Fractal growth with quenched disorder
ABSTRACT In this lecture we present an overview of the physics of irreversible fractal growth process, with particular emphasis on
a class of models characterized by quenched disorder. These models exhibit self-organization, with critical properties developing spontaneously, without the fine tuning of external
parameters. This situation is different from the usual critical phenomena, and requires the introduction of new theoretical
methods. Our approach to these problems is based on two concepts, the Fixed Scale Transformation, and the quenched-stochastic
transformation, or Run Time Statistics (RTS), which maps a dynamics with quenched disorder into a stochastic process. These
methods, combined together, allow us to understand the self-organized nature of models with quenched disorder and to compute
analytically their critical exponents. In addition, it is also possible characterize mathematically the origin of the dynamics
by avalanches and compare it with the continuous growth of other fractal models. A specific application to Invasion Percolation will be discussed. Some possible relations to glasses
will also be mentioned.
arXiv:cond-mat/9906443v1 [cond-mat.stat-mech] 30 Jun 1999
FRACTAL GROWTH WITH QUENCHED
L. PIETRONERO1, R. CAFIERO1and A. GABRIELLI2
1Dipartimento di Fisica, Universit´ a di Roma ”La Sapienza”,
P.le Aldo Moro 2, I-00185 Roma, Italy; Istituto Nazionale di Fisica della Materia,
unit´ a di Roma I
2Dipartimento di Fisica, Universit´ a di Roma ”Tor Vergata”,
Via della Ricerca Scientifica 1, I-00133 Roma, Italy
Abstract. In this lecture we present an overview of the physics of irreversible frac-
tal growth process, with particular emphasis on a class of models characterized by
quenched disorder. These models exhibit self-organization, with critical properties de-
veloping spontaneously, without the fine tuning of external parameters. This situation
is different from the usual critical phenomena, and requires the introduction of new the-
oretical methods. Our approach to these problems is based on two concepts, the Fixed
Scale Transformation, and the quenched-stochastic transformation, or Run Time Statis-
tics (RTS), which maps a dynamics with quenched disorder into a stochastic process.
These methods, combined together, allow us to understand the self-organized nature
of models with quenched disorder and to compute analytically their critical exponents.
In addition, it is also possible characterize mathematically the origin of the dynamics
by avalanches and compare it with the continuous growth of other fractal models. A
specific application to Invasion Percolation will be discussed. Some possible relations
to glasses will also be mentioned.
The introduction of the fractal geometry (Mandelbrot (1982)) has changed the
way physicists look at a vast class of natural phenomena which produce irregu-
lar structures. Many models have been introduced since the early eighties trying
to relate these structures to well defined physical phenomena. These are the
Diffusion Limited Aggregation (DLA) (Witten and Sander (1981)), the Dielec-
tric Breakdown Model (DBM) (Niemeyer and Pietronero (1984)), the Invasion
Percolation (IP) (Wilkinson and Willemsen (1983)), the Sandpile (Bak, Tang
and Wiesenfeld (1987)), the Bak and Sneppen model (BS) (Bak and Sneppen
(1993)), just to give some examples. All these models lead spontaneously (for a
broad range of parameters) to the development of critical properties and fractal
In the last years these has been a great interest on fractal models charac-
terized by quenched disorder. These models are generally characterized by an
intermittent dynamics, with bursts of activity of any size concentrated in a re-
gion of the system (avalanches), and by memory effects induced by the presence
of quenched disorder and by the dynamical rules (Paczuski, Bak and Maslov
2L. PIETRONERO, R. CAFIERO and A. GABRIELLI
(1996); Cafiero, Gabrielli, Marsili, and Pietronero (1996); Vendruscolo and Mar-
sili (1996); Marsili, Caldarelli and Vendruscolo (1996)). These memory effects
in fractal growth processes with quenched disorder, may resemble an element
characteristic of the spin glasses and glass dynamics. In fact, also in spin glasses,
memory effects (aging) are due to the presence of quenched disorder, and are
relied to the typical Kolrausch stretched exponential relaxation dynamics. How-
ever, at the moment it is hard to develop such an analogy in a more concrete
The study of physical phenomena leading to fractal structures can be classi-
fied by three different levels:
– Mathematical Level: Fractal Geometry. This is a descriptive level, at which
one simply recognizes the fractal nature of the phenomena and extimates
the fractal dimension D.
