Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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6.2 The compressible liquid 291
is maintained even for the fluctuating variables. The reversible term for the equation for ρ
is obtained as
V
ρ
=
j
dx{ρ(x), g
j
(x
)}
δF
δg
j
(x
)
=−
j
∇
j
ρ(x)
δF
δg
j
(x)
. (6.2.17)
In order to obtain the continuity eqn. (6.2.16) we require that
δF
δg
j
(x)
=
g
j
(x)
ρ(x)
, (6.2.18)
which was obtained from a detailed microscopic consideration in Appendix A6.1.Using
the Poisson-bracket relations (6.2.5), the streaming velocity for the momentum density is
given by
V
i
g
(x) =−ρ(x)∇
i
δF
U
δρ (x)
−
j
∇
j
g
i
g
j
ρ
. (6.2.19)
The remaining steps in constructing the equations of motion for the hydrodynamic modes
concern obtaining dissipative terms in the momentum equations. This requires that the ele-
ment of the dissipative matrix L corresponding to one of the elements being ρ is L
0
ρa
=0.
The matrix elements L
0
g
i
g
j
are chosen to be the same as those presented in Section 5.1.3
for the isotropic fluid (see eqn. (5.1.44)). The nonlinear equations for the components of
the momentum density g are therefore obtained in a generalized form of the Navier–Stokes
equation (Zuberav et al., 1997),
∂g
i
∂t
=−ρ∇
i
δF
U
δρ
−
j
∇
j
g
i
g
j
ρ
+
j
L
0
ij
g
j
ρ
+ f
i
. (6.2.20)
The second term on the RHS of eqn. (6.2.20) refers to the well-known Navier–Stokes
nonlinearity and is a result of coupling of convective currents. The stochastic term in the
generalized Langevin equation (6.2.20) is denoted by the noise f
i
, which is assumed to be
Gaussian. The correlation of the noise is related to to the bare damping matrix L
0
ij
through
the following fluctuation–dissipation relation:
f
i
(r, t) f
j
(r
, t
)=2k
B
TL
0
ij
(r, r
)δ(t − t
). (6.2.21)
The above set of equations (6.2.16) and (6.2.20) forms the basic set of nonlinear fluctu-
ating hydrodynamic equations for the set {ρ,g}. The presence of 1/ρ in various terms on
the RHS of (6.2.20) presents an odd kind of nonlinearity that creates an infinite set of non-
linear terms. For compressible liquids both the convective term (the second term) and the
dissipative term (the third term) of the RHS of this equation contain the ρ
−1
nonlinearity.
292 Nonlinear fluctuating hydrodynamics
Typically, dealing with the 1/ρ nonlinearity is avoided by working with the velocity field
v(x, t) defined through the relation
g(x, t) = ρ(x, t )v(x, t). (6.2.22)
For an incompressible liquid this is a trivial relation, since ρ(x, t) = ρ
0
is a constant.
However, for a compressible liquid this imposes a nonlinear constraint on the fluctuating
fields.
The density nonlinearity present in the reversible part of the equation of motion (6.2.20)
has important consequences for the dynamics of compressible liquids. This is included in
the first term representing the pressure functional. Assuming a local functional form of F
U
with respect to the density ρ,
F
U
=
dx f [ρ(x)], (6.2.23)
the first term in eqn. (6.2.20) is expressed as ∇
i
P[ρ], with the pressure functional P[ρ]
satisfying the standard thermodynamic relation
P = ρ
∂ f
∂ρ
− f. (6.2.24)
For a set of three variables {ρ,g, v}, the equations of motion are eqn. (6.2.16), eqn. (6.2.20)
expressed as
∂g
i
∂t
+
j
∇
j
(g
i
v
j
) +∇
i
P −
j
L
0
ij
v
j
= f
i
, (6.2.25)
and the nonlinear constraint (6.2.22).
By expressing the free-energy functional F
U
[ρ] as a polynomial in terms of the density
fluctuations, the corresponding nonlinear equations for the dynamics of the momentum
densities g
i
are obtained. As indicated above, in the simplest case we choose for the free-
energy functional F
U
, which corresponds to a wave-vector-independent structure factor χ.
