PHYSICAL REVIEW A V O L U M E 4 5 , N U M B E R 4 I 5 F E B R U A R Y 1 9 9 2

Avalanches and epidemic models of fracturing in earthquakes

7J. Lomnitz-Adler

Institute de Fisica, Universidad Nacional Autbnoma de Mexico, Apartado Postal 20-364,01000 Mexico D.F., Mexico

L. KnopoffDepartment of Physics and Institute of Geophysics and Planetary Physics, University of California,

Los Angeles, California 90024

G. Martihez-MeklerInstitute de Fisica, Universidad Nacional Autbnoma de Mexico, Apartado Postal 20-364,

01000 Mexico D.F., Mexico(Received 21 March 1991; revised manuscript received 31 October 1991)

The evolutionary history of elastic materials subjected to the shear stress and the high confining pressures encountered in seismological applications is simulated by a set of avalanche models. Some modelsexhibit bimodal fracture-size distributions reminiscent of a first-order phase transition, while others havescaling behavior. We argue that this scaling behavior arises as a consequence of the high degree of self-generated roughness. In all models the scaling region broadens as surface roughness is increased Anepidemic growth model, closely related to the avalanche models, is introduced to understand the statistical properties obtained in the simulations. By means of these alternative formulations we interpret thescaling in the size distributions of the avalanche models as a finite-size effect; the system self-organizestuning itself to a size-dependent steady state which exhibits an apparent criticality. A value for the scaling exponent, determined from the epidemic model by way of correlated percolation theory, is in goodagreement with the numerical result. These problems have relevance to sandpile and earthquake modelsand to aspects of self-organized criticality.

PACS number(s): 05.20.-y, 05.70.Jk, 46.30.Nz, 91.30.Bi

I. INTRODUCTION

In this paper we are concerned with fracture propagation in a regime relevant to the study of earthquakes.Shallow seismic events occur at depths of 10-50 km,with corresponding pressures of 3-15 kbar. In contrastto typical laboratory experiments, these high pressureseliminate the possibility of generating tensile cracks andfavor shear fracture. Repeated fracture and healing on agiven surface is now possible, and is in fact observed.

In the following we develop several discrete models forfracture propagation on a two-dimensional fault withhealing. Based on rock friction experiments that indicatethat the velocity dependence of the friction force is weak[1], we work with a velocity-independent dynamical friction coefficient. This assumption is frequently found inthe seismological literature. The growth rules we propose for these fractures are given in terms of a nearest-neighbor algorithm valid in the range of parameters of interest. These rules reproduce the scaling laws of elasticcracks.

Our approach is set in the context of the general problem of the statistical study of irreversible phenomena.We work within two frameworks: (i) the "time-averageframework," in which a model for the system is allowedto evolve and statistics are obtained by averaging overlong time intervals, and (ii) the "ensemble framework,"where a series of fractures are generated on a set of systems that satisfy identical "macroscopic" constraints but

differ at the "microscopic" level. Models of type (i) are,for example, the avalanche models of Bak, Tang, andWiesenfeld [2], Zhang [3], Ito and Matsuzaki [4], Carlsonand Langer [5], and Nakanishi [6]. Work along the linesof type (ii) has been previously carried out, among others,by Herrmann and collaborators [7], and Lomnitz-Adlerand Lemus-Diaz [8]. In equilibrium statistical physicsboth approaches are in general interchangeable due to anunderlying ergodic hypothesis. In our case, despite theabsence of a formal justification, we work with bothdescriptions. The model used in framework (ii) has beenconstructed to incorporate the essential features of theavalanche models. Both treatments are complementaryand provide a fuller understanding of the dynamics offracture propagation.

As far as self-organized criticality Is concerned, whileit can be argued that in the thermodynamic limit none ofthese system have critical points, for any finite lattice oneof the models tunes itself to a state which appears critical.

