5Dynamics
III Theoretical Physics of Soft Condensed Matter
5.1 A single particle
We ultimately want to think about field theories, but it is helpful to first consider
the case of a single, 1-dimensional particle. The action of the particle is given by
A =
Z
L dt, L = T − V.
The equations of motion are
δA
δx(t)
=
∂L
∂x
−
d
dt
∂L
∂x
= 0.
For example, if
L =
1
2
m ˙x
2
− V (x),
then the equation of motion is
−
δA
δx(t)
= m¨x + V
0
(x) = 0.
There are two key properties of this system
– The system is deterministic.
–
The Lagrangian is invariant under time reversal. This is a consequence of
the time reversal symmetry of microscopic laws.
We now do something different. We immerse the particle in a fluid bath,
modelling the situation of a colloid. If we were an honest physicist, we would
add new degrees of freedom for each of the individual fluid particles. However,
we are dishonest, and instead we aggregate these effects as new forcing terms in
the equation of motion:
(i) We introduce damping, F
D
= −ζ ˙x.
(ii) We introduce a noise f, with hfi = 0.
We set F
BATH
= F
D
+ f. Then we set our equation of motion to be
−
δA
δx
= F
BATH
− ζ ˙x + f.
This is the Langevin equation.
What more can we say about the noise? Physically, we expect it to be the
sum of many independent contributions from the fluid particles. So it makes
sense to assume it takes a Gaussian distribution. So the probability density of
the realization of the noise being f is
P[f(t)] = N
f
exp
−
1
σ
2
Z
f(t)
2
dt
,
where
N
f
is a normalization constant and
σ
is the variance, to be determined.
This is called white noise. This has the strong independence property that
hf(t)f(t
0
)i = σ
2
δ(t − t
0
).
In this course, we always assume we have a Gaussian white noise.
Since we have a random noise, in theory, any path we can write down is a
possible actual trajectory, but some are more likely than others. To compute
the probability density, we fixed start and end points (
x
1
, t
1
) and (
x
2
, t
2
). Given
any path
x
(
t
) between these points, we can find the noise of the trajectory to be
f = ζ ˙x −
δA
δx
.
We then can compute the probability of this trajectory happening as
P
F
[x(t)] “=” P[f] = N
x
exp
−
1
2σ
2
Z
t
2
t
1
ζ ˙x −
δA
δx
2
dt
!
.
This is slightly dodgy, since there might be some Jacobian factors we have missed
out, but it doesn’t matter at the end.
We now consider the problem of finding the value of
σ
. In the probability
above, we wrote it as
P
F
, denoting the forward probability. We can also consider
the backward probability
P
B
[
x
(
t
)], which is the probability of going along the
path in the opposite direction, from (x
2
, t
2
) to (x
1
, t
1
).
To calculate this, we use the assumption that
δA
δx
is time-reversal invariant,
whereas ˙x changes sign. So the backwards probability is
P
B
[x(t)] = N
x
exp
−
1
2σ
2
Z
t
2
t
1
−ζ ˙x −
δA
δx
2
!
dt.
The point is that at equilibrium, the probability of seeing a particle travelling
along
x
(
t
) forwards should be the same as the probability of seeing a particle
travelling along the same path backwards, since that is what equilibrium means.
This is not the same as saying
P
B
[
x
(
t
)] is the same as
P
F
[
x
(
t
)]. Instead, if at
equilibrium, there are a lot of particles at
x
2
, then it should be much less likely
for a particle to go from x
2
to x
1
than the other way round.
Thus, what we require is that
P
F
[x(t)]e
−βH
1
= P
B
[x(t)]e
−βH
2
,
where
H
i
=
H
(
x
i
, ˙x
i
). This is the principle of detailed balance. This is a
fundamental consequence of microscopic reversibility. It is a symmetry, so coarse
graining must respect it.
To see what this condition entails, we calculate
P
F
[x(t)]
P
B
[x(t)]
=
exp
−
1
2σ
2
R
t
2
t
1
(ζ ˙x)
2
− 2ζ ˙x
δA
δx
+
δA
δx
2
dt
exp
−
1
2σ
2
R
t
2
t
1
(ζ ˙x)
2
+ 2ζ ˙x
δA
δx
+
δA
δx
2
dt
= exp
−
1
2σ
2
Z
t
2
t
1
−4ζ ˙x
δA
δx
dt
.
To understand this integral, recall that the Hamiltonian is given by
H(x, ˙x) = ˙x
∂L
∂ ˙x
− L.
In our example, we can explicitly calculate this as
H =
1
2
˙x
2
+ V (x).
We then find that
dH
dt
=
d
dt
˙x
∂L
∂ ˙x
−
dL
dt
= ¨x
∂L
∂ ˙x
+ ˙x
d
dt
∂L
∂ ˙x
−
˙x
∂L
∂x
+ ¨x
∂L
∂ ˙x
= ˙x
d
dt
∂L
∂ ˙x
−
∂L
∂x
= −˙x
δA
δx
.
Therefore we get
Z
t
2
t
1
˙x
δA
δx(t)
dt = −(H(x
2
, ˙x
2
) − H(x
1
, ˙x
1
)) = −(H
2
− H
1
).
Therefore, the principle of detailed balance tells us we should pick
σ
2
= 2k
B
T ζ.
This is the simplest instance of the fluctuation dissipation theorem.
Given this, we usually write
f =
p
2k
B
T ζΛ,
where Λ is a Gaussian process and
hΛ(t)Λ(t
0
)i = δ(t − t
0
).
We call Λ a unit white noise.
In summary, for a particle under a potential V , we have an equation
m¨x + V
0
(x) = −ζ ˙x + f.
The term
−ζ ˙x
gives an arrow of time en route to equilibrium, while the noise
term resolves time reversal symmetry once equilibrium is reached. Requiring
this fixes the variance, and we have
hf(t)f(t
0
)i = σ
2
δ(t − t
0
) = 2k
B
T ζδ(t − t
0
).
In general, in the coarse grained world, suppose we have mesostates
A, B
,
with probabilities e
−βF
A
and e
−βF
B
, then we have an identical statement
e
−βF
A
P(A → B) = e
−βF
B
P(B → A).