on the modelling of real sand piles

Physica A 182 (1992) 295-319

North-Holland

On the modelling of real sand piles

H. Puhl HLRZ, KFA Jiilich, Postfach 1913, W-5170 Jiilich, Germany

Received 26 August 1991

We are interested in the behaviour of realistic sand piles. A new model belonging to the

class of critical-slope models is proposed to describe the construction of piles and the

avalanches on them. We find a scaling type of behaviour and compare the results with

classical sand pile models and experiments.

1. Introduction

Self-organized criticality (SOC) was proposed by Bak, Tang and Wiesenfeld [l, 21 some years ago. A self-organized critical system evolves into a critical state without the fine tuning normally necessary. This critical state is an attractor in the dynamics, hence initial configurations do not play any role for the critical state itself. It was then proposed that SOC could account for the behaviour in such different fields as the “game of life” [3] and earthquakes [4].

The dynamics of SOC is characterized by correlations over a wide range of timescales, so that one expects a l/f” noise power spectrum. Examples are (as given by refs. [l, lo]) the intensity of quasars, the current through resistors, sandflow in hour glasses and others. The second fingerprint of SOC, as proposed by ref. [l], is the self-similarity of the resulting avalanches. It has turned out, however, that this is usually only true for the distribution of sizes but not for the individual avalanches, i.e. their internal structures are not necessarily fractal.

The easiest toy model proposed initially (ref. [l], later extended in ref. [7]) was a cellular automaton supposed to simulate very roughly sand piles, even if its application is not at all limited to this case. The authors argued that this model should have a l/f power spectrum of the sand leaving the system, that the critical height should not depend on the way the pile was constructed, that finite size scaling should be possible and that the system should be characterized by critical exponents for the power-law distributions of lifetime D(t) and size D(s) of induced avalanches:

0378-4371/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved

296

D(t) cc t-”

and

D(s) m CT .

H. Puhl I Modeliing real sand piles

(2)

There are three major classes of sand pile models today (cf. ref. [5]). The

cited toy model belongs to the first of these, the critical-height models, which

are the best understood nowadays. It has, however, turned out, that height

models are not really useful in describing sand piles, so that two other classes

have been introduced, called the critical-slope and the Laplacian models. In

these two models the toppling condition depends not on the height itself, but

on its first and second derivatives, respectively. Although these two classes of

models have been under investigation for some time [5], not very much is

known about them. In particular, the existence of a power-law behaviour for

the size and lifetime distributions of the slope model is not clear yet.

Experimental work on sand piles has been performed [8, 91 to see if they

exhibit any power-law behaviour in lifetime and cluster size distributions, but

the results do unfortunately not indicate very clearly yet if this type of

behaviour is typical for real sand piles or not.

All models have in common, that-despite the allusion to sand piles - their

outcome is not to describe the real sand piles found in nature. So the purpose

of this work is to develop a model that is able to produce piles that behave and

look - to some extent - like natural sand piles and to check if it is still

characterized by the features of SOC. In our case we limit ourselves to the

easiest variety, namely piles consisting of grains of the same size on square and

circular supports.

After a detailed description of our model and its realization on a computer,

we study some real sand piles with some easy experiments. We then investigate

how well our model is able to produce nicely shaped sand piles on different

types of supports and see if we can detect any special kinds of behaviour.

2. Model

2.1. Real sand piles

As we want to investigate simple models which produce sand piles that are as

close as possible to real piles, we shall list the main features of real sand piles

that we want to include in our model:

(a) piles that are governed by gravitation, so the grains always move

H. Puhl I Modelling real sand piles 297

downhill (if the initial kinetic energy of grains falling on top of the pile is neglected),

(b) piles that are not homogenous, i.e. the grains do not necessarily have the same size and roughness,

(c) the piles that are built on infinite surfaces by adding grain on grain are perfect cones.

As a compromise between simplicity and realism of the model we have decided to model piles of unit grain size on regular flat surfaces.

2.2. Description

The pile is simulated on a square lattice of size L. On each lattice site (x, y) we define an integer function h(x, y) that describes the height of the pile. We assume that the grains are cubes and packed on top of each other without leaving empty spaces. The process of distributing sand only takes into account nearest-neighbour interactions.