This paper is organized as follows: In Sec. II wepresent the necessary elements of the theory of elasticcrack propagation, to justify that a two-dimensional,nearest-neighbor approximation is valid for modeling seismic fracture. We introduce a set of avalanche modelsin Sec. Ill, and interpret the generated statistics in termsof a nucleation model in Sec. IV. In Sec. V we constructan epidemic growth model that corresponds to theavalanche models under consideration. From the

45 2211 ©1992 The American Physical Society

2212 J. LOMNITZ-ADLER, L. KNOPOFF, AND G. MARTINEZ-MEKLER 45

cluster-size distribution of the epidemic model we obtainin Sec. VI, via correlated percolation theory, the exponent observed for the automaton's avalanche-size distribution. Section VII contains our conclusions and adiscussion.

II. FRACTURE-PROPAGATION MODEL

A fracture that is introduced into a prestressed solidinduces a redistribution of the stress field. The stress onthe fracture surface itself is reduced; beyond the fracturesurface and in its plane, stresses are increased,significantly so at short distances from the edge of thecrack and less so at longer range (see reviews [9]). Theamplitude of the redistributed stress fields depends on thesize of the crack and the stress drop on the fractured surface. Hence, a succession of cracks induces fluctuationsin the stress field.

We simulate a fault which supports repeated seismicevents by two-dimensional-lattice models with nearest-neighbor interactions. The conditions under which thenearest-neighbor approximation is appropriate can be derived as follows. A planar crack in the interior of astressed elastic solid has a stress on its walls equal to thesliding friction; the difference between the external stressand the friction is the stress drop r. Because we neglectinertial effects in our problem, we take the friction on thecrack wall to be zero. For a thin crack of characteristiclength L that terminates abruptly at a smoothedge, theexterior stress increment falls off as t{L/%) at shortdistance £ from the edge. At long range the stress increment falls off as the point-source Green's function, whichis rd for a rf-dimensional crack. If (L /£)1/2»1, the incremental energy density outside the crack is proportional to the square of the incremental stress, and falls off at arate proportional to L/|. In the case of a two-dimensional antiplane shear crack in an infinite, elastical-ly homogeneous medium, the exact solution for the stressis [9, 10]

o = r x / ( x 2 - L 2 ) w 2 , x > L ( 2 . D

where L is half of the width of a crack and x is measuredfrom the center; £=x -L. Thus the incremental energyin a finite interval of distance from the edge varies asL ln(|j/|2), where §, and £2 are the outer and inner distances of the interval from the edge.

Since a real elastic material with a finite yield stresscannot support the infinity at the edge implied by theshort-range dependence, a slip-weakening zone (sometimes called a plastic, transition, or damage zone) develops over the distance £c, where t(L/2£c)1/2 = o> withoY the yield strength of the material (Fig. 1). The transition zone is a tubular region that surrounds and followsthe edge; in the transition zone dense dislocations areformed that are displayed as microfracturing, twinning,development of slip lamellae, etc. Outside the slip-weakening zone we assume the material to be homogeneous and elastic.

To fix the range of validity of a nearest-neighbor modelof stress redistribution in a lattice model with lattice constant b, we compute the ratio of the energies per unitlength of the crack edge in the regions 0<£<_» and

-tL+£. L+2b

FIG. 1. Distribution of stresses in the plane of fracture (solidcurve). The shaded regions denote the slip weakening zone; thelattice spacing b is also indicated. Dotted-dashed curve illustrates the nearest-neighbor approximation.

b<£<2b. The energy density is h =o /2p, where /_ isthe shear modulus. We use the exact expression for a 2Dcrack given in (2.1). If we assume both a linear stress anda linear shear modulus profile in the slip-weakening zone(see Fig. 1) we obtain, in the limits (_»/!) «1,(£C/L)«1,

h]/h2 = [\ + \n(b/^c))An(2) . (2.2)

The logarithmic factors describe the energies in the twolattice intervals outside the slip-weakening zone while thefirst term describes the energy inside it. If the assumptions of the linearity of profiles in the slip-weakening zoneare changed, the first factor is changed slightly. Since thestress drop is related to the size of the slip-weakeningzone by

t/oy = (2$c/L A/2 (2.3)

the nearest-neighbor model will hold in the approximation £c«b, which implies r/oY«\. Since all dimensions of the redistribution are proportional to the cracksize, both for the size of the slip-weakening zone and thedecay of the stress outside it, the conditions for the validity of the nearest-neighbor model are fixed by the relations at the largest crack sizes. For a yield stress of 600bar, and a loading stress of 30 bar, r/oY- 0.05 and thenearest-neighbor approximation fails for fracture dimensions greater than L/b^ 80. We show below that thephysical system loses its mechanical stability long beforefractures of this size appear.

