Das Sh.P. Statistical Physics of Liquids at Freezing and Beyond
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1.4 Brownian motion 51
where we have chosen the mass m of the Brownian particle as unity to keep the algebra
simple. Here i denotes the corresponding Cartesian component. The first term on the RHS
represents a frictional drag term on the Brownian particle proportional to the velocity v.
ζ is a controlling kinetic coefficient, which in this case is taken to be the same for all
components i for simplicity. The second term f
i
(t), on the other hand, denotes the effect
of all the random kicks the big particle gets from the smaller particles in the surrounding
fluid. The above equation is the simplest form of the Langevin equation written in one
dimension. To solve eqn. (1.4.1) we multiply it by the factor exp(ζ t), to write
d
dt
[v
i
(t)e
ζ t
]=e
ζ t
f
i
(t). (1.4.2)
It is straightforward to integrate from t
0
to time t to obtain
v
i
(t)e
ζ t
= v
0
i
e
ζ t
0
+
t
t
0
dτ e
ζτ
f
i
(τ ), (1.4.3)
where v
i
(t
0
) = v
0
i
.Theith component of the velocity of the heavy particle is therefore
obtained as
v
i
(t) = v
0
i
e
−ζ(t−t
0
)
+
t
t
0
dτ f
i
(τ )e
−ζ(t−τ)
. (1.4.4)
The first term on the RHS of eqn. (1.4.4) gives the contribution from the initial value of
v
0
decaying exponentially, while the second term gives the integrated effect from the force
f
i
(τ ) acting over this time interval. To make further progress, we need to prescribe the
nature of the random force f
i
(t). Since it keeps pushing the big particle at random from all
directions, it is plausible to assume that, for all times t,
f
i
(t)=0. (1.4.5)
Using this result, we obtain that the average velocity at time t simply follows from the
exponential decay from the initial value, since the average noise is zero:
v
i
(t)=
+
v
0
i
,
e
−ζ(t−t
0
)
. (1.4.6)
In thermal equilibrium v
0
i
=0 and hence v
i
(t) remains zero at all times.
To compute the correlation of the velocity at different times, we need to know how
the noise values f
i
(t) at different times are correlated with each other. If the system is in
equilibrium, the correlation between the random forces at times t and t
depends only on
the difference of the two times, i.e., time translational invariance holds. For the isotropic
system we define this correlation in terms of the time-dependent function
¯
D
0
(t − t
),
f
i
(t) f
j
(t
)=δ
ij
¯
D
0
(t − t
). (1.4.7)
In Appendix A1.3 we consider some general properties of the function
¯
D
0
(t), and show
that it is symmetric around t = 0. It remains at all times bounded between limits +
¯
D(0)
and −
¯
D(0), decaying to zero in the long-time limit. As pointed out above, in the Langevin
52 Statistical physics of liquids
description of the dynamics the velocity v of the massive Brownian particle is treated as
a slow variable and the force f (t ) changes randomly over time scales much shorter than
the time scale of variation of v(t). As a simplification we will consider the extreme case in
which the correlation of the noise is defined in terms of a delta function,
f
i
(t) f
j
(t
)=δ
ij
¯
D
0
δ(t − t
), (1.4.8)
giving what is termed white noise. Furthermore, since the physical process of driving the
Brownian particle with the noise must follow causality, i.e., f
i
(t) has no influence on v
i
(t
),
where t > t
,wehave
f
i
(t)v
j
(t
)=0. (1.4.9)
The correlation of the velocities at two different times t and t
is obtained from the general
solution given by (1.4.4),
C
ij
(t, t
) =v
i
(t)v
j
(t
)=
+
v
0
i
v
0
j
,
e
−ζ(t+t
−2t
0
)
+ I
ij
, (1.4.10)
where the integral I
ij
is obtained from the relation
I
ij
=
t
t
0
dτ
t
t
0
dτ
e
−ζ(t−τ)
e
−ζ(t
−τ
)
f
i
(τ ) f
j
(τ
). (1.4.11)
On the RHS of eqn. (1.4.10) a cross term is set equal to zero using the causality relation
(1.4.9). Now, with the correlation of the noise as defined above in the diagonal form eqn.
