Part III — Theoretical Physics of Soft Condensed
Matter
Based on lectures by M. E. Cates
Notes taken by Dexter Chua
Lent 2018
These notes are not endorsed by the lecturers, and I have modified them (often
significantly) after lectures. They are nowhere near accurate representations of what
was actually lectured, and in particular, all errors are almost surely mine.
Soft Condensed Matter refers to liquid crystals, emulsions, molten p olymers and other
microstructured fluids or semisolid materials. Alongside many hightech examples,
domestic and biological instances include mayonnaise, toothpaste, engine oil, shaving
cream, and the lubricant that stops our joints scraping together. Their behaviour is
classical (~ = 0) but rarely is it deterministic: thermal noise is generally important.
The basic modelling approach therefore involves continuous classical field theories,
generally with noise so that the equations of motion are stochastic PDEs. The form
of these equations is helpfully constrained by the requirement that the Boltzmann
distribution is regained in the steady state (when this indeed holds, i.e. for systems
in contact with a heat bath but not subject to forcing). Both the dynamical and
steadystate behaviours have a natural expression in terms of path integrals, defined
as weighted sums of trajectories (for dynamics) or configurations (for steady state).
These concepts will be introduced in a relatively informal way, focusing on how they
can b e used for actual calculations.
In many cases meanfield treatments are sufficient, simplifying matters considerably.
But we will also meet examples such as the phase transition from an isotropic fluid
to a ‘smectic liquid crystal’ (a layered state which is periodic, with solidlike order,
in one direction but can flow freely in the other two). Here meanfield theory gets
the wrong answer for the order of the transition, but the right one is found in a
selfconsistent treatment that lies one step beyond meanfield (and several steps short
of the renormalization group, whose application to classical field theories is discussed
in other courses but not this one).
Imp ortant models of soft matter include diffusive
φ
4
field theory (‘Model B’), and
the noisy Navier–Stokes equation which describes fluid mechanics at colloidal scales,
where the noise term is responsible for Brownian motion of suspended particles in a
fluid. Coupling these together creates ‘Model H’, a theory that describes the physics of
fluidfluid mixtures (that is, emulsions). We will explore Model B, and then Model H,
in some depth. We will also explore the continuum theory of nematic liquid crystals,
which spontaneously break rotational but not translational symmetry, focusing on
top ological defects and their associated mathematical structure such as homotopy
classes.
Finally, the course will cover some recent extensions of the same general approach to
systems whose microscopic dynamics does not have timereversal symmetry, such as
selfpropelled colloidal swimmers. These systems do not have a Boltzmann distribution
in steady state; without that constraint, new field theories arise that are the subject of
ongoing research.
Prerequisites
Knowledge of Statistical Mechanics at an undergraduate level is essential. This course
complements the following Michaelmas Term courses although none are prerequisites:
Statistical Field Theory; Biological Physics and Complex Fluids; Slow Viscous Flow;
Quantum Field Theory.
Contents
0 Introduction
0.1 The physics
0.2 The mathematics
1 Revision of equilibrium statistical physics
1.1 Thermodynamics
1.2 Coarse Graining
2 Mean field theory
2.1 Binary fluids
2.2 Nematic liquid crystals
3 Functional derivatives and integrals
3.1 Functional derivatives
3.2 Functional integrals
4 The variational method
4.1 The variational method
4.2 Smectic liquid crystals
5 Dynamics
5.1 A single particle
5.2 The Fokker–Planck equation
5.3 Field theories
6 Model B
7 Model H
8 Liquid crystals hydrodynamics
8.1 Liquid crystal models
8.2 Coarsening dynamics for nematics
8.3 Topological defects in three dimensions
9 Active Soft Matter
0 Introduction
0.1 The physics
Unsurprisingly, in this course, we are going to study models of softcondensed
matter. Soft condensed matter of various types are ubiquitous in daily life:
Type Examples
emulsions mayonnaise pharmaceuticals
suspensions toothpaste paints and ceramics
liquid crystals wet soap displays
polymers gum plastics
The key property that makes them “soft” is that they are easy to change in
shape but not volume (except foams). To be precise,
–
They have a shear modulus
G
of
∼
10
2
–10
7
Pascals (compare with steel,
which has a shear modulus of 10
10
Pascals).
–
The bulk modulus
K
remains large, with order of magnitude
K ∼
10
10
Pascal. As K/G ∼ ∞, this is the same as the object is incompressible.
Soft condensed matter exhibit viscoelasticity, i.e. they have slow response to
a changing condition. Suppose we suddenly apply a force on the material. We
can graph the force and the response in a single graph:
t
σ
0
σ
0
/G
τ
η
−1
here the blue, solid line is the force applied and the red, dashed line is the
response. The slope displayed is
η
−1
, and
η ≈ G
0
τ
is the viscosity. Note that
the time scale for the change is of the order of a few seconds! The reason for
this is large internal length scales.
Thing Length scale
Polymer 100 nm
Colloids ∼ 1 µm
Liquid crystal domains ∼ 1 µm
These are all much much larger than the length scale of atoms.
Being mathematicians, we want to be able to model such systems. First of
all, observe that quantum fluctuations are negligible. Indeed, the time scale
τ
Q
of quantum fluctuations is given by
~ω
Q
=
~
τ
Q
' k
B
T.
At room temperature, this gives that
τ
Q
∼ 10
−13
s
, which is much much smaller
than soft matter time scales, which are of the order of seconds and minutes. So
we might as well set ~ = 0.