III. TIME-AVERAGE FRAMEWORK

We construct several avalanche models that simulatefracture propagation and in which inertial effects areneglected. As a starting point, consider a typicalavalanche model recently described in the literature[2,3,4,11]. Space is discretized into a two-dimensionallattice with coordination number z, and at every latticepoint UJ) a scalar variable h UJ) is defined which weshall refer to as the energy. Time is also discretized, andat every time step one of the lattice sites is chosen at random and a fixed quantity £ is added to it. If the h value

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45 AVALANCHES AND EPIDEMIC MODELS OF FRACTURING IN ... 2215

3.00

2.00 -

f«h»1.00 -

0.000.00 1.00

FIG. 6. Variation with e of the relative frequency of occurrence f((h)) of the mean energy (h >, in model 1. Solidcurve, £ = }; dashed curve, e = {; dotted curve, z=j£. Notethat (h )lr decreases with e.

take place until (h > lies just below (h )Ir = 0.4. The numerical values of parameters such as (h)tT,LtT depend onthe level of roughness of the fault surface which is controlled by the parameter e. Immediately after a largeevent the energy surface is set to hij=0, and during thecharging period, the distribution of energy for any givensite is given by the binomial distribution. It follows thatfor a mean energy < h > the width of the energy profile isequal to VE</.>, so that as e decreases the surface becomes smoother, (h >tr increases (see Fig. 6) and Ltr decreases (see Fig. 7), qualitative features remaining thesame.

As a final remark we point out that in these calculations the nearest-neighbor approximation is valid. Thecritical cluster length Ltr = 3 is well within the boundL/b ^80 obtained in Sec. II.

B. Model 2: Local, conservative, relaxed thresholddynamics

In this model an isolated site with energy greater thanthe threshold does not break and continues to retain itsenergy after the fracture has been completed. This sitecan only break in a future avalanche, which occurs if e isadded to it, or if it is engulfed by a future event initiatedelsewhere. This model is inconsistent with a postulate ofthreshold dynamics, wherein a site that has an energy inexcess of a critical threshold is always unstable. We expect to observe large energy variations, especially following a rupture whose size is comparable to that of the entire lattice; at this time most of the sites have h =0, but afew sites still can have large values of h. These isolated,highly charged sites can initiate a rupture at any time,and the system can generate avalanches of any size.

Model 2 is conservative, except for fractures which intersect the boundaries of the lattice. It is not a reasonable choice for the modeling of fractures; since the elasto-dynamic interactions which underlie energy transfers between sites are long range, it is difficult to justify a modelin which energy can either be transferred to nearestneighbors or not at all. Our purpose in this exploration isto investigate the role of energy conservation in suchmodels. Although physically unrealistic, the model hassome interest from the statistical-mechanics point ofview, and shows some generalizable features of collectivebehavior of extended nonlinear systems that have implications beyond the problems of fracture propagation.

The avalanche-size distribution for this model is shownin Fig. 8. Avalanches of all sizes are present with apparent scale invariance extending up to, but not including, the largest avalanches, where a maximum is onceagain observed. These simulation results are compatiblewith the nucleation theory arguments presented in connection with model 1 if the transition avalanche size Ltris of the same order as the lattice length N; in this case,

1

10 4

FIG. 7. Variation with e of the avalanche-size distribution ofnodel 1: open stars, e = i; crosses, e=i; solid stars, b=±.tote how Str increases with e.

p(S)

FIG. 8. Avalanche-size distributions of model 2. The solidline has a slope of — 1.45.

2216 J. LOMNITZ-ADLER, L. KNOPOFF, AND G. MARTINEZ-MEKLER 45

f«h»

FIG. 9. Relative frequency of occurrence f((h)) of themean energy (h ) for model 2.

the average value of < h > should scale as 1 /N, from Eq.(4.2).