The main ingredient of the model are the four thresholds z(x, y, 2;) that are assigned to each site, where .Fj (i = 1, . . ,4) vicinity vectors#‘. These thresholds are uniformly distributed random numbers on the bonds with

The z represent the maximum possible difference between the height at the site (x, y) and its neighbours in the direction F;. As for each bond two values are independently chosen at random; the interactions are asymmetrical.

During the course of the simulation sand is added to the center of the lattice (and only there). The system then behaves according to the following rules: we test the four surrounding sites. If the height difference between nearest neighbours exceeds the bond threshold, i.e.

y, ,

4x, y) +, y) - 4x, y, Ci> , (54

h(x L 1, y * 1) = h(X + 1, y +- 1) + tr(x, Y, ‘i) . (5b)

The order in which the four directions are scanned is not fixed, but chosen for

*I Which points in our case in the four directions north, east, south, west.

298 H. Puhl I Modelling real sand piles

each site randomly from a look-up table containing all possible permutations of

the four directions.

Suppose that we have a lattice of size L = 101, the central point then has the

coordinates (50, 50). We now choose an entry in the look-up table and obtain

an order of testing, say 3, 1, 2, 4, which means south, north, east, west. Now

we test whether

h(50,50) - h(50,51) > tr(50,50, south) .

If this condition is true, we will shift ~(50, 50, south) grains of sand to the site

(50, 51), else we shall do nothing. Then we shall test sites 1, 2, and 4 in the

same way.

For the test we can choose between updating the whole lattice at each sweep

or keeping a list of critical sites that need to be tested. In our case we found

that the number of critical sites at each sweep through the lattice was small

compared to the total number of sites. So we considered a list that keeps track

of each critical site. Here care has to be taken that not only those sites can

become critical where grains are added, but also the sites being around the site

where grains are taken from. At each sweep the old list is treated and a new

one constructed. The algorithm stops when the new list is empty, i.e. when

there are no critical sites left.

Two different initial conditions are used for the simulation. The first is an

empty support on which a pile is built grain by grain. In the second case we

produce piles by “erosion”. This means that a large amount of sand is put

initially on the support, which produces an overcritical pile. Then the initial list

is set up by testing each site systematically. The simulation then runs in the

same way as usual until the list is empty.

The combination of eqs. (4) and (5) allows us to simulate indirectly the

influence of gravity in our model. Quenched randomness of the thresholds

takes into account the different sizes and friction values of the grains, even if in

our model these values are attached to the bonds and not the grains. So we

account with this model for the first two real sand pile features mentioned

above.

A potential problem of the model is the anisotropy induced by the square

lattice which has no correspondence in nature. Another simplification is the

fact, that - regardless of the thresholds-the grains are regarded as cubes, all

of the same size, so that empty holes inside the pile are not possible.

All calculations for this model were made on the Alliant FX-2800 at GMD.

The operating system was the concentrix version 2.1.00, the programming

language was the FX/C-2800 versions 1.2.00.

This computer is a MIMD unix machine, containing 16 i860 processors.


Eight of these processors are able to run concurrently on one program. The program took advantage of this facility in calculating with each of these processors a different sand pile. The time loss was lower than 5% compared to single-processor runs, the speed was about 20000 updates per second. To produce the results published here, about 300 hours of CPU time were needed.

Problems arose using the standard C random-number generator “random”. Apparently this generator does not work correctly with concurrent programs, so we implemented an “add and cut” random-number generator with a ring of 250 numbers and a standard congruential generator that gave surprisingly good results. Nevertheless the problem of defining a good random-number generator remains unsolved.

3. Some experimental facts

Simulating sand piles has the major advantage that some experiments can be easily performed because the material needed is not expensive. So we have carried out some simple experiments, which we can highly recommend to the reader, since they are easy, entertaining and instructive.

When sand piles are built grain by grain on infinite flat surfaces the results are - as everyone knows - perfect cones. Our question was, how do piles behave on finite-sized supports of different shapes?

The sand used was sieved to obtain at least nearly homogenous grain sizes. Then base surfaces of different shapes were put on a support, so that the sand grains reaching the border of the surface could fall off. Building piles from scratch was obtained by pouring (a small amount of) sand through a hole on top of the center of the surface. Another way to obtain piles was to put a huge amount of sand on the base enclosing it with the help of some piece of paper forming a rim, which was then removed, allowing the sand to pour out very quickly and leaving behind a sand pile.