The course would be short if there were no fluctuations at all. The counterpart
is that thermal fluctuations do matter.
To give an example, suppose we have some hard, spherical colloids suspended
in water, each of radius
a ' 1 µm
. An important quantity that determines the
behaviour of the colloid is the volume fraction
Φ =
4
3
πa
3
N
V
,
where N is the number of colloid particles.
Experimentally, we observe that when Φ
<
0
.
49, then this behaves like fluid,
and the colloids are free to move around.
In this regime, the colloid particles undergo Brownian motion. The time scale of
the motion is determined by the diffusivity constant, which turns out to be
D =
k
B
T
6πη
s
a
,
where
η
s
is the solvent viscosity. Thus, the time
τ
it takes for the particle to
move through a distance of its own radius
a
is given by
a
2
=
Dτ
, which we can
solve to give
τ ∼
a
3
η
s
k
B
T
.
In general, this is much longer than the time scale
τ
Q
of quantum fluctuations,
since a
3
η
S
~.
When Φ > 0.55, then the colloids fall into a crystal structure:
Here the colloids don’t necessarily touch, but there is still resistance to change
in shape due to the entropy changes associated. We can find that the elasticity
is given by
G ' k
B
T
N
V
.
In both cases, we see that the elasticity and time scales are given in terms of
k
B
T
. If we ignore thermal fluctuations, then we have
G
= 0 and
τ
=
∞
, which
is extremely boring, and more importantly, is not how the real world behaves!
0.2 The mathematics
To model the systems, one might begin by looking at the microscopic laws
of physics, and build models out of them. However, this is usually infeasible,
because there are too many atoms and molecules lying around to track them
individually. The solution is to do some coarsegraining of the system. For
example, if we are modelling colloids, we can introduce a function
ψ
(
r
) that
tells us the colloid density near the point
r
. We then look for laws on how this
function
ψ
behave, which is usually phenomenological, i.e. we try to find some
equations that happen to model the real world well, as opposed to deriving these
laws from the underlying microscopic principles. In general,
ψ
will be some sort
of order parameter that describes the substance.
The first thing we want to understand is the equilibrium statistical physics.
This tell us what we expect the field
ψ
to look like after it settles down, and also
crucially, how the field is expected to fluctuate in equilibrium. Ultimately, what
we get out of this is
P
[
ψ
(
r
)], the probability (density) of seeing a particular field
configuration at equilibrium. The simplest way of understanding this is via mean
field theory, which seeks a single field
ψ
that maximizes
P
[
ψ
(
r
)]. However, this
does not take into account fluctuations, A slightly more robust way of dealing
with fluctuations is the variational method, which we will study next.
After understanding the equilibrium statistical physics, we turn to under
standing the dynamics, namely what happens when we start our system in a
nonequilibrium state. We will be interested in systems that undergo phase
transition. For example, liquid crystals tend to be in a disordered state at high
temperatures and ordered at low temperatures. What we can do is then to
prepare our liquid crystals at high temperature so that it stays at a homogeneous,
disordered state, and then rapidly decrease the temperature. We then expect
the system to evolve towards a ordered state, and we want to understand how it
does so.
The first step to understanding dynamics is to talk about the hydrodynamic
level equations, which are deterministic PDEs for how the system evolves. These
usually look like
˙
ψ(r, t) = ··· ,
These equations come from our understanding of equilibrium statistical mechan
ics, and in particular that of the free energy functional and chemical potential.
Naturally, the hydrodynamic equations do not take into account fluctuations,
but report the expected evolution in time. These equations are particularly
useful in late time behaviour where the existing movement and changes dominate
over the small fluctuations.
To factor in the fluctuations, we promote our hydrodynamic PDEs to stochas
tic PDEs of the form
˙
ψ(r, t) = ···+ noise.
Usually, the noise comes from random external influence we do not wish to
explicitly model. For example, suspensions in water are bombarded by the water
molecules all the time, and we model the effect by a noise term. Since the noise is
the contribution of a large number of largely independent factors, it is reasonable
to model it as Gaussian noise.
The mean noise will always be zero, and we must determine the variance. The
key insight is that this random noise is the mechanism by which the Boltzmann
distribution arises. Thus, the probability distribution of the field
ψ
determined
by the stochastic PDE at equilibrium must coincide with what we know from
equilibrium statistical dynamics. Since there is only one parameter we can toggle
for the random noise, this determines it completely. This is the fluctuation
dissipation theorem.
Example.
To model a onecomponent isothermal fluid such as water, we can
take
ψ
(
r, t
) to consist of the density
ρ
and velocity
v
. The hydrodynamic PDE
is exactly the Navier–Stokes equation. Assuming incompressibility, so that
˙ρ
= 0,
we get
ρ(
˙
v + v · ∇v) = η∇
2
v − ∇p,
We can promote this to a stochastic PDE, which is usually called the Navier–
Stokes–Landau–Lipschitz equation. This is given by
ρ(
˙
v + v · ∇v) = η∇
2
v − ∇p + ∇·Σ
N
,
The last term is thought of as a noise stress tensor on our fluid, and is conven
tionally treated as a Gaussian. As mentioned, this is fixed by the fluctuation
dissipation theorem, and it turns out this is given by
hΣ
N
ij
(r, t)Σ
N
k`
(r
0
, t
0
)i = 2k
B
T η(δ
i`
δ
jk
+ δ
ik
δ
j`
)δ(r − r
0
)δ(t − t
0
).
Example.