(1.4.8), the velocity correlation function at two different times is obtained. For an isotropic
system both terms on the RHS of eqn. (1.4.11) are diagonal, reducing to C
ij
(t, t
) ≡
δ
ij
C(t, t
) and I
ij
= δ
ij
I
B
. We obtain for C(t, t
) the result
C(t, t
) = e
−ζ(t+t
−2t
0
)
C(t
0
, t
0
) + I
B
, (1.4.12)
where we define the integral I
B
as
I
B
=
¯
D
0
t
t
0
dτ
t
t
0
dτ
e
−ζ(t−τ)
e
−ζ(t
−τ
)
δ(τ − τ
). (1.4.13)
The last integral is evaluated in Appendix A1.3 to obtain the following result for the
velocity correlation function:
C(t, t
) =
¯
D
0
2ζ
e
−ζ |t −t
|
+ e
−ζ(t+t
−2t
0
)
C(t
0
, t
0
) −
¯
D
0
2ζ
. (1.4.14)
The equal-time correlation C(t, t) is obtained by setting t = t
:
C(t, t) =
¯
D
0
2ζ
+ e
−2ζ(t−t
0
)
C(t
0
, t
0
) −
¯
D
0
2ζ
. (1.4.15)
The above relation implies that if the system reaches equilibrium it continues to remain at
equilibrium for all times t unless disturbed. The equal-time correlation C(t, t) should then
1.4 Brownian motion 53
remain fixed for all t as given by eqn. (1.4.15) at
C(t
0
, t
0
) =
¯
D
0
2ζ
. (1.4.16)
If the Brownian particle is in equilibrium in the liquid at temperature T at time t
0
, then
the initial distribution for the velocities is Maxwell–Boltzmann, giving the equal-time
correlation as
C(t
0
, t
0
) = k
B
T, (1.4.17)
where we have taken the mass m for the particle to be unity as above. The correlation
C(t, t) is then obtained from eqn. (1.4.14) as
C(t, t) =
¯
D
0
2ζ
+ e
−2ζ(t−t
0
)
k
B
T −
¯
D
0
2ζ
. (1.4.18)
For the noise correlation (1.4.7), the corresponding velocity correlation function C(t, t
)
for the Brownian particle follows from eqn. (1.4.14). In the equilibrium fluid at temperature
T we obtain the correlation function C(t, t
) in the time translational invariant form,
C(t, t
) ≡ C(t − t
) = k
B
Te
−ζ |t −t
|
. (1.4.19)
Finally, according to the condition (1.4.17), we have, for the Brownian particle moving in
the fluid in equilibrium at temperature T ,
¯
D
0
= 2ζ C(t
0
, t
0
) = 2ζ k
B
T. (1.4.20)
The noise correlation as defined in eqn. (1.4.7) is therefore given by
f
i
(t) f
j
(t
)=2k
B
T ζδ(t − t
)δ
ij
. (1.4.21)
1.4.2 The Stokes–Einstein relation
Using the result for the velocity correlation function for the Brownian particles, straight-
forward integration gives the displacement x
i
(t) for a particle after time t. Since v
i
(t) =
dx
i
/dt, we can integrate the above result to compute the root-mean-square displacement
of the particle. We rewrite eqn. (1.4.19) as
-
d
dt
x
i
(t)
d
dt
x
i
(t
)
.
