Mapping Self-Organized Criticality Onto Criticality Didier Sornette, Anders Johansen, Ivan Dornic

Mapping Self-Organized Criticality onto Criticality Didier Sornette, Anders Johansen, Ivan Dornic

To cite this version:

Didier Sornette, Anders Johansen, Ivan Dornic. Mapping Self-Organized Criticality onto Criticality. Journal de Physique I, EDP Sciences, 1995, 5 (3), pp.325-335. 10.1051/jp1:1995129. jpa-00247058

HAL Id: jpa-00247058 https://hal.archives-ouvertes.fr/jpa-00247058 Submitted on 1 Jan 1995

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(1995) Phys.

J. I IYance 325-335 5

1995, 325 MARCH PAGE

Classification

ysics PA Abstracts

64.60H 05.70L 05.40

CriticaEty Mapping Self.Organized CriticaEty enta

Sornette, Didier Anders Johansen Dornic and Ivan

(*), Physique Laboratoire de Sciences, de la Matière Condensée Université des 70, B-P. Parc

Valrose, 2, Nice Cedex France 06108

(Received 1994, received November accepted in final November 14 1994, November form 30 25

1994)

Résumé. stratégie générale responsable

identifier Nous mécanisme le

proposons une pour

phénomènes duto-organisés, critiques

qu'ils simplement traduction, des

basée l'idée la sont sur

paramètres choisis,

dynamique dans de

critique d'un instable standard. La point

espace un

ajusté auto-organisée résulte paramètre positive criticalité du

valeur contrôle du d'ordre à une

zéro, paramètre correspondant

qui tendant automatiquement

de contrôle

que vers ce assure

critique cale valeur

critique sous"jacente.

Ce résultat

de la de transition exactement

se sur sa

forçage

explique particulier joué le le rôle infiniment lent

caractère à

est tous

par qui commun un

auto-organisés. systèmes critiques appliquons sable,

les Nous idées modèles de de

tas

ces aux

décrochage forêts, modèles de tremblements de

feux de transitions de

terre, et

aux aux aux

qui fractale, été

proposés d'exemples caractéristiques modèles croissance

de ont autant comme

duto-organisée. criticalité de la

(SOC),

general conceptual self-organized criticality

Abstract. framework for We present a

nothing

recognition that expression, but the "unfolded" based the it is suitable

param- on m a

precisely, underlying dynarnical

SOC of unstable is shown critical More

to point. eter

space, an

value, thus

vanishingly small, positive of the order result from the but

tuning parameter to a

corresponding

under- exactly critical control value for the that lies the its parameter at ensunng

driving ail

lying This clarifies the rote of the slow transition. and rate to

nature very common

earthquakes, sandpiles,

exhibiting apply SOC. This mechanism of

shown models systems to to is

growth tires, proposed exarnples SOC. depinning, fractal and forest been of which have

Introduction 1.

insightful early witnessed clear [1-6], Following of decade has

studies number the past

a a

by acknowledgement phenomena described law statistics.

that natural be

must power many

of these developed origin understand

the Correspondingly, activity has order in to intense an

criticality' of'self-organized particular

This has led the ubiquitous law tails. in concept to power

CNRS URA190. (*

PHYSIQUE JOURNAL I N°3 DE 326

spatially extended evolve dynamically (SOC) driven according which certain systems to 8], [7,

dynamical characteristic globally with time stationary critical spontaneously

towards

state no a

length

scales- or

equilibrium phase underlying that, SOC unlike transitions statis- in is fundamental idea The

fine-tuning

of control reached without the need critical parameter, physics, the is tical state a

SOC, ideas Bak dynamics.

of illustrate the basic of the To

the critical is 1-e- attractor state an

pile of

inspired by avalanches of the creation

cellular in used and co-workers

automaton a a

slope

until local by grain

added lattice becomes models, grain "sand" is Sand. In these

a on a

pile stationary initiated. the critical reaches avalanche

this In unstable, state, is and

way, a an

pile of will fall grains slope, which sand off the via critical additional

by in characterized a

according lifetime and scale, lattice distributed grain from size in avalanches of ail sizes to to

law.

