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In-Depth Information
The mode amplitudes and their complex conjugates constitute sets of canonical vari-
ables, so their dynamics are determined by Hamilton's equations. Put the equations of
motionintotheform(
3.32
) both for the web and for the bath. Solve the bath equations
formally and substitute that solution into the web equation of motion as was done in the
text and obtain an equation of the form
da
dt
+
t
dt
K
t
)
[
t
)
+
a
∗
(
t
)
]
.
i
ω
0
a
=
f
(
t
)
+
(
t
−
a
(
0
Find explicit expressions for
f
in terms of the bath parameters; explain
how the former is noise, whereas the latter is a dissipative memory kernel, and how the
two are related to one another.
(
t
)
and
K
(
t
)
3.3 Stochastic resonance
In Section
3.2.3
we discussed SR and presented the equation of motion. Use linear per-
turbation theory to solve the equation of motion for the average signal strength given by
(
3.99
). Graph the signal strength versus the intensity of the fluctuations
D
and explain
what is significant about the resulting curve.
3.4 Complete derivation
In the text we did not include all the details in the derivation of the probability densities
(
3.155
) and (
3.169
). The key step in both cases is evaluating the characteristic function.
Go back and derive these two distributions, including all the deleted steps in the process,
and review all the important properties of Gaussian distributions.
3.5 The inverse power law
Consider the dynamical equation defined on the unit interval [0, 1]
dQ
dt
=
aQ
z
and apply the same reasoning as was used at the beginning of this section to obtain
its solution. Each time the dynamical variable encounters the unit boundary
Q
1
record the time and reinject the particle to a random location on the unit interval. The
initial condition is then a random variable
(τ)
=
, as is the time at which the reinjec-
tion occurs. Show that for a uniform distribution of initial conditions the corresponding
times are distributed as an inverse power law.
ξ
=
Q
(
0
)
References
[1] B. J. Adler and T. E. Wainwright, “Velocity autocorrelation function of hard spheres,”
Phys.
Rev. Lett.
18
, 968 (1967); “Decay of the velocity autocorrelation function,”
Phys. Rev. A
1
,
18 (1970).
[2] O. C. Akin, P. Paradisi and P. Grigolini,
Physica A
371
, 157-170 (2006).
[3] P. Allegrini, F. Barbi, P. Grigolini and P. Paradisi, “Renewal, modulation and superstatistics,”
Phys. Rev. E
73
, 046136 (2006).