= k
B
Te
−ζ |t −t
|
. (1.4.22)
Note that in the above expression and in what follows below the repeated index i is not
summed over. The above equation is integrated to obtain the following results for the
correlation between the displacements at times t and t
:
x
i
(t)x
i
(t
)=
k
B
T
ζ
t
t
0
dτ
t
t
0
dτ
e
−ζ(τ−τ
)
≡ I
B
, (1.4.23)
54 Statistical physics of liquids
where we have denoted the displacement at time t as x
i
(t) = x
i
(t) − x
i
(t
0
). We evaluate
the integral I
B
in Appendix A1.3 to obtain
x
i
(t)x
i
(t
)=k
B
T ζ
−1
t + t
−|t − t
|−2t
0
+ ζ
−1
#
e
−ζ(t−t
0
)
+ e
−ζ(t
−t
0
)
− e
−ζ |t
−t|
− 1
$
. (1.4.24)
For the case t = t
the result for the mean-square displacement is
x
2
i
(t)=2k
B
T ζ
−1
t − t
0
+ ζ
−1
{e
−ζ(t−t
0
)
− 1}
. (1.4.25)
For short times relative to the characteristic time ζ
−1
, i.e., ζ(t − t
0
) 1, we can expand
the exponential on the RHS and obtain that the root-mean-square displacement is directly
proportional to the time (t − t
0
),
#+
[x
i
(t) − x
i
(t
0
)]
2
,$
1/2
= v
0
(t − t
0
), (1.4.26)
where v
2
0
= k
B
T is the root-mean-square average velocity of the Brownian particle (its
mass is taken as unity). This is free-particle behavior. On the other hand, for long times
relative to the characteristic time ζ
−1
, i.e., ζ(t − t
0
) 1, we obtain
+
[x
i
(t) − x
i
(t
0
)]
2
,
= 2k
B
T ζ
−1
(t − t
0
) ≡ 2
¯
D
0
(t − t
0
), (1.4.27)
showing that the mean-square displacement grows linearly with time. This is termed the
Einstein relation, and the constant of proportionality is defined as the diffusion coefficient
¯
D
0
.Fromeqn. (1.4.27) we obtain a relation between the diffusion coefficient
¯
D
0
and the
friction coefficient ζ , namely
¯
D
0
= k
B
T ζ
−1
. In fact, the friction coefficient ζ is a property
related to the resistance to the motion of the Brownian particle from the surrounding liquid.
If the Brownian particle has a radius a and the surrounding fluid has shear viscosity η, then
the frictional drag on the sphere is given by Stokes’ law, i.e., ζ = 0.6πaη. The diffusion
coefficient
¯
D
0
for the Brownian particle is then inversely proportional to the viscosity of the
surrounding medium according to what is termed the Stokes–Einstein relation (Einstein,
1956):
¯
D
0
=
k
B
T
0.6πaη
. (1.4.28)
It is important to note at this point that the relation (1.4.28) is established here for the
motion of a heavy Brownian particle in the surrounding fluid. However, it has also been
found that even in the case of a liquid particle moving in a sea of identical particles
the Stokes–Einstein relation of an inverse dependence of the diffusion coefficient on the
viscosity seems to holds in liquids. We present a phenomenological argument for justi-
fying the Stokes–Einstein relation with respect to the self-diffusion process in Chapter 4,
Section 4.1.2, using the free-volume theory.
Appendix to Chapter 1
A1.1 The Gibbs inequality
For any variable x we write the identity
g ln g − g + 1 =
g
1
dx
d
dx
[
x ln x − x + 1
]
=
g
1
dx ln x. (A1.1.1)
If g > 1 then the integrand ln x is positive in the integration range and hence the integral
on the RHS is positive definite. If, on the other hand, g < 1, then with a change of variable
y = 1/x the integrand reduces to ln y/y
2
, with y as the integration variable changing from
y = 1toy = 1/g > 1. Over this range the integrand is positive and hence the integral is
again positive definite. For g = 1 the RHS is zero. Thus we have proved that the quantity
g ln g − g +1 is always positive as long as g is positive.
A1.2 The force–force correlation
We obtain here the expression for the force–force correlation function for the αth particle
for a fluid interacting through a pairwise-additive continuous interaction potential. This is
used in computing the Einstein frequency for a liquid particle as presented in eqn. (1.3.38).