The distribution of < h > becomes sharper as the latticesize is increased, with the maximum shifting to smallervalues of (h > (Fig. 9). We infer that the system is self-organizing to a steady state. The peak and averagevalues of the distributions scale with N with an exponentthat is significantly different from -1.0, namely about—0.5 (Fig. 10), this behavior is an indication that the assumptions behind the nucleation model appropriate forcase 1 cannot accommodate case 2 without somemodification. If we allow for the possibility of a geome

trically complex avalanche boundary, we rewrite Eq. (3.1)as

(A >,_!_,_,-<* >+<*>«/»). <4-41where S is the crack area and t its perimeter. If we assume that t scales with S in terms of an exponent vdefined by

r ~ S ' - v , W . 5 )

where v ___ 1 /2 is an effective dimension that measures theamount of structure in the perimeter, then (h > scales as

n.N)

FIG. 10. Scaling of (h )MP with lattice size (model 2). Blackdots are the maxima in the distribution of (h) (see Fig. 9),whereas white dots are global time averages of (h >. Straightlines have been drawn to guide the eye and have slopes of—0.51 and —0.47 respectively.

< / i ) ~ A r - 2 v (4.6;

In Fig. 10 we observe that v is equal to about j.With regard to temporal correlations, since large

amounts of energy can be accumulated at some sites,small fluctuations in energy can trigger large events atany time. Thus the periodic-relaxation oscillator modelis inadequate. In this case large events may be followedby other large events after a short time interval; thetime-interval distribution between catastrophic events isnot sharply peaked (Fig. 11).

The power-law size distribution over a wide range offracture sizes (Fig. 8) might lead one to believe that thesystem is self-organizing into a critical state. There aretwo objections to this inference: first, the maximum inthe avalanche distribution for large fractures is difficultto accommodate in a finite-size scaling treatment of theproblem, since the maxima scale with a smaller exponentthan the slope in the power-law distribution; second, themaximum in the energy distribution decreases with increasing lattice size, indicating that the "critical state" issize dependent. This latter observation can be understood if the system self-organizes into a steady state suchthat Llr^N and if avalanche perimeters are complexBased on percolation-theory results below, we shall arguethat the observed scaling can then be interpreted as afinite-size effect.

When the arguments presented above for the scaling olperimeter with fracture size are extended to Eq. (2.3), i.e..r/ay = (|-/L)v, we find from Eq. (4.6) that the size of

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of lattice size, and the time-interval distribution betweenlarge events. For this particular model we had to go tolarger lattices, with N up to 128, because the trends werenot easy to determine for the smaller lattices. With thefive lattice sizes under study we see that this model, although conservative, behaves more like model 1 than likemodel 2: it does not organize into a steady state, and thefracture-size distribution separates into a bimodal distribution which does not span all avalanche sizes. It appears that the question of self-organization into a steadystate depends on more than merely having conservativeor nonconservative dynamics.

V. ENSEMBLE FRAMEWORK: EPIDEMICS MODEL

We now develop an epidemic [13] growth model forfracture propagation, closely related to the avalanchedescription of Sec. IV, with a view to understanding theappearance of a power-law decay in the cluster-size distribution of model 2. This model is explicitly constructedto mimic the dominant features of the avalanche dynamics.

Again consider fracture propagation as a process restricted to a discretized plane. At every lattice site UJ)we have an energy hijy which we may view as an externalfield, and allow for a site-dependent nearest-neighborcoupling parameter Jtj. The energy values are assumedto be distributed according to a probability distributiontP(hij) which is the result of an ensemble of possibleconfigurations consistent with a set of external constraints. We propose a growth model following theprescription presented by Lomnitz-Adler and Lemus-Diaz [8] for fracture growth on a heterogeneous seismicfault. The basic features of their model which we shalladopt are the following.

(1) Growth is initiated at a seed site chosen at random.(2) For a given cluster, a growth site is selected on the

perimeter, and a number q, which depends on the state offracture of its neighbors, is assigned to the site.