power a

physical

general effort, which conditions the under large theoretical In of system spite

a a

established: largely Some facts however been still unknown. have exhibits SOC

are

obeys diffusion large qualify equation evolution if Some SOC their scale

systems

a as .

scale)

(possibly

global satisfies characteristic which non-linear but with time

con- no a

loi. law servation [9,

obey but of which diffusion-like do

Mass exhibit

There

systems exists

not

response a a .

il1-14].

nevertheless exhibit SOC the global In

law and these conservation

to cases, seem

deep if

for SOC still clear exist there rela- underlying mechanism is to

trot seems even a

synchronization coupled oscillators problem of of of threshold relaxation tionship the with

[8,15-18].

of the

domain occurring in parameter

space some

which, of usual from the generally, perspective feedback mechanism

More operate

must

a .

the descnbes the of the order control phenomena, action critical parameter onto parame-

dynamics for critical mechanism

This then

and the suggests to ter state.

attracts [19] a a

transforming phase We critical SOC call usual "unstable transitions" unstable into [19].

self-organized. phenomena which those critical not are

understanding fragmented

theoretical of In the SOC rather real with is present summary, no

unifying

goal general

theoretical Our here perspective. framework present attempt to to

a is

SOC,

for nothing recognition expression, based the that the

"unfolded" it but is in

on a

underlying suitable correspondence

of unstable critical This genuine parameter point.

space, an

provides, fundamental

what believe mechanism for SOC and the information

is, we more on

obtained relevant be framework. the within this critical exponents con

Self-Organized Criticality The Nature of 2.

approach

of GENERAL MECHANISM. The be few summarized 2.1. in

essence our con a sen-

Ising ferromagnet phase

Consider "standard" unstable such transition, critical the tences.

as a

percolation.

analogously, assigned Here, bond down,

spin,

each with

site

or, a up or is an

Furthermore, exchange coupling

neighbour defines J.

be sites constant to two nearest

con- one

e~~~/~B~'

probability

field, nected with if both spin have For 1

externat this

p a up. zero =

bond-density

defines order critical below which the the T~

temperature parameter

p~ mû, or a

T)P

probability

cluster, infinite

of magnetization the and behaves is

(T~

mû an or zero as oc

diverging length by characterized above. further T~)~" correlation The is

( transition

a oc

T~)~~ susceptibility

quantifying approached,

and spatial fluctua- hence the is T~

x oc as

Suppose

the of be natural the order that for under tions it parameter. turns out to system now

MAPPING SELF.ORGANIZED N°3 CRITICALITY ONTO CRITICALITY 327

controlling "operator" that, T, consideration instead of the

controls the order parameter mû

fixing

limiting

furthermore of arbitrary positive takes but and the

it small value. The to

case a

(Specifically, Tj. equivalent

condition the

T is above

mû scenano comes more - -

only strength" cluster, infinite natural the of

"zero where

in bond needs be context

to a one

ferromagnet.) broken, words,

of thon in that other the In

is the of critical value system at a

therefore fluctuations the unstable

point critical exhibit and ail scales its in must at response.

nothing underlying of This is but the hallmark the unstable critical As point. will be shown

following applies explicitly examples, the naturally

this out-of-equilibrium in

most to scenario

driven systems.

precisely, exhibiting More shall that SOC

genuine critical systems present

tran- argue we a

by forced often form generalized

suitable control the when of force sition in parameter,

a a

(torque depinning systems). earthquake models, sandpile models, for for force for Then, stress

SOC controls drives the the order of the critical via

system parameter

appears as soon as one or

(it

that, these the order also the of the transition is conjugate in systems, parameter tums out

equations).

control of mechanical Hamilton-Jacobi the The order in parameter parameter sense

general velocity form of flux. control of the hence takes the The in

order parameter,

a or a

o+ out-of-equilibrium flux,

driving natural The condition that the M is in

systems.

