# Fine Structure of Avalanches in the Abelian Sandpile Model

**ABSTRACT** We study the two-dimensional Abelian Sandpile Model on a square lattice of

linear size L. We introduce the notion of avalanche's fine structure and

compare the behavior of avalanches and waves of toppling. We show that

according to the degree of complexity in the fine structure of avalanches,

which is a direct consequence of the intricate superposition of the boundaries

of successive waves, avalanches fall into two different categories. We propose

scaling ans\"{a}tz for these avalanche types and verify them numerically. We

find that while the first type of avalanches has a simple scaling behavior, the

second (complex) type is characterized by an avalanche-size dependent scaling

exponent. This provides a framework within which one can understand the failure

of a consistent scaling behavior in this model.

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- 01/1992; Addison-Wesley.
- J. Artificial Societies and Social Simulation. 01/2001; 4.
- SourceAvailable from: citeseerx.ist.psu.edu[Show abstract] [Hide abstract]

**ABSTRACT:**The gaps in the fossil record gave rise to the hypothesis that evolution proceeded in long periods of stasis, which alternated with occasional, rapid changes that yielded evolutionary progress. One mechanism that could cause these punctuated bursts is the recolonization of changing and deserted niches after mass extinction events. Furthermore, paleontological studies have shown that there is a power law relationship between the frequency of species extinction events and the size of the extinction impact. Power law relationships of this kind are typical for complex systems, which operate at a critical state between chaos and order, known as self-organized criticality (SOC). Based on this background, we used SOC to control the size of spatial extinction zones in a diffusion model. The SOC selection process was easy to implement and implied only negligible computational costs. Our results show that the SOC spatial extinction model clearly outperforms simple evolutionary algorithms (EAs) and the diffusion model (CGA). Further, our results support the biological hypothesis that mass extinctions might play an important role in evolution. However, the success of simple EAs indicates that evolution would already be a powerful optimization process without mass extinction, though probably slower and with less perfect adaptationsEvolutionary Computation, 2001. Proceedings of the 2001 Congress on; 02/2001

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arXiv:0808.0709v1 [cond-mat.stat-mech] 5 Aug 2008

Fine Structure of Avalanches in the Abelian Sandpile Model

Amir Abdolvand and Afshin Montakhab∗

Physics Department, College of Science, Shiraz University, Shiraz 71454, Iran.

We study the two-dimensional Abelian Sandpile Model on a square lattice of linear size L. We

introduce the notion of avalanche’s fine structure and compare the behavior of avalanches and

waves of toppling. We show that according to the degree of complexity in the fine structure

of avalanches, which is a direct consequence of the intricate superposition of the boundaries of

successive waves, avalanches fall into two different categories. We propose scaling ans¨ atz for these

avalanche types and verify them numerically. We find that while the first type of avalanches has a

simple scaling behavior, the second (complex) type is characterized by an avalanche-size dependent

scaling exponent. This provides a framework within which one can understand the failure of a

consistent scaling behavior in this model.

PACS: 89.75.Fb, 45.70.Cc, 45.70.Ht, 89.75.-k

I.INTRODUCTION

Bak, Tang, and Wiesenfeld (BTW) introduced the no-

tion of Self-Organized Criticality (SOC) [1,2] as a possi-

ble mechanism for the generic emergence of spatial and

temporal power law correlations. To elucidate the con-

cept of SOC, they introduced acellular automaton known

as sandpile model which is an example of slowly driven,

spatially extended, dissipative dynamical system [3]. The

generality inherent in the basic notions of SOC has led to

its successful application in various problems in physics

as well as biology [4,5,6,7]. The common characteristics

of all these systems is that at the self-organized critical

state the microscopic details of the system are shadowed

by the collective behavior of the individual constituents

of the system. Due to the simplicity of local dynami-

cal rules, and the ease with which they are implemented

on a computer, many different models exhibiting SOC

have been introduced and studied by various authors[8,9].

However, the prototypical sandpile model of SOC and a

variant of it known as the Abelian Sandpile Model model

(ASM)[10], has resisted many clever efforts in fully un-

derstanding its dynamical behavior [11,12,13,14,15]. This

is despite the fact that the analytical tractability of this

model, which enables one to evaluate exactly many of its

static properties [10,16,17,18], had provided hope for a

better understanding of its dynamical properties. Cur-

rently, a complete description of the dynamical properties

of the ASM is still missing [19].

To highlight this important point, we concentrate on

the event-size (avalanche) distribution function. In order

to check the assumption that the characteristicproperties

of avalanches in the critical state are described by scale

free distribution functions with cutoffs limited only by

the finite size effects, BTW proposed a simple picture of

∗Corresponding author.

