays as t
−d/2
, the long-time diffusion coefficient diverges for d ≤2
(see eqn. 5.2.32). The origin of such observations lies in collective effects over semi-
hydrodynamic length scales. Indeed, by measuring the hydrodynamic flow field around
a tagged particle Alder and Wainwright (1970) showed that the initial momentum of this
particle is partially transferred to the surrounding liquid, thereby setting up vortices around
it. The flow field in turn transfers some of its momentum to the tagged particle by produc-
ing a kick from the back, which results in a positive tail in the correlation of v(t) · v(0)
extending up to hydrodynamic time scales. A simple estimation of the above cooperative
process can be carried out as follows. The initial velocity of the tagged particle is shared
with the surrounding particles producing vortices in a volume
t
of radius R
t
. After a short
time the tagged particle is moving at velocity v(t) ∼ v(0)/(n
t
), n being the number of
particles in unit volume. This length R
t
grows as t
1/2
, through diffusion of vortices, so
that
t
∼ t
d/2
, producing the t
−d/2
power-law tail. The theoretical models for the liquid
state must take into account the collective effects present in the strongly interacting fluid.
These local collective effects in a dense fluid are included in the theoretical description in
several ways.
(1) In the kinetic-theory approach the effects of correlated collision of the particles in
the dense system are taken into account in terms of repeated binary collisions. This
involves computing effects of the so-called ring- and repeated-ring-type collisions
between the particles (Dorfman and Cohen, 1972, 1975). A general formalism for
including the effects of correlated collisions was proposed through the fully renormal-
ized kinetic theory (Mazenko, 1973a, 1973b, 1974; Mazenko and Yip, 1977; Sjögren
and Sjölander, 1978). The dynamics of dense fluids is separated into a binary colli-
sion part and a contribution from ring collisions. These methods provided reasonable
explanations of the features of the dense fluid dynamics seen in computer simulations
and scattering experiments. It was demonstrated that the self-diffusion coefficient of a
tagged fluid particle in a sea of similar particles decreases as the density increases. For
very high densities, however, the traditional kinetic-theory models have not proved
to be very effective, though they acted as an important precursor to reaching more
innovative models for the dynamics in the strongly correlating liquid state.
(2) In the so-called memory-function approach the correlated dynamics in a fluid is
accounted for by introducing nonlinear coupling of the hydrodynamic modes in the
dynamic description (Kawasaki, 1970b, 1971; Ernst et al., 1971, 1976; Ernst and Dorf-
man, 1976). These theories predicted a decay of t
−d/2
in d dimensions for a variety
of correlation functions connected with viscosity, diffusion, and thermal conductivity,
6.1 Nonlinear Langevin equations 273
through the Green–Kubo relations. A practical and feasible approach to the study of
the liquid-state dynamics at high density involves focusing on a set of modes {
ˆ
φ
i
}
with characteristic time scales different from that of the rest of the degrees of freedom
for the many-particle system. The dynamics is controlled by generalized equations of
motion for the chosen set of variables {
ˆ
φ
i
}. The deduction is outlined in Appendix A7.4
using the Mori–Zwanzig projection-operator formalism. To state the result briefly:
∂
ˆ
φ
i
(t)
∂t
−i
ij
ˆ
φ
i
(t) +
t
0
M
ij
(t − s)
ˆ
φ
i
(s)ds = R
i
(t). (6.1.1)
ij
is the n × n frequency matrix obtained as
ij
=
k
+
ˆ
φ
∗
i
iL
ˆ
φ
k
,
S
−1
kj
, (6.1.2)
where L is the Liouville operator for the microscopic dynamics of the N-particle sys-
tem and S
ij
=
+
ˆ
φ
∗
i
ˆ
φ
j
,
is the static or the equal-time correlation function of the slow
variables. R
i
(i = 1,...,n) represents the noise corresponding to the remaining large
