Paths to Self-Organized Criticality

Home , Percolation

Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 27

1 2 3 4

Ronald Dickman , Miguel A. Munoz ~ , Alessandro Vespignani , and Stefano Zapp eri

Departamento de Fsica, ICEx, UniversidadeFederal de Minas Gerais,

Caixa Postal 702,

30161-970 Belo Horizonte, MG, Brazil

Institute Carlos I for Theoretical and Computational Physics

and Departamento de Electromagnetismo y Fsica de la Materia

18071 Granada, Spain.

The Abdus Salam International Centre for Theoretical Physics (ICTP) P.O. Box 586, 34100 Trieste, Italy

PMMH - Ecole de Physique et Chimie Industriel les, 10, rue Vauquelin, 75231 Paris CEDEX 05, France

Received 15 Octob er 1999

We present a p edagogical intro duction to self-organized criticality (SOC), unraveling its connections

with nonequilib riu m phase transitions. There are several paths from a conventional critical p oint

to SOC. They b egin with an absorbing-state phase transition (directed p ercolation is a familiar

example), and imp ose sup ervision or driving on the system; two commonly used metho ds are

extremal dynamics, and driving at a rate approaching zero. We illustrate this in sandpiles, where

SOC is a consequence of slow driving in a system exhibiting an absorbing-state phase transition with

a conserved density. Other paths to SOC, in driven interfaces, the Bak-Snepp en mo del, and self-

organized directed p ercolation, are also examined. We review the status of exp erimental realizations

of SOC in light of these observations.

I Intro duction how the system is maintained (or maintains itself)atthe

critical p oint. (Alternatively one can try to showthat

The lab el \self-organized" is applied indiscriminately in

there is generic scale invariance, that is, that criticality

the current literature to ordering or pattern formation

app ears over a region of parameter space with nonzero

amongst manyinteracting units. Implicit is the notion

measure [1,2].) \SOC" has b een used to describ e sp on-

that the phenomenon of interest, b e it scale invariance,

taneous scale invariance in general; this would seem to

co op eration, or supra-molecular organization (e.g., mi-

embrace random walks, as well as fractal growth [3], dif-

celles), app ears sp ontaneously. That, of course, is just

fusive annihilation (A + A ! 0 and related pro cesses),

how the magnetization app ears in the Ising mo del; but

and nonequilibrium surface dynamics [4]. Here were-

we don't sp eak of \self-organized magnetization." Af-

strict the term to systems that are attracted to a criti-

ter nearly a century of study,we've come to exp ect the

cal (scale-invariant) stationary state; the chief examples

spins to organize; the zero- eld magnetization b elow

are sandpile mo dels [5]. Another class of realizations,

T is no longer a surprise. More generally,spontaneous

exempli ed by the Bak-Snepp en mo del [6], involve ex-

organization of interacting units is precisely what we

tremal dynamics (the unit with the extreme value of

seek, to explain the emergence of order in nature. We

a certain variable is the next to change). We will see

can exp ect many more surprises in the quest to discover

that in many examples of SOC, there is a choice b e-

what kinds of order a given set of interactions lead to.

tween global sup ervision (an o dd state of a airs for a

All will b e self-organized, there b eing no outside agent

\self-organized" system), or a strictly lo cal dynamics in

on hand to imp ose order!

which the rate of one or more pro cesses must b e tuned

to zero.

\Self-organized criticality" (SOC) carries greater

sp eci city, b ecause criticality usually do es not happ en The sandpile mo dels intro duced byBak,Tang and

sp ontaneously: various parameters havetobetuned Wiesenfeld (BTW) [5], Manna [7], and others have at-

to reach the critical p oint. Scale-invariance in natural tracted great interest, as the rst and clearest examples

systems, far from equilibrium, isn't explained merely by of self-organized criticality. In these mo dels, grains of

showing that the interacting units can exhibit scale in- \sand" are injected into the system and are lost at the

variance at a p oint in parameter space; one has to show b oundaries, allowing the system to reach a stationary

28 Ronald Dickman et al.

state with a balance b etween input and output. The of the eld are not discussed.

input and loss pro cesses are linked in a sp ecial wayto

the lo cal dynamics, which consists of activated, conser-

II A simple example

vative, redistribution of sand. In the limit of in nitely

slow input, the system displays a highly uctuating,

We b egin with a simple mo del of activatedrandom

scale-invariantavalanche-like pattern of activity.One

walkers (ARW). Eachsitej of a lattice (with p erio dic

may asso ciate rates h and , resp ectively, with the ad-

b oundary conditions) harb ors a number z =0; 1; 2::: of

dition and removal pro cesses. Wehave to adjust these j

random walkers. (For purp oses of illustration the ring

parameters to realize SOC: it app ears in the limit of h

1;:::;L will do.) Initially, N walkers are distributed

and ! 0 with h= ! 0[1,8,9, 10]. (The addition

randomly amongst the sites. Eachwalker moves inde-

and removal pro cesses o ccur in nitely slowly compared

p endently, without bias, to one of the neighb oring sites

to the lo cal redistribution dynamics, which pro ceeds

(i.e., from site j to j +1 or j 1, with site L +1 1

at a rate of unity. Loss is typically restricted to the

and 0 L), the only restriction b eing that an isolated

b oundaries, so that ! 0 is implicit in the in nite-size

walker (at a site with z = 1) is paralyzed until such

limit.)

time as another walker or walkers joins it. The active

Questions ab out SOC fall into two categories. First,

sites (with z 2) followa Markovian (sequential) dy-

Why do es self-organized criticality exist? What are the

namics: each active site loses, at a rate 1, a pair of walk-

conditions for a mo del to have SOC? Second, the many

ers, which jump indep endently to one of the neighbors

questions ab out critical b ehavior (exp onents, scaling

of site j .(Thus in one dimension there is a probability

functions, p ower-sp ectra, etc.) of sp eci c mo dels, and

of 1/2 that each neighb or gains one walker, while with

whether these can b e group ed into universality classes,

probability 1/4 b oth walkers hop to the left, or to the

as for conventional phase transitions b oth in and out

right.)

of equilibrium. Answers to the second typ e of question

The mo del wehavejustde nedischaracterized by

come from exact solutions [11], simulations [12], renor-

the numb er of lattice sites, L ,andthenumber of par-

malization group analyses [13], and (one mayhope)

ticles, N .Ithastwo kinds of con gurations: active, in

eld theoretical analysis. Despite these insights, asser-

which at least one site has two or more walkers, and

tions in the literature ab out sp ontaneous or parameter-

absorbing, in whichnositeismultiply o ccupied, ren-

free criticalityhave tended to obscure the nature of the

dering all the walkers immobile [17]. For N>L only

phase transition in sandpiles, fostering the impression

active con gurations are p ossible, and since N is con-

that SOC is a phenomenon sui generis, inhabiting a dif-

served, activitycontinues forever. For N L there

ferentworld than that of standard critical phenomena.

