A  										  one-dimensional model of a rice-pile  										  is numerically studied for different  										  driving mechanisms. We found that for  										  a sufficiently large system, there is  										  a sharp transition between the trivial  										  behaviour of a 1D BTW model and  										  self-organized critical (SOC)  										  behaviour. Depending on the driving  										  mechanism, the self-organized critical  										  rice-pile model belongs to two  										  different universality classes.A  						 				  one-dimensional model of a rice-pile  										  is numerically  studied for different  										  driving mechanisms. We found that for   										  a sufficiently large system, there is  										  a sharp  transition between the trivial  										  behaviour of a 1D BTW model  and  										  self-organized critical (SOC)  										  behaviour.  Depending on the driving  										  mechanism, the self-organized  critical  										  rice-pile model belongs to two  										   different universality classes

Bengrine, M.

Benyoussef, A.

El Kenz, A.

Loulidi, M.

Mhirech, F.

English

A  						 				  one-dimensional model of a rice-pile  										  is numerically  studied for different  										  driving mechanisms. We found that for   										  a sufficiently large system, there is  										  a sharp  transition between the trivial  										  behaviour of a 1D BTW model  and  										  self-organized critical (SOC)  										  behaviour.  Depending on the driving  										  mechanism, the self-organized  critical  										  rice-pile model belongs to two  										   different universality classes.A  										  one-dimensional model of a rice-pile  										  is numerically studied for different  										  driving mechanisms. We found that for  										  a sufficiently large system, there is  										  a sharp transition between the trivial  										  behaviour of a 1D BTW model and  										  self-organized critical (SOC)  										  behaviour. Depending on the driving  										  mechanism, the self-organized critical  										  rice-pile model belongs to two  										  different universality classes