In general, if the Hamiltonian is H and the total potential energy is V , the derivatives with
respect to the position coordinates r
α
of the αth particle are equal since the kinetic part
of H is only momentum-dependent. We obtain for the force–force correlation for the αth
particle
F · F=∇
α
V · ∇
α
V =∇
α
V · ∇
α
H. (A1.2.1)
In order to compute the above average, let us first obtain for the general dynamic variable
A the following result:
55
56 Appendix to Chapter 1
A · ∇
α
H=
dr
N
dp
N
A ·
∂ H
∂r
α
f
eq
=−β
−1
dr
N
dp
N
A ·
∂
∂r
α
f
eq
= β
−1
dr
N
dp
N
∂
∂r
α
· A
f
eq
. (A1.2.2)
In reaching the last equality we ignored a surface term. In the present case, A ≡∇
α
V and
we obtain from eqn. (A1.2.1)
F · F=k
B
T
dr
N
dp
N
f
eq
∇
2
α
V. (A1.2.3)
For a fluid in which the particles are interacting through a two-body potential only the total
potential energy V is expressed as a pairwise sum of pair interactions u(r
1
, r
2
) ≡ u(12)
between particles 1 and 2 and the RHS of eqn. (A1.2.3) is obtained as
F · F=k
B
T
dr
N
dp
N
f
eq
∇
2
α
V
= k
B
T
dr
1
dr
2
∇
2
1
u(12)
3
(N − 1)
dr
3
... dr
N
exp[−β H ]
Z
N
4
=
k
B
T
N
dr
1
dr
2
∇
2
1
u(12)n
2
0
g(12)
= n
0
k
B
T
dr
#
∇
2
u(r)
$
g(r). (A1.2.4)
A1.3 Brownian motion
A1.3.1 The noise correlation
In equilibrium the correlation between the noise values f (t) in the Langevin equation at
two different times t and t
depends only on the difference of the two times, i.e., time
translational invariance holds. We define for the isotropic system
f
i
(t) f
j
(t
)=δ
ij
¯
D
0
(t − t
). (A1.3.1)
Since the correlation matrix is diagonal, in this case
¯
D
0
(t) =f
i
(t) f
i
(0)=f
i
(0) f
i
(−t)
=f
i
(0) f
i
(−t)=
¯
D
0
(−t). (A1.3.2)
Thus
¯
D
0
is a symmetric function. Also
¯
D
0
(0) =
1
f
2
i
(t)
2
is by definition a positive definite
quantity. In fact,
¯
D
0
(0) presents a limit for the function
¯
D
0
(t):
{ f
i
(t) ± f
i
(0)}
2
=
+
f
2
i
(t)
,
+
+
f
2
i
(0)
,
± 2 f
i
(t) f
i
(0)
= 2{
¯
D
0
(0) ±
¯
D
0
(t)}. (A1.3.3)
A1.3 Brownian motion 57
Since the LHS of (A1.3.3) is always greater than or equal to zero, we have that
¯
D
0
(t)
cannot go beyond the limits ±
¯
D
0
(0). Finally, for times t large relative to the correlation
time for the noise,
lim
t→∞
f
i
(t) f
i
(0)=f
i
(t) f
i
(0)=0. (A1.3.4)
The correlation of the noise represented by
¯
D
0
(t) therefore is a symmetric function
bounded between limits ±D(0) and decaying to zero in the long-time limit.
A1.3.2 Evaluation of the integrals
We present here the evaluation of some of the integrals involved in the discussion of the
Langevin dynamics. First we consider the integral I
B
defined in eqn. (1.4.13),
I
B
=
¯
D
0
t
t
0
dτ
t
t
0
dτ
e
−ζ(t−τ)
e
−ζ(t
−τ
)
δ(τ − τ
)
=
¯
D
0
θ(t − t
)
t
t
0
dτ
e
−ζ(t+t
−2τ
)
+ θ(t
− t)
t
t
0
dτ e
−ζ(t+t
−2τ)
=
¯
D
0
e
−ζ(t+t
)
2ζ
θ(t − t
){e
2t
− e
2t
0
}+θ(t
− t){e
2t
− e
2t
0
}
=
¯
D
0
2ζ
e
−ζ |t −t
|
− e
−ζ(t+t
−2t
0
)
. (A1.3.5)
Next we consider the integral I
B
defined in eqn. (1.4.23),
I
B
= θ(t
− t)
t
t
0
dτ
τ
t
0
dτ
e
−ζ(τ−τ
)
+
t
τ
dτ
e
−ζ(τ
−τ)
+ θ(t − t
)
t
t
0
dτ
τ
t
0
dτ
e
−ζ(τ−τ
)
+
t
τ
dτ
e
−ζ(τ
−τ)
= θ(t
− t)
2(t − t
0
) + ζ
−1
(
e
−ζ(t−t
0
)
+ e
−ζ(t
−t
0
)
− e
−ζ |t
−t|
− 1
)
+ θ(t − t
)
2(t
− t
0
) + ζ
−1
(
e
−ζ(t−t
0
)
+ e
−ζ(t
−t
0
)
− e
−ζ |t −t
|
− 1
)
= 2tθ(t
− t) +2t
θ(t − t
) − 2t
0
+ ζ
−1
e
−ζ(t−t
0
)
+ e
−ζ(t
−t
0
)
− e
−ζ |t
−t|
− 1
= t + t
−|t − t
|−2t
0
+ ζ
−1
e
−ζ(t−t
0
)
+ e
−ζ(t
−t
0
)
− e
−ζ |t
−t|
− 1
. (A1.3.6)
2
The freezing transition
In this chapter we discuss a theory for the freezing of an isotropic liquid into a crys-
talline solid state with long-range order. The transformation is a first-order phase transi-
tion with finite latent heat absorbed in the process. We focus on a first-principles order-
parameter theory of freezing that originated from the pioneering work of Ramakrishnan
and Yussouff (1979). The theory approaches the problem from the liquid side and views