2284594

E-mail address: montakhab@shirazu.ac.ir

Tel.: +98-711-2284609; Fax: +98-711-

finite-size scaling (FSS) in analogy with more traditional

critical phenomena [3,20,21]. Although, it is now gener-

ally accepted that simple FSS picture fails in describing

the scaling behavior of avalanches in the BTW model, the

reasons suggested for this inconsistency seem to be very

different [22,23,24,25,26]. For example, in a large-scale

simulation, Drossel [26] explained the deviations from

pure power law behavior by dividing avalanches into dis-

sipative and non-dissipative avalanches. However, such

attempts which aim at relating the deviations from pure

power law behavior to the finite-size (or boundary) effects

seem to be problematic. In fact, it is shown by Ktitarev

et. al [27], that by reducing such effects one still observes

the aforementioned deviations from the simple power law

behavior.

Due to the complex spatiotemporal behavior of

avalanches, it is reasonable to decompose them into more

elementary objects. As is shown in Ref. [28], due to

the Abelian property of the model which admits an in-

terchangeable order in the relaxation of local instabili-

ties, one can consider an avalanche as a composition of

a series of (global) instabilities which are referred to as

waves of topplings. The inconsistencies in our under-

standing of the dynamical behavior of avalanches reveal

themselves more clearly when we consider that an en-

semble of waves behaves simply and obeys FSS ans¨ atz

[27]. This is in contrast with the complex behavior of

avalanches. That is, avalanches which are presumably

simple composition of waves do not obey FSS. In view of

the aforementioned points, the following main questions

arise: While the time evolution of an avalanche differs

only in the order of the relaxation of local instabilities

with that of a wave, what makes the two events have

such different scaling behaviors? Moreover, beside pecu-

liarities imposed by boundaries of the system and finite

size effects, what possible mechanism, probably inherent

in the dynamical behavior of an avalanche itself, might

be responsible for the observed complexity in the scaling

behavior of avalanches?

In general, one expects to answer these questions by

considering the effect of “nontrivial composition” of cor-

Page 2

2

related waves [29,30]. However, to obtain a clear quan-

titative picture one needs to clarify beforehand what ex-

actly “nontrivial composition” means, how this nontrivi-

ality is related to the complexity in avalanche dynamics,

and last but not least, how this is related to the failure of

simple scaling picture [31]. In the present work, we ana-

lyze the spatiotemporal structure of avalanches in order

to investigate the above issues. The bulk of an avalanche

consists of sites which have toppled as well as their near-

est neighbors. This might lead one to naively believe that

the bulk of an avalanche consists of sites which are un-

changed, i.e. recurrent, where only sites on the boundary

of an avalanche change their states (dynamical variable).

However, avalanches can have complex internal struc-

tures. As waves of topplings occur during an avalanche,

the boundaries of these waves could interact with each

other leading to a complex internal (bulk) structure con-

sisting of both recurrent and non-recurrent states, see

for example Fig. 1(b). Using these facts, we classify

avalanches into two classes, simple and complex, and in-

vestigate their scaling behavior. We show that different

classes of avalanches have distinctly different scaling be-

havior. In particular, while the scaling behavior of the

first type is observed to be independent of avalanche-size,

in the second type an avalanche-size dependent scaling

exponent is found. We therefore argue this to be the

main cause of inconsistent scaling behavior in avalanche

statistics of the ASM.

The present article is organized as follows: In Section

II, we give a brief review of the basic concepts and def-

initions of the ASM. In Section III, we analyze the spa-

tial structures of avalanches and introduce the idea of the

avalanche’s fine structure. In Section IV, we compare the

behavior of avalanches and waves of topplings and argue

that according to the degree of complexity in their fine

structures, avalanches fall into two different categories,

type α and type β. In Section V, we propose scaling

ans¨ atz for the two types of avalanches and verify them

numerically. Finally, Section VI is devoted to a short

summary and outlook.

II. TWO-DIMENSIONAL ASM AND THE

WAVE PICTURE OF EVOLUTION

The two-dimensional ASM is a cellular automaton de-

fined on a square lattice of linear size L. To every point of

the lattice there corresponds an integer dynamical vari-

able hi, which in the language of sandpiles represents the

height of the column of sand at the ithsite. To simulate

external drive, the system is perturbed by increasing the

dynamical variable of a randomly chosen site by one,

hi→ hi+ 1. (1)

This can be interpreted as an increase in the local value of

height, energy, pressure, etc. A site is considered unsta-

ble if its dynamical variable exceeds a predefined thresh-

old value (hi> hc). An unstable site then topples, upon

which its dynamical variable is decreased by 4, whilst

each of its four nearest neighbors (nn) receive one unit

of energy:

hi → hi− 4,

hnn → hnn+ 1.

(2)

(3)

In turn, through the relaxation processes, see Eqs.(3), the

neighboring sites may become unstable themselves, lead-

ing to a series of instabilities the sum of which is referred

to as an avalanche. Since the local dynamical rules are

conservative, the dissipation can take place only at the

boundary of the system. Here we will use open boundary

conditions where, if an unstable site is on the boundary,

one or more grains of sand will leave the system.

