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.