number of degrees of freedom other than {
ˆ
φ
i
}.TheR
i
(t) remain orthogonal to the
space of
ˆ
φ
i
at all times,
+
ˆ
φ
∗
i
R
j
(t)
,
= 0. (6.1.3)
The kernel matrix M in (6.1.1) links the dynamics of
ˆ
φ
i
to its values at earlier times
and is termed the memory-function matrix. The memory function M
ij
in eqn. (6.1.1)
is obtained as an autocorrelation of the noise R
i
,
M
ij
=R
i
(t)R
k
S
−1
kj
. (6.1.4)
This is a generalized fluctuation–dissipation relation whose validity does not require
the system to be close to the equilibrium state but is dependent only on the orthogonal-
ity condition (6.1.3). The linearized dynamical equations (6.1.1) for the slow variables
are non-Markovian, i.e., have frequency-dependent transport coefficients M
ij
.The
generalized transport coefficients are then evaluated by computing the nonzero ele-
ments of the memory-function matrix M. These frequency- and wave-number-dependent
functions are obtained in terms of the corresponding microscopic time correlation
functions. The resulting model is known as the self-consistent mode-coupling the-
ory (MCT) (Götze, 1991). It has proved to be an extremely useful step in the study
of slow dynamics in the supercooled liquid state approaching the glass transition. The
deduction of the MCT using this approach is outlined in the appendix following the
discussion for the linear Langevin equations with nonlocal transport coefficients.
We formulate the dynamics of the supercooled liquid taking an approach that is inter-
mediate between the above two routes. The choice of a basic set {
ˆ
φ
i
} of slow variables
for the system is dictated by various physical considerations as indicated in Section 5.3.1.
The many remaining degrees of freedom are treated as noise fluctuating on much shorter
274 Nonlinear fluctuating hydrodynamics
time scales. The nonlinear dynamics involves in the equations of motion the products of
the slow modes, which are also treated as slow. The inclusion of the nonlinear interaction
of the slow modes in the stochastic equations brings in collective effects. To demonstrate
this, consider the following simple analysis. Take a slow variable a
i
, which, being a local
density of a globally conserved property, has the property
lim
q→0
∂
∂t
a
i
(q, t) = 0. (6.1.5)
With this criterion, can the products of the slow variables also be treated as slow variables?
The corresponding integral for a product of the slow variables, a
2
i
(r, t),is
lim
q→0
∂
∂t
dx e
iq · x
a
2
i
(x, t) = 2
dk
(2π)
3
i[k · J
a
i
(k, t)]a
i
(−k, t), (6.1.6)
where in reaching the second equality we have used the balance equation
∂
t
a
i
+∇· J
a
i
= 0 (6.1.7)
for the slow variable a
i
and the corresponding current J
a
i
. For long times t,thevalueof
the integral in (6.1.6) is dominated by the small-k regime (since a
i
(k, t) is a hydrodynamic
variable) and hence becomes small due to the factor of k in the integrand. Thus, by virtue
of the presence of the nonlinear couplings of the slow variables in the equations for the
dynamics, we essentially include collective effects in the dense systems over long length
scales.
6.1.2 Nonlinear Langevin equations
In the following, a set of nonlinear Langevin equations for the chosen set of slow modes
{
ˆ
φ} describing the dynamics of dense fluids is obtained:
∂
ˆ
φ
i
(t)
∂t
= V
i
[
ˆ
φ]−
j
L
0
ij
∂ F
∂
ˆ
φ
j
+ θ
i
(t), (6.1.8)
where (a) F is the equivalent Hamiltonian for the system expressed as a functional of
the slow modes {
ˆ
φ
i
} such that the equilibrium distribution function at temperature T is
P[φ]=Z
−1
exp[−F/(k
B
T )], (b) the streaming current V
i
represents the reversible part
of the dynamics expressed in terms of the slow variables, and (c) the correlation of the
noise θ
i
(t) is given by
θ
i
(t)θ
j
(t
)=2k
B
TL
0
ij
δ(t − t
) (6.1.9)
in terms of the matrix of bare transport coefficients L
0
ij
. We follow the treatment developed
by Ma and Mazenko (1975) and Mazenko (2006b).
