Information Technology Reference
In-Depth Information
corresponding to
q
0. However, we have to keep in mind that reality can be more
complex and that it may be a mixture of renewal and non-renewal properties.
=
3.6
Overview
In this chapter we have introduced a variety of concepts that are usually discussed in
the context of several non-overlapping phenomena. The mechanical basis of Newton's
laws, namely the conservation of energy, forms the content of a number of undergrad-
uate and graduate courses in physics and engineering. The dynamical equations for
mechanical webs establish a paradigm for understanding the motion of matter from
planets to elementary particles. The solutions to such equations make prediction pos-
sible and, although our private lives do not lend themselves to such prescriptions, the
technology of our society is predictable in this way. However, being able to predict the
operation of inanimate objects and not being able to do the same for animate objects
sets up a tension in our intellectual map of the world. We can understand the opera-
tion of the devices cluttering up our world, but we do not understand the relationships
between these devices and human beings. A simple example is the personal computer,
the engineering of which we understand, but the way it has transformed society over the
past decade was not anticipated by anyone, neither scientist nor social commentator.
To model our inability to predict the behavior of the more complex phenomena in
our world, such as human beings, we introduced uncertainty in the form of random
variables. The randomness shifted the emphasis from a prediction of the single trajec-
tory solving the deterministic equation of motion to an ensemble of such trajectories.
This change replaces the prediction of a definite outcome stemming from an exact ini-
tial condition by a distribution of possible futures with varying degrees of probability.
Certain phenomena, such as the spreading of a drop of ink in water or a rumor in the
population of a tightly coupled community, cannot be described by deterministic tra-
jectories but are well represented by probability densities. The dynamical description
therefore shifts to how the probability density changes over time, and this unfolding of
the probability suggests the possible futures for the phenomena being studied.
It is natural that such human activities as decision making lend themselves more
readily to the uncertainty associated with the probability description than to the pre-
dictability of the trajectory description. The tools from statistical physics that describe
classical diffusion lend themselves to phenomena that are a combination of determinis-
tic and random behavior. The deterministic behavior is expressed in terms of the average
of the dynamical variable and the random behavior is in terms of fluctuations. We under-
stand the world, at least in part, by smoothing over these fluctuations. Imagine looking
down from the top of a twenty-story building in New York City at the movement of
people. In the aggregated flow below you do not see the individuals moving to avoid
collisions, to let someone by, to dodge a young child; all these personal choices of mov-
ing faster or slower and changing direction are lost in the smoothed view from the top.
In the case of people the uncertainty associated with fluctuations may be the variability
in choices that psychologists try to understand, but these variations might also be the