– Physical Models. One develops a model of fractal growth based on the phys-
ical process. This level is the analogue of the Ising model in equilibrium
– Physical Theories. This level corresponds to a fully understanding of the
origin of fractals in nature, their self- organization etc. The corresponding
level for phase transitions is the Renormalization Group.
The analogy we make with phase transitions is quite natural, because, like or-
dinary critical phenomena, fractal growth models are scale invariant. However,
some profound differences, like irreversible, non-equilibrium dynamics and SOC,
make unavoidable the development of new theoretical concepts.
Fractal physical models can be classified into two main groups:
1. Irreversible stochastic models
2. Irreversible quenched models
In the next section we discuss these models in relation with standard critical
PHYSICAL MODELS FOR SELF-SIMILAR
We briefly mention below some examples of these two classes of models, with a
particular emphasis on models with quenched disorder.
1) Irreversible stochastic models
– Diffusion Limited Aggregation (DLA) (Witten and Sander (1981)). This is
the first physical model of fractal growth. Particles performing a Brownian
motion aggregate and form complex fractal structures.
– Dielectric Breakdown Model (DBM) (Niemeyer and Pietronero (1984)). Is a
generalization of DLA via the relation between potential theory and random
FRACTAL GROWTH WITH QUENCHED DISORDER3
– Sandpile Models (Bak, Tang and Wiesenfeld (1987)). These models are in-
spired by the marginal stability of sandpiles. The random addition of sand
grains drives the system into a stationary state with a scale invariant distri-
bution of avalanches.
2) Irreversible Quenched Models
– Invasion Percolation (IP) (Wilkinson and Willemsen (1983)). This model
was developed to simulate the capillary displacement of a fluid in a porous
medium. The porous medium is represented by a lattice where to each bond
i is assigned a quenched value xi of its conductance. At each time step
the dynamics of the fluid evolves by occupying the bond with the smallest
conductance between all its perimeter bonds. We call this kind of dynamics
extremal dynamics. IP is known to reproduce asymptotically the Percolation
cluster of standard critical Percolation.
The main characteristics of this model are:
1. Deterministic dynamics. Once a realization of the quenched disorder is
chosen, the dynamical rule selects in a deterministic way the bond to be
2. Self-organization. The process spontaneously develops scale-invariantstruc-
tures and critical properties. In the limit t → ∞ both long range space
and time correlations appear.
3. Avalanches. The asymptotic dynamical evolution consists of local, scale-
invariant macro-events, composed by elementary growth steps spatially
and causally connected, called avalanches. When an avalanche stops, the
activity is transferred to another region of the perimeter.
– Bak and Sneppen Model (BS) (Bak and Sneppen (1993)). This model is
similar to IP and has the same properties exposed in points 1-3. In fact,
points 1-3 are characteristic of the whole class of SOC models with quenched
disorder and extremal dynamics ( for a review see Paczuski, Bak and Maslov
(1996)). The BS model has been introduced to describe scale free events in
– Quenched models with an external modulating field. This is a particular class
of models, the prototype of which is the Quenched Dielectric Breakdown
Model (QDBM (Family and Zhang (1986); De Arcangelis, Hansen, Her-
rmann and Roux (1989)). This model is a sort of combination between IP
(quenched) and DBM (stochastic). To each bond of a lattice is assigned a
quenched random number xi, representing the local resistivity. In addition,
an external electric field E is introduced. The ratio yi = xi/Ei between
quenched disorder and local electric field is the inverse of the current flow-
ing in the bonds. At each time step the dynamics breaks the bond with the
smallest yi, that is to say with the biggest current. The QDBM modelizes
the dielectric breakdown of a disordered solid. A similar model can be for-
mulated to modelize the propagation of fractures in a inhomogeneous solid
(Hansen, Hinrichsen, Roux, Herrmann and De Arcangelis (1990)).