(3) A probability p (q) that the site fracture is defined.(4) A random number R, 0<R<\, is generated; if

R <p, the site fractures and the cluster grows.

(5) Steps (2)-(4) are iterated until no more growth sites ^* *remain.

Step (3) depends on a coupling constant, which in Ref. T , .[8] has a fixed value. We introduce a coupling that de- ,pends on the size of the cluster in accordance with the j .dependence of the stress concentration on fracture size., . K,mderived in Sec. IV. The connection with the avalanche . . nmodels can be made as follows: consider a boundary site: . a ing'UJ) at some point in the evolution of an avalanche, , . es/ • : o l a t i o n , c ow h o s e e n e r g y i s . . . '

:t o —Oanch ^ h j j + J i j , ( 5 . 1 ) t f / 0 > 0 a n c

. . . . . r o t a t i o n w h iwhere hy is the energy of the site prior to the initiation of.0jatjon rigthe avalanche and is the external field in the ensemble ap- re Qu^g ♦proach. The term /» is the energy transferred by the g,y ma^™ rbroken neighboring sites (coupling constant in the ensem-,r0pertjes mble approach). We make contact at a mean-field level in pj~ ywith Eq. (4.4) through the following identification: orrelated er

<A W,„=<'< > + <* HS/t) = {h > + <_>/<_,-) , ^f^khlwhere ( ) denotes an ensemble average defined below, • "» 1S zero-(q) is the average number of broken neighbors, and the n tnis 'lne 'effective coupling parameter J{S,t) is proportional to theower *aw «area to perimeter ratio of the crack (cluster). on value of

The mean-field approximation assumes that the ener- 0Ints» the usgies at different sites are uncorrelated prior to the initia- n tne casetion of growth of the cluster, i.e., their values have a dis-1^ cluster in>tribution T{h) which is independent of position. TheIICated °y thlaverage energy i s now g iven by - r S lze * *c suc

(h) = fh'P(h)dh (5.3) J(SC)=J,

where the integral is taken from zero to infinity in order1" whlcri theto allow for the avalanche model 2 situation. Incorporat-1)rrelated Peiing Eq. (4.5), the effective coupling parameter J (5) is re 1S a ^n!ize exceeds .!

J ( S , t ) = J { S ) = J 0 S v , h i t e p r o b a b i l

J n ^ ( h )(5.4| For a given

Equation (5.4) describes the cluster-size dependence olthe coupling parameter consistent with our avalanch(models.

It remains to specify the definition of the probability dfracture py{q) at a cluster boundary site UJ). For thlavalanche models, we have a threshold dynamics iiwhich the site breaks if /.,-,- —(hfj +/«) > 1. We now writiJjj =quJ, where J is given by (5.4). In order to mimic ththreshold dynamics, we define the probabilityp(q) thati.site on the cluster boundary with q broken neighborin|sites breaks by

fin be expect

p(q i= f" dhTih,J \ - J q

fXdhnh)=\ (5.)

Our results do not depend strongly on the specific form iP(h).

Our growth algorithm is the Leath prescription [14],which a cluster grows in successive shells, J being tbFlG. 15. Schesame in each shell, and the sites within each shell are aMel (v=0). T.sumed to fracture simultaneously. This is the same algfetion universerithm used in the avalanche simulations. *Ved during the

45 AVALANCHES AND EPIDEMIC MODELS OF FRACTURING IN . .2219

VI. CLUSTER GROWTH PHASE DIAGRAMAND PERCOLATION

In this section we utilize some results from percolationtheory to elucidate features of the power-law decay observed in model 2. We show that if v > 0 and J0 > 0, as inthis problem, the system always has a finite probability ofpercolating, i.e., there is no attainable critical point Thisresult is established through an interplay between percolation, correlated percolation and this growth model.If/0-0 and v-0 we have percolation theory [15], whileif J0>0 and v=0 we have a problem of correlated percolation which falls in the same universality class as percolation [16]. In our problem, with J0>0 and v>0, weare outside the universality class of percolation; howe'verby making reference to the previous cases, some generalproperties may be deduced.