The method generally used during the relaxation of an

avalanche is the parallel updating method, where all un-

stable sites are relaxed simultaneously during an instant

in the relaxation process. Due to the Abelian nature of

the model, the order of the toppling during an avalanche

does not affect the final state [10]. Therefore, beside the

parallel method of updating, it is possible to perform the

relaxation process by a succession of waves. There is a

simple dynamical procedure leading to such a decompo-

sition [28]. In this method, we relax the seeding site,

say i, after its first instability. This may cause further

instabilities in the neighboring sites. We then relax all

other unstable sites, except the seeding site i. The set of

all toppled sites during this process forms the first wave

of toppling. If, after the termination of the first wave,

the seeding site is still unstable, we repeat the above

procedure obtaining the second, third, etc. waves of top-

plings. This procedure continues until all sites are stable

again. Therefore, one can consider the relaxation process

of an avalanche as a sequence of waves of topplings, all of

which originate from the seeding site. While in the for-

mer method waves overlap in time, by decomposing an

avalanche to a sequence of waves, only one wave propa-

gates at a time. This method of updating enables us to

view the time evolution of the model in an ensemble of

waves.

It can be shown both analytically and numerically [27]

that the scaling property of waves is simple and obeys

FSS. However, avalanches, which might naively be con-

sidered as a simple sum of waves, do not have simple

scaling behavior and in fact do not obey FSS [22,23,27].

We believe that the reason for this discrepancy is found

in the fine structure of avalanches which is due to wave

boundary interactions.

In the present work we show that the boundaries of

waves making up a given avalanche could interact with

each other. This interaction of boundaries may lead to a

complex spatial structure within an avalanche bulk struc-

ture which substantially changes the dynamical proper-

ties of the avalanche, including its scaling behavior. To

show this, we divide avalanches into two distinct classes

based on the complexity in their (internal) fine structure:

simple (type α) avalanches that behave much like waves,

Page 3

3

and complex (type β) avalanches which have distinctly

different, size-dependent scaling behavior.

III.FINE STRUCTURE OF AN AVALANCHE

The area a of a relaxation process R (avalanche or

wave) is defined as the number of distinct sites toppled

during that process.In general, this area can be di-

vided into two different structures which make up the

fine structure of an avalanche. The first structure con-

sists of those sites for which the corresponding dynamical

variables have remained unchanged before and after the

relaxation event. In fact, these sites may have toppled

once or more during the relaxation event. But due to the

balance between the outflow (Eq. (2)), and inflow (Eq.

(3)) of particles through the relaxation of their nearest

neighbors, their states remain unchanged after the ter-

mination of the relaxation process. The second structure

consists of sites whose dynamical variables have changed

as a result of a relaxation event.

Let us denote by |hi, t?, i = 1,...,N the single site

microstates of the system at time t, where t denotes the

macroscopic time scale of the system.

|hi, t?I and |hi, t?F, the initial and final microstate of

the ithsite before and after the occurrence of the tthre-

laxation process. A Recurrent Macrostate (RM) of the

system consists of those sites for which the corresponding

single site microstates satisfy the following relation,

We denote by

i ∈ (RM) ⇔ |hi, t?F= |hi, t?I∧ i ∈ R (4)

We say that the site i belongs to the relaxation process

R, i ∈ R, if and only if i has toppled during that process.

R is the region affected by the process.

In a similar way, one can define the Non-recurrent

Macrostate (NM) of a relaxation process as a collec-

tion of sites for which the corresponding single site mi-

crostates satisfy the relation,

i ∈ (NM) ⇔ |hi, t?F?= |hi, t?I∧ i ∈ R(5)

According to this definition those sites that form the

exterior boundary of a relaxation process, whose states

change without toppling, are excluded from the corre-

sponding NM structure.

Let nRM(NM) be the size of a recurrent (non-

recurrent) macrostate, i.e.

site recurrent (non-recurrent) microstates. Then for the

area of the avalanche we have,

the total number of single

a = nRM+ nNM. (6)

When the internal fine structure of an avalanche is made

up of multiple, spatially contained waves, the NM con-

stitutes a thin layer on the boundary of each wave, while

the bulk of the relaxation process is mainly made up of a

RM structure. On the other hand, when an avalanche is

composed of multiple, interpenetrating waves, this simple

065

0

65

RM

NM

Exterior Boundary

Seeding Site

Number of Waves: 2

(a)

0 65

0

65

Number of Waves: 4

(b)

FIG. 1: (a) Fine structure and exterior boundary of a sim-

ple avalanche composed of two waves, and (b) a complex

avalanche composed of 4 waves in a 64 × 64 lattice. Here,

the (+), (•) and (◦) signs represent RM, NM, and the ex-

terior boundary of the avalanche, respectively. Note the sim-

ple pattern of the NM structure in 1(a) compared to that of

1(b).

structure might be lost, and we may observe a complex,

interweaved pattern of NM and RM structures. These

two scenarios are shown in Figs. 1(a) and 1(b), respec-

tively.

