Randomness and a step-like distribution of pile heights in avalanche models

Abstract

This paper considers a one-parameter family of sand-piles.
The family exhibits the crossover
between the models with deterministic and stochastic relaxation.
The mean pile height is used to describe the crossover.
The height densities corresponding to the models with relaxation
of both types approach one another as the parameter increases.
Relaxation is supposed to deal
with the local losses of grains by a fixed amount.
In that case the densities show a step-like behaviour
in contrast to the peaked shape
found in the models with the local loss of grains down
to a fixed level
[S.Lübeck, Phys. Rev.E 62, 6149 (2000)].
A spectral approach based on the long-run properties of the pile height
considers the models with deterministic and random relaxation
more accurately and
distinguishes between the two cases for admissible parameter values.

Full text

Page 1

arXiv:cond-mat/0605705v1 [cond-mat.stat-mech] 29 May 2006
Randmoness and Step-like Distribution of Pile Heights in Avalanche Models
A. B. Shapoval∗and M. G. Shnirman†
International Institute of Earthquake Prediction Theory and Mathematical Geophysics,
Warshavskoye sh. 79, kor. 2, Moscow, 117556, Russia.
(Dated: February 4, 2008)
The paper develops one-parametric family of the sand-piles dealing with the grains’ local losses
on the fixed amount. The family exhibits the crossover between the models with deterministic and
stochastic relaxation. The mean height of the pile is destined to describe the crossover. The height’s
densities corresponding to the models with relaxation of the both types tend one to another as the
parameter increases. The densities follow a step-like behaviour in contrast to the peaked shape found
in the models with the local loss of the grains down to the fixed level [S. L¨ ubeck, Phys. Rev. E, 62,
(2000), 6149]. A spectral approach based on the long-run properties of the pile height considers the
models with deterministic and random relaxation more accurately and distinguishes the both cases
up to admissible parameter values.
PACS numbers: 05.70Jk, 05.45Pq, 05.65+b
I. INTRODUCTION
In 1987 Bak et al (BTW) introduced their sand-pile
model [1]. The model determines a system containing
some physical quantity called sand in the original paper.
The system is slowly loaded. Extra loading results in
a local relaxation. The local relaxation releases energy
that can instantly spread out to the large distances. The
spreading mechanism is fully deterministic. The model’s
system achieves its critical state without tuning any pa-
rameter [2].
Numerous power laws describe the critical state. They
have been established theoretically [3, 4] and numerically
[5, 6, 7]. The model laws find their application for such
different fields as neural networks [8], earthquakes [9],
and solar flares [10].
A great demand for the model has resulted in its mod-
ifications. The closest versions to the original sand-pile
are, probably, Manna’s and Zhang’s models [11, 12].
Manna has defined the spreading of the local relaxation
in a stochastic way. Zhang has introduced a continuous
sand-pile. The critical behaviour of these models exhibits
a certain similarity. Since minor changes in the model
rules weakly influence the critical behaviour [13] and the
number of the changes is inexhaustible, the models need
a strict classification.
Many papers assign arbitrary two models to the same
universality class if they have the same set of the ex-
ponents determining the critical power laws [13]. The
”heat” discussion [14, 15, 16, 17, 18] based on the differ-
ent approaches has turned to the conclusion that BTW’s
and Manna’s sand-piles belong to the different universal-
ity classes [15]. Preliminary investigation suggests that
∗Finance Academy under the Government of the Russian Federa-
tion.; Electronic address: shapoval@mccme.ru
†Institut de Physique du Globe de Paris; Electronic address:
shnir@mitp.ru, shnirman@ipgp.jussieu.fr
BTW’s and Zhang’s sand-piles should represent the same
universality class [19].
The paper [20] has introduced the family of the mod-
els realizing the crossover between Manna’s and Zhang’s
sand-piles. The control parameter deals with the energy
of the local relaxation. The small values of the energy
characterized Manna model while the extremely big val-
ues lead to Zhang-type behaviour. The properties of the
sand distribution over the system reflect the crossover.
The local relaxation is defined for [20]’sand-piles as the
loss of the energy down to the fixed level. The family of
the sand-piles in [21] deals with the loss of energy on the
fixed amount.
According to [21], its family of the models makes the
crossover between BTW’s sand-pile and the random walk
crossing some modification of Manna’s sand-pile. This
crossover corresponds to the relatively small energy men-
tioned above. On the other hand, the family determines
a sophisticated limit behaviour as the energy tends to
infinity. The classification based on the power laws fails
to describe a great diversity appeared in this continuous
sand-pile family. The paper [22] has introduced an ap-
propriate global functional, whose evolution calculated in
terms of the spectrum determines the system dynamics.
The paper [23] introduces the energy propagation
mechanism that lacks the local symmetry, thus leading to
the sand-piles with the quenched disorder. These models
depend on some parameter exhibiting the value of the
asymmetry. BTW sand-pile belongs to the family of the
models and corresponds to the absense of the asymme-
try. Establishing the deterministic relaxation the [23]’s
family of the models principally differs from that of [20]
and [21]. The properties of this family are found but the
models’ crossover to BTW sand-pile is not considered.
In this paper we investigate the sand-pile family of [21].
The family’s sand distribution is found to be principally
different from that appearing in the usually discussed
models, in particular, in Zhang’s and Ref. [20]’s models.
As the control parameter is relatively big the difference
between the deterministic and stochastic relaxation al-