are b oth active and absorbing con gurations, the latter

In this pap er weshow that SOC is a phase transition to

representing a shrinking fraction of con guration space

an absorbing state, a kind of criticality that has b een

as the density N=L ! 1. Given that we start

well studied, principally in the guise of directed p ercola-

in an active con guration (a virtual certaintyforan

tion [14]. Connections b etween SOC and an underlying

initially random distribution with >0and L large),

conventional phase transition have also b een p ointed

will the system remain active inde nitely, or will it fall

out by Narayan and Middleton [15], and by Sornette,

into an absorbing con guration? For small it should

Johansen and Dornic [16].

b e easy for the latter to o ccur, but it seems reasonable

Starting with a simple example (Sec. I I), wewillsee

that for suciently large densities (still < 1), the like-

that the absorbing-state transition provides the mech-

liho o d of reaching an absorbing con guration b ecomes

anism for SOC (Sec. I I I). That is, we explain the exis-

so small that the walkers remain active inde nitely.In

tence of SOC in sandpiles on the basis of a conventional

other words, we exp ect sustained activity for densities

critical p oint. In Sec. IV we discuss the transforma-

greater than some critical value , with < 1.

tion of a conventional phase transition to SOC in the

c c

contexts of driven interfaces, a sto chastic pro cess that A simple mean- eld theory provides a preliminary

repro duces the stationary prop erties of directed p erco- check ofthisintuition. Consider activated random

lation, and the Bak-Snepp en mo del. We nd that criti- walkers in one dimension. For a site to gain parti-

cality requires tuning, or equivalently, an in nite time- cles, it must have an active(z 2) nearest neighb or.

scale separation. With this essential p oint in mind, we Since active sites release a pair of walkers at a rate of

present a brief review of the relevance of SOC mo dels unity,a given site receives a single walker from an ac-

to exp eriments in Sec. V. Sec. VI presents a summary tive neighb or at rate 1/2, and a pair of walkers at rate

of our ideas. We note that this pap er is not intended 1/4. Thus the rate of transitions that take z to z +1 is

j j

as a complete review of SOC; manyinteresting asp ects [P (z ;z 2) + P (z ;z 2)]=2; transitions from

j j +1 j j 1

Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 29

z to z + 2 o ccur at half this rate. In the mean- eld is the fraction of with o ccupation z and =

j j z a

z 2

approximation we ignore correlations b etween di erent active sites. Using this factorization, we can write a set

sites, and factorize the joint probabilityinto a pro duct: of equations for the site densities:

P (z; z 2) = , where is the fraction of sites

z a z

1 d

= ( )+ ( )+ ; (z =0; 1; 2:::); (1)

a z 1 z a z 2 z z +2 z 2 z

dt 2

where = 0 for n<0 and is one otherwise. The | 5000.) The inset shows that the active-site density

nal two terms represent active sites losing a pair of follows a p ower law, ( ) ,with =0:43(1); a

a c

walkers. It is easy to see that the total probability, and nite-size scaling analysis con rms this result [18]. In

the density = summary,activated random walkers exhibit a contin- z are conserved by the mean- eld

uous phase transition from an absorbing to an active equations. This in nite set of coupled equations can b e

state as the particle density is increased ab ove , with integrated numerically if we imp ose a cuto at large z .

strictly less than 1. (It has yet to b e shown rigor- (This is justi ed by the nding that decays exp o-

c z

ously that the active-site densityintheARWmodelis nentially for large z .) The mean- eld theory predicts

singular at , in the in nite-size limit; our numerical a continuous phase transition at =1=2. For <

c c c

results are fully consistent with the existence of sucha the only stationary state is the absorbing one, =0,

singularity.) while for the active-site densitygrows / .A

c c

two-site approximation (in whichwe write equations for

the fraction of nearest-neighb or pairs with given

z;z

I I.1 Absorbing-State Phase Transitions

heights, but factorize joint probabilities involving three

Absorbing-state phase transitions are well known

or more sites), yields =0:75.

in condensed matter physics, and p opulation and epi-

demic mo deling [19]. The simplest example, whichmay

b e thought of as the \Ising mo del" of this class of sys-

tems, is the contact process [20]. Again wehave a lattice

of L sites, eachofwhichmay b e o ccupied (active) or

vacant. Occupied sites turn vacant at a rate of unity;

vacant sites b ecome o ccupied at a rate of (=2d)n

where n is the numb er of o ccupied nearest neighbors

(the factor 2d represents the numb er of nearest neigh-

b ors). There is a unique absorbing con guration: all

sites vacant. For suciently small, the system will

eventually fall into the absorbing state, while for large

an active stationary state can b e maintained. Letting

represent the density of o ccupied sites, the mean- eld

theory analogous to the one formulated ab ove for acti-

vated random walkers reads:

Figure 1. Stationary density of active sites versus density

of walkers in one-dimensional ARW. The inset is a loga-

=( 1) : (2)

rithmic plot of the same data, where = . The slop e

of the straight line is 0.43.

This predicts a continuous phase transition (from 0

in the stationary state) at =1. to =1 The existence of a continuous phase transition is

Rigorous analyses [21, 22] con rm the existence of a con rmed in Monte Carlo simulations, which yield '

continuous phase transition at a critical value ,in 0:9486 in one dimension, and ' 0:7169 in two dimen-

c c

any dimension d 1. Simulations and series analyses sions. Fig. 1 shows how the stationary density of active

yield =3:29785(2) in one dimension. This mo del, sites dep ends on ;wesee growing continuously

c a a

and its continuous-up date counterpart, directedperco- from zero at .(Thepoints represent estimated densi-

lation (DP; see Sec. IV), have b een studied extensively. ties for L !1, based on simulation data for L =100

30 Ronald Dickman et al.

The critical exp onents are known to go o d precision for from its initial value). In a eld-theoretic description

d = 1, 2, and 3; the upp er critical dimension d =4. of ARWwe will therefore need (at least) two elds: the

There is, in addition, a well established eld theory for lo cal density (x; t) of active sites, and the lo cal parti-

this class of mo dels [23,24]: cle density (x; t); the latter is frozen in regions where

= 0. The evolution of is coupled to b ecause the

2 2

particle densitycontrols existence and level of activity

= r a b + (x; t) : (3)

in the ARWmodel.

