BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 118

and is shown to further reduce that part of the variance due to the finite number of
random walks without a commensurate increase in computing effort. The Markov
Monte Carlo formulation was extended to problems with some data uncertainties,
and a batching technique was shown to lead to further improvements in the figure
of merit. While we have not had the opportunity to make numerical comparisons
between Markov Monte Carlo and Kinetic Tree methods, which use Monte Carlo
data sampling, an observation seems in order. For equal data sampling one would
equate the number of Markov Monte Carlo batches to the number of Kinetic Tree
trials. If one then chose the batch size just large enough so that the random walk
variance could be ignored relative to the variance due to data uncertainty, a fair
comparison of computational efficiency would be the Monte Carlo time per batch
versus the Kinetic Tree time per trial. This, of course, assumes that the problem is
chosen in which component dependencies do not rule out the use of Kinetic Tree
methods.
Within the contents of this topic we have attempted to elucidate the essential
features of Monte Carlo simulations and their application to problems in statisti-
cal physics. We have attempted to give the reader practical advice as well as to
present theoretically based background for the methodology of the simulations
as well as the tools of analysis. New Monte Carlo methods will be devised and
will be used with more powerful computers, but we believe that the advice given
to the reader that will remain valid. In general terms we can expect that progress
in Monte Carlo studies in the future will take place along two different routes.
First, there will be a continued advancement towards ultra high-resolution studies
of relatively simple models in which critical temperatures and exponents, phase
boundaries, etc. will be examined with increasing precision and accuracy. As a
consequence, high numerical resolution as well as the physical interpretation of
simulational results may well provide hints to the theorist who is interested in ana-
lytic investigation. On the other hand, we expect that there will be a tendency to
increase the examination of much more complicated models, which provide a bet-
ter approximation to physical materials. As the general area of materials science
blossoms, we anticipate that Monte Carlo methods will be used to probe the often-
complex behavior of real materials. This is a challenge indeed, since there are
usually phenomena, which are occurring at different length and time scales. As
a result, it will not be surprising if multi scale methods are developed and Monte
Carlo methods will be used within multiple regions of length and time scales. We
encourage the reader to think of new problems which are amenable to Monte Car-
lo simulation but which have not yet been approached with this method. Lastly, it
is likely that an enhanced understanding of the significance of numerical results
can be obtained using techniques of scientific visualization. The general trend in
Monte Carlo simulations is to ever-larger systems studied for longer and longer
times. The mere interpretation of the data is becoming a problem of increasing
magnitude, and visual techniques for probing the system (again over different