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whose Laplace transform satisfies (
3.81
). Let us insert (
3.82
) into the double integral
(
3.76
) and assume
t
1
>
t
2
to obtain
t
2
e
−
λ(
t
1
+
t
2
−
2
t
2
)
ξ
2
τ
c
dt
2
t
2
)
=
V
(
t
1
)
V
(
0
e
−
λ(
t
1
−
t
2
)
−
e
−
λ(
t
1
+
t
2
)
2
=
ξ
τ
c
λ
.
(3.84)
The equilibrium condition for the solution is obtained for the sum
t
1
+
t
2
→∞
and the
difference
t
1
−
t
2
<
∞
so that (
3.84
) reduces to
ξ
2
τ
c
λ
e
−
λ(
t
1
−
t
2
)
.
V
(
t
1
)
V
(
t
2
)
eq
=
(3.85)
The equilibrium mean-square velocity obtained when
t
1
=
t
2
is, from (3.85),
V
2
2
eq
=
ξ
τ
c
λ
,
(3.86)
and, on repeating the same calculation with
t
1
<
t
2
, we finally obtain
e
−
λ
|
t
1
−
t
2
|
V
(
t
1
,
t
2
)
=
(3.87)
as the stationary velocity autocorrelation function.
Now let us apply a perturbation
E
(
t
)
to the web and replace the previous Langevin
equation with
dV
(
t
)
=−
λ
V
(
t
)
+
ξ(
t
)
+
E
(
t
),
(3.88)
dt
whose exact solution when the initial velocity is set to zero is given by
t
e
−
λ(
t
−
t
)
[
ξ(
t
)
+
t
)
]
dt
.
V
(
t
)
=
E
(
(3.89)
0
Again recalling that the average stochastic force vanishes, the average velocity of the
Brownian particle is
t
e
−
λ(
t
−
t
)
E
t
)
dt
,
(
)
=
(
V
t
(3.90)
0
where there is no averaging bracket on the perturbation because it is assumed to be
independent of the environmental fluctuations. We can reexpress the average velocity in
terms of the velocity autocorrelation function by using (
3.87
) to rewrite (
3.90
)as
t
0
V
(
t
)
t
)
dt
,
V
(
t
)
=
t
−
E
(
(3.91)
the well known Green-Kubo relation in which the kernel is the unperturbed velocity
autocorrelation function.
It is important to stress that in the example discussed here the linear response theory
(LRT) is an exact result, namely the response of the dynamic web variable to a pertur-
bation is determined by the autocorrelation function of the web variable in the absence