The collective variable g
φ
(t)
The choice of a basic set of collective variables {
ˆ
φ(x, t)} for the system is dictated by
various physical considerations (which are discussed in Section 5.3.1). In the case of a
6.1 Nonlinear Langevin equations 275
classical fluid these collective variables are the conserved densities of the mass, momen-
tum, and energy of the fluid. For a classical N-particle fluid, these collective variables are
functions of the 6N phase-space variables. The many remaining degrees of freedom of the
fluid are treated as noise fluctuating on a much shorter time scale. The equations for the
time evolution of the variables {
ˆ
φ
i
} with nonlinear interactions are obtained by introducing
the new variable g
φ
(t) in terms of the fields φ
i
(x, t) as
g
φ
(t) =
%
i
δ[φ
i
−
ˆ
φ
i
(t)]≡δ[φ −
ˆ
φ(t)], (6.1.10)
where the variables φ
i
(x, t) are defined as field variables on a set of lattice points denoted
by x. We suppress the spatial dependence of the fields in writing the above equations in
order to keep the notation simple. In the present notation the hatted entities {
ˆ
φ
i
} are depen-
dent on the phase-space coordinates {r
α
, p
α
} (α = 1,...,N ) while the {φ
i
} are the field
variables labeled by the space-time coordinates. The function g
φ
(t) satisfies the following
condition by definition:
D(φ)g
φ
(t) =
%
i
dφ
i
δ[φ
i
−
ˆ
φ
i
(t)]=1, (6.1.11)
where we have used the notation D(φ) for the multiple integral over the set {φ
i
}at all space
points on a grid of size ,
D(φ) ≡ lim
→0
%
i
dφ
i
, (6.1.12)
termed the functional integral in the space of φ. On taking the thermal average of (6.1.11)
we obtain the result
D(φ)P[φ]≡
D(φ)g
φ
(t)=1, (6.1.13)
where the thermal average of g
φ
(t) over the phase-space variables for the
ˆ
φ
i
is defined as
g
φ
(t)=P[φ]. From the definition (6.1.10) of g
φ
it is clear that its average represents the
probability that the set {
ˆ
φ
i
} will have the corresponding values {φ
i
} and hence P[φ] is a
positive number. We define an equivalent Hamiltonian F[φ] for the system as a functional
of the field variables φ
i
such that
P[φ]=
e
−β F[φ]
Z
, (6.1.14)
where
Z ≡
D(φ)exp(−β F[φ]) (6.1.15)
is a normalization constant. The equal-time correlation of the field variable φ is defined by
the relation
S
φφ
=g
φ
g
φ
=δ(φ − φ
)P[φ], (6.1.16)
276 Nonlinear fluctuating hydrodynamics
where in reaching the last equality we used the definition (6.1.10) of g
φ
in terms of delta
functions. The function g
φ
is useful for determining all of the equal-time couplings of the
collective variables
ˆ
φ
i
. For example,
D(φ)g
φ
(t)[φ
i
φ
j
]=
ˆ
φ
i
(t)
ˆ
φ
j
(t). (6.1.17)
We obtain below a stochastic equation for the time evolution of g
φ
(t) following methods
developed by Ma and Mazenko (1975) in the study of dynamic–critical phenomena. The
equation of motion for g
φ
(t) is then used to construct a set of nonlinear Langevin equa-
tions for the collective modes {
ˆ
φ
i
(t)}. The corresponding recipe for obtaining the nonlinear
Langevin equation is then applied in the subsequent sections for constructing the equations
of motion for the relevant slow modes in the compressible liquid. These form the basis for
the most widely studied model for slow dynamics in the high-density liquid and will be the
focus of our discussion later in this chapter. We present this general scheme for obtaining
the nonlinear Langevin equation in detail first.
