# 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.

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arXiv:cond-mat/9906443v1 [cond-mat.stat-mech] 30 Jun 1999

FRACTAL GROWTH WITH QUENCHED

DISORDER

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.

1 INTRODUCTION

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

structures.

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

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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

way.

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

statistical mechanics.

– 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

phenomena.

2

GROWTH

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

walk.

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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

invaded.

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

biological evolution.

– 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)).

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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

t1

t3

t4

t2

t2

t1

t4

t3

a)b)

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

growth interface.

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

growth models.

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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-