In this case the free-energy density is given by
f [ρ]≡
χ
−1
2
(δρ)
2
. (6.2.26)
The corresponding expression for the pressure functional P is obtained from (6.2.24) as
P = c
2
0
δρ + χ
−1
(δρ)
2
2
, (6.2.27)
where c
2
0
= ρ
0
χ
−1
gives the speed of sound in the liquid. Thus, even with a purely
Gaussian free energy, we have a nonlinearity in the dynamic equation for the momen-
tum density and hence the momentum density is termed a nonlinearity of purely dynamic
origin. For a more realistic description of the dynamics a standard choice for the interaction
6.2 The compressible liquid 293
part F
in
of the free-energy functional F
U
[ρ] is as stated above in eqn. (6.2.14) in terms of
the direct correlation functions. Using the free-energy functional given in eqn. (6.2.15),the
equations of motion for the g
i
are obtained as
∂
∂t
g
i
(x, t) +
j
∇
j
[ρv
i
(x, t)v
j
(x, t)]+∇
i
dx
U (x, x
)δρ(x
, t)
+
dx
1
dx
2
V
i
(x, x
1
, x
2
)δρ(x
1
, t)δρ(x
2
, t)
−
j
dx
L
0
ij
(x − x
)v
j
(x
, t) = f
i
(x, t). (6.2.28)
The kernels U and V
i
in the above equation are obtained as
U (x, x
) =
1
βm
[δ(x − x
) − n
0
c(x − x
)], (6.2.29)
V
i
(x, x
1
, x
2
) =
1
βm
2
[
δ(x − x
2
)∇
i
c(x − x
1
) + δ(x − x
1
)∇
i
c(x − x
2
)
]
,
(6.2.30)
where we denote ∇
i
≡ ∂/∂x
i
. The above equation for the momentum density g
i
obtained
with a quadratic form of F
in
is therefore consistent with a thermodynamic description
of the liquid used in the density-functional theory of the freezing transition (see also
Appendix A6.1). The nonlinear terms in the dynamic equations result from the special
Poisson-bracket structure of the conserved densities of mass and momentum. The pres-
ence of these nonlinearities modifies the dynamic behavior of the fluid obtained from the
linear theory. In the next chapter we present the scheme for the construction of such a
renormalized perturbation theory of the dense liquids.
6.2.2 The nonlinear diffusion equation
We have considered a set of nonlinear Langevin equations for the liquid involving the slow
modes of mass and momentum density, denoted by ρ and g, respectively. We consider
in the present section a formulation in which the dynamics is described in terms of the
equation of motion of the single conserved variable density ρ(x, t). We demonstrate here
how, under some suitable approximations, the momentum density g(x, t) can be integrated
out of the problem, with the equation of motion ρ(x, t) being obtained as a nonlinear
Langevin equation similar to a diffusion equation. Such models for the dynamics of fluids
have found many applications in the case of colloids. The microscopic-level dynamics of
the colloid particles in a surrounding medium is considered to have a random component
and is intrinsically dissipative.
294 Nonlinear fluctuating hydrodynamics
The nonlinear Langevin equations for a one-component fluid obtained in the previous
section for the set of collective modes {ρ,g} are
∂ρ
∂t
+∇· g = 0 (6.2.31)
and
∂g
i
∂t
=−ρ∇
i
δF
U
δρ
−
j
L
0
ij
g
j
ρ
+ f
i
, (6.2.32)
where we have ignored the convective nonlinearity involving coupling of momentum den-
sities g, since the primary aim here is to focus on the dynamics of the density fluctuations.
From eqns. (6.2.31) and (6.2.32) the momentum density g can be eliminated (Munakata,
1994, 1996). For this we assume that the momentum density relaxes fast compared with
the density. This is termed the adiabatic approximation. A crucial step in this elimination
involves assuming that the bare transport coefficient L
0
ij
is dependent on the fluctuating
variable ρ(x, t). With the above ad-hoc assumption of making the noise multiplicative, the
equation for the momentum density reduces to the form
∂g
i
∂t
=−ρ∇
i
δF
U
δρ
+
0
g
i
+ ρ
1/2
ζ
i
. (6.2.33)
The noise correlation given by eqn. (6.2.21) is now replaced with the FDT relation
ζ
i
(r, t)ζ
j
(r
, t
)=2k
B
T
0
ij
δ(r − r
)δ(t − t
). (6.2.34)
The noise in the equations is now therefore multiplicative. The correlation matrix is
assumed to be diagonal for simplicity:
0
ij
=
0
δ
ij
, where
0
has dimensions of inverse
time, representing a microscopic time scale of the problem considered here.