IV. COMPLEXITY IN THE FINE STRUCTURE

OF AVALANCHES

In Ref. [29] it has been shown that the boundaries of

consecutive waves are not simply related to each other.

In fact, the complexity present in the fine structure of an

avalanche is a direct consequence of the “complex super-

position” of the boundaries of successive waves making

Page 4

4

up an avalanche. By “complex superposition” we mean

that in general in an avalanche, excluding the exterior

boundary,

nNM?=

?

k

(nNM)Wk.(7)

Here and in the following we use symbols with sub-

script (W) to denote those quantities pertaining to a

wave. k is the index of the waves making up a given

avalanche.Equation (7) simply states that the non-

recurrent macrostate (NM) of an avalanche is not the

simple sum of the wave boundaries which form the

avalanche. So, the boundaries of successive waves can

interact and interpenetrate each other. Due to this in-

teraction, the fine structure of an avalanche may show a

complex pattern.

A quantity that contains valuable information about

the dynamical processes underlying the formation of the

fine structure of a relaxation process is the ratio of

the size of NM to RM, i.e. nNM/nRM.

we have compared the conditional expectation value of

this quantity for avalanches and waves of a given area,

E(nNM/nRM|a).While for avalanches of intermedi-

ate size the ratio of nNM/nRM tends to decrease, for

larger ones we observe a gradual increase in this quan-

tity, indicating a crossover in the avalanches’ behavior,

a point which we will return to later in this article. As

can be seen from the figure, the conditional expectation

value for the waves simply decreases with increasing area.

This simply shows that the boundary to bulk ratio for

waves decreases with increasing of the wave’s size, as

should be expected. However, in order to understand

the nature of the observed crossover phenomenon and

the differences between the behavior of waves of toppling

and avalanches, we must study more fundamental quanti-

ties describing dynamical properties of the critical state.

From Eq. (6) we can write the probability of having an

avalanche of area a as,

In Fig. 2,

P(a) =

′

?

nNM, nRM

P(nRM)P(nNM|nRM),(8)

where the summation goes over those values of nNM(RM)

that fulfill Eq. (6). Here, P(nRM) is the probability of

having an avalanche with a RM structure of size nRM.

P(nNM|nRM) is the conditional probability distribution

function (CPDF) of having a NM structure of size nNM

for a particular value of nRM. This equation, through

the quantity P(nNM|nRM), establishes a direct connec-

tion between properties of the critical state and dynam-

ical aspects of the relaxation processes. In Fig. 3, we

have plotted the CPDF, P(nNM|nRM) for several val-

ues of nRM. We note that this function is asymmetric

about its maximum value. However, more importantly,

we observe the emergence of another local maximum with

increasing nRM(point (b)). This suggests the emergence

of a different type of behavior as nRM(or avalanche size)

increases.

012345

x 10

4

0

1

2

3

4

5

6

7

a

E( n

NM / n

RM a )

10

0

10

5

10

−2

10

−1

10

0

10

1

Waves

Avalanches

Waves

Avalanches

256 × 256

FIG. 2:

value of nNM/nRM in a system of linear size L=256 for

avalanches and waves of a given area.

tistical fluctuation at large areas we have binned the data

logarithmically. This will result in losing a fraction of large

avalanches with E(nNM/nRM|a) ? 1, which manifest them-

selves when one considers the unbinned data. Inset shows the

same quantity on a double logarithmic graph. The bumps in

the left hand side of the inset are due to the grid properties

of the lattice.

Comparison between the conditional expectation

To reduce the sta-

What possible classification of avalanches can distin-

guish between the two peaks in Fig. 3, and to what ex-

tent is it related to the complexity of the fine structures

of relaxation processes? If we consider a similar quantity

in the simple NM, like the NM of waves, our simulation

shows that although the asymmetric form of the CPDF

persists, there is no significant change in the general form

of this function, i.e. no local maximum emerges as nRM

is increased, see Fig. 4. Therefore, in analogy with sim-

ple (NM) structure of waves, we define an avalanche

to be simple, i.e. of simple fine structure, if successive

waves making up that avalanche are spatially contained

in each other, i.e. their NM structures (boundaries) are

not interpenetrating, see Fig. 1(a). In this case we have

nNM ≈

?

nition does not exclude the possibility of a weak inter-

action (mixing) between the NM structures of a wave

and that of its predecessor, so that in general Eq. (7)

holds. However, it is the extent of the violation of the

equality which categorizes avalanches into simple or com-

plex. In mathematical notation, for a sequence of waves,

Wk, k = 1,...,n, making up an avalanche we have,

k(nNM)Wk. Restrictly speaking, this defi-

Simple Fine Structure ⇔ ∀i ∈ Wk⇒ i ∈ Wk−1.

On the other hand, we consider a fine structure as com-

plex, if the boundaries of successive waves exceed the

confines of their predecessors, see Fig. 1(b).