the crystal as a liquid with grossly inhomogeneous density characterized by a lower sym-
metry of the corresponding lattice. This is in contrast to description of the crystal in
terms of phonons. The crucial quantity characterizing the physical state of the system
in this non-phonon-based model is the average one-particle density function n(x).The
thermodynamic description of either phase involves a corresponding extremum principle
for a relevant potential. The latter, obtained as a functional of the one-particle density
n(x) and the stable thermodynamic state of the system, is identified by the corresponding
density required for invoking the extremum principle. This approach, which is generally
referred to as the density-functional theory (DFT) of freezing (Haymet, 1987; Baus, 1987,
1990; Singh, 1991; Löwen, 1994; Ashcroft 1996), has been improved over the years and
successfully applied for the study of liquid-to-crystal transitions in various simple liq-
uids, the solid–liquid interface, two-dimensional systems, metastable glassy states, etc.
For applications of density-functional methods in statistical mechanics there exist general
reviews (Evans, 1979; Henderson, 1992). The theoretical formulation presented here also
becomes the starting point of consideration of the dynamics of the dense liquid near freez-
ing point and beyond that in the supercooled region. In the present chapter we focus on
the treatment of the thermodynamic phase transition from the liquid to the crystalline state
using DFT.
2.1 The density-functional approach
In the liquid state of matter the constituent particles move randomly so that the time-
averaged density is the same at all points. As the temperature of the liquid is lowered
(or equivalently the average density increased), the particles abandon the random spatial
arrangement and form the crystalline state having a spatially varying density n(x) accord-
ing to the symmetry of the corresponding lattice. At zero temperature, minimizing the
58
2.1 The density-functional approach 59
potential energy of the system corresponding to the optimum distribution of particles pro-
vides the choice of the relevant lattice structure. Understanding the transformation at finite
temperature is more subtle. Here the phase transformation is a result of the competition
between the energetic and entropic contributions. Predicting (in a quantitative manner) the
thermodynamic condition under which such a transformation occurs in a many-particle
system is the primary issue in the theory of freezing. This refers to the temperature of
freezing as well as the volume change in the freezing process. Subsequent to some ini-
tial attempts (Lindemann, 1910; Born, 1939; Kirkwood and Monroe, 1941; Jancovici,
1965; Brout, 1963, 1966; Thouless and Kosterlitz, 1973) at understanding this very general
phenomena of freezing, an order-parameter theory using the equilibrium density as the rel-
evant variable was proposed by Ramakrishnan and Yussouff (1979). This theory adopts an
approach that is intermediate between one of purely microscopic origin and a phenomeno-
logical description of the liquid near the transition. The thermodynamic properties of the
inhomogeneous crystalline state are obtained in terms of the corresponding quantities for
the homogeneous liquid state. The thermodynamics of the dense uniform liquid is well
understood through integral-equation theories or simulations, which are used as an input
in the theory described below.
In the density-functional theory for the freezing transition the interaction potential
between the liquid particles constitutes the microscopic-level description of the many-
particle system. The basic characteristics of the two-body potential for which a crys-
talline state appears (under appropriate conditions of density and temperature) include (a)
a strongly repulsive part at short range and (b) an attractive part effective at long range.
