5Dynamics
III Theoretical Physics of Soft Condensed Matter
5.2 The Fokker–Planck equation
So far, we have considered a single particle, and considered how it evolved over
time. We then asked how likely certain trajectories are. An alternative question
we can ask is if we have a probability density for the position of
x
, how does
this evolve over time?
It is convenient to consider the overdamped limit, where
m
= 0. Our equation
then becomes
ζ ˙x = −∇V +
p
2k
B
T ζΛ.
Dividing by ζ and setting
˜
M = ζ
−1
, we get
˙x = −
˜
M∇V +
q
2k
B
T
˜
MΛ.
This
˜
M is the mobility, which is the velocity per unit force.
We define the probability density function
P (x, t) = probability density at x at time t.
We can look at the probability of moving by a distance ∆
x
in a time interval
∆t. Equivalently, we are asking Λ to change by
∆Λ =
1
√
2k
B
T ζ
(ζ∆x + ∇V · ∆t).
Thus, the probability of this happening is
W (∆x, x) ≡ P
∆t
(∆x) = N exp
−
1
4ζk
B
T ∆t
(ζ∆x + ∇V ∆t)
2
.
We will write
u
for ∆
x
. Note that
W
(
u, x
) is just a normal, finite-dimensional
Gaussian distribution in
u
. We can then calculate that after time ∆
t
, the
expectation and variance of u are
hui = −
∇V
ζ
∆t, hu
2
i − hui
2
=
2k
B
T
ζ
∆t + O(∆t
2
).
We can find a deterministic equation for P (x, t), given in integral form by
P (x, t + ∆t) =
Z
P (x − u, t)W (u, x − u) du.
To obtain a differential form, we Taylor expand the integrand as
P (x − u, t)W (u, x − u)
=
P − u∇P +
1
2
u
2
∇
2
P
W (u, x) − u∇W +
1
2
u
2
∇
2
W
,
where all the gradients act on
x
, not
u
. Applying the integration to the expanded
equation, we get
P (x, t + ∆t) = P (x, t) − hui∇P +
1
2
hu
2
i∇
2
P − P ∇hui,
Substituting in our computations for hui and hu
2
i gives
˙
P (x, t)∆t =
∇V
ζ
∇P +
k
B
T
ζ
∇
2
P +
1
ζ
P ∇
2
V
∆t.
Dividing by ∆t, we get
˙
P =
k
B
T
ζ
∇
2
P +
1
ζ
∇(P ∇V )
= D∇
2
P +
˜
M∇(P ∇V ),
where
D =
k
B
T
ζ
,
˜
M =
D
k
B
T
=
1
ζ
are the diffusivity and the mobility respectively.
Putting a subscript
1
to emphasize that we are working with one particle,
the structure of this is
˙
P
1
= −∇ · J
1
J
1
= −P
1
D∇(log P
1
+ βV )
= −P
1
˜
M∇µ(x),
where
µ = k
B
T log P
1
+ V
is the chemical potential of a particle in V (x), as promised. Observe that
– This is deterministic for P
1
.
–
This has the same information as the Langevin equation, which gives the
statistics for paths x(t).
–
This was “derived” for a constant
ζ
, independent of position. However,
the equations in the final form turn out to be correct even for
ζ
=
ζ
(
x
) as
long as the temperature is constant, i.e.
˜
M
=
˜
M
(
x
) =
D(x)
k
B
T
. In this case,
the Langevin equation says
˙x = −
˜
M(x)∇V +
p
2D(x)Λ.
The multiplicative (i.e. non-constant) noise term is problematic. To
understand multiplicative noise, we need advanced stochastic calculus
(Itˆo/Stratonovich). In this course (and in many research papers), we avoid
multiplicative noise.