Sandpile Models and Power-Law Distributions

Publication Numbers: 19,31,32,34,39,41.

Critical phenomena and phase transitions under
conditions of thermal equilibrium are now well understood.
The renormalization group (RG) theory provides a classification
of the systems according to relevant symmetry properties, and
a framework in which the critical properties can be calculated.
However, for collective phenomena away from thermal equilibrium
there is no such theoretical framework.
Non-equilibrium systems exhibit complex dynamical
behavior characterized by pattern formation, long range correlations
and power-law distributions,
which resemble equilibrium critical behavior.
About ten years ago Bak, Tang and Wiesenfeld
proposed that under certain conditions non-equilibrium
systems are dynamically driven into a critical point.
They introduced
a class of {\it sand pile} models
which exhibit this phenomenon, and described it as
self-organized criticality (SOC).
It was observed that under slow
random deposition of new grains these sandpile models
spontaneously exhibit
a power-law distribution of avalanche sizes,
typical of second order phase transitions.
In spite of the enormous amount of work done in this field
during the past decade,
there is no theory which can explain all aspects of SOC.

To understand the critical behavior one may use
RG type analysis, in which it is essential to identify the
relevant variables and to classify these systems into universality
Recently, we have made a significant contribution in this
direction when we identified the relevant symmetry properties
that govern the nature of the critical behavior
showed that there are two universality classes
of sandpile models[19,31,41]: (a) Models with deterministic rules which
maintain the abelian symmetry;
(b) Models with stochastic rules, for which the abelian symmetry is irrelevant.
In addition we identified a class of non-universal models which
includes the deterministic, non-abelian models.
To this end we have analyzed a complete set of critical exponents
and introduced new
scaling functions [31].

Our results open the way for a better theoretical
understanding of SOC.
To this end we plan to
develop an RG approach
that maintains these relevant symmetries.
Such approach is expected to
distinguish between the two
universality classes,
unlike previous approaches in which
the relevant symmetries were not maintained by
the RG recursion equations.

In order to obtain deeper understanding of the dynamical
mechanisms which generate power-law distributions we have
studied a class of models
in which power-law distributions emerge due to
multiplicative noise [32,39].
In these systems the exponents are non-universal and depend
on parameters such as the imposed cutoffs. We currently
examine these mechanisms in conjunction with
selected empirical systems in order to identify
the dynamic mechanisms which generate power-law distributions
in these systems.

Power-law distributions in geometric objects gives rise to fractal
structures. We have studied the formation of fractal clusters
on surfaces in the case when the sticking coefficient of a deposited particles
decays as a power-law of its distance from already attached particles [34].
We observed a
{\it clustering transition}
which separates between a weakly correlated phase
in which
the particles are distributed homogeneously and a strongly
correlated phase in which they form clusters.
We found that the clustering transition is a second order transition
and obtained the critical exponents, analytically for the one dimensional
case and numerically for the two dimensional case.
The approach we have developed provides much insight into the nature
of clustered structures, and a set of solvable models which can be
used as a testing ground for different techniques of fractal analysis.