Information Technology Reference
In-Depth Information
fluctuations in the numbers of people making a particular decision, a vote, say; and that
concerns the sociologist. So we have tools for handling fluctuations, but to understand
the phenomenon we must be able to associate the properties of the fluctuations with the
web's complexity.
The simplicity of physical webs even when they are not deterministic is described by
equally simple statistical processes, those of Poisson and Gauss. The increasing com-
plexity of phenomena of interest to us is reflected in the increasing complexity of the
random fluctuations in such properties as long-termmemory and the diverging of central
moments. Two ways we have discussed in which these properties are manifest is through
inverse power-law correlations and hyperbolic probability densities that asymptotically
become inverse power-law probabilities. This discussion established the groundwork
for introducing the crucial events described by inverse power laws, which we take up in
more detail in the next chapter.
3.7
Problems
3.1 Slowly modulated oscillators
Consider a set of independent linear harmonic oscillators coupled to a slowly varying
external harmonic perturbation. The Hamiltonian for this web is
j = 1 ω j a j a j +
N
j = 1 j (
N
H (
a ) =
a j )
a
,
a j +
cos
(
t
),
where the natural frequencies of the oscillators are much higher than that of the per-
turbation
ω j
.
The notion of perturbation implies that the strength of the external
j <
driver is small,
1, for all j . Construct the solution to Hamilton's equations and
discuss the frequencies that naturally enter into the web's dynamics. Also discuss the
case of resonance when the external frequency falls within the spectral range of the
oscillators.
3.2 A Hamiltonian heat bath
The analysis of linear dynamical equations performed in this section can be simplified
to that of a single harmonic oscillator coupled to a heat bath, which in turn is modeled as
a network of harmonic oscillators. Consider the linear oscillator Hamiltonian in terms of
mode amplitudes H 0 = ω 0 a a , the Hamiltonian for the environment, also in the mode-
amplitude representation
1 ω k b k b k ,
H B =
k
=
and the interaction Hamiltonian
k = 1 k b k +
b k a +
a .
H int =
Search WWH ::




Custom Search