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The linear dynamics of Hamiltonian webs were made more complex by coupling the
web of interest to the environment. The early models of classical diffusion were con-
structed in this way and Brownian motion was successfully explained. The dynamics
were made incrementally more complex by introducing memory into the fluctuations
produced by the environment, and of particular interest was the inverse power-law
memory and the response of the dynamical web to such fluctuations. In this way the
workhorse of statistical physics, linear response theory (LRT), was introduced, laying
the groundwork for its use in nonlinear webs whose dynamics could be described per-
turbatively. An exemplar of such a nonlinear dynamical network is the double-well
potential and Kramers' estimate of the transition rate from one well to the other. This
example was used to explain the phenomenon of stochastic resonance, that is, whether a
relatively small-amplitude sinusoidal signal can be amplified by internal nonlinear inter-
actions in the presence of noise. This was subsequently shown to be the harbinger of
the more general phenomenon of complexity-matching discovered by the authors [
1
,
3
]
and taken up in later chapters.
Of course, stochastic dynamical equations are only one way to describe a class
of complex webs. Another is to replace the dynamical variable with a phase-space
variable and the dynamical equations by a phase-space equation for the probability
density, namely the Fokker-Planck equation (FPE). One particularly important solution
to an FPE is the Gaussian probability distribution that results from a linear dynami-
cal equation with additive random fluctuations (white noise). Such a strategy with the
appropriate boundary conditions produced the drift-diffusion model (DDM) of decision-
making, which is one of the most popular models of decision-making. We review
the DDM in order to set the stage for the more general decision-making model we
developed [
7
] involving the dynamical interaction among members of a nonlinear web.
An important asymptotic solution to a particular FPE is the inverse power law
resulting from a process in which the fluctuations are multiplicative, not additive. The
distribution is normalizable, but it has a diverging first moment when the power-law
index is in the interval 1
2. For a power-law index in this interval the statistics
of the underlying multiplicative dynamical process are not ergodic.
The most significant concept introduced in Chapter
3
is the replacement of the con-
tinuous trajectories of Hamiltonian dynamics with the dynamics of discrete events. The
dynamics of these events is described using renewal theory and the formalism was orig-
inally developed to solve problems in operations research. We show that this formalism
gives rise to hyperbolic distributions of events, rather than the probability density of
ensembles of trajectories that solve the FPE. It is natural that such human activities
as decision-making lend themselves more readily to the uncertainty associated with
probabilities than to the predictability of the trajectory description. The details of this
difference were introduced in the next chapter.
The way physical scientists are trained to understand randomness is through random-
walk theory and its generalizations. This is the starting point of Chapter
4
, where the
briefest of histories of random walks is used to introduce the notion of a random web.
The randomness of the web provides a readily understood paradigm against which to
test the more mathematically sophisticated theories that are subsequently brought to
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