Note that, from the structure of the above equation of motion for the momentum density
g, it is apparent that the latter is no longer being treated like a conserved mode.Now,
in the adiabatic approximation we put the LHS of eqn. (6.2.33) equal to zero, thereby
obtaining
g
i
=
−1
0
−ρ∇
i
δF
U
δρ
+ f
i
. (6.2.35)
By substituting this into the continuity equation (6.2.31) we obtain the equation of motion
for the density variable as
∂ρ
∂t
=
−1
0
∇ ·
ρ ∇
δF
U
δρ
+
√
ρθ. (6.2.36)
The noise θ(x, t) has the correlation
θ(r, t)θ(r
, t
)=2D
s
∇ · ∇
δ(r − r
)δ(t − t
), (6.2.37)
6.2 The compressible liquid 295
where we have the bare diffusion coefficient D
s
in terms of the kinetic coefficient
0
as
D
s
=
k
B
T
0
. (6.2.38)
Using
−1
0
as the basic unit of time, we obtain the basic equation for the density fluctuation,
∂ρ
∂t
=∇·
ρ ∇
δF
U
δρ
+
√
ρθ. (6.2.39)
Note that the deduction outlined above involves taking the ad-hoc step of making the noise
multiplicative. Indeed, this removes the 1/ρ nonlinearity present in the dynamic equa-
tion (6.2.32), reducing it to the form (6.2.33).InSection 8.1.6 we will further discuss
these interchanges between the 1/ρ nonlinearity and the multiplicative noise. Kawasaki
and Miyazima (1997) demonstrated from a proper MSR field-theoretic analysis that, on
integrating out the momentum density g from the problem in the adiabatic approximation,
the multiplicative noise follows in a natural way.
6.2.3 A two-component fluid
Let us now consider the equations of nonlinear fluctuating hydrodynamics for a binary
mixture. The slow variables in this system are the two partial densities ρ
s
(x), s = 1, 2, and
the total momentum density g(x). The two densities representing conservation of the two
species are defined microscopically as
ρ
s
(x) = m
s
N
s
α=1
δ
x − r
α
s
(t)
, (6.2.40)
where m
s
and r
α
s
(t) denote the mass and the position of the αth particle of the sth species,
respectively. The momentum densities g
s
for the two species are represented similarly as
g
s
(x) =
N
s
α=1
p
α
s
δ
x − r
α
s
(t)
, s = 1, 2, (6.2.41)
where p
s
α
is the momentum of the αth particle of species s (s = 1 or 2) and N
s
is the
number of particles of the sth species in the mixture. The total number of particles is
given by
N =
2
s=1
N
s
. (6.2.42)
The relative abundance of each of the two species is given by x
s
= N
s
/N for s = 1, 2,
such that x
1
+ x
2
= 1. The number of particles of species s per unit volume V is defined
as n
s0
= N
s
/V and the total number density as
296 Nonlinear fluctuating hydrodynamics
n
0
=
N
V
= n
10
+ n
20
. (6.2.43)
Note that the average mass density ρ
0
is obtained as
ρ
0
= m
1
n
10
+ m
2
n
20
. (6.2.44)
The total momentum density,
g = g
1
+ g
2
, (6.2.45)
in a binary mixture is a conserved property and hence a hydrodynamic variable. The equa-
tions for the dynamics of the slow variables consisting of the set {ρ
1
,ρ
2
, g} are obtained
using the standard procedure outlined above. We present the equations of nonlinear fluctu-
ating hydrodynamics here with an alternative choice of slow modes. The theory is formu-
lated in terms of the total density
ρ(x, t) = ρ
1
(x, t) + ρ
2
(x, t) (6.2.46)
and the concentration variable defined as
c(x, t) = x
2
ρ
1
(x, t) − x
1
ρ
2
(x, t). (6.2.47)
The average of c(x, t) is obtained as
c
0
= n
0
(m
1
− m
2
)x
1
x
2
. (6.2.48)
The set of slow modes in terms of which the dynamics is formulated is ψ ≡{δρ, δc, g},
where δρ = ρ − ρ
0
and δc = c − c
0
. The fluctuating equations for this set of slow vari-
ables are obtained using the standard procedure outlined in Section 6.1.2. The generalized
Langevin equations for the slow mode ψ(x, t) are obtained in the standard form
∂ψ
i
(t)
∂t
+
Q
ij
+ L
0
ij
δF
δψ
j
= θ
i
(t), (6.2.49)
where Q
ij
={ψ
i
,ψ
j
} denotes the Poisson bracket between the slow variables ψ
i
and ψ
j
.