4 L. PIETRONERO, R. CAFIERO and A. GABRIELLI
All the models in these classes share some important properties, that differen-
tiates them from ordinary critical phenomena. First of all they are characterized
by an irreversible dynamics, so that they cannot be described in hamiltonian
terms, and the statistical weight of a configuration depends on the complete
growth history (see. fig. 1). Quenched models have an additional problem, in
Fig.1. For systems with irreversible dynamics, the statistical weight of a configuration
depends on the whole growth history. The two identical configuration shown here have,
in general, different statistical weights because their histories, indicated by the times
ti i = 1,2,3,4, are different.
that they have a deterministic dynamics, and the stochasticity enters only in
the choice of the realization of the disorder. So, it is difficult to define transition
probabilities for the dynamics.
Another fundamental difference, is that these models are self- organized.
Their dynamics evolves spontaneously in the phase space towards an attractive
fixed point. No fine tuning of any parameter is needed. In addition, quenched
models with extremal dynamics have an avalanche dynamics, that is to say
the system tends to concentrate its activity in a well localized region of the
perimeter, during an avalanche. When an avalanche stops, the activity transfers
to another region of the perimeter and a new avalanche starts. On the contrary,
in stochastic models, like DLA, there is a continuous growth process, that is
to say for large systems the probability to have two nearby subsequent growth
events tends to zero. The dynamical activity is diffused at each time on the whole
In table. 1 we propose a scheme of comparison between the properties of or-
dinary critical phenomena and the most popular stochastic and quenched fractal
FRACTAL GROWTH WITH QUENCHED DISORDER5
3NEW THEORETICAL CONCEPTS
The application of the standard theoretical methods of statistical physics (field
theory and renormalization group) is, in general, not possible for the main fractal
growth problems, like DLA, DBM, Invasion Percolation, the sandpile model,
which are characterized by an intrinsically irreversible dynamics.
Here we discuss two theoretical methods we have developed in the past few
years, as a step towards the construction of a physical theory for self-organized
fractal growth processes. The exposition will be colloquial, and mainly devoted
to the theoretical analysis of quenched models. For more details readers should
refer to the bibliography.
3.1FIXED SCALE TRANSFORMATION (FST)
This approach combines a technique of lattice path integral, to take into account
the irreversible dynamics, with the study of the scale invariant dynamics inspired
by the RG theory. It permits a description of the scale invariant properties of
fractal growth models.
The method focuses on the dynamics at a given scale and analyzes the nearest
neighbours correlations at this scale using lattice path integral approach by which
one can calculate the elements of a probability matrix, the FST matrix. The fixed
point of the FST matrix gives the nearest neighbour correlations, at that scale. If
one uses the scale invariant dynamics of the system, that can be obtained by Real
Space Renormalization Group approaches (Cafiero, Vespignani and Pietronero
(1993)), one can generalize these correlations to all scales and compute the fractal
dimension (Erzan, Pietronero and Vespignani (1995)).
The basic point of the FST is the separation of the long time limit (t → ∞)
for the dynamical process at a given scale, from the large scale limit (r → ∞),
that defines the scale invariant dynamics. The interesting feature of FST is that it
works at a fixed scale, so it is possible to include the fluctuation of the boundary
conditions, that in systems with long range interactions, like Diffusion Limited
Aggregation (DLA), have a great influence on the fractal properties. For this
reason, the FST approach allows to reach a remarkable level of accuracy in the
calculation of the fractal dimension.
At the moment, the FST framework, eventually combined with the RTS
method that we discuss below, seems to be the only general approach to un-
derstand the self-organized critical nature of a broad class of models going from
DLA, to Percolation, to sandpile, to Invasion Percolation (Erzan, Pietronero
and Vespignani (1995); Cafiero, Gabrielli, Marsili, and Pietronero (1996)). This
situation therefore supports the idea that these models pose new questions for
which one would like to develop a common theoretical scheme.
3.2QUENCHED-STOCHASTIC TRANSFORMATION (RTS)
As we mentioned above, quenched models with extremal dynamics, like Invasion
Percolation (IP), have a deterministic dynamics. This makes it impossible to ad-