Building a pile from scratch by throwing grains at the center of a support gives a cone growing as long as the pile does not touch the border. When the pile reaches the border, a runway forms and most of the grains move down that way to the edge#*.

The pile seems not to depend on the surface used; as soon as one point of the cone reached the border, the runway appeared and the growth of the pile almost stopped. Only a few grains took any other path down the pile#3.

#* To avoid this, it would have been necessary to add only one grain at a time. Our experimental

set-up was not sophisticated enough to control this.

x3 We cannot tell if this would eventually lead to filled corners, as the process is very slow.


Fig. 1.

The piles obtained by putting a huge amount of sand on a support, and

letting it evolve by itself, did depend strongly on the shape of the support. For

circular supports we obtained perfect cones. The photo shown in fig. 1 shows

the clearly shaped pyramids with quite sharp edges that we get with square

supports. When an octagonal support is used, we obtain pyramids with a

corresponding base. The sharpness of the edges that appear did depend on the

uniformity of the grain sizes.

Apparently there are strong correlations inside a collapsing sand pile that


lead to the fact that the top of the pile has some “knowledge” about the

underlying surface.

4. Piles without disorder

Returning to simulations, we first looked at a simplified version, where the

disorder introduced by the thresholds is removed. All the thresholds thus have

the same value, namely tr( , , ) = 1.

Of interest was the influence of the support on the final shape of the pile.

Both empty and prefilled supports have been used; the size of the lattice was

L = 101.

Assigning another uniform value IZ to the thresholds, i.e. tr( , , ) = n, does

not change the behaviour of the piles, not even in the case where the

thresholds are increased, but at each step only one grain of sand is toppled.

The only fact of importance is that the threshold values are uniform.

Avalanches on these piles are not really interesting, since in our simulation it

turns out that they all have the same length and size and that avalanches always

go down runways. This is similar as in the experiment on short timescales.

4.1. Initially empty support

To simulate a support, we marked each site belonging to the border. For a

square support, this meant marking only the sites at the border of the lattice. A

circular support was produced by inscribing a circle on the square lattice and

marking all the sites outside this circle as belonging to the border. The sand

height at the border sites was maintained at zero.

For fig. 2 about 8 x 10’ topplings of sand have been effected. It shows no

difference between the different types of supports. In both cases the pile is a

pyramid, with its angles at the middle of the edges of the supports. After a

closer look we see that the edges of these pyramids are not sharp, but three

lattice sites broad. These edges are the runways the grains use to travel from

the top to the border of the support.

In fact, as soon as a runway has completely formed, the grains do not have

any other possibility than to slide down on it. Thus, once all four runways

exist, the evolution of the pyramid stops. The corners can never be filled.

As the pile touches the border at the same moment for the square and for

the circular support, there can be no difference between the two of them.

By looking, for instance, at the shape of the base of the pyramid, we can see

clearly the strong influence of the anisotropy of the lattice of our model. The

resulting pile has four privileged directions, it grows much faster in the

direction (0, l), than in the direction (1,l).


Fig. 2. Piles on an initially empty support (no disorder): (a) square; (b) circular.

4.2. Prejilled support

When we take prefilled supports, which are then eroded, the influence of the

anisotropy is not as strong as the border effects. As we see in fig. 3, for which

about 2.8 x lo6 topplings were necessary, the pile (a), which is built on a

square support, is a pyramid that has as base the underlying support. The pile

(b), built on a circular support, has the shape of a cone, with a weak regularity

in the discretization due to the underlying square lattice. These results coincide

surprisingly well with our experiments.

For collapsing piles we find, that border effects are more important than

anisotropy. The shape of the pile is determined by the underlying surface.

Some anisotropy can, however, still be seen in the structure of the pile.


Fig. 3. Piles on prefilled supports (no disorder): (a) square; (b) circular.

All avalanches on piles with uniform thresholds have exactly the same size and lifetime. They are independent of using either initially empty or prefilled supports.

The height of the piles is given by h(tr) = [(L - 1) /2] tr; this depends linearly on the uniform thresholds assigned to the bonds.