Page 2

2
most disappears. Then the sand distribution over the
system is proved to become similar for the both types of
the models. However the spectral properties select the
sand-piles with the deterministic relaxation.
II.MODEL
The model deals with a two-dimensional square lattice
L × L. Each cell contains hij grains, where hij is less
than an integer threshold H. At each time moment a
cell (i,j) is chosen at random. Its number of the grains
(further, height) hijincreases on 1:
hij−→ hij+ 1.
If the resulting height hijremains less than H, then noth-
ing more happens at the moment. Otherwise, the cell
(i,j) becomes unstable and relaxes. Relaxation depends
on an integer control parameter n. The unstable cell dis-
tributes its n grains “in equal parts” among its 4 nearest
neighbours. Namely, as n = 4k each neighbour gets ex-
actly k grains.
Naturally, there exist the numbers n = 4k + r, where
the residue r < 4 is not equal to zero. Then each neigh-
bour gets k grains and the rest r grains are distributed
to four different neighbours at random. The following
formula expresses the idea of the construction:
hij−→ hij− n,
hneighbour(i,j)−→ hneighbour(i,j)+ k
hneighbour(i,j)−→ hneighbour(i,j)+ k + 1.
or
During this relaxation other cells can achieve the
threshold H and become unstable. Then they relax ac-
cording to the same rules. If a boundary cell relaxes,
[n/4] or [n/4] + 1 grains leave the lattice and dissipate,
where [x] is the integer part of x. (The dissipation is
bigger for the corner cells).
The sequential acts of relaxation are called an avala-
nche. The size of any avalanche is the number of the
unstable cells during the avalanche counting according
to their multiplicity. The dissipation at the boundary
assures that the avalanches are defined correctly and their
size is finite.
The case of n = H = 4 corresponds to the original
sand-pile of [1]. It worth noting that the paper [20] intro-
duces relaxation with the loss of any unstable cell down
to the fixed level. Namely, hij −→ H − n with this hij
grains passed one-by-one to the nearest neighbours; the
receiver for each grain is found at random. These changes
in the rules qualitatively influence the systemThe limit
behaviour (n → ∞) of [20]’s family and the models in
question is different.
The family really depends only on parameter n. The
values of H does not influence the system dynamics.
They determine the admissible interval [H −n,H) of the
heights.
0.25 0.50.75
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
h i j n
ρnn
n = 4, k = 1.990
n = 12, k = 1.865
n = 40, k = 1.863
FIG. 1: Normalized density of the heights; the dashed lines
are the linear fits, L = 256.
III.PILE HEIGHT DENSITIES
Following the ideas of [20] we establish some features of
the sand distribution over the lattice. Let the normalized
heights hijnbe hijn= (hij−H+n)/n. Then hijn∈ [0,1).
Let a function ρn(·) be the density of the normalized
heights hijn: ρn(k/n) is the number of hijnbeing equal to
k/n. Then the normalized densities ρn(·)n follow four flat
steps with a high accuracy (fig. 1) for n = 4k. The steps
correspond to the values of the density ρ4 for BTW’s
sand-pile. (The points of ρ4shown in fig. 1 are in good
agreement with the analytical values found in [24]).
Each density is fitted by a linear function (fig. 1). The
slopes slightly increase as n goes up and must be satu-
rating as n tends to infinity.
The following construction manages to compare quan-
titatively four steps of ρn(·) as n > 4 with the four values
of ρ4(·) representing BTW sand-pile. Given n = 4,5,...,
the values of the density ρn(·) is sampled into four
bins [0,0.25), [0.25,0.5), [0.5,0.75), and [0.75,1) with re-
ported values ψn(·) at 0, 0.25, 0.5, and 0.75 respectively.
For example, ψn(0) = ρn(0) + ρn(1/n) + ... + ρn(j0/n),
where j0is the biggest integer with j0/n < 0.25. In par-
ticular, ψ4(·) coincides with ρ4(·). Then ψn(·) is defined
in 4 points. Each value ψ(k/4), k = 0,1,2,3, represents
one step of ρn(·).
Let
σn=


