Deterministic Abelian Sandpile Models And Patterns (springer Theses)
by Guglielmo Paoletti /
2013 / English / PDF
7.9 MB Download
The model investigated in this work, a particular cellular
automaton with stochastic evolution, was introduced as the simplest
case of self-organized-criticality, that is, a dynamical system
which shows algebraic long-range correlations without any tuning of
parameters. The author derives exact results which are potentially
also interesting outside the area of critical phenomena. Exact
means also site-by-site and not only ensemble average or coarse
graining. Very complex and amazingly beautiful periodic patterns
are often generated by the dynamics involved, especially in
deterministic protocols in which the sand is added at chosen sites.
For example, the author studies the appearance of allometric
structures, that is, patterns which grow in the same way in their
whole body, and not only near their boundaries, as commonly occurs.
The local conservation laws which govern the evolution of these
patterns are also presented. This work has already attracted
interest, not only in non-equilibrium statistical mechanics, but
also in mathematics, both in probability and in combinatorics.
There are also interesting connections with number theory. Lastly,
it also poses new questions about an old subject. As such, it will
be of interest to computer practitioners, demonstrating the
simplicity with which charming patterns can be obtained, as well as
to researchers working in many other areas.
The model investigated in this work, a particular cellular
automaton with stochastic evolution, was introduced as the simplest
case of self-organized-criticality, that is, a dynamical system
which shows algebraic long-range correlations without any tuning of
parameters. The author derives exact results which are potentially
also interesting outside the area of critical phenomena. Exact
means also site-by-site and not only ensemble average or coarse
graining. Very complex and amazingly beautiful periodic patterns
are often generated by the dynamics involved, especially in
deterministic protocols in which the sand is added at chosen sites.
For example, the author studies the appearance of allometric
structures, that is, patterns which grow in the same way in their
whole body, and not only near their boundaries, as commonly occurs.
The local conservation laws which govern the evolution of these
patterns are also presented. This work has already attracted
interest, not only in non-equilibrium statistical mechanics, but
also in mathematics, both in probability and in combinatorics.
There are also interesting connections with number theory. Lastly,
it also poses new questions about an old subject. As such, it will
be of interest to computer practitioners, demonstrating the
simplicity with which charming patterns can be obtained, as well as
to researchers working in many other areas.