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.