Here (x; t) is a lo cal particle density,and (x; t)isa

Given that absorbing-state phase transitions fall

Gaussian noise with auto correlation

generically in the universality class of directed p erco-

lation, it is natural to ask whether this is the case for

activated random walkersaswell. The answer, appar-

0 0 0 0

h (x; t) (x ;t )i =(x; t) (x x ) (t t ) : (4)

ently, is \No." The critical exp onent for ARW is,

That h i is linear in the lo cal density follows from the as we noted ab ove, 0.43, while for one-dimensional DP

fact that the numb ers of events (creation and annihila- =0:2765 [29]; the other critical exp onents di er as

tion) in a given region are Poissonian random variables, well [18]. While the reason for this di erence is not un-

so that the variance equals the exp ected value. (The dersto o d, it app ears, at least, to b e consistent with the

noise must vanish when = 0 for the latter to b e an ab- existence of a conserved eld in ARW.

sorbing state!) This eld theory serves as the basis for a

To summarize, our simple mo del of activated ran-

strong claim of universality[23,25]: Continuous phase

dom walkers has an absorbing-state phase transition, as

transitions to an absorbing state fal l generical ly in the

do es the contact pro cess, directed p ercolation and the

universality class of directedpercolation. (It is under-

PCP. All p ossess the same basic phase diagram: active

sto o d that the mo dels for whichwe exp ect DP-likebe-

and inactive phases separated bya continuous phase

havior have short-range interactions, and are not sub-

transition at a critical value of a \temp erature-like"

ject to sp ecial symmetries or conservation laws b eyond

parameter ( in ARW, in the CP). But ARWpos-

the simple translation-invariance of the contact pro cess.

sesses an in nite numb er of absorbing con gurations,

Mo dels sub ject to a conservation law are known to have

and the evolution of its order parameter (the active-

a di erent critical b ehavior [26].)

site density) is coupled to a conserved density . The

The activated random walkers mo del resembles the

latter presumably underlies its b elonging to a di erent

contact pro cess in having an absorbing-state phase

universality class than DP.

transition. We should note, however, two imp ortant

di erences b etween the mo dels. First, ARW presents

III Activated Random Walkers

an in nite number (2 , to b e more precise) of ab-

sorbing con gurations, while the CP has but one. In

and Sandpiles

fact, particle mo dels in whichthenumber of absorb-

ing con gurations grows exp onentially with the system The activated random walkers mo del p ossesses a con-

size have also b een studied intensively. The simplest ventional critical p oint: wehave to tune the parame-

example is the pair contact process,inwhichbothel- ter to its critical value. What has it got to do with

ementary pro cesses (creation and annihilation) require self-organized criticality? The answer is that ARWhas

the presence of a nearest-neighb or pair of particles [27]. essentially the same lo cal dynamics as a mo del known

In one dimension, a pair at sites i and i + 1 can either to exhibit SOC, namely, the Manna sandpile [7]. In

annihilate, at rate p, or pro duce a new particle at ei- Manna's sandpile, the redistribution dynamics runs in

ther i 1ori +2,atrate1 p (provided the selected parallel: at each time step, all of the sites with z 2

site is vacant). This mo del shows a continuous phase simultaneously lib erate twowalkers, which jump ran-

transition from an active state for p< p to an absorb- domly to nearest neighb or sites. This mayresultina

ing state ab ove p . The static critical b ehavior again new set of active sites, which relax at the next time

b elongs to the DP universality class, but the critical ex- step, and so on. (Time advances by one unit at each

p onents asso ciated with spreading of activity from an lattice up date, equivalent to the unit relaxation rate

initially lo calized region are nonuniversal, varying con- of an activesiteinARW.) We de ned ARW with se-

tinuously (in one dimension) with the particle density quential dynamics as this makes it a Markov pro cess

in the surrounding region [28]. with lo cal transitions in con guration space, likea ki-

A second imp ortant di erence b etween ARW and netic Ising mo del. There is of course nothing wrong

the CP and PCP is that the former is sub ject to a in de ning ARW with parallel dynamics; it to o has an

conservation law (the number of walkers cannot change absorbing-state phase transition.

Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 31

There is a much more fundamental di erence b e- That the Manna sandpile, in two or three dimen-

tween the Manna sandpile and the ARW mo del: the for- sions, with parallel dynamics, has a scale-invariant

mer allows addition and loss of walkers. Recall that we avalanche distribution is well known [7]. Here wenote

de ned the ARW with p erio dic b oundary conditions; that the same holds for the one-dimensional version,

walkers can never leave the system. In the sandpile with random sequential dynamics. Fig. 2 shows the

walkers may exit from one of the b oundary sites. (On probability distribution for the avalanche size (the total

the square lattice, for example, a walker at an edge site numb er of topplings) when wemodifyARW to include

has a probability of 1/4 to leave the system at the next loss of walkers at the b oundaries, and addition at a

step.) If we allowwalkers to leave, then eventually the randomly chosen site, when the system falls into an ab-

system will reach an absorbing con guration. When sorbing con guration. The distribution follows a p ower

this happ ens, we add a new walker at a randomly cho- law, P (s) s ,over a wide range of avalanche sizes

sen site. This inno cent-sounding prescription | add and durations; there is, as exp ected, an exp onential

awalker when and only when all other activity ceases cuto s L for events larger than a characteristic

| carries the in nite time scale separation essential to value asso ciated with the nite size of the lattice. (Our

the app earance of SOC in sandpiles. The sequence of b est estimates are =1:10(2) and D = 2.21(1).) The

active con gurations b etween two successive additions upp er inset of Fig. 2 shows that the stationary density

is known as an avalanche;avalanches mayinvolveany approaches , the lo cation of the absorbing-state phase

numb er of sites, from zero (no topplings) up to the en- transition, as L !1. It is also interesting to note that,

tire system. in contrast with certain deterministic one-dimensional

sandpile mo dels [30,31], the present example app ears

Manna showed that his mo del reaches a stationary

to exhibit nite-size scaling, as shown in the lower inset

state in whichavalanches o ccur on all scales, up to the

of Fig. 2.

size of the system, and followa power-law distribution,

P (s) s , for s s . (Here s is the numb er of trans-

fer or toppling events in a given avalanche, and s L

is a cuto asso ciated with the nite system size.) In

other words, the Manna sandpile, like the mo dels de-

vised by Bak, Tang and Wiesenfeld and others, exhibits

scale invariance in the stationary state.

WeknowthatARW, which has the same lo cal dy-

namics as the Manna sandpile, shows scale invariance

when (and only when) the density = . Sointhe

stationary state of the Manna mo del, the densityis

somehow attracted to its critical value. Howdoesit

happ en? The mechanism of SOC dep ends up on a par-

ticular relation b etween the input and loss pro cesses,

and the conventional absorbing-state phase transition

in the mo del with a xed numb er of particles. Walkers

cannot enter the system while it is active, though they

may of course leaveuponreaching the b oundary.Inthe

presence of activity,then, > and d =dt < 0. In the

Figure 2. Stationary avalanche-size distribution in the one-

absence of activity there is addition, but no loss of walk-

dimensional Manna sandpile with sequential dynamics, for

L = 500, 1000, 2000, and 5000 (left to right) . Lower inset:

ers, so < implies d =dt > 0. Evidently, the only

nite-size scaling plot of the data in the main graph, ln P

p ossible stationary value for the density in the sandpile

2:21 2:43

versus ln s , with s L s and P L P . Upp er

is ! Of course, it is p ossible to havealow level of

inset: stationary density in the inner 10% of the system,

plotted versus 1=L. The diamond on the axis is the critical

activity lo cally, in a region with < , but under such

densityofARW.

conditions activity cannot propagate or b e sustained.