5. Piles with disorder

Going one step further we now allow the thresholds to take random values in a certain range: 1 s tr( , , ) 4 trmax. The value of tr,,, determines the degree of discretization of the uniform threshold distribution. The higher the value of

304 H. Puhl I Model&g real sand piles

x103 1.20 r 1.00

%

.9 2 0.80

5 0.60

E

‘3 0.40

E

0.00 0.20 0.40 0.60 0.80

maximum threshold : trmax Fig. 4. Height of the pile depending on the value of bnlax for piles with disorder on a square support.

trmex is, the nearer we get to a continuous distribution of thresholds. We then

repeat the simulations already performed without disorder.

The question to ask here is, if the randomness in the thresholds is capable of

introducing enough noise that the piles have a more realistic look (weaker

anisotropy effects) and dynamics and avalanches that correspond more to

self-organized critical behaviour.

The first thing we look at, is the height a pile reaches, which depends on the

value of trmax. Fig. 4 shows that the dependency is simply linear. This is not

surprising when we think about it that the mean threshold is assigned a value of

tr,,,/2 as the thresholds ranging from zero to trmax.

For the plot, the mean values of the heights of the piles were calculated over

16 piles. They were built on prefilled circular supports.

5.1. Initially empty support

We start by choosing a value of tr,,, = 4, i.e. the thresholds take randomly

assigned values between 1 and 4. Taking a look at the pile (a) of fig. 5, which

has been built on a square support and stopped before the boundary was

touched, we can see, that the pyramid has softened edges, and the base

becomes rounder. The smoothening effect is even stronger for the pile (b) of

fig. 5, where tr,,, = 8. Here we needed about 5 x 10h topplings for the

construction of the piles.


Fig. 5. Piles on initially empty square support before touching the boundary: (a) tr_ = 4; (b)

Wm._ = 8.

The quenched randomness in the thresholds seems to be able to overcome to

some degree the anisotropy of the underlying grid as far as the shape of the

pile is concerned. The higher the threshold, the better the result will be.

The result for piles on circular supports is exactly the same. As the four axes

grow at the highest speed, the pile will not feel the border earlier than on a

square support.

One can ask now, whether this smoothening is systematic and if we can hope

to end up with a perfect cone for infinite thresholds. Another question to ask is

whether the corners of piles built from scratch on square surfaces will ever be

306 H. Puhl I Model&g real sand piles

filled or not, since all the results reported up to now are on short timescales. We shall try to answer these questions later.

5.2. Prejilled support

The results in fig. 6 are not surprising. We show here two piles, (a) on a

square support, (b) on a circular one, fr,,, = 8 for both of them. We obtain roughly the same shape of envolope as in the deterministic case, only the surface starts to get rough, and the edges are not straight lines anymore. This corresponds fairly well to the situation in our experiment.

Nevertheless it should be noted that the influence of the grid has not totally disappeared. One can distinguish some regular structures on the circular pile, being a consequence of the underlying lattice.

Little more than 7 x 10’ topplings were necessary to build up these piles. This number seems high compared to the 5 x 10” needed for building a pile from an empty support, but one should bear in mind that it depends on the amount of sand put on the support (here we initially started with 4.5 x 10’ grains of sand).

To be sure that the round piles do not exhibit any concavities in the horizontal directions and have a constant slope, we show in fig. 7 the average slopes in the (1, 0) and (1, 1) directions over 200 piles for trmax = 4. The slope around the pile is constant within the limit of accuracy, nevertheless the top of the pile is not sharp so that at the top the slope goes to zero smoothly.

5.3. Circular piles?

We have seen in the last section that for low thresholds one does not find round piles. Nevertheless, piles seem to evolve from this unphysical result when the thresholds are all the same, to a more realistic one, when disorder in the thresholds increases.

It might well be worth having a closer look at this phenomenon and to evaluate if the pile really would be an isotropic cone in the limit of tr,,, + 30. It would also be interesting to know how fast this would happen.

So we have calculated the ratio of the slopes in the directions (1, 0) and (1, 1). In order to obtain better statistics we decided to suppose that a cut through the pile would give a triangular cross section. Rather than to measure only the extension of the pile on the support in a given direction, we counted the mass in a slice through the pile following these directions. This makes the data less susceptible to statistical fluctuations on the surface.