is -

dTiuing special played the of slow role illuminates the

constraint

Tate to veTy a common

exhibiting

being SOC, the control ail the order condition

systems parameter to exact at as

of positioning which the critical value the control parameter. at exact ensures

general

develop by idea,

detailed shall this We illustrate discussion of rive and

now a ex-

pinned-depinned Charge-Density-Waves amples: sandpiles, earthquake models,

lines the or

models, forest tires. growth fractal and

processes

SANDPILE THE 2.2.

sandpile

Model cellular consider trie

of Unstable Transition. Let 2.2.1. automaton

us an

mspired namely rotating and from that of

model it in geometry [20], experiments

put

[7] a a

cylinder. cylinder cylinder Trie horizontal and trie trie Trie axis is rotation is axis.

same as

partially initially flat filled horizontal interface below trie

presenting with "sand" axis. is an

cylinder fixed frame which Suppose spring

trie that of trie held torsion

a on one axis is a

cylinder position surface

If trie takes trie such that trie 0, controlled T. T

torque exert

a can =

non-vanishing

angle If

horizontal, 9 of trie sand trie rotation o. is to starts

1-e-, exert

one a =

sandpile

angle by trie tilted cylinder

T, such 9 that trie exerted trie torque

rotates to up an

finally exactly which T, cntical

reaches value applied Increasing balances trie T. T~ at

a one

J, flow corresponding triggering of sand trie sandpile instability of slope trie reaches its to 9~

magnitude with whose T~. T increases >

of for

T active sliding from 0 transition, T~)

We cntical to witness

state

< repose

an a a =

cylinder

steady (J

of trie corresponding

sliding rotation

for T

o to T~) state > > average an

(d9/dt).

T, trie mcreasing

velocity This angular

rotation since

at occurs, average non-zero a

applied trie longer

and balance

by sand increase exerted trie slope and therefore

torque can no

overlap

avalanches and dynamical

which trie re#me

One thus in

construct torque. enters a a

fluctuating flow.

non-zero

above flow from by

characterized trie trie

critical This

transition

zero average increases way is

of burst temporal (J (T trie local spatial correlations by

trie and T~)P), in well

as as ~J

increasing by small when T of these bursts

amount below Trie size J maximum

T~. a non-zero

addition)

grain-hole

(or

allows perturbation such local by small mserting

below

T~ one as a a

trie of conservation, grain that, from trie length f~ constraint define trie Note correlation to

(d9/dt) describe proportional simply angular velocity and cylinder sand flux trie J and are

PHYSIQUE N°3 JOURNAL DE I 328

increasing of

control T trie ordered trie physically trie parameter system

to response saine

(d9/dt)

(d6/dt)

(J for For T~). and J T it is for T o

> >

< purpose,

our

now

= ~J

(d9/dt). speak of

of trie order illuminating

in parameter terms to more

controlling

However, sliding when trie T. critical transition Trie above

torque suppose occurs

o+, d9/dt d9/dt

angular vanishingly

velocity small impose and controls trie 1-e- that

a one =

corresponds interchange mechanics, of rote of language of trie trie this value. In positive to an

d9/dt (defined product by gives fact their trie mechanical variables T and trie that conjugate trie

system). adjusts

then dear critical

that value and trie of It is T its trie T~ to response power

previo1÷sly

of law distribution sandpile will be doc1÷mented that of in trie terms [7], j1÷st power

of avalanches.

specific paradigrn sandpile by describing of briefly

this trie how conclude section Let

a us

sliding instability implemented, knowing trie initial rules of trie critical be model for trie can

configuration by of given A sand charactenzed trie column sandpile cellular is automaton. set

being by

slopes, just total ail of trie exerted heights and trie trie sites torque torque

sum over

configuration

cylinder Then, given grain of trie column with rotation. each axis respect to a

(it

configurations similarly global

corresponds that have the T dear T is

torque

to many same a

finding

correspond study

trie SOC found trie that micrc-states in

macrc-state, to

to many as a

sandpile corresponds Trier,

cellular small of T of Abelian increment ). automata to

[22] a a

leading Starting slopes possibly instabilities, global of trie local local avalanches. increase 1-e-, to

regime only

T, the avalanche from small initial When reaches and is transient.