If we want to describe a binary fluid, i.e. a mixture of two fluids, we
introduce a further composition function
φ
that describes the (local) proportion
of the fluids present.
If we think about liquid crystals, then we need to add the molecular orienta
tion.
1 Revision of equilibrium statistical physics
1.1 Thermodynamics
A central concept in statistical physics is entropy.
Definition (Entropy). The entropy of a system is
S = −k
B
X
i
p
i
log p
i
,
where
k
B
is Boltzmann’s constant,
i
is a microstate — a complete specification
of the microscopics (e.g. the list of all particle coordinates and velocities) — and
p
i
is the probability of being in a certain microstate.
The axiom of Gibbs is that a system in thermal equilibrium maximizes
S
subject to applicable constraints.
Example.
In an isolated system, the number of particles
N
, the energy
E
and
the volume
V
are all fixed. Our microstates then range over all microstates
that have this prescribed number of particles, energy and volume only. After
restricting to such states, the only constraint is
X
i
p
i
= 1.
Gibbs says we should maximize
S
. Writing
λ
for the Lagrange multiplier
maintaining this constraint, we require
∂
∂p
i
S − λ
X
i
p
i
!
= 0.
So we find that
−k
B
log p
i
+ 1 − λ = 0
for all i. Thus, we see that all p
i
are equal.
The above example does not give rise to the Boltzmann distribution, since
our system is completely isolated. In the Boltzmann distribution, instead of
fixing E, we fix the average value of E instead.
Example.
Consider a system of fixed
N, V
in contact with a heat bath. So
E
is no longer fixed, and fluctuates around some average
hEi
=
¯
E
. So we can
apply Gibbs’ principle again, where we now sum over all states of all
E
, with
the restrictions
X
p
i
E
i
=
¯
E,
X
p
i
= 1.
So our equation is
∂
∂p
i
S − λ
I
X
p
i
− λ
E
X
p
i
E
i
= 0.
Differentiating this with respect to p
i
, we get
−k
B
(log p
i
+ 1) − λ
I
− λ
E
E
i
= 0.
So it follows that
p
i
=
1
Z
e
−βE
i
,
where Z =
P
i
e
−βE
i
and β = λ
E
/k
B
. This is the Boltzmann distribution.
What is this mysterious
β
? Recall that the Lagrange multiplier
λ
E
measures
how S reacts to a change in
¯
E. In other words,
∂S
∂E
= λ
E
= k
B
β.
Moreover, by definition of temperature, we have
∂S
∂E
V,N,...
=
1
T
.
So it follows that
β =
1
k
B
T
.
Recall that the first law of thermodynamics says
dE = T dS − P dV + µ dN + ··· .
This is a natural object to deal with when we have fixed
S, V, N
, etc. However,
often, it is temperature that is fixed, and it is more natural to consider the free
energy:
Definition
(Helmholtz free energy)
.
The Helmholtz free energy of a system at
fixed temperature, volume and particle number is defined by
F (T, V, N ) = U − T S =
¯
E − T S = −k
B
T log Z.
This satisfies
dF = −S dT − P dV + µ dN + ··· ,
and is minimized at equilibrium for fixed T, V, N.
1.2 Coarse Graining
Usually, in statistical mechanics, we distinguish between two types of objects
— microstates, namely the exact configuration of the system, and macrostates,
which are variables that describe the overall behaviour of the system, such that
pressure and temperature. Here we would like to consider something in between.
For example, if we have a system of magnets as in the Ising model, we the
microstate would be the magnetization at each site, and the macrostate would
be the overall magnetization. A coarsegraining of this would be a function
m
(
r
)
of space that describes the “average magnetization around
r
”. There is no fixed
prescription on how large an area we average over, and usually it does not matter
much.
In general, the coarsegrained variable would be called
ψ
. We can define a
coarsegrained partition function
Z[ψ(r)] =
X
i∈ψ
e
−βE
i
,
where we sum over all states that coarsegrain to
ψ
. We can similarly define the
energy and entropy by restricting to all such ψ, and get
F [ψ] = E[ψ] − T S[ψ].
The probability of being in a state ψ is then
P[ψ] =
e
−βF [ψ]
Z
TOT
, Z
TOT
=
Z
e
−βF [ψ]
D[ψ].
What we have on the end is a functional integral, where we integrate over all
possible values of ψ. We shall go into details later. We then have
F
TOT
= −k
B
T log Z
TOT
.
In theory, one can obtain
F
[
ψ
] by explicitly doing a coarse graining of the
macroscopic laws.
Example.
Consider an interacting gas with
N
particles. We can think of
the energy as a sum of two components, the ideal gas part (
d
2
NkT
), and an
interaction part, given by
E
int
=
1
2
X
i6j
U(r
i
− r
j
),
where
i, j
range over all particles with positions
r
i
, r
j
respectively, and
U
is
some potential function. When we do coarsegraining, we introduce a function
ρ
that describes the local density of particles. The interaction energy can then be
written as
E
int
=
1
2
ZZ
U(r − r
0
)ρ(r)ρ(r
0
) dr dr
0
.
Similarly, up to a constant, we can write the entropy as
S[ρ] = −k
B
Z
ρ(r) log ρ(r) dr.
In practice, since the microscopic laws aren’t always accessible anyway, what
is more common is to take a phenomenological approach, namely we write down
a Taylor expansion of
F
[
ψ
], and then empirically figure out what the coefficients
should be, as a function of temperature and other parameters. In many cases, the
signs of the first few coefficients dictate the overall behaviour of the system, and
phase transition occurs when the change in temperature causes the coefficients
to switch signs.
