5Dynamics
III Theoretical Physics of Soft Condensed Matter
5.3 Field theories
Suppose now that we have
N
non-interacting colloids under the same potential
V
(
x
). As usual, we model this by some coarse-grained density field
ρ
(
x, t
). If
we assume that the particles do not interact, then the expected value of this
density is just given by
hρi = NP
1
, h˙ρi = N
˙
P
1
.
Then our previous discussion entails ˙ρ evolves by
˙ρ = −∇ · J,
where
hJi = −ρ
˜
M∇µ, µ = k
B
T log ρ + V (x).
If we wish to consider a general, interacting field, then we can take the same
equations, but set
µ =
δF
δρ
instead.
Note that these are hydrodynamic level equations for
ρ
, i.e. they only tell
us what happens to
hρi
. If we put
J
=
hJi
, then we get a mean field solution
that evolves to the minimum of the free energy. To understand the stochastic
evolution of ρ itself, we put
J = −ρ
˜
M∇µ + j,
where
j
is a noise current. This is the Langevin equation for a fluctuating field
ρ(r, t).
We can fix the distribution of
j
by requiring detailed balance as before. We
will implement this for a constant
M
=
ρ
˜
M
, called the collective mobility. This
is what we have to do to avoid having multiplicative noise in our system. While
this doesn’t seem very physical, this is reasonable in situations where we are
looking at small fluctuations about a fixed density, for example.
As before, we assume j(r, t) is Gaussian white noise, so
P[j(r, t)] = N exp
−
1
2σ
2
Z
t
2
t
1
dt
Z
dr |j(r, t)|
2
.
This corresponds to
hj
k
(r, t)j
`
(r
0
, t
0
)i = σ
2
δ
k`
δ(r −r
0
)δ(t −t
0
).
We now repeat the detailed balance argument to find σ
2
. We start from
J + M∇µ = j.
Using F to mean forward path, we have
P
F
[J(r, t)] = N exp
−
1
2σ
2
Z
t
2
t
1
dt
Z
dr |J + M∇µ|
2
,
where
µ =
δF [ρ]
δρ
.
We consider the backwards part and get
P
B
[J(r, t)] = N exp
−
1
2σ
2
Z
t
2
t
1
dt
Z
dr | − J + M∇µ|
2
,
Then
log
P
F
P
B
= −
2M
σ
2
Z
t
2
t
1
dt
Z
dr J · ∇µ.
We integrate by parts in space to write
Z
dr J · ∇µ = −
Z
dr (∇ · J)µ =
Z
dr
˙ρ
δF
δρ
=
dF [ρ]
dt
.
So we get
log
P
F
P
B
= −
2M
σ
2
(F
2
− F
1
).
So we need
2M
σ
2
= β,
or equivalently
σ
2
= 2k
B
T M.
So our final many-body Langevin equation is
˙ρ = −∇ · J
J = −M∇
δF
δρ
+
p
2k
B
T MΛ,
where Λ is spatiotemporal unit white noise. As previously mentioned, a constant
M avoids multiplicative white noise.
In general, we get the same structure for any other diffusive system, such as
φ(r, t) in a binary fluid.
We might want to get a Fokker–Planck equation for our field theory. First
recap what we did. For one particle, we had the Langevin equation
˙x = −
˜
M∇V +
q
2k
B
T
˜
MΛ,
and we turned this into a Fokker–Planck equation
˙
P = −∇ · J
J = −P
˜
M∇µ
µ = k
B
T log P + V (x).
We then write this as
˙
P = ∇ ·
˜
Mk
B
T (∇ + β∇V ) P
,
where P (x, t) is the time dependent probability density for x.
A similar equation can be derived for the multi-particle case, which we will
write down but not derive. We replace
x
(
t
) with
ρ
(
r, t
), and we replace
P
(
x, t
)
with
P
[
ρ
(
r
);
t
]. We then replace
∇
with
δ
δρ(r)
. So the Fokker–Planck equation
becomes
˙
P [ρ(t); t] =
Z
dr
δ
δρ
k
B
T ∇ ·
˜
M∇
δ
δρ
+ β
δF
δρ
P
.
This is the Fokker–Planck equation for fields ρ.
As one can imagine, it is not very easy to solve. Note that in both cases, the
quantities
∇
+
β∇V
and
δ
δρ
+
β
δF
δρ
annihilate the Boltzmann distribution. So
the Boltzmann distribution is invariant.
The advantage of the Langevin equation is that it is easy to understand the
mean field theory/deterministic limit
ρ
=
ρ
hydro
(
r, t
). However, it is difficult to
work with multiplicative noise. In the Fokker–Planck equation, multiplicative
noise is okay, but the deterministic limit may be singular. Schematically, we
have
P [ρ(r), t] = δ(ρ(r, t) −ρ
hydro
(r, t)).
In this course, we take the compromise and use the Langevin equation with
constant M.