NANOSCALE DEVICES: APPLICATIONS AND MODELING - Bio-Inspired and Nanoscale Integrated Computing - page 51

themselves models at the most basic and general level, and they attempt to
describe all of nature and its phenomena in one shot. But as long as these models
are not derived from more general models and are at the root of our modeling
hierarchy, we call them fundamental laws.
Before the advent of fast calculating machines and computers, researchers had
to rely mainly on solving these equations analytically (as much as possible) before
attempting to plug in numbers. In the recent times, modeling is often accompanied
by heavy numerical simulations; that is, once a good model is chosen for a given
system, predictions on the behavior of the system under various circumstances are
done by numerically solving the equations using fast computers.
One may ask that if we know the fundamental laws of nature, and if we have a
fast computer to solve equations, why can we not always start from those
fundamental laws (whether classical or quantum mechanical) and solve the relevant
equations for any system in the most general sense? The answer is that this ''first-
principles'' approach usually leads to a complex system of interrelated equations that
even the most advanced supercomputers cannot handle numerically for a simple
physical system. As a consequence, regardless of how good our computers have
become, modeling has remained an integral part of studying natural systems. For
example, think about an inert gas. The sciences of statistical mechanics and
thermodynamics have provided us with an elegant model that describes an ''ideal''
gas that results in a very simple equation relating its pressure, volume, and
temperature. For many a practical application, this model provides a description
that is accurate enough to find those parameters, without the need for heavy
calculations. In contrast, imagine the practically impossible task of solving the
equations of motion for all the atoms in a given volume of gas (trillions of trillions of
atoms) every time one needs to study the gas' expansion as a function of temperature.
With the introduction of every new branch of science and engineering, models
start being developed (and gradually improved) to describe systems relevant to the
discipline. There are well established models for gases, solids, and fluids that
engineers use every day to design new devices and systems. We have been very
successful so far in coming up with good physical models to describe systems
ranging from giant spacecraft all the way down to microelectronic devices. In other
words, when designing a spacecraft, an engineer does not start with the behavior
of the individual atoms in it, but rather uses the much more high level models of
mechanical engineering that describe the system containing a very large number
of atoms. Similarly, even in microelectronic devices, a designer/engineer does not
look at individual atoms or electrons, but uses the well established equations of
microscopic electronic transport in a system containing a large number of atoms
and electrons. These cases (spacecraft and microelectronic device) have in common
the fact that in both systems there exists such a large collection of atoms that the
details of the exact behavior of each individual atom is of little significance to the
overall system, and it is the statistical averages of these behaviors that determine
what happens. Thus, the fact that there is a large number of atoms to be accounted
for is not only not a threat to modeling, but is actually the very reason why such
successful macroscopic models can be developed and used.