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when the initial velocity is zero. For each realization of the stochastic force (
3.74
)pro-
vides a trajectory solving the Langevin equation. As we learned in our discussion of the
law of frequency of errors, the best representation of all these realizations is the average
value, but we recall that the average of the
ξ
-fluctuations is zero, so the average velocity
vanishes,
V
(
t
)
=
0
,
(3.75)
when the initial velocity is set to zero.
The lowest-order non-vanishing moment of the velocity of the Brownian particle is
therefore the autocovariance function
t
1
t
2
e
−
λ
(
t
1
)
e
−
λ(
t
2
−
t
2
)
ξ(
t
1
−
t
1
)ξ(
t
2
)
dt
1
dt
2
,
(
t
1
)
(
t
2
)
=
V
V
(3.76)
0
0
which is averaged over an equilibrium ensemble of realizations of the fluctuations.
Note that we no longer subscript the averaging bracket when there is little danger of
confusion. The equilibrium autocorrelation function for the stochastic force is given by
t
2
)
=
ξ(
t
1
)ξ(
t
2
)
ξ
2
,
ξ
(
t
1
,
(3.77)
where
ξ
2
is independent of time and, if we make the stationarity assumption,
ξ
(
t
1
,
t
2
)
=
ξ
(
|
t
1
−
t
2
|
).
(3.78)
An analytic form of the autocorrelation function that is often used is the exponential
e
−
λ
|
t
1
−
t
2
|
.
ξ
(
|
t
1
−
t
2
|
)
=
(3.79)
Note that the Laplace transform of the exponential
ξ
(
t
)
is
∞
1
ˆ
ξ
(
e
−
ut
u
)
=
LT
[
ξ
(
t
),
u
]≡
ξ
(
t
)
dt
=
+
λ
,
(3.80)
u
0
so that
1
λ
≡
τ
c
.
ˆ
ξ
(
0
)
=
(3.81)
τ
c
.
To simplify the calculations we assume that the random force is delta-correlated in
time,
We denote the correlation time of the noise by
2
ξ(
t
1
)ξ(
t
2
)
=
2
ξ
τ
c
δ(
t
1
−
t
2
),
(3.82)
so that the stationary autocorrelation function reduces to
)
=
ξ(
)
t
)ξ(
0
ξ
(
ξ
2
=
τ
c
δ(
),
t
2
t
(3.83)
