0Introduction
III Theoretical Physics of Soft Condensed Matter
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 coarse-graining 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
non-equilibrium 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 one-component 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.