Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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6.2 The compressible liquid 301
The dissipative parts of the dynamical equations are expressed in terms of the bare transport
coefficients and are assumed to be the same as those in the case of the linear dynamics.
For the momentum-density equation, the irreversible part has the standard form with the
dissipative coefficient in eqn. (6.2.1) given by
L
g
i
g
j
≡ L
0
ij
=−η
0
1
3
∇
i
∇
j
+ δ
ij
∇
2
− ζ
0
∇
i
∇
j
, (6.2.80)
where ζ
0
is the bare bulk viscosity and η
0
is the bare shear viscosity. We define the longi-
tudinal viscosity as
0
= ζ
0
+4η
0
/3. The dissipative term in the equation for the u field is
assumed to have the simple time-dependent Ginzburg–Landau (TDGL) form,
L
u
i
u
j
≡
ij
=
0
δ
ij
. (6.2.81)
We require that all other L
ij
= 0. In the fluctuating-hydrodynamics description the bare
transport coefficients ζ
0
, η
0
, and
0
act as external parameters. For the bare viscosities
Green–Kubo-type relations can be evaluated in a kinetic-theory approach with suitable
models. The time scale corresponding to the quantity
0
which relates to the vacancy
diffusion is, however, much longer.
The construction of the nonlinear Langevin equation (6.2.1) involves the free energy.
The effective Hamiltonian F contains, in addition to the usual terms for an isotropic liquid,
the energy cost due to distortion in the elastic solid. The elastic part is constructed in terms
of the gradient of the displacement field u using the nonlinear strain field
s
ij
=
1
2
(∇
i
u
j
+∇
j
u
i
) −
1
2
∇
i
u
m
∇
j
u
m
. (6.2.82)
λ
0
and μ
0
are related to the bulk and shear modulus of the amorphous solid. The longitudi-
nal modulus is given by ϑ
0
= λ
0
+ μ
0
. We also consider a term representing the coupling
of the density fluctuations to the elastic strain field s
ij
. In the simplest form it is a coupling
between the density and the trace S of s
ij
. At the linear level in u, S is simply ∇· u, which
in our discussion of the linear dynamics we have seen to be related to the defect density
(ρ
D
) in the solid state. The free energy F is obtained as the sum of two parts as in the case
of the liquid in eqn. (6.2.6). The kinetic part F
K
is given by (6.2.7) as before. The potential-
energy part F
U
[ρ,u] is constructed preserving the rotational and translational invariance
of the system,
F
U
=
1
2
dx
χ
−1
(δρ)
2
+ 2ς
0
S δρ + λ
0
S
2
+ 2μ
0
(s
ij
s
ji
)
. (6.2.83)
302 Nonlinear fluctuating hydrodynamics
The couplings χ and ς
0
in the effective Hamiltonian relate to the static structure factor
for the system. Using eqn. (6.2.83), the following set of fluctuating nonlinear equations is
obtained:
∂ρ
∂t
+∇· g = 0,
∂g
i
∂t
+
j
∇
j
σ
ij
= θ
i
,
∂u
i
∂t
+ v · ∇u
i
= v
i
−
ij
δF
δu
j
+ f
i
, (6.2.84)
together with the nonlinear constraint
g = ρv. (6.2.85)
The stress-energy tensor σ
ij
has reversible and dissipative (irreversible) parts, denoted by
σ
R
ij
and σ
D
ij
, respectively, such that
σ
ij
= σ
R
ij
+ σ
D
ij
, (6.2.86)
where
σ
R
ij
= ρV
i
V
j
+
c
2
0
δρ +
χ
−1
2
(δρ)
2
+ ρ
0
ς
0
S −
λ
0
2
S
2
−
μ
0
2
(s
lm
s
ml
)
δ
ij
−
[
λ
0
S + ς
0
δρ
]
∂ S
∂∇
i
u
m
∂ S
∂∇
j
u
m
− μ
0
!
s
ij
− 2s
im
s
jm
"
. (6.2.87)
The dissipative part is related to gradients of the velocity field,
σ
D
ij
=−η
0
∇
i
V
j
+∇
j
V
i
−
2
3
δ
ij
(∇ · v)
− ζ
0
δ
ij
(∇ · v). (6.2.88)
The stress tensor satisfies σ
ij
= σ
ji
, which guarantees conservation of angular momentum
in the system. The random parts in the above fluctuating equations are Gaussian noise
terms and are related to the bare transport coefficients by
θ
i
(x, t)θ
j
(x
, t
)=2ρ
0
k
B
TL
0
ij
δ(x − x
)δ(t − t
),
f
i
(x, t)θ
j
(x
, t
)=0, (6.2.89)
f
i
(x, t) f
j
(x
, t
)=2k
B
T
ij
δ(x − x
)δ(t − t
).
