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.