2 Mean field theory
In this chapter, we explore the mean field theory of two physical systems —
binary fluids and nematic liquid crystals. In mean field theory, what we do is we
write down the free energy of the system, and then find a state
φ
that minimizes
the free energy. By the Boltzmann distribution, this would be the “most likely
state” of the system, and we can pretend F
TOT
= F [φ].
This is actually not a very robust system, since it ignores all the fluctuations
about the minimum, but gives a good starting point for understanding the
system.
2.1 Binary fluids
Consider a binary fluid, consisting of a mixture of two fluids
A
and
B
. For
simplicity, we assume we are in the symmetric case, where
A
and
B
are the same
“type” of fluids. In other words, the potentials between the fluids are such that
U
AA
(r) = U
BB
(r) 6= U
AB
(r).
We consider the case where
A
and
B
repulse each other (or rather, repulse each
other more than the
A

A
and
B

B
repulsions). Thus, we expect that at high
temperatures, entropy dominates, and the two fluids are mixed together well. At
low temperatures, energy dominates, and the two fluids would be wellseparated.
We let
ρ
A
(
r
) and
ρ
B
(
r
) be the coarsegrained particle density of each fluid,
and we set our order parameter to be
φ(r) =
ρ
A
(r) − ρ
B
(r)
(N
A
+ N
B
)/V
,
with
N
A
and
N
B
, the total amount of fluids
A
and
B
, and
V
the volume. This
is normalized so that φ(r) ∈ [−1, 1].
We model our system with Landau–Ginzburg theory, with free energy given
by
βF =
Z
a
2
φ
2
+
b
4
φ
4
 {z }
f(φ)
+
κ
2
(∇φ)
2
dr,
where a, b, κ are functions of temperature.
Why did we pick such a model? Symmetry suggests the free energy should
be even, and if we Taylor expand any even free energy functional, the first few
terms will be of this form. For small
φ
and certain values of
a, b, κ
, we shall see
there is no need to look further into higher order terms.
Observe that even without symmetry, we can always assume we do not have
a linear term, since a
cφ
term will integrate out to give
cV
¯
φ
, and
¯
φ
, the average
composition of the fluid, is a fixed number. So this just leads to a constant shift.
The role of the gradient term
R
κ
2
(
∇φ
)
2
d
r
captures at order
∇
(2)
the non
locality of E
int
,
E
int
=
X
i,j∈{A,B}
Z
ρ
i
(r)ρ
j
(r
0
)U
ij
(r − r
0
) dr dr
0
,
If we assume
φ
(
r
) is slowly varying on the scale of interactions, then we can
Taylor expand this E
int
and obtain a (∇φ)
2
term.
Now what are the coefficients
a, b, κ
? For the model to make sense, we want
the free energy to be suppressed for large fluctuating
φ
. Thus, we want
b, κ >
0,
while
a
can take either sign. In general, the sign of
a
is what determines the
behaviour of the system, so for simplicity, we suppose
b
and
κ
are fixed, and let
a vary with temperature.
To do mean field theory, we find a single
φ
that minimizes
F
. Since the
gradient term
R
κ
2
(
∇φ
)
2
d
x ≥
0, a naive guess would be that we should pick a
uniform φ,
φ(r) =
¯
φ.
Note that
¯
φ
is fixed by the constraint of the system, namely how much fluid of
each type we have. So we do not have any choice. In this configuration, the free
energy per unit volume is
F
V
= f(
¯
φ) =
a
2
¯
φ
2
+
b
4
¯
φ
4
.
The global of this function depends only on the sign of
a
. For
a >
0 and
a <
0
respectively, the plots look like this:
¯
φ
f
a > 0
a < 0
a > 0
a < 0
We first think about the a > 0 part. The key point is that the function f(φ) is
a convex function. Thus, for a fixed average value of
φ
, the way to minimize
f(φ) is to take φ to be constant. Thus, since
βF =
Z
f(φ(r)) +
κ
2
(∇φ)
2
dr,
even considering the first term alone tells us we must take
φ
to be constant, and
the gradient term reinforces this further.
The
a <
0 case is more interesting. The function
f
(
φ
) has two minima,
φ
1,2
= ±φ
B
, where
φ
B
=
r
−a
b
.
Now suppose
¯
φ
lies between
±φ
B
. Then it might be advantageous to have some
parts of the fluid being at
−φ
B
and the others at
φ
B
, and join them smoothly
in between to control the gradient term. Mathematically, this is due to the
concavity of the function f in the region [−φ
B
, φ
B
].
Suppose there is
V
1
many fluid with
φ
=
φ
1
, and
V
2
many fluid with
φ
=
φ
2
.
Then these quantities must obey
V
1
φ
1
+ V
2
φ
2
= V
¯
φ,
V
1
+ V
2
= V.
Concavity tells us we must have
V
1
f(φ
1
) + V
2
f(φ
2
) < (V
1
+ V
2
)f(
¯
φ).
Thus, if we only consider the
f
part of the free energy, it is advantageous to
have this phase separated state. If our system is very large in size, since the
interface between the two regions is concentrated in a surface of finite thickness,
the gradient cost will be small compared to the gain due to phase separation.
We can be a bit more precise about the effects of the interface. In the first
example sheet, we will explicitly solve for the actual minimizer of the free energy
subject to the boundary condition
φ
(
x
)
→ ±φ
B
as
x → ±∞
, as in our above
scenario. We then find that the thickness of the interface is (of the order)
ξ
0
=
−2κ
a
,
and the cost per unit area of this interface is
σ =
−8κa
3
9b
2
1/2
.