The set of nonlinear Langevin equations obtained above gives the dynamics of the slow
modes for a solid with elastic properties. It is generally applicable to a crystal in which
long-range translational symmetry is broken. In this case the free energy defined in eqn.
(6.2.83) involves the rank-four elastic tensor C
ijkl
. Similarly, the viscosity tensor η
ijkl
appears in the dissipative stress tensor σ
D
ij
. These will respectively involve more elastic
6.3 Stochastic balance equations 303
and transport coefficients as required by the symmetry of the corresponding crystal, mak-
ing the formulation extremely complicated. The simplified description given above for
the isotropic system is more relevant to the case of amorphous solids in which freezing
has occurred at the scale of local structure but overall translational invariance is main-
tained over longer distances. However, this inherently brings into the above description
an underlying time scale over which the definition of the displacement field is applicable.
The particles vibrate around their mean positions, forming a random lattice structure that
corresponds to a metastable minimum of the potential energy of the many-particle system.
In the deeply supercooled glassy state the solid-like behavior persists for very long times
over which the above solid-like description applies.
6.3 Stochastic balance equations
In the previous sections of this chapter we have considered the dynamics for a system in
which the individual particles follow time-reversible equations of Hamiltonian mechan-
ics. We discussed the formulation of linear as well as nonlinear Langevin equations in
terms of collective densities like the mass density ρ(x, t) and g(x, t). These coarse-grained
densities or averaged quantities whose space and time variations are smooth have dynam-
ics described in terms of partial differential equations, which have in general reversible
as well as irreversible parts. On the other hand, we have non-coarse-grained densities
corresponding (see definitions (5.1.1)–(5.1.3) in Chapter 5)toexact balance equations
(see eqns. (5.1.4)–(5.1.6) with the respective currents). These equations are reversible
since the microscopic dynamics is reversible. Irreversibility in the Langevin equations for
the coarse-grained densities in the case of Newtonian dynamics is phenomenologically
introduced.
In the present section we will discuss the situation in which the dynamics at the micro-
scopic level is chosen to be irreversible. This can represent the dynamics of the colloid
particles in a solvent. We describe the dynamics of the individual particle approximately
with an equation of motion that is a Langevin equation with a dissipative as well as fluctuat-
ing term representing noise. This approximates the effect of the solvent molecules of much
smaller inertia constantly colliding with the bigger colloidal particles. With this dissipa-
tive microscopic dynamics we obtain here a corresponding set of exact balance equations,
similarly to the case of Newtonian dynamics discussed in the earlier sections.
6.3.1 Smoluchowski dynamics
In the present section we consider an N -particle system in which the microscopic dynamics
of the constituent particles is described with the Smoluchowski equations (Smoluchowski,
1915) involving only the particle coordinates. The momentum dependence of the particles
is ignored in the over-damped limit. A colloidal system with heavy particles in a solution
is a typical example of such a system. Let each of the N particles be of mass m, with their
304 Nonlinear fluctuating hydrodynamics
position coordinates given by {x
α
} for α = 1,...,N . The density at position x at time t is
formally defined as
1
ˆρ(x, t) =
N
α=1
δ(x − x
α
(t)) ≡
N
α=1
ˆρ
α
(x, t), (6.3.1)
where we put the hat on ρ to indicate that it is a microscopic function dependent on
the phase-space densities {x
α
}. Since the number N of particles is constant, ˆρ(x, t) is a
microscopically conserved quantity. The microscopic dynamics of the N-particle system
is assumed to follow Brownian dynamics. The momentum of the individual particles does
not enter the description of the dynamics in the so-called over-damped limit. The equation
of motion for each particle is given by the stochastic equation representing the irreversible
dynamics,
ζ
B
dx
α
(t)
dt
=−
N
β=1
∇
α
U (x
α
(t) − x
β
(t)) + f
α
(t), (6.3.2)
where we have used the following notation to represent the derivative operator ∇
α
=∂/∂x
α
.