overpasses one a

("sand"

flow) flow and the becomes the positive, the order continuous T~, parameter appears

being Thus, fluctuations. of avalanches avalanches the characteristic order the parameter size

length. that,

previous Note discussion of the trie correlation correction in is to contrast a

phenomena naturally framework standard critical

SOC and introduces trie between [21], our

slope

dynamical adjusts itself

Trie just trie order variable which is parameter.

current

as as a

(control

parameter). function

of trie externat torque a

corresponding

Scaling Viewed cntical derme transition, trie Laws. 2.2.2.

power as a one con

laws:

(1) J T~)P

(order parameter), for trie flow current

Tj-~

(21

jT~ x ~

susceptibility by) for function trie trie flow induced small

pertur-

(1.e. to current response or a

bation close trie critical and transition to [21] very

Tl~" f (3)

lTc ~

length correlation by

for trie linear of trie which trie domain

is local

sensitive

to given size a

perturbation.

Note that that and here

both of trie T~. sides

assume we rl are same on v

general, should

interdependent these three scaling In obey be

and expect

exponents to one a

expressing by perturbation induced

trie local flux relation that function

of suscepti- trie is

a a

bility trie and of

here, correlated

volume of infer trie domain. From properties trie of

we cari

0+,

SOC driving

1.e., trie by when it with J trie

"avalanches" system, system

reacts upon -

according

distributed law

to power size a

s~~~+")

P(s) (4)

The ( by fractal of avalanche maximum related where trie dimension size is D is

s~utooE ~

by Then, flow trie avalanches. simply perturbation trie caused local below trie is

T~ average a

N°3 SELF.ORGANIZED MAPPING CRITICALITY CRITICALITY ONTO 329

s)j$~,

sP(s)ds rl/(vD). yielding avalanches, of size

This

1 expression

ÎÎ~~~°~~ ~' = ~J

previously

bas checked derived

been and with

yields and numerical simulations o.l

[21] p ct

(D general,

and trie avalanches 3D. In d, in 2D 0 in where d

trie is compact

p are ci space =

dimension),

showing from

that trie be determined properties of trie critical

transition p can

v). dynamical

(rl and Trie determmation of trie by defined scaling

trie of exponent

z, an

(1). (~, fl avalanche duration involves trie equation simplest

in In trie picture t exponent

dilfusive, dynamics flux proportional diffusion where trie trie sand coefficient trie is is which to

(~/t.

proportional expression itself This that renormalization there is of trie

is to

assumes no

coefficient definition microscopic of trie diffusion coefficient trie entenng and transport leads to

general simple pu. expression This last diffusion is uncertain since trie in approximation 2+

z =

question. is in

Sandpile provides simple Our framework

Non.Conservative Models. 2.2.3. and natural a

by sandpile observation explanation for trie that models be SOC. non-conservative [11] may

earthquakes.

Trie model model for Trie local variable their trie total authors is toy present

as a

model, slowly force their initial trie force In increasing

exerted each site. site is

very on a on

threshold,

given

trie force reaches

When

site it is while constant rate. reset to at

on a zero a a

neighbors. redistributed from Trie fraction trie trie fact

is non-conservation nearest stems a on

which that total distributed trie initial less thon value. trie is is amount

global

force,

that fixed that trie

impose work ail 1.e. We at sites

over we sum now assume we

global larger

Increasing force value local forces

this of the is

constant. not

to may may a or

global

force value for the

readj1÷stments. again The that there exists cntical point trigger

is a

Again, velocity characterizing

avalanches above

which trie

stop,

average non-zero a never v.

o+ adj1÷sting global force places by

point critical trie trie trie its ass1÷ring system to

v -

crucial for trie why then dear condition of critical value. It trie conservation is is not appearance

sliding

underlying

rely fundamentally

trie of critical

SOC existence of SOC: is

an seen on

dynamics

of critical

need well-documented st1÷dies trie conservation, which does point, in not as

phase second order of1÷nstable transitions [23].

