Information Technology Reference
In-Depth Information
In this subsection, for simplicity's sake, we focus on the case when we observe the
response of the variable that is directly perturbed by the external stimulus. This is the
reason why in (
7.73
) we do not assign subscripts to the linear response function: we
are assuming that the observed variable coincides with the perturbed variable. To make
evident to the reader that in this case the ordinary LRT yields (
7.70
) with the linear
response function (
7.73
), let us consider the case described by
dQ
(
t
)
=
V
(
t
)
(7.123)
dt
and
dV
(
t
)
2
Q
=−
λ
V
(
t
)
+
F
(
t
)
+
E
(
t
)
−
ω
(
t
).
(7.124)
dt
This is the same Brownian particle as the one described in Section 3.2.2 with the new
condition that the perturbation process occurs in the presence of the harmonic potential
U
(
q
)
given by
1
2
ω
2
q
2
(
)
=
.
U
q
(7.125)
The harmonic potential produces a confinement effect, yielding for the dynamical space
variable
Q
the finite second moment
Q
2
k
B
T
abs
ω
eq
=
.
(7.126)
2
Let us consider the condition
ω
λ,
(7.127)
which allows us to use the Smoluchosky approximation, namely, to set
dV
/
dt
=
0in
(
7.124
), so as to express
V
(
t
)
in terms of
Q
(
t
)
,
E
(
t
)
and
F
(
t
)
. We then insert this
expression for
V
(
t
)
into (
7.123
), thereby obtaining
dQ
(
t
)
=−
γ
Q
(
t
)
+
η(
t
)
+
f
(
t
),
(7.128)
dt
where
)
λ
,
(
E
t
η(
)
≡
t
(7.129)
F
(
t
)
f
(
t
)
≡
(7.130)
λ
and
2
λ
.
γ
≡
ω
(7.131)