We adopt the notation that the β = α contribution of the summation in the second term on
the RHS is zero. The stochastic part or the noise f is white and Gaussian, with its correlation
given by
+
f
i
α
(t) f
j
β
(t
)
,
= 2k
B
T δ
αβ
δ
ij
δ(t − t
). (6.3.3)
In the dimensionality-correct form of the equation of motion (6.3.2), we note that the factor
ζ
B
on the LHS denotes a friction coefficient associated with the microscopic dynamics. It
has the dimension of inverse time. To keep the notation simple, we take ζ
B
= 1inthe
following.
To obtain an equation for the density ˆρ(x, t) defined above, we make use of the fol-
lowing chain rule of stochastic differential equations in Itô calculus (Oksendal, 1992).
Let us consider a set of stochastic variables x
i
(t) (i = 1,..., m) satisfying the stochas-
tic equation ˙x
i
= h
i
+ g
ij
ξ
j
, where the correlation of the white noise ξ
i
is defined as
1
ξ
l
(t)ξ
m
(t
)
2
= δ
lm
δ(t −t
). The Itô chain rule gives the stochastic differential equation for
the variable y({x
i
}) in the form
˙y =
i
∂y
∂x
i
˙x
i
+
i, j,k
1
2
∂
2
y
∂x
i
∂x
j
g
ik
g
kj
. (6.3.4)
This above result follows directly on taking the deviation of the function y(x
i
) in terms
of the corresponding variation in the x
i
. We apply the above chain rule for the stochastic
differential equations for the variables x
α
(t) for α = 1,..., N with the respective differ-
ential equations being given by (6.3.2). Here we adopt the notation that the Greek indices
1
In dealing with the nonlinear dynamics of the density field we take the mass m of the particles as unity in order to keep the
notation simple. This makes the number density
n(x) and mass density ρ(x) the same.
6.3 Stochastic balance equations 305
correspond to the particle labels while the Roman indices are for the Cartesian components.
In this case we define the function y as
y({x
α
}) ≡ˆρ(x, t) =
N
α=1
δ(x − x
α
). (6.3.5)
Note that the coordinate x acts as a label on ˆρ. In this case the matrix g
ij
= δ
ij
is diagonal.
The corresponding stochastic differential equation for the density variable ˆρ(x, t) is then
obtained as
∂
∂t
ˆρ(x, t) =
α
{∇
α
ˆρ
α
(x, t)}·
⎡
⎣
−
β
∇
α
U (x
α
− x
β
) + f
α
(t)
⎤
⎦
+ k
B
T
α
∇
α
· ∇
α
ˆρ
α
(x, t)
=−∇·
α
ˆρ
α
(x, t)
⎡
⎣
f
α
(t) −∇
α
dx
U (x
α
− x
)
β
δ(x
− x
β
)
⎤
⎦
+ k
B
T ∇
2
ˆρ(x, t)
=−∇·
α
ˆρ
α
(x, t)
f
α
(t) −∇
α
dx
U (x
α
− x
) ˆρ(x
, t)
+ k
B
T ∇
2
ˆρ(x, t)
= k
B
T ∇
2
ˆρ(x, t) +∇· ˆρ(x, t)∇
3
dx
U (x − x
) ˆρ(x
, t)
4
−∇·
&
α
ˆρ
α
(x, t)f
α
(t)
'
. (6.3.6)
In reaching the above equation, we have generally used the trick for the derivative of the
function that
∇
α
ˆρ
α
≡∇
α
δ(x − x
α
) =−∇δ(x − x
α
). (6.3.7)
Hence we obtain an equation for the time evolution of the fluctuating density ˆρ(x, t):
∂ ˆρ(x, t )
∂t
= k
B
T ∇
2
ˆρ(x, t) +∇·
3
ˆρ(x, t)∇
dx
U (x − x
) ˆρ(x
, t)
4
− η
(x, t). (6.3.8)
306 Nonlinear fluctuating hydrodynamics
In eqn. (6.3.8) we have defined the random force denoted by η
(x, t) as
η
(x, t) =−
N
α=1
∇ · {ˆρ
α
(x, t)f
α
(t)}. (6.3.9)
The correlation of the noise η
is obtained as