(One can similarly construct absorbing con gurations

with > , but these are unstable to addition of walk-

I I I.1 A Recip e for SOC

ers, or the propagation of activity from outside.) In the

in nite-size limit, the stationary activity densityiszero The connection b etween activated random walkers

for <, and p ositivefor >, ensuring that is and the Manna sandpile suggests the following recip e

c c

pinned at , when loss is contingent up on activity, and for SOC. Start with a system having a continuous

addition up on its absence. absorbing-state phase transition at a critical value of

32 Ronald Dickman et al.

as a suitable candidate for SOC b ecause changing the a density . This density should represent the global

parameter on the basis of the current state (activeor value of a lo cal dynamical variable conserved by the dy-

frozen) amounts to tuning. Cannot the same b e said namics. Add to the conservative lo cal dynamics (1) a

for adding walkers in the Manna sandpile? Somehow, pro cess for increasing the density in in nitesimal steps

adynamicsofwalkers entering and leaving the system ( ! + d ) when the lo cal dynamics reaches an ab-

seems more \natural" than wholesale ddling with a sorbing con guration, and (2) a pro cess for decreasing

parameter. But who is going to watch for activity,to the density at an in nitesimal rate while the system is

know when to add a particle? A system managed by active. Run the system until it reaches the stationary

a sup ervisor can hardly b e called \self-organized!" If state; it is now ready to display scale invariance.

wewanttoavoid building a sup ervisor or baby-sitter Let's see how these elements op erate in the Manna

into the mo del, we had b etter say that addition go es sandpile. We started with activated random walkers,

on continuously, at rate h, and that SOCisrealizedin which do es indeed displayacontinuous absorbing-state

the limit h ! 0 [9,10]. (The original sandpile de - transition as a function the density of walkers; this

nitions have a baby-sitter. Simulations, in particular, density, moreover, is conserved. To this we added the

have a live-in baby-sitter to decide the next move. Ad- input of one walker ( ! +1=L in d dimensions),

dition at rate h ! 0 is a sup ervisor-free interpretation when the system is inactive. We then broke the transla-

of the dynamics [33].) In the recip e for SOC without tional symmetry of the ARW mo del to de ne b oundary

baby-sitters, we replace addition (1) ab ove with (1'): sites, and allowed walkers at the b oundary to leavethe

allow addition at rate h, independent of the state of the system. The latter implies a loss rate d =dt /L ,

system, and take h ! 0 . (There is no problem with where is the activity density at the b oundary sites.

the removal step: dissipation is asso ciated with activ- The conditions of our recip e are satis ed when L !1,

ity,which is lo cal.) Wepay a price when we rethe whichweneededanyway,tohave a prop er phase tran-

baby-sitter: there is now a parameter h in the mo del, sition in the original mo del.

which has to betunedtozero.Evidently, sandpiles don't Nowwe can examine the ingredients one byone.

exhibit generic scale invariance, but rather, scale invari- First, the phase transition in the original mo del should

ance at a point in parameter space. This is consistent be to an absorbing state, b ecause our input and loss

with Grinstein's de nition of SOC, which requires an steps are conditioned on the absence or presence of ac-

in nite separation of time scales from the outset [1]. tivity. Second, the temp erature-like parameter control-

ling the transition should b e a conserved density.Sothe

contact pro cess and PCP aren't suitable starting p oints

III.3 Variations

for SOC, b ecause the control parameter isn't a dy-

namical variable. (To self-organize criticalityintheCP,

In certain resp ects, our recip e allows greater free-

we'd havetochange itself, dep ending on the absence

dom than was explored in the initial sandpile mo dels.

of presence of activity. But this is tuning the param-

There is no sp ecial reason, for example, why loss of

eter by hand!) Third, weneedtochange the density

walkers has to o ccur at the b oundaries. Wesimply

in in nitesimal steps, else we will always b e jump-

require that activity b e attended by dissipation at an

ing b etween values ab ove or b elow without actually

in nitesimal rate. SOC has, indeed, b een demonstrated

hitting the critical density. The same thing will hap-

in translation-invariant mo dels with a uniform dissipa-

p en, incidentally,ifwe start out with a mo del that has

tion rate when ! 0 [9,34]. In the original sandpile

a discontinuous transition (with attendanthysteresis)

mo dels, addition takes place with equal probabilityat

between an active and an absorbing state; this yields

any site, but restricting addition to a subset of the lat-

self-organized stick-slip b ehavior.

tice will still yield SOC.

The basic ingredients of our recip e are an absorbing-

state phase transition, and a metho d for forcing the

Our recip e allows a tremendous amount of freedom

mo del to its critical p oint, by adding (removing) par-

for the starting mo del; the only restriction is that it

ticles when the system is frozen (active). Following

p ossess an absorbing-state critical p oint as a function

the recip e, the transformation of a conventional critical

of a conserved density. The dynamical variables can b e

p oint to a self-organized one do es not seem surprising

continuous or discrete. The hopping pro cess do es not

[32].

have to b e symmetric, as in ARW. (In fact, directed hop-

ping yields an exactly-soluble sandpile [35].) The mo del

I I I.2 Firing the Baby-Sitter

need not b e de ned on a regular lattice; any structure

The reader mayhave noted a subtle inconsistency in with a well de ned in nite-size limit should do. The

the ab ove discussion. We rejected the contact pro cess dynamics, moreover, can b e deterministic. Consider

Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 33

avariant of the ARW mo del (on a d-dimensional cu- is the xed-energy Manna sandpile, and the variantde-

bic lattice) in which a site is active if it has z 2d scrib ed in the preceding subsection is the BTW FES.

walkers. At each lattice up date (p erformed here with Now the essential feature of the xed-energy sandpile

parallel dynamics), every active site `topples, transfer- is an absorbing-state phase transition. SOC app ears

ring a single walker to each of the 2d nearest-neighbor when we rig up the addition and removal pro cesses to

sites. In this case the only randomness resides in the drive the lo cal FES dynamics to .To understand the

initial con guration. But the mo del again exhibits a details of SOC, then, we ought to try to understand

continuous absorbing-state phase transition as wetune the conventional phase transition in the corresp onding

the numberofwalkers p er site, . Starting with this xed-energy sandpile. This is our program for address-

deterministic mo del, our recip e yields the celebrated ing the second class of questions (ab out critical exp o-