In Fig. 15 we show a schematic phase diagram forcorrelated epidemic models with v=0. There is a line ofcritical points Jc((h )) which separates two regions: onefor which the order parameter, the percolation probability (Q, is zero, and one for which it is finite. At any pointon this line the cluster-size distribution p(S) follows thepower law S r+1 where the exponent r takes its percolation value of 2.05. In the neighborhood of any of thesepoints, the usual percolation scaling relations hold.

In the case v>0 the effective 7(5) [Eq. (5.4)] of a growing cluster increases with size following the trajectory indicated by the arrows in Fig. 15. There is a critical cluster size Sc such that

J ( S c ) = J 0 S ; = J c ( ( h ) , v = 0 ) , ( 6 . Dat which the coupling constant appears to be that ofcorrelated percolation at criticality. For any value of J0,there is a finite probability of growing a cluster whosesize exceeds Sc; further, for any such cluster there is afinite probability of additional growth and percolation

For a given pair (</. ),J0) the cluster-size distributioncan be expected to resemble the schematic diagram of

7Fig. 16. For cluster sizes in the neighborhood of Sc thecoupling constant is approximately equal to that o/per-colation theory because at Sc, dJ/dS =J0S~l+v is small,and consequently J(S) changes slowly with cluster size.'Hence, we expect to find a range of 5 values for whichthe cluster-size distribution scales as in percolationtheory. Recall that there is a finite probability 0 ofgrowing a percolating cluster of infinite size. If Sc is ofthe order of the lattice size, Sc ^N2, this percolating cluster will appear to be finite and the cluster-size distribution will resemble the dashed curve in Fig. 16. Underthese circumstances, from Eqs. (5.2), (5.4), and (6.1), wehave that J0N2v~(h )N2^Jc((h ),v=0) from whichwe recover Eq. (4.6),

( h ) ~ N (6.2)We now interpret the scaling behavior of model 2 as an

apparent percolation-class critical behavior due to thefinite size of the system. The system self-organizes bytuning its average energy through Eq. (6.2) in order toachieve the critical parameters of the v=0 epidemicsmodel near the largest possible fractures.

A quantitative test of this interpretation is an estimateof the slope of the avalanche-size distribution (Fig. 8) inthe region near the largest avalanches in terms of percolation theory in the neighborhood of pc. Let p(S) bethe correlated percolation cluster-size distribution function. Near the line of critical points of Fig. 15, thecluster-size distribution associated to a set ((h ),J) mapsonto that of percolation theory for an effective probability p. In the avalanche models we observe fluctuations inthe average energy (h ); variation in the perimeter to surface ratio of the clusters also induces variations in J,which result in a distribution g(p) of probabilities in theassociated percolation problem. We write

p(S)=fig(p)ns(p)Sdp , (6.3)

where ns(p) is the number of S clusters per unit site. In-

Q > 0

mJ?. Schematic Phase diagram for correlated epidemicsodel (v-0). The curve is a line of critical points in the per

flation universality class. Arrowed line is the trajectory folded during the growth of a cluster in the v > 0 caseFIG. 16. Schematic diagram of the cluster size distribution

for a v>0 correlated epidemics model. The cluster at the farright is infinite and occurs with finite nrnhahiiitv ro

2220 J. LOMNITZ-ADLER, L. KNOPOFF, AND G. MARTINEZ-MEKLER 45 45

troducing the usual percolation-theory scaling expressionns(p)=f(z)/ST, with z=(p —pe )Sa, into (6.3), we have

p(S)=S-T-°+l / f(z)[g(pc) + (z/Sa)g'(pc) + ■••]<&^ A S - r - c r + l + BS - T - l a - (6.4)

where we have expanded g(p) around pc and A, B areconstants. Substituting the percolation values 7=2.05and a =0.4 in (6.4), the leading term for the cluster-growth-size distribution in our epidemic model shouldscale with an exponent — 1.45. For comparison, we havedrawn a straight line with slope - 1.45 near the distribution for the largest avalanches in Fig. 8.