?
?
?
?
3
?
k=0
(ψn(k/4) − ρ4(k/4))2


?
3.
Then σnmeasures the difference between the steps and
the function ρ4(·) corresponding to BTW sandpile.
The difference from BTW sand-pile increases as σn
goes up (fig. 2). The reported values of σnindicate that
there exist a certain maximum of the difference corre-
sponding to n ≈ 60. This observation agrees with the
change of the tendency exhibited by the sand-pile family

Page 3

3
2040 6080
0.002
0.004
0.006
0.008
n
σn
FIG. 2: Difference between the steps of ρn and the function
ρ4 measured by the functional σn.
0.250.50.75
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
h i j n
ρnn
n = 3, k = 1.959
n = 10, k = 1.918
n = 30, k = 1.898
FIG. 3: Normalized density of the heights; the dashed lines
are the linear fits, L = 256.
of [20] for the intermediate values of the control param-
eter.
In the same way, the normalized densities are intro-
duced to develop the models with n ?= 4k starting with
n = 3. The computer experiment proves (fig. 3) that the
densities ρnquickly (as n goes up) achieves four steps of
ρ4. The linear fits have slopes appearing close to that for
ρ4k.
The following reasoning gives a rough explanation of
the step-like behaviour of the densities. Given the pa-
rameter value n, the heights are naturally divided into
four intervals [H − n,H − 3n/4), [H − 3n/4,H − n/2),
[H −n/2,H −n/4), [H −n/4,H) since each act of relax-
ation maps one interval into another (possibly, except-
ing the boundary values due to the random effect). The
normalization (on n) transforms these intervals to the
domain of definition of four steps in fig. 1, and 3.
It worth noting that the sand-piles of [12, 20] demon-
strate peaked densities of the average height in contrast
to our ρn.
So, in terms of the sand distribution the model family
exhibits a certain similarity. There exist possibilities to
select the limit behaviour (n → ∞) and define an inter-
mediate deviation from the extreme cases. The densities
ρnfor n = 4k and n ?= 4k become almost indistinguish-
able as n has the order of dozens.
IV.SPECTRAL APPROACH
Another approach gives evidence of the diversity of the
two cases (n = 4k and n ?= 4k). It deals with the spec-
trum of the average height h = L−2?L
and treated as the function on time, h(t). The paper [22]
has successfully used h(t)’s spectrum to describe the pa-
per’s sand-pile’s dynamics. Our h(t)’s spectrum appears
to be noisy, therefore it is averaged over its several real-
izations. Besides, each h(t)’s realization is stored in the
bins of some length ∆. Then the Fourier transform de-
termines the spectrum ξ. The highest frequency is 1/∆
in the case.
A formal procedure applied for the spectrum calcula-
tion consists of four steps.
i,j=1hij. The av-
erage height is calculated at the end of each time moment
1. For some fixed N the time moments, which h(t)
is catalogued at, is divided into N non-intersecting
intervals of the same length T.
2. Given any fixed interval of step 1, the averaging
of h(t)-values over relatively small sub-intervals is
applied to reduce the number of data for further
numerical application of the fast Fourier transform.
Let ∆ be the length of the small sub-intervals and
r be their quantity, r∆ = T. Then a signal xj is
defined as the arithmetic mean of all the h(t)-values
in the j-th small sub-interval, j = 1,2,...,r.
3. The Fourier transform determines the spectrum of
xj. Let
ˆ xk=
1
√r
r
?
j=1
?xj− ?x??e−2πikj
r ,
where ?x? is the mean of xj, j = 1,...,r, i =√−1.
Then the frequencies fkare k/(r∆), k = 1,2,...,r,
and the power spectrum ξ(fk) of the signal xj is
defined as ξ(fk) = |ˆ xk|2.
4. Each of N intervals defined in step 1 generates its
own spectrum ξ(fk). The averaging of ξ(fk) results
in the stabilized spectrum, which the construction
aims at. The stabilized spectrum depends on the
model parameter n. Thus, the notation ξn(fk) is
keeping further for the stabilized spectrum.
According to [21], the family of the spectra admits a
certain normalization. If ∆ is proportional to n, the exhi-
bition of ξn(fk)/n versus dimensionless frequencies fk∆