argued by several authors that

TECTONICS. It lias been EARTHQUAKES [24, 25] 2.3. AND

SOC. consider thus phenomenology

geology signature trie of Let earthquake trie in

us a is

perform

plate laboratory mechanical scaled down trie model elastic

tectonic 27],

[26, to a m

for Trie namely imposed border

deformation thought its experiment, shear instance. at a

(trie

imposed borders other

opposite

force simplest shear where is F is two situation

two on a

similarly

compressive experiment (One

consider free),

could being by spring

set-up.

a say, a

applied

force F As trie change

discussion. triaxial without in done in

[28], tests any our as

faults)

and (which damage

cracks such plate pre-existing increases, trie contain starts to

as can

during sufliciently F, alter transient low damage

For increasing deform trie internai [29]. some

and applied force, static becomes itself trie adjusts deforms the the system and which system to

(here

neglect

velocity deformation becomes

of nothing

happens: trie strain

rate any we zero or

longer longer behavior).

trie becomes and ductile increases, As F transient

additional

creep or

plate.

There develop will within trie larger exists plastic-like deformations larger and since a

globally "plastic", trie that in plate

threshold F~ trie becomes plasticity which critical at sense

de/dt

F~,

As F above under fixed F. increases

flow with strain it rate

starts to non-zero a

plastic

this experiments, de/dt laboratory and models in increases. In trie shear strain rate

control trie F

usual

critical

trie in from brittle ductile behavior

transition to

sense. a is a

/dt

/dt for

and de (de de/dt for

F~ F qualifies o o trie order

and

parameter < > parameter at =

F~). F >

(very small)

apply

[et controlling exerted trie of force trie

constant Instead system,

us a on

for boundary condition "natural"

trie plate. of thus

border We trie trie shear at rate recover

PHYSIQUE JOURNAL N°3 DE I 330

(the typical plate velocity plate deformations relative of order earthquakes and is tectonic 1

/s). during velocity earthquake /year

of order

compared the fast Such km 2

rupture

to an cm

SOC, showing shape studied works the of both the been in has

in existence situation

many a

by earthquake fractal distribution the fault selected the and of law in

geometry size power

nothing dynamics. It becomes dear that this "natural" condition but earthquake is now

de/dt

controlling by order plate point the the critical F driving the

F~ parameter

to at =

ensuring value, thus the critical of Note that SOC will

properties the small systems.

very a

corresponding specific

for if be the transition critical. This the is

not

may some case appear

spring-block the models

models such [30]. as

lying of

within TRANSITIONS. Consider elastic fine random system DEPINNING 2.4.

a an

length

pulled by density and force

impurities asperities

its pinning unit at two

per or or a

perpendicular direction. stretched directed

direction This be ends its

to average can a or a

charges

electric electric field 32], polymer with ends which submitted its [31, at to

are an

superconductor superficial the

electric of II with which

line in in type current

vortex a a a

magnetic

field force ends of the line

of will Lorentz the

create vortex at two

presence a a on

sample displacing by

borders interface created fluid another fluid the in [33],

porous a a an

quenched

disorder.

magnetic domain wall the of The electric medium in

[34] presence a or or

(the

on) plays

field

field of E from magnetic given the role the control For parameter.

a now

(small)

overdamped dynamics conditions, initial and the will the relaxation E

ensure some

configuration.

by well-defined of from value the line line When small in E

mcreasing a some

fixed

readjust increment, small itself of the hne

localized

via stay

may may sequence some or a

sliding conformation. been another increases, has static As E well-documented it into events

begins

E~ critical value point E~ which the fine The that there above E is is

move. a =

pinned-depinned

dynamical point points, critical similar critical endowed usual with

to very a

corresponding

velocity

properties. the of ail the The order is the fine and parameter

average v

(E E~)P.

scales

as v ~J

controlling Suppose of

that E,

instead vanishingly

drives the line ends

at constant

now one a

velocity,

field, small the controls

namely By conjugate the order 1e. the the parameter. to one

above,

automatically the fine

pinned-depinned critical is then arguments its

at tran- same as

long-range point. As spatial fluctuations sition

large correlations

and consequence, a appear,

distribution reflected sliding the of burst-like along in the line. occurring events

One

"avalanche" the overdamped from distribution

simulations numeric extract

can on an

pulled

elastic string by medium

random length Using

force fact in

trie that

[31]. unit a per a

typical trie

fluctuation

along ( implies

scale of hne that

trie order

transverse

< a over is z z

jump

trie characteristic length proportional Then, portion for of line of

the is

size a z z.