Supposing that the ratio r of the masses of the two different slices tends to one with tr+ m, we plot 1 - r in fig. 8 on a log-log plot against tr,,,.


Fig. 6. Both piles constructed with disorder with a prefilled support, fr,,, = 8; support (a) was

square, support (b) was circular.


X10’ 0.50

0.40

& o.30 ,o m 0.20

2

.s 0.10

0.00

-0.10

q (1,o) (l,l)

I (b) 1

1

0.00 0.10 0.20 0.30

x position 0.40 0.50x10

Fig. 7. Local slope on the pile shown in (a) is plotted as a function of the height (b). The slopes on

the pile are shown following a cut in the (1, 0) direction (solid line) and the (1, 1) direction (dashed line).


lod loo

I I 10’ lo2 lo3

maximum threshold : trmax lo4

Fig. 8. The ratio of the pile’s slope in the (0, 1) and (1, 1) directions are plotted against the value

of rr,,,*x.

The values plotted here are the mean values over 100 to 300 different piles for each value of tr,,,, each pile being measured several hundred times during the build-upS4. Each pile is built from scratch on an initially empty square support of size L = 101 and the ratios are measured in regular intervals. Care is taken that the piles do not reach the horde?.

It is clear from the data that the piles grow rounder with increasing thresholds. It cannot be decided from the data if the dependence is a power law. Better statistics on larger lattices would be necessary to clarify this.

In particular for the high thresholds it is not easy to get good data, since the height of the piles becomes very large compared to the lattice size, which produces larger error bars.

5.4. Filling of corners

Another interesting question is whether the corners will be filled or not after a long enough time. When we build a pile on an empty support we see an approximate cone growing until it reaches the border of the support. Now there are two possibilities: either the corners will be filled or not. If sand is

” We make here the implicit supposition that the slope of a pile does not change when the total

size of the pile changes. In nature this seems evident, we can nevertheless not be sure that the

same is valid for our model. None of the calculations made to clarify this, however, showed any

indication that the supposition was not valid.

*’ Reaching a border would probably influence the slope.

310 H. Puhl I Mode&g real sand piles

Fig. 9. Time evolution of the mass of a pile with tr,,_ = 4.

distributed in all directions with the same probability, then the corners should

fill. However, the slope then decreases towards the corners and sand ava-

lanches will then tend to become anisotropic.

The time evolution of a pile’s mass is shown in fig. 9. To produce this figure,

we monitored the mass m of the pile while throwing six million sand grains on

top of it. The pile was build on an initially empty square support with trmax = 4.

In fact we plot the difference between the full pyramid and the growing pile.

We calculated the mass of the pyramid mp by simulating several piles on

prefilled supports and taking the mean value of their masses.

We then plotted the difference mP - m against the time.

We see in the figure that we mainly have two different regimes. The first one

corresponds to t c 2 x 10’. Here no sand leaves the system, m 3~ t. Above that

we find a transitory regime where more and more of the added particles leave

the system. For t 3 5 x 10” only a very few particles contribute to the growth of

the pile. The mass increase seems to be of the power-law type.

6. Avalanches

To study avalanches, we use “eroded” piles, i.e. constructed by taking

prefilled supports. Avalanches are induced by just adding one grain on top of

the pile. We then wait for the avalanche to die out, before a new one is started.

So we can gather data for each avalanche separately and get the lifetime and

size distributions.


The size of the avalanche clusters was determined by counting all those sites that toppled at least once during the avalanche. The lifetime is the total number of times the list of critical sites was processed until it was empty.

It is not straightforward to plot avalanche data (i.e. the lifetime and cluster size distributions) for the critical-slope model. As already pointed out by other authors [5], these distributions contain humps, which make it difficult to see scaling.

Since there could be a qualitative difference between avalanches that do not touch the border and those which do, we followed the approach of dividing the avalanches into these two groups.

6.1. General remarks

To see the dynamics on the surface of a pile, we look at the plot of the distribution of topplings during one simulation run. The height on this plot gives the number of times a site toppled. As the heights in the center of the plot are very high#h, the top has been cut off so that the fine structures at the bottom can be seen.