Bak-Tang-Wiesenfeld sandpile. nents and universality classes) mentioned in the Intro-

As a further variation, we can even relax the condi- duction. Since xed-energy sandpiles have a simple dy-

tion that the order parameter is coupled to a conserved namics (Markovian or deterministic) without loss or ad-

eld [36]. The price is the intro duction of an additional dition, and are translation-invariant (when de ned on

driving rate. This situation is exempli ed by the forest- a regular lattice), they should b e easier to study than

re mo del [37,38]. The mo del is de ned on a lattice their SOC counterparts. The relation to absorbing-

in which each site can b e in one of three states: empty, state phase transitions leads to a prop er identi cation

or o ccupied by a tree, either live or burning. Burning of the order parameter [9], and suggests a strategy for

trees turn into empty sites, and set re to the trees at constructing a eld theory of sandpiles [41]. Spreading

nearest-neighb or sites, at a rate of unity. Itiseasyto exp onents, conventionally measured in absorbing-state

recognize that burning trees are the active sites: any phase transitions, are related through scaling laws to

con guration without them is absorbing. In an in nite avalanche exp onents, usually measured in slowly driven

system, there will b e a critical tree density that sep- systems [42, 43].

arates a phase in which res spread inde nitely from

an absorbing phase with no burning trees. In a nite

IV Other Paths to SOC

system we can study this critical p ointby xing the

density of trees at its critical value [39].

IV.1 Driven Interfaces

So far wehave no pro cess for growing new trees.

The forest- re propagates like an epidemic with immu-

In this section we illustrate the central idea of the

nity: a site can only b e active once, and there is no

preceding section | the transformation of a conven-

prop er steady state [40]. As in sandpiles, to obtain a

tional phase transition to a self-organized one | in a

SOC state wemust intro duce an external driving eld f

di erent, though related, context. We b egin with a sin-

that intro duces a small probability for each tree to catch

gle p oint mass undergoing driven, dissipative motion in

re sp ontaneously. This driving eld allows the system

one dimension. Its p osition H (t)follows the equation

to jump b etween absorbing con gurations through the

of motion

spreading of res. The latter, however, are completely

dissipative, i.e., the numb er of trees is not conserved.

dH d H

= F F (H ); (5) + M

Thus, if wewant to reach a stationary state wemust

dt dt

intro duce a second external driving eld p that causes

where M is the mass, H represents viscous dissipa-

new trees to app ear. (Empty sites b ecome o ccupied

tion, F is the applied force, and F (H ) is a p osition-

by a living tree at rate p.) In this case criticalityis

dep endent pinning force. In many cases of interest

reached by the double slow driving condition f; p ! 0

(i.e., domain walls or ux-lines) the motion is over-

and f=p ! 0. In practice, this slow driving condition is

damp ed and wemay safely set M = 0. The pinning

achieved by the usual sup ervisor, that stops re ignition

force has mean zero (hF (h)i = 0) and its auto corre-

and tree growth during activeintervals.

lation hF (h)F (h + y )i(jy j)decays rapidly with

p p

I I I.4 Fixed-Energy Sandpiles

jy j; the statistical prop erties of F are indep endentof

H . Assuming, as is reasonable, that F is b ounded

If someone hands us a sandpile displaying SOC, we

(F F ), we exp ect the motion to continue if the

can identify the initial mo del in our recip e; it has the

p M

driving force F exceeds F . Otherwise the particle

same lo cal dynamics as the SOC sandpile. Thinking

gets stuck somewhere.

of the conserved as an energy density,wecallthe

Now consider an elastic interface (or a ux line) starting mo del a xed-energy sandpile (FES). Thus the

sub ject to an external force, viscous damping, and a activated random walkers mo del intro duced in Sec. I I

34 Ronald Dickman et al.

pinning force asso ciated with irregularities in the sur- v = hdH =dti.

rounding medium. If we discretize our interface, using

Toreach the absorbing-state phase transition in the

H (t) to represent the p osition, along the direction of

driven interface mo del we need to adjust the applied

the driving force, of the i-th segment , the equation of

force F to its critical value F .Canwe mo dify this sys-

motion is

tem so that it will b e attracted to the critical state?

Note that F is not a dynamical variable, anymore

than is ,inthecontact pro cess. Our sandpile recip e

= H + H 2H (t)+F F (H ); (6)

i+1 i1 i p;i i

dt do esn't seem to apply here. The crucial observation is

that wemaychange the nature of the driving, replacing

where the F (H ) are a set of indep endent pinning

p;i i

the constant force F with a constraintof xed velocity,

forces with statistical prop erties as ab ove. This driven

dH =dt = v . A nite v corresp onds to a state in the

interface mo del has a depinning transition at a criti-

active phase: the mean driving force hF i >F for

i v c

cal value, F , of the driving force [44]. (Eq. (6) de-

v>0. When we allow v to tend to zero from ab ove,

scrib es a linear driven interface, so-called b ecause it

we approach the depinning transition. This limit can

lacks the nonlinear term / (rh) , familiar from the

b e attained through an extremal dynamics in whichwe

KPZ equation [4,45].)For F

advance, at a given step, only the element sub ject to

tually arrested (dH =dt = 0 for all i), while for F>F

i c

the smallest pinning force [46, 47]. (Notice that in ex-

movementcontinues inde nitely. Close to F there are

tremal dynamics we are directly adjusting the order

avalanche-like bursts of movement on all scales, in-

parameter[16].)

tersp ersed with intervals of near-standstill. The cor-

relation length and relaxation time diverge at F ,as Toavoid the global sup ervision implicit in extremal

in the other examples of absorbing-state phase tran- dynamics wemayattacheach element of the interface

sitions we've discussed ab ove. Wemay take the or- to a spring, and move the other end of eachspringat

der parameter for this transition as the mean velo city, sp eed V .Now the equations of motion read

= H + H 2H (t)+ k (Vt H ) F (H ); (7)

i+1 i1 i i p;i

where k is the spring constant. For high applied velo c- driven interface, the order parameter is coupled to a

ities, the interface will in general move smo othly,with conserved density. The sandpile, moreover, do es not

velo city H = V , while for low V stick-slip motion is involve a quenched random eld as do es the driven in-

likely. In the overdamp ed regime, the amplitudes of terface. Despite these apparent di erences, close con-

the slips are controlled by V and k , and the statistics nections have b een suggested b etween the two kinds of

of the p otential. In the limit V ! 0, the interface mo- mo del [15, 49, 50, 51]. We review this corresp ondence

tion exhibits scale invariance; V plays a role analogous in the next subsection, following Ref. [51].

to h in the sandpile. (The limits V ! 0andk ! 0

IV.2 Sandpiles and Driven Interfaces

have a particular signi cance, since the blo ckcanex-

plore the pinning-force landscap e quasistatically.) The

Consider the BTW xed-energy sandpile in two di-

ne tuning of F to F in the constant-force driving has

mensions; let H (t)bethenumb er of times site i has

b een replaced by ne tuning V to zero. This parame-

toppled since time zero. To write a dynamics for H ,we

ter tuning corresp onds, once again, to an in nite time-

observe that the o ccupation z (t) of site i di ers from

scale separation. Finally,we note that restoring inertia

its initial value, z (0), due to the in ow and the out-

(M>0) results in a discontinuous depinning transition

ow of particles at this site. The out owisgiven by

with hysteresis, resulting in stick-slip motion of the sort

4H (t), since each toppling exp els four particles. The

asso ciated with friction [48].

in ow can b e expressed as H (t): site i gains a

particle each time one of its nearest neighb ors topples.