The noise θ
i
is assumed to be white and Gaussian, with its correlation defining the bare
transport matrix L
0
as
θ
i
(x, t)θ
j
(x
, t
)=2k
B
TL
0
ij
δ(x − x
)δ(t − t
). (6.2.50)
The deduction of the above nonlinear equations for the binary mixture therefore requires
as inputs
(a) the Poisson bracket Q
ij
between the slow modes {ψ};
(b) the free-energy functional F, which specifies the equilibrium state of the binary liquid;
and
(c) the dissipative matrix L
0
defined by the corresponding noise correlations.
6.2 The compressible liquid 297
We consider these quantities in the following.
(A) The Poisson brackets. The reversible, or Euler, part of the equations of motion (6.2.49)
is obtained in terms of the Poisson brackets Q
ij
between the slow variables. Using the
basic relation
#
r
i
s
, p
j
s
$
= δ
ss
δ
αβ
, (6.2.51)
we obtain the following Poisson brackets between the slow variables:
{ρ(x), g
i
(x
)}=−∇
i
[δ(x − x
)ρ(x)], (6.2.52)
{c(x), g
i
(x
)}=−∇
i
[δ(x − x
)c(x)], (6.2.53)
{c(x), ρ(x
)}=0. (6.2.54)
The corresponding Poisson-bracket relations between the partial densities {ρ
s
} are
obtained as
{ρ
s
(x), g
i
(x
)}=−∇
i
[δ(x − x
)ρ
s
(x)], (6.2.55)
{g
i
(x), g
j
(x
)}=−∇
j
[δ(x − x
)g
i
(x)]+∇
i
[δ(x − x
)g
j
(x
)]. (6.2.56)
(B) The free-energy functional. The equilibrium behavior of a system with Langevin
dynamics is determined by the functional F. Similarly to the one-component case,
it has two parts, the kinetic and the “potential,” denoted by F
K
and F
U
, respectively,
F = F
K
+ F
U
. (6.2.57)
The kinetic part F
K
of the free-energy functional is expressed in terms of the slow
variables g and ρ following the same standard procedure (Langer and Turski, 1973)
as that described for the one-component system in Appendix A6.1. The details of
this deduction, beginning from a microscopic Hamiltonian for the binary fluid, are
available in Harbola and Das (2003). We obtain
F
K
=
1
2
dx
g
2
(x)
ρ(x)
. (6.2.58)
Note that the total density ρ = ρ
1
+ ρ
2
appears in the denominator of (6.2.58).
This term is essential for generating the Gallelian invariant form of the generalized
Navier–Stokes equation obtained for the momentum density g discussed below. For
the potential part F
U
of the free energy, there is an ideal-gas contribution together
with the interaction term,
F
U
= F
id
[ρ,c]+F
in
[ρ,c]. (6.2.59)
The ideal-gas part F
id
is obtained as a generalization of the corresponding one-
component result in the form
F
id
[ρ
1
,ρ
2
]=
s
1
m
s
dx ρ
s
(x)
ln
ρ
s
(x)
m
s
3
s
− 1
, (6.2.60)
298 Nonlinear fluctuating hydrodynamics
where
s
= h
√
β/(2πm
s
) denotes the corresponding thermal de Broglie wavelength.
