*Sandpile Models and Power-Law
Distributions*

(1996-Present)

*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

classes.

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

and

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.