Once again, wehave transformed an absorbing-state

Summing the ab ovecontributions we obtain:

phase transition (F = F )into SOC by driving the sys-

z (t) = z (0) + H (t) 4H (t)

tem at a rate approaching zero (V ! 0). But there

i i j i

jN N i

app ear to b e fundamental di erences b etween sandpiles

and driven interfaces. In the sandpile, but not in the = z (0) + r H (t); (8)

i i D

Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 35

where r stands for the discretized Laplacian. Since if b oth its neighb ors in row i 1 are dry; otherwise,

sites with z (t) 4 topple at unit rate, the dynamics of it is wet with probability p, and dry with probability

H is given by 1 p. This sto chastic cellular automaton is called site

directedpercolation.Like the contact pro cess, it p os-

sesses an absorbing state: all sites dry in row k implies

= [z (0) + r H (t) 3]

i i

all dry in all subsequentrows. The dynamics of site

DP can b e expressed in a compact form if we de ne the

= [r H (t)+ F F ]; (9)

i p;i

site variable x to b e zero (one) if site j in row i is wet

where dH =dt is shorthand for the rate at whichthe

(dry). The variables in the next row are given by

integer-valued variable H (t) jumps to H (t)+1,and

i i

(x)=1 forx>0 and is zero otherwise. In the sec-

i+1 i i i

ond line, F 3andF z (0) . (Recall that

P;i i x = [maxf ; minfx ;x gg p] ; (10)

j j 1 j +1

= hz (t)i for all t.) Thinking of H (t) as a discretized

i i

interface height, Eq. (9) represents an overdamp ed,

where the are indep endent random variables, uni-

driven interface in the presence of columnar noise, F ,

form on [0,1]. If b oth neighb ors in the preceding roware

p;i

i+1 i+1

whichtakes indep endentvalues at each site, but do es

in state 1, x must also equal 1; otherwise x =0

j j

not dep end up on H ,asitdoesintheinterface mo del

with probability p. Thinking of the rows as time slices,

discussed in the preceding subsection. We see from this

we see that site DP is a parallel-up date version of the

equation that tuning to its critical value is anal-

contact pro cess: increasing p renders the survival and

ogous to tuning the driving force to F .Ifwe replace

propagation of the wet state more probable, and is anal-

the discrete height H in Eq. (9) with a continuous

ogous to increasing in the CP. Just as the CP has a

eld, H (x; t) (and similarly for F ), and replace the

phase transition at , site DP has a transition from

-function by its argument, we obtain the Edwards-

the absorbing to the active phase at p ' 0:7054.

Wilkinson surface-growth mo del with columnar disor-

We've already dismissed the contact pro cess (and by

der, which has b een studied extensively [52]. The simi-

extension DP) as starting mo dels for realizing SOC via

laritybetween the present height representation and the

the recip e of Sec. I I I. Remarkably,however, it is p ossi-

dynamics of a driven interface suggests that the criti-

ble to de ne a parameter-free sto chastic pro cess whose

cal p oint of the BTW xed-energy sandpile b elongs to

stationary state repro duces the prop erties of critical DP

the universality class of linear interface depinning with

[53, 54, 55]. This pro cess, self-organized directed p er-

columnar noise, if the rather violent nonlinearityofthe

colation (SODP), is obtained by replacing the discrete

-function is irrelevant. (The latter remains an op en

variables in Eq. (10) byrealvariables which store the

question. A height representation for the Manna sand-

. In place of Eq. (10) we value of one of the previous

pile is also p ossible, but is complicated by the sto chastic

have simply

nature of the dynamics.)

Applying the recip e of Sec. I I I to the driven in-

i+1

i i i

gg ; (11) ;x ; minfx x =maxf

j +1 j 1 j

terface, wewould imp ose op en b oundaries, which drag

b ehind the interior as they havefewer neighb ors pulling

Notice that parameter p has disapp eared, along with

on them; eventually the interface gets stuck. When this

the function. Starting from a distribution with

happ ens, we ratchet up the \force" at a randomly cho-

x < 1 for at least one site (but otherwise arbitrary),

sen site (in e ect, F ! F 1 at the chosen site).

this pro cess eventually reaches a stationary state, char-

p;j p;j

The dynamics is then attracted to the critical p oint.

acterized by the probabilitydensity (x). One nds

Once again, wemay trade sup ervision (checking if the

that (x)iszerofor x< p (the critical value of site

interface is stuck) for a constant drive(F ! F + ht)in

DP), jumps to a nonzero value (in nity, in the ther-

the limit h ! 0.

mo dynamic limit), at p , and decreases smo othly with

x for x>p. The pro cess has discovered the critical

IV.3 Self-Organized Directed Percolation

value of site directed p ercolation!

and the Bak-Snepp en Mo del

Hansen and Roux explained howthisworks [53]: for

any p 2 [0; 1] the probabilitythat x

Take the square lattice and rotate it by45 , so that

or b oth of the neighb ors in the previous time slice have

each site has two nearest neighb ors in the rowabove,

i1 i1

values less that p (i.e., if the smaller of x and x is

and two b elow. The sites exist in one of two states,

j 1 j +1

i1 i1

\wet" and \dry." The states of the sites in the zeroth b oth exceed p. This and x

j +1 j 1

(top) row can b e assigned at will; this de nes the ini- is exactly howthe\wet" state propagates in site DP,

tial condition. A site in row i 1 is obliged to b e dry with parameter p,ifwe equate the events `site j in row

