Information Technology Reference
In-Depth Information
SHM. Finally, we have an explanation as to why the uninterrupted drone of the lecturer
puts even the most dedicated student to sleep.
7.6
Problems
7.1 Stochastic resonance
Show that (
7.100
) yields a curve that agrees qualitatively with Figure
7.6
.
7.2 A harmonic oscillator in a heat bath
Consider the situation in which a simple harmonic oscillator is in contact with a heat
bath so that the FP operator is determined by the potential function
U
1
2
q
2
The
solution to the eigenvalue equation allows an exact evaluation of the sum in (
3.134
)to
obtain a closed-form solution to the FPE. Obtain this solution for the case in which the
oscillator is initially displaced to
q
0
.
(
q
)
=
2
ω
.
7.3 Linear response
It is interesting to notice that it is possible to establish a connection between the results
of this and the preceding section, by assuming the web variable to be
Q
(
t
)
and the
perturbed variable too to be
Q
(
t
)
so that (
7.114
) becomes
t
Q
E
·
Q
t
)
t
)
dt
.
Q
(
t
)
=
β
(
t
)
(
(
0
Use the equipartition theorem (
7.126
) and the chain condition on derivatives to obtain
Q
·
Q
d
dt
e
−
γ(
t
−
t
)
t
)
t
)
=
γ
Q
2
(
)
(
=
(
)
(
t
Q
t
Q
and show that this yields
t
e
−
γ(
t
−
t
)
γ
ω
t
)
dt
,
(
)
=
(
Q
t
2
E
0
which, using the definition (
7.129
), allows us to recover (
7.132
).
7.4 Habituation to arbitrary stimulus
The behavioral response to a single frequency is the basis for determining the response
to a much richer stimulus using the principle of superposition for linear processes. For
example, consider the road noise from the Interstate highway outside the window of
your motel room, or the gasping air conditioning unit below that window. These sig-
nals, acoustic, olfactory, tactile and of many other types, can be represented as the
superposition of
M
modes
M
S
(
t
)
=
A
k
cos
(ω
k
t
+
φ
k
),
(7.186)
k
=
1
