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”.