VII. DISCUSSION

In this paper we have constructed several nearest-neighbor planar models of seismic shear crack propagation with the property that broken sites are incapable ofaccumulating energy for the duration of the fracture.The models differ only according to the rule that must beinvoked for the seemingly rare case in which the energyof a site exceeds the threshold for rupture, and yet has nonearest neighbors to which this energy can be transferredbecause they are all broken sites. In the first model theenergy was transferred directly out of the system; in thesecond, the site did not rupture until some later fractureevent, and in the third, the site ruptured and its energywas transferred to the entire crack perimeter. The firstand third models generate bimodal fracture distributionfunctions; model 2 generates fractures of all sizes whichsatisfy a power-law distribution except for fractureswhose size are almost those of the lattice itself.

Our results can be understood in terms of a classicalnucleation process, provided that in model 2 the perimeter is allowed to scale in a not-trivial way with clustersize. The scaling behavior observed in model 2 is a consequence of an unusual form of self-organization. The system tunes itself into a steady state for which the criticalcluster size, i.e., the largest avalanche which does not result in mechanical instability, is comparable to the latticedimensions. This tuning is manifested in the size dependence of the system's mean energy.

The power laws generated by the model can be understood, via an epidemic model, in terms of finite-sizeeffects and percolation theory. In most systems which exhibit critical behavior, finite size tends to soften thesingularities, in this case they produce an apparentsecond-order phase-transition behavior of the percolationuniversality class. The avalanche-size distribution has apower-law decay that is very close to that expected frompercolation theory (see Fig. 8), while the exponentv = 0.25 in the perimeter to area ratio t/S~S~v obtained from Fig. 9, is incompatible with the percolation

providing a mechanism of dynamic selection. Othermechanisms capable of such a selection appear in invasion percolation [17] and the two-state models of Carlson et al. [18]. In model 2, as in [18], finite-size effectsplay a central role in generating the observed "critical"behavior. In our case even the location of the critical region is size dependent.

While universal agreement as to what constitutes self-organized criticality does not exist, one of the signaturesof this phenomenon is considered to be the appearance ofpower laws and scaling relations. Although our modelsshare some features in common with previous automatonmodels, the principal differences are that all three of ourmodels preserve a memory of rupture for the duration ofa fracture, i.e., healing of the frictional bonds in the fracture takes place only after the entire fracture has beencompleted, whereas in Bak, Tang, and Wiesenfeld [2] andZhang [3] all sites have exactly the same propertieswhether they have ruptured or not. For this reason wedescribe our models as being temporally nonlocal. Model1 is not conservative. Model 2 is conservative and all energy transfers are between nearest neighbors. Model 3 isconservative but is not spatially local. Our results a-esummarized in Table I.

Hwa and Kardar [12] state that a necessary conditionfor a system to self-organize into a critical state is that iibe conservative. Model 3, while being conservative, doesnot settle into a steady state, indicating the relevance oflocality. It is tempting to think that a conservative localdynamics is a sufficient condition for SOC. Howeveimodel 2, though capable of generating power-law decays.does not sit at a critical point. This fact is brought outwhen the tail of the size distribution is considered: thereis a large probability of generating avalanches whose sizeis comparable to the lattice dimensions themselves andwhich do not allow for a simple finite-size scalinganalysis. We conjecture that such spikes in theavalanche-size distribution function can be used as a signature to distinguish between avalanche dynamics whichare analogous to systems with first-order phase transitions and systems at criticality. All of our models presenibehavior reminiscent of first-order phase transitions and.we believe, the same applies to the Burridge-Knopoff [19.nonconservative model with nearest-neighbor dynamicsstudied in detail by Carlson and Langer [5]. Though thequestion of what is a sufficient condition for SOC remainsopen, much effort is currently being applied in this direction [18].

Avalanche models such as these have been proposed todescribe earthquake dynamics [4,5,6,20], the principal at-

TABLE I. Dependence of self-organized criticality on general features of the models. Zhang's results are those of Ref. [-1For more details see text.

value v =0. This can be understood i f we accept that, Model Local Memory Conservative SOCalthough the system does not belong to the universalityclass of percolation, the largest finite cracks grow in amanner analogous to an epidemic model of the percola-*:— „i-.-- -,_.„.- fV,_. /-ritJ^oi r»nint Tt iq in fart the nonzero

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