Complex Fine Structure ⇔ ∃i ∈ Wk∋ i ?∈ Wk−1.

Page 5

5

20406080100120

0

0.02

0.04

P( nNM nRM=32 )

50100150 200

0

0.01

0.02

0.03

P( nNM nRM=64 )

50100150 200250300

0

5

10

15

20x 10

−3

P( nNM nRM=128 )

100 200300400500

0

5

10

x 10

−3

P( nNM nRM=256 )

100200 300 400

nNM

500600700800

0

2

4

6

x 10

−3

P( nNM nRM=512 )

200 400 600800

nNM

1000 1200 1400 1600

0

1

2

3

4

x 10

−3

P( nNM nRM=1024 )

Type α

Type β

Type α & β

(a)

(b)

(b)

(a)

(b)

(a)

(a)

(b)

(a)

(b)

(a)

(b)

FIG. 3: Normalized CPDF P(nNM|nRM) of avalanches as a

function of nNM for different values of nRM on a 512 × 512

lattice. Note the emergence of a local maximum with increas-

ing nRM. This indicates the emergence of a different sort

of behavior with increasing avalanche size. Here the dashed

and dash-dotted lines show the decomposition of the CPDF

to that of type α and type β avalanches.

020406080 100120

0

0.02

0.04

0.06

0.08

P(w)( nNM nRM=32 )

20406080 100 120

0

0.02

0.04

0.06

P(w)( nNM nRM=64 )

0 50100 150 200

0

0.01

0.02

0.03

0.04

P(w)( nNM nRM=128 )

50 100 150 200250

0

0.01

0.02

0.03

P(w)( nNM nRM=256 )

50 100150200 250 300

0

0.005

0.01

0.015

0.02

nNM

P(w)( nNM nRM=512 )

100150200250

nNM

300350400

0

0.005

0.01

0.015

P(w)( nNM nRM=1024 )

FIG. 4: Normalized CPDF P(W)(nNM|nRM) of waves as a

function of nNM for different values of nRM. No local maxi-

mum emerges as nRM is increased.

We call these two classes of avalanches type α and type

β. As shown in Fig. 3, our numerical simulation indi-

cates that the CPDF P(nNM|nRM) is a superposition

of CPDF’s of type-α and type-β avalanches, in which

the first and second maxima correspond to type α (Fig.

3, point (a)), and type-β avalanches (Fig. 3, point (b)),

respectively.

At this point, we need to make a connection between

the properties of the critical state and those of the type

α and type β avalanches. Let N be the total number of

avalanches. Then, N = Nα+ Nβ where Nαand Nβ are

the number of type α and type β avalanches, respectively.

Using the abbreviated notations nNM?→ y and nRM?→

x, we have the following definitions:

• Nα(β): Total number of type α (or β) avalanches.

• Nα(β)

yx

with NM of size y and RM of size x.

: Total number of type α (or β) avalanches

• Nα(β)

y

with NM of size y.

: Total number of type α (or β) avalanches

• Nα(β)

x

with RM of size x.

: Total number of type α (or β) avalanches

• Nx: Total number of avalanches with RM of size

x.

• Nyx: Total number of avalanches with NM of size

y and RM of size x.

Using these definitions, we can rewrite Eq. (8) in terms

of the properties of type α and type β avalanches. We

start with:

P(y|x) =

Nyx

Nx

=Nα

yx+ Nβ

Nx

+ Pβ(y|x)Nβ

yx

=Nα

yx

Nα

x

Nα

Nx

x

+Nβ

yx

Nβ

x

Nβ

Nx

x

= Pα(y|x)Nα

x

Nx

x

Nx.(9)

Substituting Eq. (9) in (8), and using the fact that

P(x) = Nx/N we have,

P(a) =

′

?

x, y

P(x)P(y|x) =

′

?

x, y

?

Pα(y|x)Nα

x

N

+ Pβ(y|x)Nβ

x

N

?

,

(10)

where the summation goes over those values of x and y

that fulfill Eq. (6). Finally, using the relation,

Nα(β)

x

/N =Nα(β)

x

Nα(β)

Nα(β)

N

= Pα(β)Pα(β)(x),

we have,

P(a) =

′

?

x, y

[PαPα(x)Pα(y|x) + PβPβ(x)Pβ(y|x)]. (11)

According to Eq. (11), an avalanche size distribution

function can be written as a separate combination of

type-α and type-β distribution functions. Now, if type-

α and type-β avalanches have similar scaling properties,

one can expect the avalanche-size probability distribu-

tion functions to scale accordingly. However, if these two

types of avalanches have different and distinct scaling

properties, one cannot find a consistent scaling behavior

Page 6

6

for the overall probability distribution function. This is

in fact the key message of the present Article. In the next

section we will give analytical as well as numerical evi-

dence on how the scaling properties of these two types of

avalanches differ from each other. The key difference, as

we will see, is the size-dependence of the scaling behavior

in type-β avalanches.