Dynamics of g
φ
(t)
We obtain a linear Langevin equation for the generalized set of variables denoted as {g
φ
(t)}
following the scheme described in Section 5.3.1 of Chapter 5. We work with the Laplace
transform defined as
g
φ
(z) =−i
∞
0
dt e
izt
g
φ
(t) (6.1.18)
with Im(z)>0. The equation of motion for g
φ
is obtained as
∂g
φ
(t)
∂t
=−{H, g
φ
}≡iLg
φ
(t), (6.1.19)
which can be formally integrated to obtain g
φ
(t) = exp(iLt)g
φ
, where g
φ
= g
φ
(t = 0).
On taking the Laplace transform of eqn. (6.1.19) we obtain the relation
g
φ
(z) = R(z)g
φ
, (6.1.20)
where the resolvent operator R(z) is given by
R(z) =[z + L]
−1
. (6.1.21)
Equivalently, we can express the above relation in the form
zg
φ
(z) = g
φ
− Lg
φ
(z). (6.1.22)
Following steps analogous to the earlier case, from (6.1.22) we obtain a Langevin equation
for g
φ
by approximating the quantity Lg
φ
(z) as a sum of two parts: (a) the first part is
linearly proportional to the g
φ
and (b) the rest is treated as a noise f
φ
(z). Thus we write,
schematically,
− Lg
φ
(z) = K
φ
¯
φ
(z)g
¯
φ
(z) +if
φ
(z), (6.1.23)
6.1 Nonlinear Langevin equations 277
where K
φ
¯
φ
(z) is a kernel function that represents the projection of Lg
φ
(z) on g
φ
(z).In
eqn. (6.1.23) and in the rest of this section we adopt the Einstein summation convention
that repeated (barred) functions are integrated over in the function space, i.e.,
K
φ
¯
φ
(z)g
¯
φ
(z) =
D(
¯
φ)K
φ
¯
φ
(z)g
¯
φ
(z). (6.1.24)
The leftover contribution in (6.1.23) denoted by f
φ
(z) is the noise, which behaves like a
dynamical variable similar to g
φ
(z). The noise f
φ
(z) has no projection on the slow mode
g
φ
(t) at t = 0,
g
φ
f
φ
(z)=0, (6.1.25)
or, equivalently, we obtain in terms of time correlations
g
φ
f
φ
(t)=0fort ≥ 0. (6.1.26)
Integration of (6.1.26) with
6
Dφ
gives the equivalent condition for the average of the
noise f
φ
,
f
φ
(t)=0. (6.1.27)
Using (6.1.23) in eqn. (6.1.22), we obtain the equation of motion for the Laplace transform
of g
φ
(t) in the frequency space,
zg
φ
(z) −K
φ
¯
φ
(z)g
¯
φ
(z) = g
φ
+if
φ
(z). (6.1.28)
Equivalently, eqn. (6.1.28) reduces to a linear Langevin equation in the time space,
∂
∂t
g
φ
(t) + i K
φ
¯
φ
g
¯
φ
(t) = f
φ
(t). (6.1.29)
Let us focus now on the memory function K
φ
¯
φ
.