1
η
(x, t)η
(x
, t
)
2
= 2k
B
T δ(t − t
)∇
x
· ∇
x
ˆρ(x, t) ˆρ(x
, t
)
= 2k
B
T δ(t − t
)∇
x
· ∇
x
[δ(x − x
) ˆρ(x, t)], (6.3.10)
where the angular brackets on the LHS indicate the average over the noise f
α
in the micro-
scopic equations of motion. We have used in obtaining eqn. (6.3.10) the following property
of the delta function:
∇
x
· ∇
x
{ˆρ(x, t) ˆρ(x
, t)}=∇
x
· ∇
x
δ(x − x
) ˆρ(x, t). (6.3.11)
We now redefine the noise η(x, t) as
η(x, t) =∇
i
{ˆρ
1/2
(x, t)θ
i
(x, t)}, (6.3.12)
where θ
i
(x, t) is a global noise with the correlation
1
θ
i
(x, t)θ
j
(x
, t
)
2
= 2k
B
T δ(t − t
)δ
ij
δ(x − x
). (6.3.13)
It then easily follows that the two noises η
and η are statistically identical. The equation
of motion (6.3.8) for the density field then takes the form
∂ ˆρ(x, t )
∂t
= D
B
∇
i
ˆρ(x, t)∇
i
δ
ˆ
F
BD
δ ˆρ(x, t )
+∇
i
{ˆρ
1/2
(x, t)θ
i
(x, t)}, (6.3.14)
where we have written ζ
B
(previously set equal to 1) explicitly in terms of the
constant D
B
,
D
B
=
k
B
T
ζ
B
. (6.3.15)
D
B
has the dimension of a diffusion constant. The noise ∇ · {
√
ρθ} in the Langevin equa-
tion is multiplicative. The quantity
ˆ
F
BD
inthefirsttermontheRHSofeqn. (6.3.14) is a
functional of the density ˆρ(x, t),
ˆ
F
BD
=
ˆ
F
id
BD
+
1
2
dx
dx
ˆρ(x)
˜
U (x − x
) ˆρ(x
), (6.3.16)
where
˜
U = βU is the interaction scaled with k
B
T and
ˆ
F
id
BD
is the corresponding functional
for the noninteracting system (ideal gas),
ˆ
F
id
BD
= β
−1
dx ˆρ(x)[ln( ˆρ(x)) − 1]. (6.3.17)
6.3 Stochastic balance equations 307
The functional dependence of
ˆ
F
id
BD
on ˆρ is identical to that of the free energy obtained
as a functional of the coarse-grained density ρ in eqn. (2.1.27) in Chapter 2. The equilib-
rium distribution for the system follows from the stationary solution of the Fokker–Planck
equation corresponding to the Langevin equation (6.3.14) and is given by
P
EQ
=
exp[−F
BD
]
Z
, (6.3.18)
where Z is a normalizing factor. Equation (6.3.14) for the microscopically conserved
quantity density follows the form of a continuity equation,
∂ ˆρ(x, t )
∂t
+∇· j
ˆρ
= 0. (6.3.19)
The current
j
ˆρ
=−D
B
ˆρ ∇
δF
BD
δ ˆρ
−
;
ˆρθ (6.3.20)
has a regular part as well as a random part. Even with a Gaussian or quadratic interaction
part for the free-energy functional
ˆ
F
BD
, eqn. (6.3.14) is nonlinear. However, the ideal gas
part of F
BD
, which involves the non-Gaussian term ln ˆρ, produces a linear term in the
equation of motion.
The functional
ˆ
F
BD
[ˆρ] in eqn. (6.3.16) is similar to the free energy F
U
[ρ] expressed in
eqn. (6.2.15) as a functional of the corresponding coarse-grained density ρ in a Newtonian
fluid. This similarity holds provided that we identify the bare interaction
˜
U (r) with the
direct correlation functions −c(r) at the lowest order. With this identification, eqn. (6.3.14)
for the density ˆρ(x, t) (Dean, 1996) is identical to eqn. (6.2.39) for the coarse-grained
density for ρ(x, t) (Kawasaki, 1998). Microscopic derivations (Yo sh imo ri, 2005) of similar
equations using projection-operator methods have also been done.