V.SCALING OF CPDF FOR DIFFERENT

TYPES OF AVALANCHES

In order to investigate the properties of P(a) it is im-

portant to study the properties of Pα(y|x) and Pβ(y|x).

We have carried out an extensive study of such CPDF’s.

We find that the scaling behavior of these distributions

are distinctly different. Accordingly, we propose the fol-

lowing scaling ans¨ atz for the above distribution functions:

• Type α avalanches:

Pα(y|x) = x−γαUα(y − Eα(y|x)

xγα

). (12)

• Type β avalanches:

Pβ(y|x) = x−γβUβ(y + x + Eβ(y|x)

xγβ

), (13)

where γβis a size-dependent exponent, i.e. γβ= γβ(x).

Here Uαand Uβare universal functions, and Eα(β)(y|x)

is the mean value of the given CPDF, defined through

the relation Eα(β)(y|x) =?yPα(β)(y|x)dy.

From Eqs. (12) and (13), we can readily calculate the

scaling behavior of the first and second moments of y.

This provides a suitable way via which one can con-

firm the proposed scaling ans¨ atz for the corresponding

CPDF’s.

Let us first consider the case of type β avalanches. Us-

ing the suggested form in Eq. (13), we have

Eβ(y|x) =

?∞

0

yPβ(y|x) dy (14)

= x−γβ

?∞

0

yUβ((y + x + Eβ(y|x))/xγβ) dy.

Performing the change of variable z = y+x+Eβ(y|x) in

Eq. (14) and integrating we obtain,

Eβ(y|x) =1

2Cxγβ−x

2,(15)

where C =?ξ Uβ(ξ) dξ; ξ = z x−γβ. So, after the addi-

tion of the linear term x/2 to Eβ(y|x), it must scale as

xγβ(x)for different scaling regions defined by the area or

the size of the RM structure of an avalanche. In the case

of type α avalanches, we cannot obtain the scaling behav-

ior of the corresponding conditional expectation value,

Eα(y|x) from Pα(y|x), as we did in the case of type β

−10 −50510 152025

0

0.05

0.1

0.15

0.2

0.25

(y − Eα (yx)) / xγα

xγα P(y x)

512 × 512

γα = 0.423 ± 0.002

x = 32

x = 64

x = 128

x = 256

x = 512

x = 1024

1.52 2.533.5

1

1.2

1.4

1.6

1.8

2

log10 (x)

log10 ( σα )

σα ~ xγα

FIG. 5: CPDF data collapse using the suggested form, Eq.

(12), for type-α avalanches. Here, we obtain a good collapse

for different values of nRM with γα = 0.423 ± 0.002. Inset

shows the scaling of σα with x for different values of x ranging

from x = 32 to x = 4096. The slope of the dashed line is equal

to γα = 0.423

avalanches. So, to obtain any further information, we

must look at higher moments of y, e.g. Eα(y2|x).

Eα(y2|x) =

?∞

0

y2Pα(y|x)dy(16)

= x−γα

?∞

0

y2Uα((y − Eα(y|x))/xγα)dy.

Performing the change of variable z = y − Eα(y|x), we

can rewrite Eq. (16) as,

Eα(y2|x) = C x2 γα+ [Eα(y|x)]2,(17)

where C =?ξ2Uα(ξ) dξ; ξ = z x−γα. Therefore,

σα=?Eα(y2|x) − (Eα(y|x))2?1/2∼ xγα,

where σαis the standard deviation of the given distribu-

tion. In obtaining the last relation we have used the fact

that?ξUα(ξ)dξ = 0. So, in the case of type α avalanches

σαscales as xγα.

To verify the proposed scaling forms and also to ex-

tract the scaling exponents γαand γβ, we implement the

method of data collapse in line with Ref. [25]. In Fig.

5, we have applied this technique to type α avalanches

for different values of x(= nRM). As can be seen, we

obtain a reasonable collapse with the scaling exponent

γα= 0.423. The inset shows the scaling of standard de-

viation, σα, with x for the corresponding CPDF’s. There

the dashed line shows xγα. This verifies in a straight-

forward manner, the validity of Eq. (18), thus lending

support to our scaling ans¨ atz, Eq.(12). By performing a

similar analysis on an ensemble of waves, we could verify

that the CPDF P(W)(y | x) of waves of toppling also pos-

sesses a similar scaling to type α avalanches, Eq. (12),

(18)

Page 7

7

0123456

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

log10 x

10

1

10

−4

10

−3

10

−2

10

−1

10

0

(x + y + <y>x) / xγβ

xγβ P(y x)

x0.7

log10(Eβ(yx) + x/2)

512 × 512

γβ = 0.7; x ∈ [32, 64]

x = 32

x = 64

x = 32

x = 64

0123456

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

log10 x

log10(Eβ(yx) + x/2)