In such a condition the Hamiltonian can be written in a harmonic expansion around the
equilibrium sites which correspond to the minimum-potential-energy configuration. The
attractive part of the potential seemingly appears to play an important role in stabiliz-
ing the solid in a crystalline state in which each of the individual particles is localized
around its mean position. However, even for the hard-sphere interaction, the liquid-to-
crystal transition occurs, as was shown by computer simulations (Alder and Wainwright,
1962; Hoover and Ree, 1968). In this case the system must be stabilized externally by
application of appropriate pressure. The hard-sphere liquid transforms into an f.c.c. solid
at the reduced density n
0
σ
3
= 0.96 while the close-packed f.c.c. structure corresponds
to a much higher value of n
0
σ
3
= 1.41. There are thus aspects of the freezing transition
that go beyond the details of the interaction potential. Indeed, the geometric nature of the
freezing transition is apparent in the structural similarity of classical liquids at high den-
sities. Close to the freezing-transition point the static structure factor S(q) (see Chapter 1
for its definition) of many liquids near the first peak (wave vector q → q
m
) has approxi-
mately the value S(q
m
) = 2.85 (Ve rl et, 1974). This includes computer-simulation results
for a Lennard-Jones liquid (Lennard-Jones and Devonshire, 1939) along the melting curve
(Hansen and Verlet, 1969), a hard-sphere fluid (Ve rlet , 1968), and a one-component plasma
(Hansen, 1973), and experimental results for materials such as Ar (Page et al., 1969), Na
(Greenfield et al., 1971), and Pb (North et al., 1968). The value of S(q
m
) lies in the range
2.8–3.1. This is the analogue of the corresponding result in the solid phase, namely the
60 The freezing transition
Lindemann (1910) criterion of melting, i.e., the mean-square displacement u
2
scaled by
the square of the interatomic separation is 0.01 at melting (Ross and Alder, 1966).
Let us briefly outline the classical density-functional scheme adopted in the following.
In the order-parameter theory for the freezing transition presented here the free energy of
an inhomogeneous system is uniquely determined from the properties of its homogeneous
counterpart. The thermodynamic property is obtained as a functional of the one-particle
density n(x). The latter is expressed in terms of a suitable set of parameters, treated as the
order parameters of the transition. The liquid and the crystalline states correspond to order-
parameter values from which one obtains the appropriate density functions representing the
corresponding phase. The thermodynamic properties are computed assuming the system to
be in a single phase (either liquid or crystal) and fluctuations of the order parameter are
not taken into account. The density-functional approach is mean-field-like in this respect
and it ignores the effects of fluctuations which make the crystalline state unstable in lower
dimensions. In what follows, we first describe the formulation of an extremum principle
for the appropriate thermodynamic potential. This is followed by the construction of the
appropriate free-energy functional in terms of the inhomogeneous density function. Testing
the optimization of the thermodynamic property then naturally leads to identification of the
corresponding equilibrium phase.
2.1.1 A thermodynamic extremum principle
The equilibrium state of the many-particle system is identified from the minimization
of the appropriate thermodynamic free energy. For uniform systems the minimization is
done with respect to the variation of the average density. For example, in the liquid-to-
vapor transformation both phases are characterized by their respective densities. In the
grand-canonical-ensemble description of the uniform system the equilibrium state is char-
acterized by the minimum value of the thermodynamic potential
EQ
=−PV, where P
and V , respectively, denote the thermodynamic pressure and volume (McQuarie, 2000)
of the fluid. In the case of the freezing transition, on the other hand, the crystalline state
is characterized by a density function n(x) that is space-dependent and hence the corre-
sponding free energy is treated as a functional of the density function. The appropriate
free-energy functional required for the identification of the equilibrium state depends on
the type of ensemble chosen in the statistical averaging. In the present case of solid–liquid
transition it is appropriate to consider the grand-canonical ensemble so that the number of
particles in a given volume can change abruptly. We present here, following Haymet and
Oxtoby (1986) and Oxtoby (1991), the analysis for the proper extremum principle and the
construction of the free-energy functional for the inhomogeneous state.
Let us consider a system of N identical particles of mass m and the position and momen-
tum coordinates {r
α
, p
α
}, for α = 1,...,N . The whole set of phase-space variables is to
be denoted as {r
N
, p
N
}. Its Hamiltonian is given by
H
N
=
N
α=1
p
2
α
2m
+U +U
ext
≡ H
0
N
+U
ext
, (2.1.1)