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description and can therefore have dynamics whose description is not obvious. The
state of a real web cannot be determined to arbitrary accuracy; finite-temperature effects
ultimately limit the reliability of a measurement and therefore limit any specification of
the displacement and momentum of the constituent particles. Ultimately the displace-
ments and momenta of the particles are random, and are described by a probability
density function (pdf), not a trajectory. Even without introducing restrictions imposed
by thermodynamics, the results of experiments are not reproducible: the observed out-
put changes from experiment to experiment, regardless of how careful the preparation
of the initial state is or how skilled the experimenter is. The initial state is not exactly
reproducible and consequently experimental observation is slightly altered. The collec-
tion of results from a series of ostensibly identical experiments gives rise to the notion
of a bundle of experimental values, characterized by an ensemble distribution function.
The classical notion of the existence of an ensemble distribution of a collection of exp-
erimental results historically led to the law of frequency of errors and the functional
form of the normal distribution discussed in Chapter
1
.
Historically, the law of errors was the first attempt to characterize complex physical
webs in a formal way and through it one of the most subtle concepts in physics was
introduced: the existence and role of randomness in measurement. Randomness is asso-
ciated with our inability to predict the outcome of an experiment, say the flipping of a
coin or the rolling of a die. It also applies to more complicated phenomena, for exam-
ple, the outcome of an athletic contest or, more profoundly, the outcome of a medical
procedure such as the removal of a tumor. From one perspective the unknowability of
such events has to do with the large number of elements in the web; there are so many,
in fact, that the behavior of the web ceases to be predictable. On the other hand, the
scientific community now knows that having only a few dynamical elements in a web
does not insure predictability or knowability. It has been demonstrated that the irregu-
lar time series observed in such disciplines as biology, chemical kinetics, economics,
meteorology, physics, physiology and on and on, are at least in part due to chaos. Tech-
nically the term chaos refers to the sensitive dependence of the solution to a set of
nonlinear, deterministic, dynamical equations on initial conditions. The practical mean-
ing of chaos is that the solutions to deterministic nonlinear equations look erratic and
may pass all the traditional tests for randomness even though the solutions are deter-
ministic. In subsequent chapters we examine more closely these ideas having to do with
nonlinear dynamics and statistics. For the time being let us examine the properties of
the non-normal statistics of interest.
2.3.1
Inverse power laws of Zipf and Pareto
Zipf's law, an empirical law formulated using mathematical statistics [
104
], refers to
the fact that many types of data studied in the physical and social sciences can be
approximated with an inverse power-law distribution, one of a family of related discrete
power-law probability distributions. The law is named after the linguist George Kings-
ley Zipf, who first proposed it in 1949, though J. B. Estoup appears to have noticed the
regularity before Zipf [
46
]. This law is most easily observed by scatterplotting the data,
