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arXiv:cond-mat/9707210v1 19 Jul 1997

Damage spreading in the ’sandpile’

model of SOC

Ajanta Bhowal∗

Saha Institute of Nuclear Physics

1/AF Bidhannagar, Calcutta-700064, India

Abstract: We have studied the damage spreading (defined in the text) in the ’sand-

pile’ model of self organised criticality. We have studied the variations of the critical

time (defined in the text) and the total number of sites damaged at critical time as

a function of system size. Both shows the power law variation.

Keywords: Sandpile model of SOC, Damage spreading

PACS Numbers: 05.50 +q

—————————-

∗E-mail:ajanta@tnp.saha.ernet.in

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I. Introduction

The damage spreading [1-3] studies the dynamic behaviour of cooperative sys-

tems. The main idea of this problem is to study how a small perturbation, called a

damage, in a cooperative system changes with the evolution of time. This is studied

by observing the time evolution of the two copies of the system with slightly different

initial configuration under the same dynamics and measuring the damage by count-

ing the number of elements which are different in the two copies of the system. The

damage spreading has been studied exhaustively, in the Kauffman model [1] and the

spin systems [2,3].

In this paper, we have studied, by computer simulation, how damage spreads,

in the ’sandpile’ model, over the whole lattice during the course of time evolution.

’sandpile’ model [4] is a lattice automata model which describes the appearance of

long range spatio-temporal correlations observed in extended, dissipative dynamical

systems in nature. The essential feature of this model is the occurence of fractal

structure in space and ’1/f’ noise in time, which is so called self-organised criticality

(SOC) [4]. Substantial developments have been made on the study of ’sandpile’

model. But all these studies have been made in the steady (SOC) state, reached by

the system.

II. The model and simulation

The lattice automata ’sandpile’ model [4] of SOC evolves to a stationary state in

a self-organised (having no tunable parameter) way. This state has no scale of length

and time, hence is called critical. Altogether the state is called self-organised critical

state. The description of the lattice automata model is as follows: At each site of this

lattice, a variable (automaton) z(i,j) is associated which can take positive integer

values. Starting from the initial condition (at every site z(i,j) = 0), the value of

z(i,j) is increased (so called addition of one ’sand’ particle) at a randomly chosen

site (i,j) of the lattice in steps of unity as,

z(i,j) = z(i,j) + 1.

When the value of z at any site reaches a maximum zm, its value decreases by four

units (i.e., it topples) and each of the four nearest neighbours gets one unit of z

(maintaining local conservation) as follows:

z(i,j) = z(i,j) − 4

z(i ± 1,j ± 1) = z(i ± 1,j ± 1) + 1(1)

for z(i,j) ≥ zm. At the boundary sites z = 0 (dissipative; open boundary condition).

In this simulation, a square lattice of size L × L has been considered. The value

of zm = 4 here. It has been observed that, as the time goes on the average value

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(¯ z) of z(i,j), over the space, increases and ultimately reaches a steady value (¯ zc)

characterising the SOC state.

We study here, by computer simulation, how a small perturbation spreads in time

in the ’sandplie’ model. We have considered a square lattice and allowed it to evolve

under the dynamics until the average value (¯ z) of z(i,j) reaches a steady value (¯ zc).

We also have considered a 2nd lattice, which is a replica of the 1st lattice. After

reaching the steady (SOC) state (characterised by the steady value of ¯ z) we have

perturbed suddenly one of the system (say the 1st lattice) by adding unity to the

automaton value at the central site of the lattice, i.e, z1(l/2,l/2) = z1(l/2,l/2) + 1 .

Then we allowed the two lattices to evolve in time by the specified dynamics in the

same way (i.e, by using the same sequence of random numbers). It will be observed

that both lattices (perturbed and unperturbed) give the same macroscopic behavior

(the same ¯ zcand same scale invariant (power law) distribution of the avalanches size).

But the microscopic details (the z(i,j) at any site i,j at any time) of the two lattices

are different. The differences in microscopic details of the two lattice are described

here in terms of the ”damaged sites”, i.e., the sites of the perturbed lattice which

are different from the unperturbed one. The damaged lattice is characterised by a

variable (say d(i,j)), which is zero if the two lattices have the same z (for any site i,j)

value and 1 otherwise. More precisely, d(i,j) = 0 if z1(i,j) = z2(i,j) and d(i,j) = 1

otherwise. The non-zero sites of the damaged lattice (i.e., d(i,j) = 1) indicates the

damage in this model.

III. Results

It has been observed that, with the evolution of time, the damage (cloud formed

by the sites having d(i,j) = 1) spreads and touches any one of the boundary line of the

lattice at time τ (starting from the initial time when the central site of lattice 1 was

perturbed by adding unity). Here we study the spreading of damage by measuring

the following quantities:

(1) Minimum time (τ) taken by the damage to touch any one of the boundaries (upper

or lower) of the lattice.

(2) The number (Mτ) of damaged sites at τ-th instant.

(3) The total number (MTot) of sites damaged during the time τ, starting from the

initial time when the perturbation was added.

Fig.1 shows the variation of critical time (τ) with the sytem size L in the log scale

indicating the power law variation, following, τ ∼ Lawith a = 1.97. Fig.2 shows the

variations of Mτ, the number of sites damaged at critical time (τ) and Mtot, the total

number of sites damaged up to critical time with the syatem size L in the log scale,

indicating the power law variations, Mτ ∼ Lbwith b = 1.47. and Mtot∼ Lcwith

c = 1.66. Thus we see that τ, Mτ and Mtot, all these quantities follow a power law

variation with the system size. All these data have been obtained by sampling over

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100 different seeds of random number generator for L = 30,40,···150 and over 10

samples for L = 200,300.

IV. Summary

In the case of damage spreading in Ising model the spreading of damage is con-

trolled by tuning the temperature. But due to the absence of any tunable parameter

in the ’sandpile’ model it has been observed that in this case there is number spread-

ing transition contrast to the other cases of damage spreading (like Ising system etc).

There is a power law variation of critical time (τ) and Mτ, Mtotwith length as in the

other cases (though the exponents are different).

References

[1] A. Kauffman, J. Theor. Biol. 22, 437 (1969); See also B. Derrida and Y. Pomeau,

Europhys. Lett. 1, 45 (1986); B. Derrida and G. Weisbuch, J. Phys. (Paris) 47,

1297 (1986); B. Derrida and D. Stauffer, Europhys. Lett. 2, 739 (1986).

[2] N. Jan and L. de Arcangelis, in Annual Reviews of Computational Physics, Vol

1., Ed. D. Stauffer, World-Scientific, Singapore, (1994) 1.

[3] P. Grassberger, J. Stat. Phys. 79, 13 (1995).

[4] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381; Phys. Rev.

A. 38 (1988) 364.

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1000

10000

100000

1e+06

10100

L

1000

τ

3

3

333333

3

3

3

Fig. 1. Variation of critical time (τ) with system size L.

100

1000

10000

100000

10100

L

1000

3

3

333333

3

3

3

+

+

++++++

+

+

+

Fig. 2. Variations of Mτ(3) and Mtot(+) with system size L.

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