3Functional derivatives and integrals

III Theoretical Physics of Soft Condensed Matter

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 coarse-grained 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 non-conserved 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”.