This is known as the interfacial tension. When calculating the free energy of a
phase separated state, we will just multiply the interfacial tension by the area,
instead of going back to explicit free energy calculations.
In general the meanfield phase diagram looks like
a
−1 1
¯
φ
a(T ) = 0
Within the solid lines, we have phase separation, where the ground state of the
system for the given
a
and
¯
φ
is given by the state described above. The inner
curve denotes spinodal instability, where we in fact have local instability, as
opposed to global instability. This is given by the condition
f
00
(
¯
φ
)
<
0, which
we solve to be
φ
S
=
r
−a
3b
.
What happens if our fluid is no longer symmetric? In this case, we should
add odd terms as well. As we previously discussed, a linear term has no effect.
How about a cubic term
R
c
3
φ
(
r
)
3
d
r
to our
βF
? It turns out we can remove
the
φ
(
r
) term by a linear shift of
φ
and
a
, which is a simple algebraic maneuver.
So we have a shift of axes on the phase diagram, and nothing interesting really
happens.
2.2 Nematic liquid crystals
For our purposes, we can imagine liquid crystals as being made of rodlike
molecules
We are interested in the transition between two phases:
– The isotropic phase, where the rods are pointed in random directions.
–
The nematic phase, where the rods all point in the same direction, so that
there is a longrange orientation order, but there is no long range positional
order.
In general, there can be two different sorts of liquid crystals — the rods can
either be symmetric in both ends or have “direction”. Thus, in the first case,
rotating the rod by 180
◦
does not change the configuration, and in the second
case, it does. We shall focus on the first case in this section.
The first problem we have to solve is to pick an order parameter. We want
to take the direction of the rod
n
, but mod it out by the relation
n ∼ −n
. One
way to do so is to consider the secondrank traceless tensor
n
i
n
j
. This has the
property that
A
i
n
i
n
j
is the component of a vector
A
in the direction of
n
, and
is invariant under
n
i
7→ n
j
. Observe that if we normalize
n
to be a unit vector,
then
n
i
n
j
has trace 1. Thus, if we have isotropic rods in
d
dimensions, then we
have
hn
i
n
j
i =
δ
ij
d
.
In general, we can defined a coarsegrained order parameter to be
Q
ij
(r) = hn
i
n
j
i
local
−
1
d
δ
ij
.
This is then a traceless symmetric secondrank tensor that vanishes in the
isotropic phase.
One main difference from the case of the binary fluid is that
Q
ij
is no longer
conserved., i.e. the “total Q”
Z
Q
ij
(r) dr
is not constant in time. This will have consequences for equilibrium statistical
mechanics, but also the dynamics.
We now want to construct the leadingorder terms of the “most general” free
energy functional. We start with the local part
f
(
Q
), which has to be a scalar
built on Q. The possible terms are as follows:
(i) There is only one linear one, namely Q
ii
= Tr(Q), but this vanishes.
(ii)
We can construct a quadratic term
Q
ij
Q
ji
=
Tr
(
Q
2
), and this is in general
nonzero.
(iii)
There is a cubic term
Q
ij
Q
jk
Q
ki
=
Tr
(
Q
3
), and is also in general nonzero.
(iv) There are two possible quartic terms, namely Tr(Q
2
)
2
and Tr(Q
4
).
So we can write
f(Q) = a Tr(Q
2
) + c Tr(Q
3
) + b
1
Tr(Q
2
)
2
+ b
2
Tr(Q
4
).
This is the local part of the free energy up to fourth order in
Q
. We can go on,
and in certain conditions we have to, but if these coefficients
b
i
are sufficiently
positive in an appropriate sense, this is enough.
How can we think about this functional? Observe that if all of the rods point
tend to point in a fixed direction, say
z
, and are agnostic about the other two
directions, then Q will be given by
Q
ij
=
−λ/2 0 0
0 −λ/2 0
0 0 λ
, λ > 0.
If the rod is agnostic about the
x
and
y
directions, but instead avoids the
z
direction, then
Q
ij
takes the same form but with
λ <
0. For the purposes of
f
(
Q
), we can locally diagonalize
Q
, and it should somewhat look like this form.
So this seeminglyspecial case is actually quite general.
The
λ >
0 and
λ <
0 cases are physically very different scenarios, but the
difference is only detected in the odd terms. Hence the cubic term is extremely
important here. To see this more explicitly, we compute f in terms of λ as
f(Q) = a
3
2
λ
2
+ c
3
4
λ
3
+ b
1
9
4
λ
4
+ b
2
9
8
λ
4
= ¯aλ
2
+ ¯cλ
3
+
¯
bλ
4
.
We can think of this in a way similar to the binary fluid, where
λ
is are sole
order parameter. We fix
¯
b
and
¯c <
0, and then vary
¯a
. In different situations,
we get
λ
f
α < α
c
α = α
c
α > α
c
Here the cubic term gives a discontinuous transition, which is a firstorder
transition. If we had ¯c > 0 instead, then the minima are on the other side.
We now move on to the gradient terms. The possible gradient terms up to
order ∇
(2)
and Q
(2)
are
κ
1
∇
i
∇
i
Q
j`
Q
j`
= κ
1
∇
2
Tr(Q
2
)
κ
2
(∇
i
Q
im
)(∇
j
Q
jm
) = κ
2
(∇ · Q)
2
κ
3
(∇
i
Q
jm
)(∇
j
Q
im
) = yuck.