36 Ronald Dickman et al.

i is wet' and `x

state, with its neighb or to pro duce an o spring that inherits

the tness of the less- t parent. This o spring sur-

vives if her tness exceeds that of an interlop er, whose

Pr[x

tness is random. (It is, to put it crudely, asifanes-

tablished p opulation were constantly challenged bya

equals the probability P (p) that a randomly chosen site

ux of outsiders.) Seen in this light, SODP b ears some

is wet, in the stationary state of site DP with param-

resemblance to the evolutionary dynamics represented,

eter p. This explains why (x)= 0forx< p,and

again in very abstract form, in the Bak-Snepp en mo del

why (p ) is in nite in the in nite-size limit (dP =dp is

[6]. Here, the global ly minimum tness variable, along

in nite at p ). The spatio-temp oral distribution of DP

with its nearest neighb ors, is replaced by a [0,1] random

is also repro duced; for example, the joint probability

i i

numb er at each time step. (If the x are asso ciated with

Pr [x p ;x p ] decays as a p ower law for large

c c

j k

di erent sp ecies, then the app earance of a new sp ecies

separations jj k j. The pro cess e ectively studies all

at site i a ects the tness of the \neighboring" species

values of p at once, greatly improving eciency in sim-

in the community in an unpredictable way.) This is a

ulations. Sto chastic pro cesses corresp onding to other

kind of extremal dynamics, a scheme we've already en-

mo dels (DP on other lattices, b ond instead of site DP,

countered in the driven interface mo del; another famil-

epidemic pro cesses) have also b een devised [54,56]. It

iar example is invasion p ercolation [46]. Interestingly,

seems unlikely, on the other hand, that such a real-

the Bak-Snepp en mo del shows the same qualitativebe-

valued sto chastic pro cess exists for activated random

havior as SODP: a singular stationary distribution of

walkers or other xed-energy sandpiles. (Of course,

tness values x . The mo del exhibits avalanches in

such a pro cess would b e of great help in studying sand-

which replacement of a single sp ecies provokes a large

piles!)

numb er of extinctions.

SODP do esn't t into the same scheme as sand-

In the interface under extremal dynamics, the height

piles or driven interfaces. It is a real-valued sto chastic

H (t) cannot decrease. In the Bak-Snepp en mo del mo-

pro cess that generates, by construction, the probability

mentary setbacks are allowed (x can decrease in a

distribution of DP for al l parameter values, including

p . The pro cess itself do es not have a phase transi- given step), but individuals of low tness will even-

tion; all sites are active (except those inside a sequence tually b e culled. This is likeaninterface mo del with

of 1's | a con guration that will never arise sp onta- quenched noise such that, on advancing to a new p osi-

neously), since there is a nite probabilityfor x to tion, an elementmayencounter a force that throws it

change. SODP is self-organized in the sense that its backward, for a net negative displacement. The Bak-

stationary probability density has a critical singularity, Snepp en mo del is equivalent to a driven interface in

which the least-stable site and its neighb ors are up-

without the need to adjust parameters. If wecho ose

dated at the same moment; we can, as b efore, trade

to regard SODP as an instance of SOC, wemust rec-

extremal dynamics for a limit of in nitely slow driving.

ognize that the path in this case is very di erent from

that in sandpiles or driven interfaces; the system is not Another way of obtaining the extremal dynamics of

b eing forced to its critical p ointby external sup ervi- the Bak-Snepp en mo del as the limit of a sto chastic pro-

sion or driving. Rather, SODP is directed p ercolation cess with purely lo cal dynamics is as follows [57]. Take

implemented in a di erent (parameter-free) way.Fur- a one-dimensional lattice (with p erio dic b oundaries, for

thermore, the dynamics emb o died in Eq. (11) seems a de niteness), and assign random numbers x , indep en-

much less realistic description of a physical system than dent and uniform on [0,1], to each site j =1; :::; L. The

is driven-interface motion, or even the rather arti cial con guration evolves via a series of \ ips," which reset

dynamics of a sandpile mo del. In the rather unlikely the variables at three consecutive sites. That is, when

event that SODP were realized in a natural system, site j ips, we replace x , x ,andx with three

j 1 j j +1

it would not immediately yield a scale-invariant \sig- indep endent random numb ers again drawn uniformly

nal" suchasavalanches or fractal patterns. The latter from [0,1]. Let the rate of ipping at site j be e ,

would require a second pro cess (or an observer) capa- where is a characteristic time, irrelevant to station-

ary prop erties. The Bak-Snepp en mo del is the !1

ble of making ne distinctions among values of x in the

limit of this pro cess.

neighb orho o d of p . So the kind of SOC represented

by SODP do es not app ear a likely explanation of scale We can get some insightinto the stationary b ehav-

invariance in nature. ior via a simple analysis. Let p(x)dx b e the probability

A (fanciful) interpretation of Eq. (11) is that x that x 2 [x; x + dx]. The probability density satis es

j j

Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 37

Z Z

1 1

dp(x)

x y y

= e p(x) 2 e p(x; y )dy +3 e p(y )dy (13)

0 0

Grassb erger and Zhang observed that the exis- where p(x; y ) is the joint density for a pair of nearest-

tence of SODP \casts doubt on the signi cance of self- neighb or sites. If weinvoke a mean- eld factorization,

organized as opp osed to ordinary criticality." A similar p(x; y )=p(x)p(y ), then

doubt might b e prompted by our recip e for turning a

conventional critical p oint self-organized. Of course,

dp(x)

even if it is p ossible to explain all instances of SOC in

= p(x) e +2I ( ) +3I ( ); (14)

terms of an underlying conventional critical p oint, the

details of the critical b ehavior remain to b e understo o d where

[63]. Numerical results indicate that sandpiles, driven

interfaces, and the Bak-Snepp en mo del de ne a series

e p(y )dy : (15) I ( )

of new universalityclasses.Furthermore, no one has

b een able to derive the critical exp onents of avalanches

The stationary solution is

in SOC sandpiles, even in the ab elian case, where quite

a lot is known ab out the stationary prop erties [64].

2 =3

3 1 e

: (16) p (x)=

2 =3 x =3

1 e + e (e 1)

V SOC and the Real World

The solution is uniform on [0,1] for =0,aswe'd ex-

Since SOC has b een claimed to b e the way \nature

p ect, but in the !1limit wehave p =(3=2)(x

works" [65], wewould exp ect to nd a multitude of ex-

1=3)(1 x). The probability densitydevelops a step-

p erimental examples where this concept is useful. Orig-

function singularity, as in the Bak-Snepp en mo del.

inally,SOCwas considered an explanation of p ower

Not surprisingly, the mean- eld approximation yields

laws, that it provided a means whereby a system could

a rather p o or prediction for the lo cation of the singu-

self-tune its parameters. So once wesawapower lawwe

larity, which actually falls at 0.6670(1) [58]. (A two-site

couldclaimthatitwas self-generated and \explained"

approximation places the singularityatx =1=2.) The

by SOC. The previous sections should have convinced

main p oint is that to realize singular b ehavior from a

the reader that there are no self-tuning critical p oints,

lo cal dynamics, wehave to tune a parameter asso ci-

although sometimes the ne tuning is hidden, as in

ated with the rates. Alternative mean- eld treatments

of the Bak-Snepp en mo del may b e found in Refs. [59] sandpile mo dels. Therefore, an \explanation" of ex-

and [60] p erimentally observed p ower laws requires the identi -

cation of the tuning parameters controlling the scaling,

We can construct a mo del with the same lo cal dy-

as in any other ordinary critical p oint.