The memory function
The Laplace transform of the correlation function of the g
φ
(t) at two different times is
defined as
G
φφ
(z) =g
φ
g
φ
(z)=g
φ
R(z)g
φ
. (6.1.30)
The correlation function G
φφ
(z) satisfies the memory-function equation
zG
φφ
(z) −K
φ
¯
φ
G
¯
φφ
(z) = S
φφ
, (6.1.31)
where S
φφ
is the static or the equal-time correlation function of the slow variables as
defined above in eqn. (6.1.16). The memory function K
φφ
is divided into two parts,
K
φφ
= K
(s)
φφ
+ K
(d)
φφ
, (6.1.32)
where K
(s)
φφ
is the static part and K
(d)
φφ
is the dynamic part. Using the ansatz (6.1.32) in
(6.1.28), we obtain the following version of the Langevin equation (6.1.29) for g
φ
(t):
∂
∂t
g
φ
(t) + i K
(s)
φ
¯
φ
g
¯
φ
(t) + i K
(d)
φ
¯
φ
g
¯
φ
(t) = f
φ
(t). (6.1.33)
278 Nonlinear fluctuating hydrodynamics
Following steps similar to those used in obtaining the static and dynamic parts of the mem-
ory function in the case of linearized dynamics (respectively given by eqns. (5.3.31) and
(5.3.32) in Chapter 6), we obtain
K
(s)
φ
¯
φ
S
¯
φφ
=−g
φ
Lg
φ
, (6.1.34)
K
(d)
φ
¯
φ
(z)S
¯
φφ
=−{Lg
φ
}R(z){Lg
φ
}
+{Lg
φ
}R(z)g
¯
φ
G
−1
¯
φ
¯
φ
(z)g
¯
φ
R(z){Lg
φ
}. (6.1.35)
The complicated expressions for the memory functions presented above finally lead to the
Langevin equation for the variable g
φ
. To analyze the memory function further, we use a
special property of the Liouville operator L due to its derivative form,
L ≡ i
α
∂ H
∂p
α
·
∂
∂r
α
−
∂ H
∂r
α
·
∂
∂p
α
. (6.1.36)
Since delta functions are involved in the definition (6.1.10) of g
φ
, the derivative of the latter
with respect to
ˆ
φ
i
is the negative of that with respect to φ
i
. Therefore, on replacing the
derivative of g
φ
with respect to the phase-space variables r
α
or p
α
in terms of a derivative
with respect to the phase-space variable
ˆ
φ
i
, i.e.,
∂ H
∂p
α
·
∂g
φ
∂r
α
→
i
∂g
φ
∂
ˆ
φ
i
∂
ˆ
φ
i
∂r
α
·
∂ H
∂p
α
=−
i
∂
∂φ
i
g
φ
∂
ˆ
φ
i
∂r
α
·
∂ H
∂p
α
, (6.1.37)
we obtain the result
Lg
φ
=−
i
∂
∂φ
i
{g
φ
L
ˆ
φ
i
}, (6.1.38)
since L
ˆ
φ
i
in the curly bracket is a function of phase-space variables only and does not
involve φ
i
. Equation (6.1.38) is a key result, which we use below in our analysis of the
static and dynamic parts of the memory function.
The static part
Let us first focus on the static part of the memory function, which is frequency-independent
and hence considerably simpler. Since the equal-time correlation S
φφ
defined in (6.1.16)
is diagonal in φ, we obtain using (6.1.38) on the RHS of the expression (6.1.34) for the
static part of the memory function
K
(s)
φ
¯
φ
S
¯
φ
φ
≡
D(
¯
φ)K
(s)
φ
¯
φ
δ(
¯
φ − φ
)P[φ
]
= K
(s)
φφ
P[φ
]=
i
∂
∂φ
i
g
φ
g
φ
L
ˆ
φ
i
. (6.1.39)
6.1 Nonlinear Langevin equations 279
Now, making use of the definition (6.1.10) of g
φ
(t), the RHS of the above expression for
the static part of the memory function simplifies to
K
(s)
φφ
P[φ
]=
i
∂
∂φ
i
!