(~~",

zP(z)dz

the length proportional correlation motion is

transverse to

average over a

z~~~+"1

P(z)

the where sliding corresponding problem. distribution of is the SOC

in events

reasonable

It that this

scale motion time

transverse

assume seems average occurs over a

(it

proportional ( scaling),

velocity

for is

other which leads to to

correct to easy any average an

(E scaling plu threshold E~)P just above prediction

J #ving (~", the

o.25

as p ci

= ~J ~J

using

the

results and o.25 1 [31].

ct v m

(say

of charged-density-wave reasoning applies

The pulled

elastic line type in

same a an

direction)

parallel direction

depinning exhibit well-known which critical is

to its

to average a a

driving of critical value Again,

field the driving transition the CDW

at [35]. constant at some a

velocity

SOC small sliding with

distribution of

law

system creates

events. very an power a

GROWTH PRocEssEs. FRACTAL 2.5. been

has It proposed growth that fractal processes,

(DLA), diffusion-limited-aggregation

such

fluid exemplified imbibition by for

instance as as

SELF.ORGANIZED N°3 MAPPING CRITICALITY ONTO CRITICALITY 331

growth, percolation,

dendritic breakdown, dielectric invasion media,

random in rupture con-

exhibiting another Mass of SOC [36-38]. stitute systems

growth

problems however

These belong

diiferent phenomena of Mass thon

to to seem a

sandpile models trie geometrical (and internai since

of quenched

is aggregate structure e-g- an

active) only

whereas perimeter

trie is sandpile critical of continuously is rearranging its state a

forcing. under trie of externat action

Here,

similarity point that proposed when using

stronger trie out want to

we a emerges

self-organization

fractal framework trie and growth of these

be linked trie

to processes con

underlying point. of critical

We shall illustrate existence ideas trie annealed dielectric

an on our

(best

cylindrical regime),

breakdown model steady

suited define in which geometry

to state a a

equivalent cylinder Trie known

be DLA potential base of trie fixed

and is [39]. o to to at is

fixed

growth value other end V. Trie of from its is base assumed trie is at

rate some non-zero

by probability stochastic controlled proportional

be electric field trie trie to

to process a a on

growth surface. DLA base Trie model growth recovered trie limit where the is quasi- done in is

fixing (equivalently statically. corresponds growth This particle trie flux trie situation to rate

o+. model) DLA trie J

m -

-dV/dz,

gradient impose potential consider trie where

Let

constant average us now case we a

along cylmder E, Furthermore,

electric field

trie [et that dielectric break- axis. 1.e.

assume us

growth

only quenched down, threshold,

above which certain is 1.e. site,

on a can occur a a

according

trie variable distributed given random distribution. This condition that to

ensures a

growth pinned-depinned problem

those above becomes similar transition. For sites to

now a

growth

again by threshold, of their assumed trie be stochastic controlled is rate

to process a

proportional small, field. probability applied trie electric If field few

trie electric is

very a a

growth trie duster threshold. evolve will break down and will until all below their sites sites are

Increasing applied

bound trie field threshold E~ This trie electric above certain is state. a

distribution),

begms

threshold trie finite function unbound trie

in system

to at

grow way an a

growth larger. velocity by

which finite determined the details E Trie increases is gets rate as

(which here).

dynamical finite of the breakdown do describe Note that

not processes a we

corresponds particle velocity transition, limite flux DLA trie critical trie trie model. For in to a

order) (resp.