Revues Scientifiques Marocaines

M. J. CONDENSED MATTER                                                  VOLUME 2, NUMBER 1                                              1 JULLY 1999* e-mail: benyous@fsr.ac.ma                                           2        15                   © 1999 The Moroccan Statistical PhysicalSociety Numerical study of rice-pile model.M. Bengrine, A. Benyoussef*, A. El Kenz, M. Loulidi and  F. Mhirech.Laboratoire de Magnétisme et de Physique des Hautes Energies, Département de Physique, Faculté des Sciences, Rabat, Moroc.  A one-dimensional model of a rice-pile is numerically studied for different driving mechanisms. We found thatfor a sufficiently large system, there is a sharp transition between the trivial behaviour of a 1D BTW model andself-organized critical (SOC) behaviour. Depending on the driving mechanism, the self-organized critical rice-pile model belongs to two different universality classes.PACS  number(s): 64.60.Lx. 05.40+jI. INTRODUCTION   In their pioneer work Bak, Tang and Wiensenfeld [1]introduced the concept of self-organized criticality (SOC) todescribe the behaviour of extended dissipative systems. Theparadigm of SOC is an idealized sandpile where grainsadded to a pile dissipate their potential energy throughavalanches with no characteristic scale [1-10]. Besides thenumerical simulations many different methods were alsoused to treat the SOC problems. Dynamical mean-fieldtheory [11] gives a unified description of some stochasticSOC systems including the BTW sandpile model and theforest fire model [12]. Langevin type approaches [13] havebeen used on a phenomenological basis. Furthermore, a realspace renormalization group method [14] provided goodestimates of the exponents. Early experimental studies ofreal sandpiles lead to clear disagreement with the numericalmodels: Bounded distributions of avalanche sizes wereobserved instead of the expected power-law behavior [15-19]. Using grains of rice, Frette et al. [20] showed that thedynamics exhibit self-organized critical behaviour in onecase (for grains with a large aspect ratio) but not in another(for less elongated grains). To take into account the changesin the local slopes observed in the rice-pile experiment,Christensen et al. proposed a rice-pile model, hereaftercalled Oslo model [21-23], where the critical slope for eachsite is a dynamical variable. Here we propose a model for apile of granular material where we introduce randomness inthe relaxation rule and use two types of drivingmechanisms: fixed-position driving where the grains areadded on the top of the pile, and random-position drivingwhere the grains are added on randomly chosen sites. Onthe one hand, this model permits to investigate the transitionbetween the 1D BTW model and the rice-pile model, and onthe other hand to study the effect of the driving mechanismson the size and transit-time avalanche exponents.This paper is a review of studies published in references 24and 25.II. MODEL   Our model is defined as follows: Consider the one-dimensional system formed by a set of L sites, labeled byintegers i=1 to L, with a wall at i=0 and an open boundaryat i=L+1. An integer variable h(i) is assigned to each site iand called the local height of the rice-pile at site i. Its localslope is defined as z(i)=h(i)-h(i+1) for i=1,2,...,L. Initially,the system is empty, i.e., h i i( ) = ∀0 .0,0 0,2 0,4 0,6 0,8 1,00,10,20,30,40,50,60,70,80,91,01,1  L=25  L=50  L=100  L=200<v>p1Fig. 1: The average transportation <v> as a function of the probability p1for several values of the system size and in the case of random drivingmechanism.The profile of the pile evolves through two mechanisms,perturbation and relaxation, with a separation in time scalesi.e. the rate of deposition is slow enough that any avalanche,triggered by a deposited grain, will have ended before a newgrain is added. At each time step, a grain is added to acolumn i:h i h i( ) ( ) ,→ + 1 (1)if at a certain site i, the local slope z(i) is greater than athreshold z ic ( )  ( z i z ic( ) ( )> ), where z ic ( )  takesrandomly a value 1 or 2, then this site topples with aprobability which depends on its slope z(i), namely: Ifz i( ) = 2 , the site i topples with a probability p1 and ifz(i)>2, it topples with probability p2, then one grain of rice16                                      M. BENGRINE et al.                         2will be transferred from a column i to its right neighboraccording to the following equationh i h ih i h i( ) ( )( ) ( ) ,→ −+ → + +11 1 1(2)102 1030,11slope = -0.23slope = -0.18p1=0.8<v>LFig. 2: The average velocity <v> as a function of the system size L, forp1 0 8= . . The open squares correspond to the fixed-position driving.The open circles correspond to the random-position driving.and all unstable sites topple in parallel. Notice that themodel connects the probabilities of toppling, p1 and p2, tothe actual slope z i z ic( ) ( )>  assigned to each site i.Consequently, the model dœsn’t allow for storage ofexternally imposed particles and so it can not display asubcritical behavior. If we set p10=  and p21= , themodel becomes the BTW model and if we putp p1 21= = , it is just the rice-pile model or Oslo rice-pilemodel if the driving mechanism is at the top of the pile.Furthermore, as in Ref. 21, at each grain is associated atransit time T defined by T T Tout in= − , where Tout  andTin  denote the output and the input time of the grain; thetime is measured in the unit of grains addition. Thus, wemeasure the avalanche size (i.e. the number of topplingsites), the transit-time T and their correspondingdistributions. In the next section, we will consider two waysof