10

1

10

−2

10

−1

(x + y + <y>x) / xγβ

xγβ P(y x)

x0.73

512 × 512

γβ = 0.73; x ∈ [128, 256]

x = 128

x = 256

x=256

x=128

0123456

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

log10 x

log10(Eβ(yx) + x/2)

x0.80

x = 512

512 × 512

γβ = 0.80; x ∈ [512, 1024]

10

1

10

−2

10

−1

10

0

(x + y + <y>x) / xγβ

xγβ P(y x)

x = 512

x = 1024

x = 1024

FIG. 6: Scaling behavior of the conditional expectation value,

Eβ(y|x), after the addition of the linear term x/2 in a system

of linear size L = 512 for (a) small avalanches (32 ≤ x ≤

64),(b) intermediate avalanches (128 ≤ x ≤ 256), (c) and

large avalanches (512 ≤ x ≤ 1024). Insets show the collapse

of CPDF, Pβ(y|x), using the suggested form, Eq. (13), for

different scaling regions, which determines the value of the

scaling exponent γβ(x). Note how γβincreases with avalanche

size.

with the scaling exponent γ(W)= 0.47. This indicates

an interesting similarity between the scaling behavior of

these two types of relaxation events. However, the ob-

vious difference in the values of the scaling exponents,

γαand γ(W), reveals the essential differences in the na-

ture of these two relaxation processes. In other words,

one cannot consider type-α avalanches as a simple sum

of waves making up that avalanche.

Now, the situation is not as simple for type-β

avalanches. We observe that we cannot find a unique

exponent γβ, which collapses our data for all values of

x(= nRM). Instead, we find that Eβ(y|x)+x/2 does not

have a unique slope (on a logarithmic plot) and shows an

obvious curvature as can be seen in Fig. 6. Here instead,

we find different scaling exponents for different scaling

regions. We divide our scaling region into small, interme-

diate, and large avalanches and perform our collapse (the

insets in Fig. 6) with different exponent for each of these

regions. This is shown in different parts of Fig. 6. The

important and fundamental difference here in the case of

β-avalanches is that γβ is size-dependent and increases

with avalanche’s size. However, this increase cannot be

unbound and γβshould eventually saturate. Our numer-

ical results show that this exponents saturates at γβ= 1,

indicating a range of 0.7 ≤ γβ ≤ 1.0 for this exponent,

see Fig. 7. Since we do not expect that the boundary of

a relaxation process grows larger than its bulk, the value

γβ(x >> 1) = 1 is in fact the physical upper limit for

this exponent. Another way to see that γβ saturates at

γβ= 1.0 is to plot Eβ(y

scaling ans¨ atz should scale as xγβ(x)−1. This is shown

as an inset in Fig. 7. One can see this saturation as the

eventual x-independence of the plot for large avalanches.

We therefore conclude that γβ, unlike γα, does not have

a fixed value and in fact varies between 0.7 and 1.0 with

increasing avalanche size. Such form of avalanche-size de-

pendent exponent is a result of the complexity inherent

in the dynamical behavior of type-β avalanches.

It is now clear why simple FSS fails in the 2-D BTW

ASM. Avalanches can be categorized in two classes each

of which has distinctly different scaling properties. The

combination of these two cannot exhibit a consistent scal-

ing behavior. Moreover, the culprit is identified as type-β

avalanches for which the scaling exponent depends on the

avalanche size. The coexistence of two different types of

avalanches in the critical state, with distinctly different

scaling behavior, and the failure of FSS picture, has been

already observed and analytically proved in 1-D ASM

[32]. However, and to the best of our knowledge, this is

for the first time that this phenomenon is reported and

numerically verified in the 2-D BTW ASM. Note that this

classification is irrespective of the boundary and system

size effects and is inherent in the dynamical properties of

an avalanche.

Here, one might wonder if our distinction of type-α and

-β avalanches is merely an analysis of avalanches made

up of a simple wave versus a collection of waves. In or-

der to address this issue we have performed the same

x|x)+1/2, which according to our

Page 8

8

0123456

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

log10 x

log10(Eβ(yx) + x/2)

0246

−0.5

0

0.5

1

1.5

log10 x

log10(Eβ(y/xx) + 1/2)

512 × 512

x0.7

x1.0

FIG. 7: Conditional expectation value, Eβ(y|x), after the

addition of the linear term x/2 for a system of linear size

L = 512. Note the curvature in the graph for different values

of x. Here the dashed lines have slopes 0.7 and 1.0 respec-

tively. Inset shows the scaling behavior of Eβ(y

different values of x. The gradual increase of γβ(x) to the

value γβ(x) = 1 is clear. The horizontal dashed line is as a

reference for sight.

x|x) + 1/2 for

0100200300400500600700

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

xγα

(w>1)

P( y x )

−10010 20 30

0

0.05

0.1

0.15

0.2

xγα

(w>1)

P( y x )