Collectively, these three terms describe the energy costs of the following three
things:
splay
twisting
bend
In general, each of these modes correspond to linear combination of the three
terms, and it is difficult to pin down how exactly these correspondences work.
Assuming these linear combinations are sufficiently generic, a sensible choice is
to set
κ
1
=
κ
3
= 0 (for example), and then the elastic costs of these deformations
will all be comparable.
3 Functional derivatives and integrals
We shall be concerned with two objects — functional derivatives and integrals.
Functional derivatives are (hopefully) familiar from variational calculus, and
functional integrals might be something new. They will be central to what we
are going to do next.
3.1 Functional derivatives
Consider a scalar field φ(r), and consider a functional
A[φ] =
Z
L(φ, ∇φ) dr.
Under a small change
φ 7→ φ
+
δφ
(
r
) with
δφ
= 0 on the boundary, our functional
becomes
A[φ + δφ] =
Z
L(φ, ∇φ) + δφ
∂L
∂φ
+ ∇dφ ·
∂L
∂φ
dr
= A[φ] +
Z
δφ
∂L
∂φ
− ∇ ·
∂L
∂∇φ
dr,
where we integrated by parts using the boundary condition. This suggests the
definition
δA
δφ(r)
=
∂L
∂φ(r)
− ∇ ·
∂L
∂∇φ
.
Example.
In classical mechanics, we replace
r
by the single variable
t
, and
φ
by position x. We then have
A =
Z
L(x, ˙x) dt.
Then we have
δA
δx(t)
=
∂L
∂x
−
d
dt
∂L
∂ ˙x
.
The equations of classical mechanics are
δA
δx(t)
= 0.
The example more relevant to us is perhaps Landau–Ginzburg theory:
Example. Consider a coarsegrained free energy
F [φ] =
Z
a
2
φ
2
+
b
4
φ
4
+
κ
2
(∇φ)
2
dr.
Then
δF
δφ(r)
= aφ + bφ
3
− κ∇
2
φ.
In mean field theory, we set this to zero, since by definition, we are choosing
a single
φ
(
r
) that minimizes
F
. In the first example sheet, we find that the
minimum is given by
φ(x) = φ
B
tanh
x − x
0
ξ
0
,
where ξ
0
is the interface thickness we previously described.
In general, we can think of
δF
δφ(r)
as a “generalized force”, telling us how we
should change
φ
to reduce the free energy, since for a small change
δφ
(
r
)), the
corresponding change in F is
δF =
Z
δF
δφ(r)
δφ(r) dr.
Compare this with the equation
dF = −S dT − p dV + µ dN + h · dM + ··· .
Under the analogy, we can think of
δF
δφ(r)
as the intensive variable, and
δφ
(
r
) as
the extensive variable. If
φ
is a conserved scalar density such as particle density,
then we usually write this as
µ(r) =
δF
δφ(r)
,
and call it the chemical potential. If instead
φ
is not conserved, e.g. the
Q
we
had before, then we write
H
ij
=
δF
δQ
ij
and call it the molecular field.
We will later see that in the case where
φ
is conserved,
φ
evolves according
to the equation
˙
φ = −∇ · J, J ∝ −D∇µ,
where
D
is the diffusivity. The nonconserved case is simpler, with equation of
motion given by.
˙
Q = −ΓH.
Let us go back to the scalar field φ(r). Consider a small displacement
r 7→ r + u(r).
We take this to be incompressible, so that ∇ · u = 0. Then
φ 7→ φ
0
= φ
0
(r) = φ(r − u).
Then
δφ(r) = φ
0
(r) − φ(r) = −u · ∇φ(r) + O(u
2
).
Then
δF =
Z
δφ
δF
δφ
dr = −
Z
µu · ∇φ dr
=
Z
φ∇ · (µu) dr =
Z
(φ∇µ) · u dr =
Z
(φ∇
j
µ)u
j
dr.
using incompressibility.
We can think of the free energy change as the work done by stress,
δF =
Z
σ
ij
(r)ε
ij
(r) dr,
where
ε
ij
=
∇
i
u
j
is the strain tensor, and
σ
ij
is the stress tensor. So we can
write this as
δF =
Z
σ
ij
∇
i
u
j
dr = −
Z
(∇
i
σ
ij
)u
j
dr.
So we can identify
∇
i
σ
ij
= −φ∇
j
µ.
So µ also contains the “mechanical information”.
3.2 Functional integrals
Given a coarsegrained ψ, we have can define the total partition function
e
−βF
TOT
= Z
TOT
=
Z
e
−βF [ψ]
D[ψ],
where D[
ψ
] is the “sum over all field configurations”. In mean field theory, we
approximate this
F
TOT
by replacing the functional integral by the value of the
integrand at its maximum, i.e. taking the minimum value of
F
[
ψ
]. What we
are going to do now is to evaluate the functional integral “honestly”, and this
amounts to taking into account fluctuations around the minimum (since those
far away from the minimum should contribute very little).
To make sense of the integral, we use the fact that the space of all
ψ
has a
countable orthonormal basis. We assume we work in [0
, L
]
q
of volume
V
=
L
q
with periodic boundary conditions. We can define the Fourier modes
ψ
q
=
1
√
V
Z
ψ(r)e
−iq·r
dr,
Since we have periodic boundary conditions,
q
can only take on a set of discrete
values. Moreover, molecular physics or the nature of coarsegraining usually
implies there is some “maximum momentum”
q
max
, above which the wavelengths
are too short to make physical sense (e.g. vibrations in a lattice of atoms cannot
have wavelengths shorter than the lattice spacing). Thus, we assume
ψ
q
= 0 for
q > q
max
. This leaves us with finitely many degrees of freedom.