In fig. 10 we have plotted for two different thresholds data from 128 conical piles, constructed by erosion, averaged over around 1000 avalanches each. The avalanches clearly behave very differently when the thresholds are changed. For low thresholds they go down on almost straight lines along the axes of the square lattice, whereas for larger thresholds this is not true anymore.

Fig. 11 contains a subset of the data of fig. 10. Here we have considered only those avalanches that do not reach the border of the system. Though large avalanches occur much less often than small ones, they account for the largest part of the topplings. The influence of the grid on short avalanches seems to be the same as on large ones.

We have seen that on piles built on square supports that the avalanches mostly follow the direct path down the surface. On round piles avalanches are more spread over the total surface.

6.2. Distributions

In the following we shall calculate the exponents for cluster size and lifetime distributions for avalanches that do not reach the border. All calculations have been made for piles on prefilled circular supports. The curves are smoothed for better visibility#‘, the lattice size L increases from left to right.

*’ This is due to the fact that there are many more small avalanches than large ones.

“The smoothing algorithm used is described in ref. [12]. Briefly, a Fourier transform is

performed, variations with high frequencies are cut off, and the data are transformed back.

312 H. Puhl I Modelling real sund piles

Fig. LO. The total number of topplings while avalanches are going down: (a) IT,,,, = 4; (b)

trl?Mx = 16. The sites with a number of topplings larger than 128 are given the value 128.

We see three different regimes in fig. 12, where trmax = 8. In the first one (up

to sizes of ten) there is no scaling according to a power law. The second regime

covers the medium range of 10 to over 100 sites, depending on the lattice size.

In this regime we see scaling following a power law. In the third regime the

influence of the finite size of the lattice becomes noticeable. Distributions bend

away from the ideal line.

The exponents calculated in the scaling regime by linear least-squares fit are

T= 1.9620.1,

b = 2.03 k 0.1 .

H. Puhl J Modelling real sand piles 313

Fig. 11. Topplings while only short avalanches are going down: (a) or,,,;_ = 4; (b) tr,,i, = 16. A

cutoff is applied in the same way as discussed in the caption of fig. 10.

The same plots for higher thresholds (fig. 13) show that there is no qualitative difference between low and high thresholds. The exponents obtained here are

7 = 2.02 -+ 0.2 )

b = 1.86 k 0.2 .

Apparently the exponents do not vary within the error bars when the thresholds are changed**. We propose that the exponents are equal to two.

xX There seem to be differences in exponents when compared to the other definition of cluster size (counting the total number of topplings). This will be discussed in a later paper.


loo t I I 3

I I

lOI IO2

avalanche clustersize : s

(a) :

IO3

avalanche lifetime : t Fig. 12. (a) Cluster size and (b) lifetime distributions of avalanches that do not reach the border

for round piles on prefilled lattices for tr,,,X = 8. Lattice size L increases from L = 31 to L = 241.

128 different initial configurations were used, 3000 sand grains were dropped on each.

The data for the higher threshold are statistically not as good as those for the

low threshold, as fewer avalanches are initiated#‘.

#“We drop one grain of sand at each step. Thus for the higher threshold the addition of more

grains is needed before the site becomes critical. As for each run the number of added grains is fixed, not the number of avalanches, the data are better for lower thresholds.

H. Puhl I Mode&g real sand piles 315

IO0

10 ’ 3 “a . . h 10-l

.Z rrl

S U3

%

L d

1o-5 10”


r “““’ I I 1

tr,,= 64 -

iI..

I M

I 9 1 1 10’ lo- 10.

avalanche lifetime : t

lo-

Fig. 13. (a) Cluster size and (b) lifetime distributions for round piles on prefilled lattices for

avalanches that do not reach the boundary. The value for fr,,, was 64. Lattice size L increases from L = 31 to L = 241. 128 different initial configurations were used, 3000 sand grains were

dropped on each.

Fig. 14 shows the same plots of cluster size and lifetime dependency for

those avalanches that reach the border. We see humps that reach their maxima1

height very quickly and fall off more slowly.