δ(φ − φ
)g
φ
L
ˆ
φ
i
"
. (6.1.40)
The equilibrium average g
φ
L
ˆ
φ
i
on the RHS of eqn. (6.1.40) is expressed in terms of the
Poisson bracket of the corresponding quantities. This is similar to the steps that led to the
relation (5.3.39) obtained earlier in the case of linearized dynamics,
g
φ
L
ˆ
φ
i
=−iβ
−1
{
ˆ
φ
i
, g
φ
}. (6.1.41)
Using the properties of the linear operator L as discussed in the deduction of (6.1.38),we
obtain
{
ˆ
φ
i
, g
φ
}=−
j
∂
∂φ
j
g
φ
{
ˆ
φ
i
,
ˆ
φ
j
}. (6.1.42)
The RHS of (6.1.41) is further simplified to obtain
g
φ
L
ˆ
φ
i
=iβ
−1
∂
∂φ
j
g
φ
Q
ij
(
ˆ
φ)=iβ
−1
∂
∂φ
j
Q
ij
(φ)g
φ
= iβ
−1
∂
∂φ
j
Q
ij
P[φ]
, (6.1.43)
where Q
ij
is as defined in eqn. (5.3.40) in the previous chapter to denote the Poisson
bracket as a function of the collective modes. Thus eqn. (6.1.39) reduces to the form
K
(s)
φφ
P[φ
]=iβ
−1
i, j
∂
∂φ
i
δ(φ − φ
)
∂
∂φ
j
Q
ij
P[φ]
= iβ
−1
i, j
∂
∂φ
i
δ(φ − φ
)
∂ P[φ]
∂φ
j
Q
ij
+ P[φ]
∂ Q
ij
∂φ
j
= i
i, j
∂
∂φ
i
δ(φ − φ
)
−Q
ij
∂ F[φ]
∂φ
j
+ β
−1
∂ Q
ij
∂φ
j
P[φ]
, (6.1.44)
where we have used the definition (6.1.14) for P[φ] to make the substitution
∂ P[φ]
∂φ
j
=−β P[φ]
∂ F[φ]
∂φ
j
. (6.1.45)
The above relation implies that the equilibrium probability distribution P[φ] satisfies the
following condition defined in terms of an operator
˜
D
( j)
φ
:
∂
∂φ
j
+ β
∂ F[φ]
∂φ
j
P[φ]≡
˜
D
( j)
φ
P[φ]=0. (6.1.46)
280 Nonlinear fluctuating hydrodynamics
On the RHS of (6.1.44) the factor P[φ] can be replaced with P[φ
] due to the presence of
δ(φ − φ
) and this cancels out with the same term on the LHS, leading to the following
expression for the static part of the memory function:
K
(s)
φφ
=−i
i
∂
∂φ
i
!
δ(φ − φ
)V
i
[φ]
"
. (6.1.47)
The quantity V
i
[φ] within the square brackets is termed the streaming velocity term in the
space of the functions {φ
i
} and is given by
V
i
[φ]=
j
Q
ij
∂ F[φ]
∂φ
j
− β
−1
∂ Q
ij
∂φ
j
. (6.1.48)
V
i
[φ] satisfies an important divergence property,
i
∂
∂φ
i
{
V
i
(φ)P[φ]
}
= 0, (6.1.49)
which follows directly on substituting the expression (6.1.48) for V
i
[φ] into (6.1.49):
i
∂
∂φ
i
⎡
⎣
j
Q
ij
∂ F[φ]
∂φ
j
− β
−1
∂ Q
ij
∂φ
j
P[φ]
⎤
⎦
=−β
−1
ij
∂
∂φ
i
Q
ij
∂ P[φ]
∂φ
j
+
∂ Q
ij
∂φ
j
P[φ]
=−β
−1
ij
∂
∂φ
i
∂
∂φ
j
Q
ij
P[φ]
= 0. (6.1.50)
We have used in reaching the first equality above eqn. (6.1.45) for (∂ F [φ]/∂φ
j
), while the
last equality follows from the antisymmetric property of the Poisson bracket, Q
ij
[φ]=
−Q
ji
[φ]. The condition (6.1.49) ensures that the probability current vector, which is equal
to the product of the probability P[φ] and the streaming velocity V
i
, is divergence-free in
the function space of {φ
i
}. We make use of this divergence condition further below.
Using the results (6.1.47) for the static part of the memory function in the second terms
on the LHS of the linear Langevin equation (6.1.33) for g
φ
(t), the latter reduces to the form
iK
(s)
φ
¯
φ
g
¯
φ
(t) =
i
∂
∂φ
i
δ(φ −
¯
φ)V
i
[φ]g
¯
φ
(t)
=
i
∂
∂φ
i
V
i
[φ]g
φ
(t), (6.1.51)
where the streaming current V
i
[φ]is given by eqn. (6.1.48) in terms of the Poisson brackets
of the slow variables {φ
i
}.