(resp. gradient partiale

trie electric field trie control is concentration parameter or

flux). growth velocity particle

exhibit growing E~, above trie clusters do In trie re#me not or

decreasing

length self-similarity scales, finite which function

all but correlation is

appears a a

velocity

growth language interfaces, pushed of fluid trie

trie with of trie In

rate. average c, an

Laplace velocity pv~ Bemoulli trie trie dimensionless number of trie is ratio relevant to pressure

a/b number), (similar pushing

density, trie surface

Bond fluid where trie is

to pressure p a a

thickness interface radius

of and b trie Hele-Shaw

cell trie curvature. tension or

quasi- of

trie

of viewed trie result fractal DLA be the dusters In

structure summary, as can

rate) critical (growth corresponding of the control of the order the regime, 1.e. parameter static

(positive)

value. infinitesimal transition at an

problems. problem percolation defined for and

be the mappings Similar invasion rupture can

extensively

models quasi-static realize interesting that the rupture this is In respect, it to

of elements by characterized the literature the studied

rupture in [40] one, i-e- one are

dynamical

Truly (order parameter)

vanishingly small value. controlling of

the rupture rate to a

stable the separating beyond point, critical

the unstable this of models in rupture

[41] are sense

phase. fully phase from the damage finite rupture

introduced forest-tire model self-organized critical MODEL. The FOREST-FIRE THE 2.6.

model proposed is mechanism. The example of the another by and Schwabel is Drossel [13, 42]

lattice,

tire. The lattice

defined d-dimensional where each either is

empty,

site tree is or a on

PHYSIQUE DE I JOURNAL N°3 332

following

four A synchronously according rules:

will updated 1) trie empty

tree

to grow on an

4) 3)

spreads

catches Fire and A probability becomes 2) Fire empty;

with site tree to trees u-n p;

f/p f. of critical

spontaneously probability point Trie trie limit existence

with o is in tire

f/p)~l

(si pt)/Pt (1-

lightning destroyed by of expected, number is

trie

trees

a average smce =

1/T(s) T(s)

density)

f provided trie where (pt

time is trie is

[42], < < tree average p mean

nothing of again separation of

This scales but

cluster time is takes bum size it tree

to s. a

(si

Furthermore,

SOC.models.

dTiuing for order for of slow in condition 1

< pt common a

for control

trie critical diverge. trie This then transition in agreement parameter suggests

pt as

Changing

first rule of trie model proposed framework. trie allows trie its

with to tune to one pt

keeping thus down random

Each bumed critical value: time is put tree

tree pt a new is a

density

tires then trie of trie condition fixed controlled value. Trie order is

parameter

to pf, a

o+. f/p only f

corresponding o, tire trie of trie forest In above

case

pf con

= - =

defining clusters,

density thus dynamically below

critical connected propagate

tree tree a on

finally itself forever extinguish and above. It then trie

bum forest tire will is dear that which a

modifying

explaining field thus f plays of external trie and trie role transition

parameter an

simulations

law systematic deviations from

in [42, trie 43].

power pure seen a

Concluding Remarks 3.

self-organized conceptual criticality,

framework for general proposed which We bave 1. a

driving by

mapping points, SOC unstable critical controlled trie

consists onto

corre- m

sponding

infinitesimal value. order parameter at an

previously theory recognized of clarifies extends feed- trie

Our and mechanism present a

being

for SOC

back of trie order trie control essential parameter parameter [19].

on as

mapping

novel Trie between SOC and usual critical oifers but natural transitions route a

SOC, characterizing study namely by associated further with properties trie critical trie to

only

underlying point of itself.

displaced properties bave trie critical In trie

way, a we

underlying SOC of for of search trie mechanism for that trie critical unstable to appearance

significant

However,

improvement points. this for

feel that is 1)

two

a we reasons: some

underlying

of trie points unstable critical known and documented bave been

are per se.

knowledge light

Their

shed trie SOC thus particular, models. In trie

present

on can new

framework driving physical

illummates trie of meaning

the slow all systems to common

exhibiting underlying

SOC. Even if 2) the critical probably study point

is its

new, con

employing

by quite be efficient developed large the toolbox

trie last in twenty

or more

this field. in years

bifuTcations: SupeTcTitical

applies

framework directly 2. Note supercritical that and to our

Hopf bifurcations, considering trie when order situation where trie is controlled parameter

o+.