The rapid increase is not astonishing, since it is very rare that short

avalanches reach the border. The fall-off could follow a power law, though we

cannot discriminate that with our data. If one wants to assign an exponent,

316 H. Puhl i Model@ real sand piles

L=31 L-l L=61 . L= 121 ----_--_

I I I 10’ lo* lo3


(4 1 lo4

(b) - I I I I

10’ lo2 IO3 lo4

avalanche lifetime : t

Fig. 14. (a) Cluster size and (b) lifetime distributions for round piles on prefilled lattices only for

avalanches that do reach the boundary. The value for tr,,,,_ was eight. The lattice size L increases

from L = 31 (left-hand) to L = 121 (right-hand hump). 128 different initial configurations were

used, 3000 sand grains were dropped on each.

then T could be equal to four. The center of mass of the humps seems to follow

closely the size of the lattice. More accurate data are needed to be able to

decide whether avalanches belonging to this second class do or do not scale

with a power law and if the x-axis scales with the lattice size L or with the

lattice size raised to some power L”.


Let us next focus on the scaling behaviour. As proposed in ref. [6] a finite size scaling should be effective like

D(s) = s-‘F( 1 lL”) . (6)

We therefore calculate the difference between the fitted line and the actual data. Introducing a finite size scaling, we can superpose the curves for different lattice sizes.

We find that our data fit well with a value of

w=3/2.

7. Conclusions

We found that with our modified-slope model we are able to simulate piles that do indeed resemble real sand piles compared to experiments.

We find that the anisotropy of the underlying grid can only be overcome in the limit of infinite thresholds. This means that anisotropy effects disappear with decreasing discreteness of the thresholds.

We see that there are differences between empty and prefilled supports, we think the pile constructed on an empty lattice will eventually evolve to a pyramid.

We find that the lifetime and cluster size distributions on sand piles are probably of the power-law type when only avalanches that do not reach the border are considered. The scaling behaviour of avalanches that reach the border is not yet clear.

We calculate scaling exponents and find that they are independent of the disorder in the thresholds. We suggest that the exponents for the lifetime and cluster size distributions are the same and probably that

r=b=2.

We make a finite size scaling of these two distributions and find that they scale with

(~=3/2.

For each of the sand pile models, critical height, slope and Laplacian model, the attempt was made to calculate exponents of cluster size and lifetime distributions [5]. For the critical height and Laplacian model exponents could


be determined by several authors with high accuracy (e.g. ref. [5] gives for the

critical-height and Laplacian models T = 1.201 and T = 1.288, respectively for

the cluster size and b = 1.316 and b = 1.508, respectively, for the lifetime

distribution).

Though exponents for the critical-slope model have also been searched for

[5], no scaling has yet been found. With our model, by taking into account only

avalanches that do not reach the border, we are able to calculate stable

exponents. These exponents are independent of our main parameter trmax and

we think that they are general for slope models.

We have found that our model probably does exhibit SOC behaviour if one

discards avalanches that reach the border of the system. It would be interesting

to test whether this removes the problems to finding scaling behaviour in other

slope models.

Big avalanches, which reach the border, seem to behave differently. It

should be investigated further if their distributions exhibit a power-law-type

dependency and if their maxima scale with the lattice size raised to some

exponent L”.

Anisotropy has a major influence on the resulting piles, though it does not

seem to change the exponents. It should be investigated further if the

independence of the exponents on the anisotropy effects remains valid for the

other models that have been proposed up to now. To overcome lattice

anisotropy one might introduce random lattices for the simulation of sand

piles.

The parameter trmax seems to play an important role. Although the dis-

tribution function of the thresholds remains the same, its discreteness di-

minishes with increasing trmax. The anisotropy in space, introduced by the

discrete square lattice seems to weaken when the discretization in the

thresholds is removed. This effect should be tested further. A simple way to do

this would be to repeat the simulations with thresholds and sand pile heights

that are represented by floating point values rather than by integers.

Acknowledgements

I am indebted - among others - to the following persons for fruitful discus-

sions and valuable help: S.S. Manna for many ideas about sand piles and

scaling, H.J. Herrmann for many explanations and discussions, as well as

reading the manuscript, D. Dhar for discussions and the idea to look at the

topplings, D. Stauffer for his suspicion about the exactness of my cones, T.

Brandes for help with the Alliant computer, J. Hemmingsson for help with the

photos, as well as all the HLRZ members for help with sand, computers and

physics.


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