Owing to the apparent similarity in structure between the nonlinear diffusion equation
(6.2.39) for ρ(x, t) and eqn. (6.3.14) for ˆρ(x, t ), they have often been conflated as the
Dean–Kawasaki equation. However, it is important to note that Kawasaki’s eqn. (6.2.39) is
for the coarse-grained density and involves making the adiabatic approximation regarding
the dynamics of the momentum fluctuations. On the other hand, the microscopic dynamics
itself is chosen as dissipative in the case of Dean’s eqn. (6.3.14). The latter is an exact bal-
ance equation for the microscopic density function ˆρ similar to the Euler equations (5.1.4)–
(5.1.6) obtained in the Newtonian case. The idea of mixing these two different equations
should therefore be treated with caution. We refer to the formulation of the dynamics in
terms of the stochastic eqn. (6.2.39) as dynamic density-functional theory (DDFT). In Sec-
tion 8.1.6 we discuss a possible ergodic–nonergodic transition in this so-called dynamic
density-functional theory using density as the single variable.
The tagged-particle density ρ
s
(x, t) is also a conserved quantity, and a similar balance
equation for it follows as well. The density ρ
s
is defined as
ˆρ
s
(x, t) ≡ˆρ
α
= δ(x − x
α
(t)), (6.3.21)
308 Nonlinear fluctuating hydrodynamics
where the αth particle is tagged. Starting from (6.3.6), we obtain
∂ ˆρ
s
(x, t)
∂t
= D
B
∇
2
ˆρ
s
(x, t) +∇·
3
ˆρ
s
(x, t)∇
dx
˜
U (x − x
)ρ(x
, t)
4
+ η
s
(x, t).
(6.3.22)
The correlation of the noise η
s
is obtained as
1
η
s
(x, t)η
s
(x
, t
)
2
= 2D
B
δ(t − t
)∇
x
∇
x
{δ(x − x
)ρ
s
(x, t)}. (6.3.23)
Equation (6.3.22) obtained above is similar to the equations for the tagged-particle dynam-
ics in the Newtonian-dynamics case to be discussed later, in Section 8.1 of Chapter 8.
However, for the Newtonian-dynamics case approximations have to be made in dealing
with the momentum density of the tagged particle, which is not a conserved quantity.
6.3.2 Fokker–Planck dynamics
The exact balance equations or Dean equations discussed above are not peculiar to the case
of microscopic Smoluchowski dynamics only. Other types of microscopic dynamics, such
as Fokker–Planck dynamics in terms of the position x
α
and momentum x
α
of the N -particle
system have also been considered,
dx
α
(t)
dt
= p
α
, (6.3.24)
dp
α
(t)
dt
=−
N
β=1
∇
α
U (x
α
(t) − x
β
(t)) −¯γ
0
p
α
+ ξ
α
(t), (6.3.25)
where ¯γ
0
is a dissipative coefficient with the dimension of inverse time and U (x
α
− x
β
) is
the interaction potential between the particles α and β. The particle mass m is chosen as
unity. The correlation of the noise ξ
α
is related to the dissipative coefficient ¯γ
0
by
+
ξ
i
α
(t)ξ
j
β
(t
)
,
= 2k
B
T ¯γ
0
δ
αβ
δ
ij
δ(t − t
). (6.3.26)
The microscopic density ˆρ(x, t) and the momentum density ˆg(x, t) are defined in this case
as in eqns. (5.1.1) and (5.1.2), respectively. We take the mass m of the fluid particle to
be unity in order to keep the notation simple. Using the above definitions, the following
balance equations for ˆρ and ˆg have been obtained by Nakamura and Yoshimori (2009):
∂
∂t
ˆρ(x, t) +∇·
ˆ
g(x, t) = 0, (6.3.27)
∂
∂t
ˆg
i
(x, t) +∇
j
ˆ
#
ij
(x, t) +ˆρ(x, t)∇
i
dx
U (x
α
− x
) ˆρ(x
, t)
−¯γ
0
ˆg
i
(x, t) = ξ
i
(x, t), (6.3.28)
6.3 Stochastic balance equations 309