The normalization of ψ
q
is chosen so that Parseval’s theorem holds:
Z
ψ
2
dr =
X
q
ψ
q

2
.
We can then define
D[ψ] =
Y
q
dψ
q
.
Since we imposed a
q
max
, this is a finite product of measures, and is welldefined.
In some sense,
q
max
is arbitrary, but for most cases, it doesn’t really matter
what
q
max
we choose. Roughly speaking, at really short wavelengths, the
behaviour of
ψ
no longer depends on what actually is going on in the system,
so these modes only give a constant shift to
F
, independent of interesting,
macroscopic properties of the system. Thus, we will mostly leave the cutoff
implicit, but it’s existence is important to keep our sums convergent.
It is often the case that after doing calculations, we end up with some
expression that sums over the
q
’s. In such cases, it is convenient to take the
limit V → ∞ so that the sum becomes an integral, which is easier to evaluate.
An infinite product is still bad, but usually molecular physics or the nature
of coarse graining imposes a maximum
q
max
, and we take the product up to
there. In most of our calculations, we need such a
q
max
to make sense of our
integrals, and that will be left implicit. Most of the time, the results will be
independent of
q
max
(for example, it may give rise to a constant shift to
F
that
is independent of all the variables of interest).
Before we start computing, note that a significant notational annoyance is
that if
ψ
is a real variable, then
ψ
q
will still be complex in general, but they will
not be independent. Instead, we always have
ψ
q
= ψ
∗
−q
.
Thus, we should only multiply over half of the possible
q
’s, and we usually
denote this by something like
Q
+
q
.
In practice, there is only one path integral we are able to compute, namely
when βF is a quadratic form, i.e.
βF =
1
2
Z
φ(r)G(r − r
0
)φ(r
0
) dr dr
0
−
Z
h(r)φ(r) dr.
Note that this expression is nonlocal, but has no gradient terms. We can think
of the gradient terms we’ve had as localizations of firstorder approximations to
the nonlocal interactions. Taking the Fourier transform, we get
βF [ψ
q
] =
1
2
X
q
G(q)φ
q
φ
−q
−
X
q
h
q
φ
q
.
Example. We take Landau–Ginzburg theory and consider terms of the form
βF [φ] =
Z
n
ξφ
2
− hφ +
κ
2
(∇φ)
2
+
γ
2
(∇
2
φ)
2
o
dr
The
γ
term is new, and is necessary because we will be interested in the case
where κ is negative.
We can now take the Fourier transform to get
βF {φ
q
} =
1
2
X
q
+
(a + κq
2
+ γq
4
)φ
q
φ
−q
−
X
q
+
h
q
φ
q
.
=
X
q
+
(a + κq
2
+ γq
4
)φ
q
φ
−q
−
X
q
+
h
q
φ
q
.
So our G(q) is given by
G(q) = a + kq
2
+ γq
4
.
To actually perform the functional integral, first note that if
h 6
= 0, then we
can complete the square so that the
h
term goes away. So we may assume
h
= 0.
We then have
Z
TOT
=
Z
"
Y
q
+
dφ
q
#
e
−βF {φ
q
}
=
Y
q
+
Z
dφ
q
e
−φ
q

2
G(q)
Each individual integral can be evaluated as
Z
dφ
q
e
−φ
q

2
G(q)
=
Z
ρ dρ dθ e
−G(q)ρ
2
=
π
G(q)
,
where φ
q
= ρe
iθ
. So we find that
Z
TOT
=
Y
q
+
π
G(q)
,
and so
βF
T
= −log Z
T
=
X
q
+
log
G(q)
π
.
We now take large V limit, and replace the sum of the integral. Then we get
βF
T
=
1
2
V
(2π)
d
Z
q
max
dq log
G(q)
π
.
There are many quantities we can compute from the free energy.
Example. The structure factor is defined to be
S(k) = hφ
k
φ
−k
i =
1
Z
T
Z
φ
k
φ
−k
e
−
P
+
q
φ
q
φ
−q
G(q)
Y
q
+
dφ
q
.
We see that this is equal to
1
Z
T
∂Z
T
∂G(k)
= −
∂ log Z
T
∂G(k)
=
1
G(k)
.
We could also have done this explicitly using the product expansion.
This
S
(
k
) is measured in scattering experiments. In our previous example,
for small k and κ > 0, we have
S(q) =
1
a + κk
2
+ γk
4
≈
a
−1
1 + k
2
ξ
2
, ξ =
r
κ
a
.
where ξ is the correlation length. We can return to real space by
hφ
2
(r)i =
1
V
Z
φ(r)
2
dr
=
1
V
X
q
hφ
q

2
i
=
1
(2π)
d
Z
q
max
dq
a + κq
2
+ γq
4
.
4 The variational method
4.1 The variational method
The variational method is a method to estimate the partition function
e
−βF
TOT
=
Z
e
−βF [φ]
D[φ]
when
F
is not Gaussian. To simplify notation, we will set
β
= 1. It is common
to make a notational change, where we replace
F
TOT
with
F
and
F
[
φ
] with
H
[
φ
].
We then want to estimate
e
−F
TOT
=
Z
e
−F [φ]
D[φ].
We now make a notation change, where we write
F
TOT
as
F
, and
F
[
φ
] as
H
[
φ
]
instead, called the effective Hamiltonian. In this notation, we write
e
−F
=
Z
e
−H[φ]
D[φ].