In terms of variables {ρ,c}ρ
1
and ρ
2
are expressed as
ρ
s
(x, t) = x
s
ρ(x, t) + a
s
c(x, t), (6.2.61)
where the constant a
s
is defined as a
s
=±1fors = 1 and s = 2, respectively. We
express
F
id
as
F
id
[ρ,c]=
2
s=1
dx
x
s
ρ +a
s
c
m
s
ln
x
s
ρ +a
s
c
m
s
3
s
− 1
. (6.2.62)
The interaction part F
in
is obtained in the standard form in terms of the Ornstein–
Zernike direct correlation functions for a two-component system. The latter are denoted
by c
ss
between species s and s
. F
in
for the two-component system is written as a
generalization of the corresponding result (6.2.14) for the one-component case,
F
in
(ρ) =−
1
2m
s
m
s
dx dx
c
ss
(x − x
)δρ
s
(x)δρ
s
(x
). (6.2.63)
Using the transformation (6.2.61), the free energy F
U
is expressed as an expansion in
terms of the corresponding fluctuations δρ and δc:
F
U
(ρ) = F
id
[ρ,c]−
1
2
dx dx
!
c
ρρ
(x − x
)δρ(x)δc(x)
+ c
cc
(x − x
)δc(x)δc(x) + 2c
cρ
(x − x
)δc(x)δρ(x)
"
. (6.2.64)
The coefficients c
ρρ
, c
cc
, and c
ρc
in the expansion (6.2.64) are obtained in terms of the
Ornstein–Zernike direct correlation functions c
ss
for a binary mixture defined in eqn.
(6.2.63) as follows:
c
ρρ
= x
2
1
˜c
11
+ x
2
2
˜c
22
+ 2x
1
x
2
˜c
12
, (6.2.65)
c
ρc
= x
1
˜c
11
− x
2
˜c
22
+ (x
2
− x
1
) ˜c
12
, (6.2.66)
c
cc
=˜c
11
+˜c
22
− 2˜c
12
, (6.2.67)
where we have defined
˜c
ss
=
1
m
s
m
s
c
ss
, for s, s
= 1, 2. (6.2.68)
The nonlinearities in the fluctuating equations relevant for producing slow dynamics
(as we will see in the next chapter) are present even for a purely quadratic or Gaussian
free-energy functional F
U
of δρ and δc. This is similar to the corresponding result
for the one-component system. The origin of the nonlinearities is therefore purely
6.2 The compressible liquid 299
Gaussian in nature. If we consider the binary mixture for which m
1
= m
2
and the
two species are identical, we have c
ss
≡ c, where c is the direct correlation function
of the one-component system. It follows from (6.2.65)–(6.2.67) that c
ρρ
= c and
c
cc
= c
ρc
= 0, and the expression (6.2.64) for F
U
reduces to the one-component
result (6.2.14).
(C) The bare transport coefficients. The bare-transport-coefficient matrix L
0
is constructed
using simple physical considerations. Since the current of the total density, i.e., the
momentum density itself, is a conserved quantity, there is no dissipative part in the
equation for the total density ρ and the form of the continuity equation is maintained.
Thus we have L
0
ρα
= L
0
αρ
= 0forα ≡{ρ,c, g
i
}. The matrix elements L
0
g
i
g
j
≡ L
0
ij
corresponding to the momentum current density involve the viscosity tensor given by
L
0
ij
(x) =−η
0
1
3
∇
i
∇
j
+ δ
ij
∇
2
− ζ
0
∇
i
∇
j
, (6.2.69)
where η
0
and ζ
0
are the bare shear and bulk viscosities, respectively. The longitudinal
viscosity is given by
0
= (4η
0
/3 + ζ
0
). On the other hand, L
0
cc
= γ
0
∇
2
, with γ
0
representing the inter-diffusion constant. For the matrix elements L
0
cg
i
, we note from
the relation (6.1.77) that, using the time-reversal property of the fields c and g
i
,
L
0
cg
i
(t) =−L
0
cg
i
(−t). (6.2.70)
Since we have white noise, in order to satisfy the above condition we must have L
0
cg
i
=
L
0
g
i
c
= 0.
