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
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 . 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), 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 .
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
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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  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 , that by reducing such effects one still observes
the aforementioned deviations from the simple power law
Due to the complex spatiotemporal behavior of
avalanches, it is reasonable to decompose them into more
elementary objects. As is shown in Ref. , 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
. 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-
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 . 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
hi → hi− 4,
hnn → hnn+ 1.
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 . 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 . 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
It can be shown both analytically and numerically 
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
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,
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
Number of Waves: 2
Number of Waves: 4
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
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-
IV. COMPLEXITY IN THE FINE STRUCTURE
In Ref.  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
up an avalanche. By “complex superposition” we mean
that in general in an avalanche, excluding the exterior
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
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,
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)
NM / n
RM a )
256 × 256
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
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.
P( nNM nRM=32 )
P( nNM nRM=64 )
P( nNM nRM=128 )
P( nNM nRM=256 )
100200 300 400
P( nNM nRM=512 )
200 400 600800
1000 1200 1400 1600
P( nNM nRM=1024 )
Type α & β
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.
P(w)( nNM nRM=32 )
20406080 100 120
P(w)( nNM nRM=64 )
0 50100 150 200
P(w)( nNM nRM=128 )
50 100 150 200250
P(w)( nNM nRM=256 )
50 100150200 250 300
P(w)( nNM nRM=512 )
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)),
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.
with NM of size y and RM of size x.
: Total number of type α (or β) avalanches
with NM of size y.
: Total number of type α (or β) avalanches
with RM of size x.
: Total number of type α (or β) avalanches
• Nx: Total number of avalanches with RM of size
• 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
Substituting Eq. (9) in (8), and using the fact that
P(x) = Nx/N we have,
where the summation goes over those values of x and y
that fulfill Eq. (6). Finally, using the relation,
[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
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)
• Type β avalanches:
Pβ(y|x) = x−γβUβ(y + x + Eβ(y|x)
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
Let us first consider the case of type β avalanches. Us-
ing the suggested form in Eq. (13), we have
yPβ(y|x) dy (14)
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,
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
(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
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).
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. . 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),
(x + y + <y>x) / xγβ
xγβ P(y x)
log10(Eβ(yx) + x/2)
512 × 512
γβ = 0.7; x ∈ [32, 64]
x = 32
x = 64
x = 32
x = 64
log10(Eβ(yx) + x/2)
(x + y + <y>x) / xγβ
xγβ P(y x)
512 × 512
γβ = 0.73; x ∈ [128, 256]
x = 128
x = 256
log10(Eβ(yx) + x/2)
x = 512
512 × 512
γβ = 0.80; x ∈ [512, 1024]
(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
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
. 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
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
log10(Eβ(yx) + x/2)
log10(Eβ(y/xx) + 1/2)
512 × 512
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
P( y x )
−10010 20 30
P( y x )
( y − E( yx ) ) / xγα
x = 32
x = 64
x = 128
x = 256
x = 512
x = 1024
(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)
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) . 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 .
To summarize, in this Article, we have shown that
avalanches in the ASM can have complex fine structures
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
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
The authors would like to thank M.M. Golshan for
many useful conversations. Partial financial support of
Shiraz University Research Counsel is kindly acknowl-
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