namics as that of Bak and Snepp en by replacing x ,

j 1

x , and x at rate 1, if and only if x

j j +1 j

with x >r mayonlychange if they have a nearest examples of avalanche b ehavior, leaving aside fractals

neighb or b elow the cuto .) In other words, only sites and 1=f noise whose connection with SOC is rather

with x

probability r . There is an absorbing phase for small r , tion of self-organized criticality|withoutavalanches,

separated from an active phase by a critical p ointat as far as is known | has b een identi ed in liquid He at

some r [60, 61, 62]. To get the Bak-Snepp en mo del the point[66].) Following the intro duction of SOC,

we forget ab out r , and declare the unique activesite there were many exp erimental studies of avalanches,

in the system to b e the one with the smallest value of which sometimes yielded p ower-law distributions over

r . In the in nite-size limit, the probabilityto nda a few decades, leading to endless discussions ab out the

site with r

see once again that in extremal dynamics wetunethe cal p oints don't exist, then these controversies haveno

order parameter itself to zero: at each instant there is basis: wehave only to understand how far the system

exactly one active site, so =1=L. is from the critical p oint, and why. This task has only a

38 Ronald Dickman et al.

b een accomplished in a few cases; several examples re- H = ct NM = ct ky(t), where y (t)istheaverage

int

quire further study, b oth exp erimental and theoretical. p osition of the domain wall and k / N .We recognize

here the recip e for SOC given in section I I I.1: in the

So on after the sandpile mo del was intro duced, sev-

limit c ! 0 and k ! 0we exp ect to reach the criti-

eral exp erimental groups measured the size-distribution

cal p oint. This fact was indeed veri ed in exp eriments,

of avalanches in granular materials. Unfortunately,real

where k can b e controlled by mo difying the asp ect ratio

sandpiles do not seem to b e b ehave as the SOC sand-

of the sample [76].

pile mo del. Exp eriments show large p erio dic avalanches

separated by quiescent states with only limited activ-

In typ e I I sup erconductors, when the external eld

ity [67]. While for small piles one could try to t the

is increased, ux lines are nucleated at the b order of

avalanche distribution with a p ower lawover a limited

the sample and pushed inside by their mutual repul-

range [68], the b ehavior would eventually cross over, on

sion. The resulting ux density gradient, known as

increasing the system size, to the one describ ed ab ove,

the Bean state [77], b ears some analogy with sand-

which is not scale-invariant. The reason sand do es not

piles, as p ointed out by De Gennes over 30 years ago

b ehave like an ideal sandpile is the inertia of the rolling

[78]. Unlike sand grains, ux lines have little inertia,

grains. As grains are added, the inclination of the pile

and exhibit p ower-law distributed avalanches [79]. It is

increases untilitreaches the angle of maximal stability

still unclear whether in this system a mechanism sim-

, at which p oint grains start to ow. Due to iner-

ilar to the demagnetizing eld maintains a stationary

tia, the ow do es not stop when the inclination falls to

avalanche state, as in ferromagnets. Simulations of ux

, but continues until the inclination attains the an-

line motion [80]have repro duced exp erimental results

gle of rep ose < [69]. Since the \constant force"

s c

in part, but a complete quantitative explanation of the

(i.e., with controlled) version of the system has a rst-

phenomenon is lacking.

order transition, it is no wonder that criticalityisnot

Another broad class of phenomena where SOC has

observed in the slowly driven case. So if wewantto

b een invoked on several o ccasions is that of mechani-

see p ower-lawavalanches wehave to get rid of the in-

cal instabilities: fracture, plasticity and dislo cation dy-

ertia of the grains. Grains with small inertia exist and

namics. Materials sub ject to an external stress release

can b e b oughtinany gro cery store: rice! A ricepile

acoustic signals that are often distributed as p ower laws

was carefully studied in Oslo: elongated grains p oured

over a limited range: examples are the fracturing of

at very small rate gave rise to a convincing p ower-law

wood [81], cellular glass [82] and concrete [83], in hy-

avalanche distribution [70].

drogen precipitation [84], and in dislo cation motion in

The previous discussion tells us that in order to ob-

ice crystals [85]. While it has often b een claimed that

serveapower-lawavalanche distribution, inertia should

these exp eriments provided a direct evidence of SOC,

b e negligible. As discussed in Sec. IV, the motion of do-

this is far from b eing established. In fact, fracture is an

main walls in ferromagnets and ux lines in typ e I I su-

irreversible phenomenon and often the acoustic emis-

p erconductors is overdamp ed, due to eddy-current dis-

sion increases with the applied stress [81] with a sharp

sipation; these systems are probably the cleanest exp er-

p eak at the failure p oint. There is thus no stationary

imental examples of p ower-law distributed avalanches.

state in fracture, and it is debated whether the failure

The noise pro duced bydomainwall motion is known

pointcaneven b e describ ed as a critical p oint[86]ora

as the Barkhausen e ect, rst detected in 1919 [71].

rst-order transition [87]. The situation might b e dif-

Since then, it has b ecome a common non-destructive

ferent in plastic deformation, where a steady state is

metho d for testing magnetic materials, and its statis-

p ossible [88]; recent exp erimental measurements of dis-

tical prop erties have b een studied in detail. When the

lo cation motion app ear promising [85]. Wemaymen-

external magnetic eld is increased slowly, it is p ossible

tion some related phenomena in whichavalanches have

to observewell separated avalanches, whose size dis-

b een observed, and a theoretical interpretation is still

tribution is a p ower-lawover more than three decades

debated: martensitic transformations [89], sliding sys-

[72-76]. Domain walls are pushed through a disordered

tems [90] and sheared foams [91].

medium by the magnetic eld, so wewould exp ect a de-

pinning transition at some critical eld H = H .One Finally,itisworth mentioning that SOC has b een

should note, however, that the \internal eld" acting claimed to apply to several other situations in geo-

on the domains is not the external eld, but is cor- physics, biology and economics. Wehave delib erately

rected by the demagnetizing eld H 'NM where chosen to discuss only those examples for whichex-

M is the magnetization [75,76]andN the demagne- p erimental observations are accurate and repro ducible.

tizing factor. Therefore, if we increase the external Even in these cases, it is often hard to distinguish b e-

eld at constant rate c, the internal eld is given by tween SOC-likebehavior and other mechanisms for gen-

Brazilian Journal of Physics, vol. 30, no. 1, Marco, 2000 39

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