The idea of the variational method is to find some upper bounds on
F
in
terms of path integrals we can do, and then take the best upper bound as our
approximation to F .
Thus, we introduce a trial Hamiltonian H
0
[φ], and similarly define
e
−F
0
=
Z
e
−H
0
[φ]
D[φ].
We can then write
e
−F
=
e
−F
0
R
e
−H
0
D[φ]
Z
e
−H
0
e
−(H−H
0
)
D[φ] = e
−F
0
he
−(H−H
0
)
i
0
,
where the subscript 0 denotes the average over the trial distribution. Taking the
logarithm, we end up with
F = F
0
− loghe
−(H−H
0
)
i
0
.
So far, everything is exact. It would be nice if we can move the logarithm inside
the expectation to cancel out the exponential. While the result won’t be exactly
equal, the fact that log is concave, i.e.
log(αA + (1 − α)B) ≥ α log A + (1 − α) log B.
Thus Jensen’s inequality tells us
loghY i
0
≥ hlog Y i
0
.
Applying this to our situation gives us an inequality
F ≤ F
0
− hH
0
− Hi
0
= F
0
− hH
0
i
0
+ hHi
0
= S
0
+ hHi
0
.
This is the Feynman–Bogoliubov inequality.
To use this, we have to choose the trial distribution
H
0
simple enough to
actually do calculations (i.e. Gaussian), but include variational parameters in
H
0
. We then minimize the quantity
F
0
− hH
0
i
+
hHi
0
over our variational
parameters, and this gives us an upper bound on
F
. We then take this to be
our best estimate of
F
. If we are brave, we can take this minimizing
H
0
as an
approximation of H, at least for some purposes.
4.2 Smectic liquid crystals
We use this to talk about the isotropic to smectic transition in liquid crystals.
The molecules involved often have two distinct segments. For example, we may
have soap molecules that look like this:
The key property of soap molecules is that the tail hates water while the head
likes water. So we expect these molecules to group together like
In general, we can imagine our molecules look like
and like attracts like. As in the binary fluid, we shall assume the two heads are
symmetric, so
U
AA
=
U
BB
6
=
U
AB
. If we simply want the different parts to stay
away from each other, then we can have a configuration that looks like
z
In general, we expect that there is such an order along the
z
direction, as
indicated, while there is no restriction on the alignments in the other directions.
So the system is a lattice in the
z
direction, and a fluid in the remaining two
directions. This is known as a smectic liquid crystal, and is also known as the
lamellar phase. This is an example of microphase separation.
As before, we let
φ
be a coarse grained relative density. The above ordered
phase would then look like
φ(x) = cos q
0
z.
for some
q
0
that comes from the molecular length. If our system is not perfectly
ordered, then we may expect it to look roughly like A cos q
0
z for some A.
We again use Landau–Ginzburg model, which, in our old notation, has
βF =
Z
a
2
φ
2
+
b
4
φ
4
+
κ
2
(∇φ)
2
+
γ
2
(∇
2
φ)
2
dr.
If we write this in Fourier space, then we get
βF =
1
2
X
q
(a + κq
2
+ γq
4
)φ
q
φ
−q
+
b
4V
X
q
1
,q
2
,q
3
φ
q
1
φ
q
2
φ
q
3
φ
−(q
1
+q
2
+q
3
)
.
Notice that the quartic term results in the rather messy sum at the end. For the
isosmectic transition, we choose κ < 0, γ > 0.
Again for simplicity, we first consider the case where
b
= 0. Then this is a
Gaussian model with
G(q) = a + κq
2
+ γq
4
= τ + α(q − q
0
)
2
,
Varying
a
gives a linear shift in
G
(
q
). As we change
a
, we get multiple different
curves.
q
G(q)
a = a
c
q
0
Thus, as a decreases, S(q) = hφ
q

2
i =
1
G(q)
blows up at some finite
q = q
0
=
r
−κ
2γ
, a
c
=
κ
2
4γ
.
We should take this blowing up as saying that the
q
=
q
0
states are highly
desired, and this results in an ordered phase. Note that any
q
with
q
=
q
0
is
highly desired. When the system actually settles to an ordered state, it has to
pick one such
q
and let the system align in that direction. This is spontaneous
symmetry breaking.
It is convenient to complete to square, and expand
G
about
q
=
q
0
and
a = a
c
. Then we have
G(q) = τ + α(q − q
0
)
2
,
where
τ = a − a
c
, α =
1
2
G
00
(q
0
) = −2κ.
Then the transition we saw above happens when τ = 0.
We now put back the quartic term. We first do this with mean field theory,
and then later return to the variational method.
Mean Field Theory
In mean field theory, it is easier to work in real space. We look for a single field
configuration that minimizes
F
. As suggested before, we try a solution of the
form
φ = A cos q
0
z,
which is smectic along
z
. We can then evaluate the free energy (per unit volume)
to be
βF
V
= βF [φ] =
a
2
φ
2
+
κ
2
(∇φ)
2
+
γ
2
(∇
2
φ)
2
+
b
4
φ
4
where the bar means we average over one period of the periodic structure. It is
an exercise to directly compute
φ
2
=
1
2
A
2
, (∇φ)
2
=
1
2
A
2
q
2
0
, (∇
2
φ)
2
=
1
2
A
2
q
4
0
, φ
4
=
3
8
A
4
.
This gives
βF
V
=
1
4
aA
2
+ κA
2
q
2