(possibly

trie analytically by value This

situation modelled general writing be at

can a

complex) Landau.Ginzburg fluctuations, for trie order equation conditioned parameter

vanishingly

order left for bave small trie future. This is parameter.

to average a

fized-scale Controlling tTansfoTmation:

RenoTmalization and trie order 3. parameter gToup

J of allow for

critical does standard renormalization point unstable

not

group an a -

procedure. critical Indeed,

there control and trie situation, this is in

parameter expo- no

procedure of of trie standard be trie derivatives by obtained in trie nents cannot terms

transforma- renormalization of trie control under scale transformation parameters group

Thus, point from trie critical do exist. tion. trie trie distance related not exponents to

SELF.ORGANIZED N°3 MAPPING CRITICALITY CRITICALITY ONTO 333

special this Let that due

of is point, trie critical attractive property not to

stress us a

solely claimed but driving special results from trie conditions

in which [44],

as ensure

dynamics

positioning unstable of trie trie point. critical trie this

situation, In exact

on a

generalized procedure reflect

renormalization should

positioning the critical

exact

on a

point, give should signature

trie of 1-e- point. critical attTactiue boni This indeed is an

procedure trie renormalization finding introduced in We believe that in [44]. out group

right general theory renormalization for

provide their critical point could systems at a

general of tools for phenomena.

class conformai critical This could be related

to new a

theory invariance [45].

clarify approach Relation Goldstone Our

proposai allows trie with modes:

that 4. to [46] us

dynamics by modes, non-linear of Goldstone however SOC from trie returning trie stems

logic criticality of Obukhov's SOC eifect of of Gold- is trie interaction in argument: not

underlying gapless claimed;

gapless the from modes modes rather result the stone as

driving by special that,

point, Let critical stabilized unstable the recall condition. in us

long-wavelength

homogeneous displacement limit, Goldstone

trie modes reduce of to a

sample

trie uniform of trie Since SOC whole spin rotation is system. system to

or an a

right phase

point trie critical of standard unstable cntical characterized transition at a

breaking,

by avalanches Goldstone be viewed the trie spontaneous symmetry

as a cari

large displacements) attempting fluctuations scale trie broken (1.e. restore to symme-

breaking, correspond droplet of trie avalanches In trie discrete

try. symmetry to

case a

fluctuations [47].

singulaT diffusion: of singular equations diffusion The Relation with continuous property 5.

straightforwardly by taking hydrodynamic trie of SOC derives obtained limit models [48]

framework, direct of localization unstable trie signature from

since it is

precise an our

only

singular Thus,

of SOC of diffusion trie theories point. critical ail in [48, 49] terms are

precisely critical

equation sitting of goveming that trie that expression is system at

a a

approach

generally, singular that, diffusion

recall trie point. Let

any occurs on us more

Rayleigh-Bénard instability. being bifurcation, example supercritical best known trie trie

velocity fluctuations

order and trie this trie trie

In convection parameter

average case, is

occurring patches trie critical velocity of

below associated streaks with

or non-zero are

spatial coefficient diffusion Rayleigh global off. Trie number which R convection starts at

velocity

D(R) R)~~/2 large D(R) reflects of diverges and trie existence (R~

very as

Similarly, scaling

powerlaw

A this fluctuations. be written [Soi. argument get to can

scaling derived from similar

singular diffusion SOC also be trie found models in can

showing singular diffusion.

of trie ongin trie reasoning [51], common

by several recognition proposed trie framework contradiction with Our is in recent 6. not

partial synchro- [8,15-18] deeply mechanism of that connected with trie authors SOC is

driving order

of trie slow thresholds. Under of relaxation oscillators with nization a

sandpile) isola- (for taken

trie

column each threshold element instance parameter,

a m

corresponds problem complete The periodic undergoes to oscillations of relaxation. tion

competition

in coupling between these oscillations result of the descnbe the terms a

dynamics of words, the desynchronization eifects. other In synchronization and between

slow the

critical under point of the detailed the oscillators describes relaxation response

driving of the order parameter.

PHYSIQUE JOURNAL DE N°3 I 334

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