where
ˆ
#
ij
(x, t) =
5
α
p
i
α
p
j
α
δ(x−x
α
(t)). The noise in eqn. (6.3.28) is defined as ξ
i
(x, t) =
5
α
δ(x −x
α
(t))ξ
i
α
(t) =
;
ˆρ(x, t)θ
i
(x, t) and is multiplicative in nature. θ
i
is the Gaussian
white noise with correlation,
1
θ
i
(x, t)θ
j
(x
, t
)
2
= 2k
B
T ¯γ
0
δ(x − x
)δ(t − t
). (6.3.29)
ThefirsttermontheRHSofeqn. (6.3.28) can be adjusted, allowing us to write
ˆ
#
ij
(x, t) =
5
α
p
i
α
p
j
α
δ(x − x
α
(t))
5
ν
δ(x − x
ν
(t))
5
ν
δ(x − x
ν
(t))
=
5
α
p
i
α
δ(x − x
α
(t))
5
ν
p
j
ν
δ(x − x
ν
(t))
5
ν
δ(x − x
ν
(t))
=
ˆg
i
(x, t) ˆg
j
(x, t)
ˆρ(x, t)
, (6.3.30)
where we have used the presence of delta functions in the numerator to make the α and ν
particles the same when they are both at the same position x. Thus the balance equation
(6.3.28) reduces to
∂ ˆg
i
∂t
+∇
j
ˆg
i
ˆg
j
ˆρ
+ˆρ∇
i
δF
U
δ ˆρ
−¯γ
0
ˆg
i
=
√
ρθ
i
. (6.3.31)
F
U
[ρ] is related to the bare interaction potential through the relation
F
U
=
1
2
dx
dx
ˆρ(x)U (x − x
) ˆρ(x
). (6.3.32)
The exact balance equations discussed above are similar in form to the correspond-
ing fluctuating-nonlinear-hydrodynamics (FNH) equations (see eqns. (6.2.16)–(6.2.20))
for reversible microscopic dynamics, but with some subtle differences. First, the functional
F
U
in the microscopic balance equation (6.3.31) does not contain an entropic (ideal-gas)
term, unlike eqn. (6.3.16) in the case of microscopic Brownian dynamics. The
ˆ
#
ij
term,
averaged over the p
α
with a Maxwell distribution, gives rise to such a term in F
U
. Sec-
ond, the stochastic part in eqn. (6.3.31) involves multiplicative noise. It is related to the
noise in the microscopic equations and is different from the phenomenological frictional
term in (6.2.20) involving bare transport coefficients in FNH (with reversible microscopic
dynamics).
Appendix to Chapter 6
A6.1 The coarse-grained free energy
We present here the calculation of the kinetic part, F
k
[ρ,g], of the coarse-grained free-
energy functional of the one-component system. The macroscopic system is divided into
cells of volume
cl
much larger than the average volume per particle. The linear dimen-
sions of a cell (
cl
) are much smaller than any correlation length in the system. The
characteristic function ψ
a
(r) for the cell labeled a is defined as follows:
ψ
a
(r) =
⎧
⎨
⎩
1, if r ∈ a,
0, if r a.
(A6.1.1)
The mass density ρ
a
and the total momentum g
a
corresponding to the ath cell are given in
terms of the characteristic function ψ
a
as
ρ
a
=
N
α=1
mψ
a
(r
α
), g
a
=
N
α=1
p
α
ψ
a
(r
α
). (A6.1.2)
In the limit
cl
→ 0 and a →∞, the cell values of the densities of mass and momentum
can be treated as continuous fields,
ρ
a
→ ρ(x)
cl
, (A6.1.3)
g
a
→ g(x)
cl
. (A6.1.4)
On the other hand, the microscopic density ˆρ of the particles at every given space-time
point is defined with eqn. (1.3.15), considering the particles to be point-like. The process
of coarse graining therefore can be considered as a mapping T [ρ|ˆρ] from the microscopic
density ˆρ to the smooth field ρ at every point.
The coarse-grained free-energy functional F in terms of the ρ
a
and g
a
is defined using
the statistical-mechanical partition function as
e
−β F
= Tr e
−β H
%
a
δ
k
ρ
a
−
N
α=1
mψ
a
(r
α
)
%
a
δ
g
a
−
N
α=1
p
α
ψ
a
(r
α
)
. (A6.1.5)
310