Page 4

4
10
−3
10
−2
10
−1
10
−9
10
−8
10
−7
10
−6
10
−5
f∆
ξn/n
n=4
n=12
n=60
n=100
FIG. 4: Normalized spectrum ξn/n vs normalized frequencies
f = fk. The inset contains the graphs in the boxed part of
the figure; ∆ = 8n, N = 16, r = 8192, L = 256.
collapses the graphs on the interval of the high frequen-
cies and remains reasonable on the other interval (fig. 4).
Leaving the main part of the spectrum as it is we stress
attention on some interval of the low frequencies. The
corresponding part of the graphs is boxed in fig. 4 and
put in its inset. In this part the graphs (for n = 4k)
establish a noticeable rise as n goes up to the value being
close to 60. Then the tendency principally changes. The
graphs turn down to the spectrum ξ4 at their left part
while demonstrating a definite peak (fig. 4, inset). It
agrees with the behaviour of σnreported earlier.
The spectrum’s propagation to the left needs much
bigger h(t)’s domain of definition that is hardly possi-
ble. On the other hand, the highest achieved frequencies
correspond to the avalanches of the biggest size observed
during computer simulation. Bigger avalanches are prac-
tically not observable even for much longer simulation.
Then the computer experiment must be producing the
constant spectrum just at the left of the obtained graphs’
part in fig. 4.
The horizontal coordinates of the investigated box are
not absolute constants. They depend on the lattice
length L and the simulation time. As L increases the
box moves to the left becoming invisible in fig. 4.
The influence of the simulation time is essential. The
investigated part of the spectrum reflects the frequen-
cies of the rare, big, and strongly dissipative avalanches.
The lack of the data limits the results covering these
avalanches. It concerns the partial conclusions about the
rare avalanches in BTW’s and Manna’s sand-pile [25].
In contrast to the quick convergence of ρnn, n ?= 4k, to
their four values, the family ξn/n of the spectra exhibits
quite a different tendency. Since the high-frequency spec-
tra are rather similar (fig. 5a), the analysis is focused on
the low frequencies (fig. 5b) corresponding to the boxed
part in fig. 5a.
As n = 3 the normalized spectrum has its horizontal
interval. While n increasing, ξn/n changes on this inter-
val. Fig. 5 demonstrates the minor changes as n = 30
and completely another behaviour as n is equal to 150
and 301. So, for the simulated big ns the low-frequency
spectrum essentially deviates from ξ3as well as from ξ4k
for big k.
According to fig. 4 and 5 the long-time evolution of
the sand-pile exhibits a great complexity. In contrast to
the space features, several patterns do not exhaust the
spectrum behaviour. Up to the developed experiments
the sand-piles with the deterministic and stochastic re-
laxation have the different spectra.
V.CONCLUSION
Summarizing, we develop the family of the sand-piles.
The control parameter n is the number of the grains that
any unstable cell passes to its neighbours. As n = 4k, the
propagation of the grains through the lattice is fully de-
terministic despite the models with n ?= 4k involve some
random effect. In terms of the sand distribution the ran-
dom effect disappears as n is sufficiently big. However
the trace of the deterministic relaxation remains visible
in terms of the average height’s spectrum, selecting some
peak at low-frequency spectrum. With their peculiar evo-
lutionary properties the sand-piles corresponding to the
deterministic relaxation may admit a certain prediction.
This hypothesis agrees with the effective precursors found
for BTW’s sand-pile in [26].
[1] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett.
59, 381 (1987).
[2] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38,
364 (1988).
[3] D. Dhar, Phys. Rev. Lett. 64, 1613 (1990).
[4] D. Dhar, Physica A 263, 4 (1999).
[5] P. Grassberger and S. S. Manna, J. Phys. (France) 51,
1077 (1990).
[6] O. Biham, E. Milshtein, and O. Malcai, Phys. Rev. E 63,
061309 (2001).
[7] D. V. Ktitarev, S. L¨ ubeck, P. Grassberger, and V. B.
Priezzhev, Phys. Rev. E 61, 81 (2000).
[8] E. N. Miranda and H. J.Herrman, Phys. A 175, 339
(1991).
[9] P. Bak and C. Tang, J. Geophysical Res. 94, 15635
(1987).
[10] A. Bershadskii and K. R. Sreenivasan, Eur. Phys. J. B
35, 513 (2003).
[11] S. S. Manna, J. Stat.Phys. 59, 509 (1990).
[12] Y.-C. Zhang, Phys. Rev. Lett. 63, 470 (1989).
[13] A. Ben-Hur and O. Biham, Phys. Rev. E 53, R1317
(1996).