( y − E( yx ) ) / xγα

(w>1)

x = 32

x = 64

x = 128

x = 256

x = 512

x = 1024

y

γα

(w>1) = 0.423

FIG. 8: CPDF of type α avalanches with more than one wave

for different values of nRM in a 512×512 lattice. Inset shows

the collapse of CPDF of these avalanches using the suggested

form of the CPDF of type α avalanches. Here again one ob-

tains a reasonable collapse with γ(W>1)

α

= 0.423.

analysis on the properties of type-α avalanches made up

of more than one wave. Our numerical analysis shows

that, despite the existence of multiple toppling events

(waves) in this class of avalanches, there is no depen-

dency in their scaling behavior on the NM structure (or

area) of the avalanches. In fact, as it is shown in Fig. 8,

this type of avalanches collapse with the same exponent

as that of type-α avalanches shown in Fig. 5. We now

can assert that what distinguishes α- and β-avalanches

is the interaction (mixing) of wave boundaries and not

the mere existence of multiple waves. In other words,

simple waves make up avalanches of type-α and complex

(mixing) waves constitute complex or β-avalanches.

Finally, of particular importance is the relation be-

tween the behavior of the scaling exponents γα(β)and the

deviation from simple power law behavior in the quanti-

ties such as P(s), where s is the size (or total number of

topplings) in an avalanche, in the 2-D BTW model. In

order to gain a better understanding of this general be-

havior, we have also looked at some other SOC models in

this regard. Our preliminary results in the case of Manna

model, which obeys FSS, show that the scaling exponent

γ maintains a constant value close to 0.5 [31, 33, 34].

Such observation, as well as the similarities between the

behavior of waves and type-α avalanches, makes us be-

lieve that the constancy of the γ exponent indicates that

the scaling behavior of the corresponding relaxation pro-

cess follows a simple power law behavior. In the case of

size dependency of the exponent γ, it is the rate of change

of γ which determines whether the power law behavior

emerges or not. For example, in the particular case of

the type-β avalanches in the present model, as γβ ap-

proaches unity for large avalanches, the rate of change of

γβbecomes very small, as can be seen from Figs. 6 and 7.

Therefore, we expect that the power law behavior is re-

covered for large type-β avalanches. While this reasoning

holds true for quantities like P(s) in which multiple top-

pling events play an important role, one should take much

more care in interpreting the behavior of other quantities

such as P(a). In fact, due to the large fluctuations in the

size of a NM structure of a β-avalanche (see Fig. 3), the

delicate size dependency of the exponent γβis effectively

blurred by performing the summation in Eq. (11). In

such situation, the effect of averaging, as well as ignor-

ing the fundamental differences between the two types of

avalanches, will result in a seemingly power law behavior

in the case of P(a) [23]. Indeed, by gathering separate

statistics for α- and β-avalanches, one can clearly observe

how the scaling behavior of quantities such as Pβ(a) and

Pβ(s) deviates from a simple power law for small and

intermediate avalanche sizes. However, the power law

behavior is recovered for large β-avalanches [31].

VI.CONCLUSION

To summarize, in this Article, we have shown that

avalanches in the ASM can have complex fine structures

Page 9

9

as a result of interaction of wave boundaries within a

given avalanche. We used these interactions to define

simple (type-α) and complex (type-β) avalanches. We

have studied the scaling behavior of these two avalanche

types in detail and have highlighted their differences. We

have shown how one can view the general dynamical scal-

ing properties of this model in terms of scaling properties

of the combined type-α and type-β avalanches. We have

proposed scaling ans¨ atz for these two types of avalanches

and have verified them numerically, thus showing how

these two types of avalanches have distinctly different

scaling behavior. In particular, while type-α avalanches

are characterized by a constant-value scaling exponent,

type-β avalanches are characterized by an avalanche-size

dependent exponent.We believe, this distinction be-

tween type-α and type-β avalanches underlies the failure

of consistent (finite-size) scaling in this model. Moreover,

we argued how the size dependency of the scaling expo-

nent in β-avalanches leads to the deviation from power

law behavior in important quantities describing the dy-

namical behavior of the avalanches, such as P(s).

addition, our results indicate that due to the coexistence

of two distinctly different relaxation events in the critical

In

state of the BTW model, one must separate these two

events in gathering any reliable statistics from the sys-

tem. We hope that our analysis is helpful in answering

some of the long standing problems on the behavior of

the prototype model of SOC in two dimensions. More-

over, we note that our classification of avalanches opens

up many questions as well. For example, how are the dy-

namical behavior of these avalanches different from each

other? In particular, what are the essential character-

istics of wave boundary interactions? Can such classifi-

cations be useful in other models of SOC? We plan to

address some of these issues in a forthcoming publication

[31].

VII. ACKNOWLEDGEMENT

The authors would like to thank M.M. Golshan for

many useful conversations. Partial financial support of

Shiraz University Research Counsel is kindly acknowl-

edged.

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