driving mechanisms: One way is to add grains to the topof the pile (site i=1) as it is usually made in the Oslo rice-pile model as well as in the experiment. Another way is toadd grains to randomly chosen positions (random drivingmechanism). This later way introduces an externalstochasticity in our model. Before going ahead, let usremind some differences between the 1D BTW model andthe Oslo rice-pile model. In the Oslo model, the randomnessis internal and inherent in the dynamics. For arbitrary initialconditions, the system reaches a stationary statecharacterized by power laws. However, the 1D BTW, wherethe randomness is external and the critical slope z ic ( )  isconstant, the system reaches a stationary trivial behavior.101 102 10310-710-610-510-410-310-210-1 (a)slope = -1.2L=100L=80L=40L=20P(S,L)S10-2 10-1 100 10110-310-210-1100(b)  L=20  L=40  L=80  L=100P(S,L)SτS/LDFig. 3: (a) Log-Log plot of the avalanche size distribution for severalvalues of the system size L with p1 0 8= .  and in the case of randomdriving position. (b) Data collapse of the curves displayed in (a) accordingto eq. 3 with the exponents τ = =1 20 1 25. , .D .III. NUMERICAL SIMULATIONS   We have performed extensive numerical simulations andinvestigated the effects of the parameter p1 and of drivingmechanisms on the behaviour of the system. In thefollowing we will take p21=  and p1 less than p2 becausehigher is the slope, higher is the jump probability. So byvarying p1 from 0 to 1, we can -in a continuous manner-change the model from the 1D BTW sandpile model to therice-pile model. Let us first study the transport properties ofthe model when the external grains are added to randomlychosen positions. In Fig. 1, we show the averagetransportation velocity of grains, defined as <v>=1/<T>, asa function of p1. For the case p10=  (the 1D BTWmodel), after a certain time a stationary state is reachedwhere every newly-added grain will slip out the pileinstantly, thus the transit time is T=0, and then the averagevelocity is infinite. By increasing p1 larger than a some2                                     NUMERICAL STUDY OF RICE-PILE MODEL             17value, pc1 , where pc1  depends on the system size, thevelocity <v> becomes constant and tend to 0 when Lbecomes sufficiently large. The numerical results make us toconsider that pc1 0→  as L → ∞ . For p1 0=+ ,< >=v 1 , independent of the system size. So there is asharp transition from < >= ∞v  for p10=  to < >=v 1  forp10= + . This transition can be understood by thefollowing argument. It is clear that when p1 is exactly 0, nonewly-added grain will stay in the pile as100 101 102 103 10410-710-610-510-410-310-2(a)slope = -2.35  L=20  L=40  L=80  L=100P(T,L)T10-2 10-1 100 101 10210-510-410-310-210-1100 (b)slope = -2.35   L=20   L=40   L=80   L=100LβP(T,L)T/LνFig. 4: Log-Log plot of the transit-time distribution for several values ofthe system size L with p1 0 8= .  and in the case of random drivingposition. (b) Data collapse of the curves displayed in (a) according to eq.4. The best fit to the numerical data gives the slope α = 2 35. .long the stationary state is reached. So T=0 for every grainand hence < >=T 0 . When p10= +  some grains can beburied in the surface layer of the pile. These grains will stayin the pile for a very long time. Once they slip out of thepile, these grains, although very few in number, will make asignificant contribution to < >T  since their transit times areextremely large. It is the existence of these grains thatmakes < >T  assume finite value for p10= + . Betweenp10= +  and p pc1 1= , there is a crossover behaviour of< >v , which is due to finite size effects. Since we expectpc1 0→  when L → ∞ , we can also expect that for infinitesystem the transition takes place at p10=  from < >= ∞vto < >=v 0 . Thus the sharp transition here is induced bytiny disorder. In Fig. 2, we plot the average velocity as afunction of the system size for p10 8= .  and for the twotypes of driving mechanisms. It is clear that for large system(L>100), the average velocity < >v  scales as L−γ . Whenthe grains are added on the top of the pile γ = 0 23.  [24,25],while γ = 0 18.  when the grains are added on a randomlychosen sites. Therefore, the average velocity decreases withthe system size, which is due to the increase in the activezone depth with system size, as explained by Christensen etal. [21].101 102 10310-510-410-310-210-1 (a)slope = -1.2  p1=0.5  p1=1P(S,L=50)S100 101 102 10310-610-510-410-310-2 (b)slope = -2.35  p1=0.5  p1=1P(T,L=50)TFig. 5: (a) Log-Log plot of the avalanche size distribution in the case ofrandom driving position. The best fit gives the slope τ = 1 20. . (b) Log-log plot of the transit time distribution in the case of random drivingposition. The best fit gives the slope α = 2 35. .18                                      M. BENGRINE et al.                         2We have also studied the avalanche size and the transit-timedistributions for different values of the probabilityp pc1 1>  and for a random driving mechanism. In Fig. 3awe plot our simulation data for p10 8= .  and for differentsystem sizes. The distribution is a power law with thepresence of a peak close to the cutoff size S LcD∝ . This isa finite-size effect which is due to the possibility to form asupercritical state which then relaxes through a very largeavalanche. The distribution follows the scaling form:P S L S G S LD( , ) ( / )= −τ , (3)the best collapse is obtained with the exponentsτ = ±1 20 0 05. .  and D = ±1 25 0 05. . , cf. Fig. 3b. Whenthe externally grains are added on the top of the pile,τ = ±1 53 0 05. .  and D = ±2 20 0 05. .  [21,24].100 101 102 103 10410-710-610-510-410-310-210-1100slope = -1.53(a)  p1=1  p1=0.8  p1=0.5  p1=0.3P(S,p 1)S100 101 102 103 10410-610-510-4slope = -2.4(b)  p1=1  p1=0.8  p1=0.5  p1=0.3P(T,p 1)TFig. 6: (a) Log-log plot of the avalanche size distribution in the case offixed-position driving. The best fit gives the slope τ = 1 53. . (b) Log-logplot of the transit time distribution in the case of fixed-position driving.The best fit gives the slope α = 2 40. . In both (a) and (b) the system sizeis L=400.Thus by adding an external stochasticity (random drivingmechanism) to the internal randomness (critical slope is adynamical variable), the system belongs to anotheruniversality calss characterized by τ = ±1 20 0 05. . . Byusing that the average number of toppling is < >=S L  inthe critical state, it follows from eq. 3 that: The relationτ =−2 1DD, (4)is in agreement with our numerical results. Furthermore, thedistribution functions of transit times P(T,L) for severalvalues of system sizes are shown in Fig. 4a. A data collapsefor different system size L is obtained when plottingL P T Lβ ( , )  against the rescaled variable T L/ υ  usingυ = ±1 20 0 15. .  and β = ±1 20 0 15. . ; see Fig. 4b. Thuswe can write:P T L L F T L( , ) ( / )= −β υ , (5)the scaling function F is of the form F(x)=constant for smallx and F x x( ) ∝ −α  for larger x, υ  is a critical exponentexpressing how the crossover transit time Tc  scales with thesystem size. The power-law exponent α  for large transittime is obtained as α = 2 35. . Fig. 5a shows the avalanche-size distribution for the two values of probability p1 greaterthan a critical value pc1. Fig. 5b gives the correspondingtransit time distribution. It is clear that the size and thetransit time exponents are insensitive to the values of theprobability p1 ( p1> pc1).101 10210-510-410-310-210-1L=80p1=0.8slope = -1.53slope = -1.20P(S,L)SFig. 7: Comparison of the avalanche size distributions for p1 0 8= . .Solid line correspond to the fixed-position driving. Dashed line correspondto the random-position driving.The same conclusion can be drawn in the case where thegrains are added on the top of the pile, see Fig. 6a and Fig.6b. Thus, the SOC state is insensitive to the variation in thejumping probability p10> . Finally, we could2                                     NUMERICAL STUDY OF RICE-PILE MODEL             19expect that a random external perturbation would lead to adecrease of the value of τ  because by random driving wegive less chance to big avalanches to occur. Fig. 7 gives thesize distributions for the two types of driving mechanisms.From this figure one can clearly see that the exponents fordifferents driving mechanisms are different from each other.IV CONCLUSION   In summary, we have investigated a one-dimensional ricepile model where the sites with higher slopes have morechance to topple (with a probability p21= ) while the siteswith lower slopes topple with a probability p p1 2≤ . It isfound that for a sufficiently large system, there is a sharptransition between the trivial behaviour and the SOCbehaviour at p10= . In the case where the external grainsare added to the top of the pile, the self-organized criticalmodel belongs to the known universality class that ischaracterized by an avalanche exponent τ = ±1 53 0 05. . ,whereas the model with random driving mechanism belongsto a new universality class characterized byτ = ±1 20 0 05. . .ACKOWLEDGMENTS This work was supported by the program PARS Physique035.[1] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59,381 (1987); Phys. Rev. A 38, 364 (1988).[2] L. Kadanoff, S. R. Nagel, L. Wu, and S-m Zhou, Phys.Rev. A 39, 6524 (1989).[3] J. M. Carlson and J. S. Langer, Phys. Rev. Lett. 62,2632 (1989).[4] H. J. S. Feder and J. Feder, Phys. Rev. Lett. 66, 2669(1991).[5] J. Toner, Phys. Rev. Lett. 66, 679 (1991).[6] J. Krug, J. E. S. Socolar and G. Grinstein, Phys. Rev. A46, R4479 (1992).[7] C. P. C. Prado and Z. Olami , Phys. Rev. A 45, 665(1992).[8] V. Frette, Phys. Rev. Lett. 70, 2762 (1993).[9] T. Hwa and M. Kardar, Phys. Rev. A 45, 7002 (1992).[10] S. S. Manna, Physica A 179, 249 (1991).[11] A. Vespignani, and S. Zapperi, Phys. Rev. E 57, 6345(1998).[12] P. Bak, K Chen and C. Tang, Phys. Rev. Lett. A 147,297 (1990).[13] A. Diaz-Guilera, Phys. Rev. A 45, 8551 (1992).[14] A. Vespignani, S. Zapperi and L. Pietronero, Phys.Rev. E 51, 1711 (1995); E. V. Ivashkevich, Phys. Rev. Lett.76, 3368 (1996).[15] H. M. Jaeger, C. H. Liu, and S. R. Nagel, Phys. Rev.Lett. 62, 40 (1989).[16] G. A. Held, D. H. Solina, D. T. Kaene, W. J. Haag, P.M. Horn and G. Grinstein, Phys. Rev. Lett. 65, 1120 (1990).[17] J. Rosendahl, M. Vekic and J. Kelly, Phys. Rev. E 47,1401 (1993).[18] J. Rosendahl, M. Vekic and J. E. Rutledge, Phys. Rev.Lett. 73, 537 (1994).[19] J. Feder, Fractals 3, 431 (1995).[20] V. Frette, K. Christensen, A. Malthe-Sorenssen, J.Feder, T. Jossang, and P. Meakin,Nature(London) 379, 49 (1996).[21] K. Christensen,A. Corral, V. Frette, J. Feder and T.Jossang, Phys. Rev. Lett. 77, 107 (1996).[22] L. A. N. Amaral, and K. B. Lauritsen, Phys. Rev. E 54,R4512 (1996), \PRE 56, 231 (1997); Physica A 231,608(1996).[23] M. Paczuski and S. Boettcher, Phys. Rev.Lett. 77, 111(1996).[24] M. Bengrine, A. Benyoussef, A. El Kenz, M. Loulidiand F. Mhirech, to be published in the European PhysicalJounal B (1999).[25] M. Bengrine, A. Benyoussef, F. Mhirech and S. D.Zhang, Preprint IC/98/221 and submitted to Phys. Rev. E.

Numerical study of rice-pile model

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Numerical study of rice-pile model

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Revues Scientifiques Marocaines

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