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In order to use a mesoscopic view we have to convert existing microscopic
or macroscopic theories into a mesoscopic description. This means that we have
to find equations describing the evolution of probabilities. It turns out that this
problem is much more difficult than we initially thought. But the reward of this
research seems great.
2
The Importance of a Stochastic Analysis?
In a recent book about cellular automata [7] we find the following remark: “One
should not be surprised that it is precisely that feature which makes CA so ap-
pealing as “toy” models of physical phenomena - namely their propensity to
exhibit a remarkable wide range of possible behavior- that lies at the heart of
the problem what makes CA so difficult to study analytically.”
Wolfram [16] was one of the first to recognize the importance of a stochastic
analysis of cellular automata. He proposed as Problem 10: What is the corre-
spondence between cellular automata and stochastic systems ? Closely related
is Problem 11: How are cellular automata affected by noise and other imper-
fections ?
Wolfram noted [16]: “The problems are intended to be broad in scope, and
are probably not easy to solve. To solve any of them completely will probably
require a multitude of subsidiary questions to be asked and answered. But when
they are solved, substantially progress towards a theory of cellular automata and
perhaps of complex systems in general should have been made.”
This remark is absolutely true. The problem we want to solve is very diffi-
cult. We approach both problems with the same technique - the approximation
of probability distributions by products of low dimensional distributions. The
method can be used for deterministic automata given a distribution as input, or
stochastic automata, where the transition rules have stochastic components. The
approximation of distributions has recently been advanced in such diverse fields
as Bayesian networks [8], graphical models in statistics [9], and optimization by
search distributions [13].
In [7] we found one approach which is at least similarly in spirit to our
approach. It is called the local structure theory , developed by Gutowitz [4, 5].
His approximations for 1-D CA are very similar to ours, but he did not come
very far with 2-D CA.
Our theory is not restricted to cellular automata. It can also be used for
general spatial distributions which arise for instance in ecological problems [2].
The importance of space for evolution was already recognized by Darwin in his
”Origin of Species”. We investigated Darwin's conjecture by micro-simulations
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