With the above choice of the free-energy functional, and the Q
ij
and L
ij
matri-
ces, we obtain the equations of motion for {ρ,c, g
i
}. The equation corresponding to ρ
is just the continuity equation (6.2.16) with g as the current,
∂ρ
∂t
+∇· g = 0. (6.2.71)
The Langevin equation for c is a diffusive equation with a coupling to the total den-
sity ρ,
∂c
∂t
+∇
i
c
g
i
ρ
+ γ
0
∇
2
δF
U
δc
= f. (6.2.72)
The corresponding equation for the momentum density g
i
is obtained as
∂g
i
∂t
+∇
j
g
i
g
j
ρ
+ ρ∇
i
δF
U
δρ
+ c∇
i
δF
U
δc
+ L
0
ij
g
j
ρ
= θ
i
. (6.2.73)
The noise terms in the Langevin equations (6.2.73) and (6.2.72) for g
i
and c are
denoted by θ
i
and f , respectively. These are assumed to be white Gaussian and are
related to the corresponding bare transport coefficients by
θ
i
(x, t)θ
j
(x
, t
)=2k
B
TL
0
ij
δ(x − x
)δ(t − t
), (6.2.74)
300 Nonlinear fluctuating hydrodynamics
f (x, t) f (x
, t
)=2k
B
T γ
0
∇
2
δ(x − x
)δ(t − t
), (6.2.75)
θ
i
(x, t) f (x
, t
)=0. (6.2.76)
The functional derivatives of the free energy F
U
with respect to ρ or c needed in the
construction of the equations of motion are obtained from the expression (6.2.64) for
this quantity as an expansion in terms of δρ and δc.
6.2.4 The solid state
The extended set of slow variables of the solid includes the conserved modes as well as the
Nambu–Goldstone modes discussed in the previous chapter (see Section 5.4) in the con-
text of linear dynamics. The nonlinear Langevin equations for these collective modes are
obtained using the standard techniques outlined above. The reversible part of the dynam-
ics is expressed in terms of the Poisson brackets Q
ij
={ψ
i
,ψ
j
} between the different
slow modes of the set {ψ
i
}. For evaluating these Poisson brackets mentioned above we
need the microscopic definitions for the variables
ˆ
ψ
i
(x) corresponding to fields ψ
i
(x).We
consider a system of N classical particles each of mass m. r
α
(t) and p
α
(t), respectively,
are the position and the momentum of the αth particle at time t . For the density ˆρ and
the momentum density
ˆ
g we use the standard prescription (5.3.34). The extra slow modes
or the Nambu–Goldstone modes to be added to the set of conserved densities take into
account the broken symmetry of the solid state with elastic properties. We define the fluc-
tuating variable u(x, t) by considering the displacements of the individual particles u
α
(t)
(α = 1,...,N ) from their respective mean positions denoted by r
0
α
,
ˆρ(x, t) ˆu
i
(x, t) = m
N
α=1
u
i
α
(t)δ(x − r
α
(t)), (6.2.77)
such that r
α
(t) = r
0
α
+ u
α
(t). Using the canonical Poisson-bracket relation
#
r
i
α
, p
j
β
$
=
δ
ij
δ
αβ
we obtain the following relations for the hydrodynamic variables:
{ˆg
i
(x), ˆu
j
(x
)}=−δ(x − x
)
δ
ij
−∇
i
x
u
j
(x
)
,
{ˆu
i
(x), ˆu
j
(x
)}=0, (6.2.78)
{ˆu
i
(x), ˆρ(x
)}=0.
Using eqns. (6.2.78) and (6.2.58), we obtain the following forms for the streaming velocity
V [ψ] in the nonlinear Langevin eqn. (6.2.1):
V
ρ
=−∇· g,
V
i
g
=−∇
j
g
i
g
j
ρ
− ρ∇
i
x
δF
U
δρ
−
(
δ
ij
−∇
i
x
u
j
(x)
)
δF
U
δu
j
(x)
,
V
i
u
=
g
i
ρ
−
g
ρ
· [∇
x
u
i
]. (6.2.79)