Page 5

5
10
−3
10
−2
10
−1
10
−9
10
−8
10
−7
10
−6
10
−5
f∆
ξn/n
n=3
n=30
n=150
n=301
10
−3
10
−6
10
−5
f∆
ξn/n
n=3
n=30
n=150
n=301
FIG. 5: Normalized spectrum ξn/n vs normalized frequencies f = fk; ∆ = 8n, NR = 16, r = 8192, L = 256 (a) full graph, (b)
box of fig. a.
[14] L. Pietronero, A. Vespignani, and S.Zapperi, Phys. Rev.
Lett. 72, 1690 (1994).
[15] A. Chessa, H. E. Stanley, A. Vespignani, and S. Zapperi,
Phys. Rev. E 59, R12 (1999).
[16] E. V. Ivashkevich, A. M. Povolotsky, A. Vespignani, and
S. Zapperi, Phys. Rev. E 60, 1239 (1999).
[17] C. Tebaldi, M. DeMenech, and A. L. Stella, Phys. Rev.
Lett. 83, 3952 (1999).
[18] S. L¨ ubeck, Phys. Rev. E 61, 204 (2000).
[19] S. L¨ ubeck and K. D. Usadel, Phys. Rev. E 55, 4095
(1997).
[20] S. L¨ ubeck, Phys. Rev. E 62, 6149 (2000).
[21] A. B. Shapoval and M. G. Shnirman, IJMPC 16 (2005).
[22] P. de los Rios and Y. Zhang, Phys. Rev. 82, 472 (1999).
[23] R. Karmakar, S. S. Manna, and A. L. Stella, Phys. Rev.
Lett. 94, 088002 (2005).
[24] V. B. Priezzhev, J. Stat. Phys. 74, 955 (1994).
[25] M. DeMenech, A. L. Stella, and C. Tebaldi, Phys. Rev. E
58, R2677 (1998).
[26] A. B. Shapoval and M. G. Shnirman, IJMPC 15, 279
(2004).