Mapping Self-Organized Criticality onto Criticality

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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�

Classification PAysics Abstracts

64.60H 05.70L 05.40

Mapping Self.Organized CriticaEty _enta CriticaEty

Didier

Sornette,

Anders Johansen and Ivan Dornic

Laboratoire de Physique de la Matière Condensée

(*),

Université des Sciences, B-P. 70, Parc

Valrose, 06108 Nice Cedex 2, France

(Received

14 November 1994, received in final form 25 November 1994, accepted 30 November 1994)

Résumé. Nous proposons une stratégie générale _pour identifier le mécanisme responsable

des phénomènes critiques duto-organisés, basée

sur l'idée qu'ils sont simplement la traduction,

dans _{un espace} de paramètres choisis, d'un point critique dynamique instable standard. La

criticalité auto-organisée résulte du contrôle du paramètre d'ordre ajusté à

une valeur positive tendant _vers zéro, _ce qui _assure automatiquement _que le paramètre de contrôle correspondant

se cale exactement _{sur sa} valeur critique de la transition de critique sous"jacente. Ce résultat explique le rôle particulier joué _par le forçage infiniment lent _{qui est}

un caractère _commun à tous les systèmes critiques auto-organisés. Nous appliquons _ces idées _aux modèles de tas de sable,

aux modèles de tremblements de terre, de feux de forêts, _aux transitions de décrochage et _aux modèles de croissance fractale, qui ont été proposés _comme autant d'exemples caractéristiques

de la criticalité duto-organisée.

Abstract. We present _a general conceptual framework for self-organized criticality

(SOC),

based _on the recognition that it is

nothing

but the expression, "unfolded" _{m a} suitable param-

eter space, of _an underlying unstable dynarnical critical point. More precisely, SOC is shown to

result from the tuning of the order parameter to _a vanishingly small, but positive value, thus ensunng that the corresponding control parameter lies exactly at its critical value for the under-

lying transition. This clarifies the rote and nature of the _very slow driving rate _common to ail systems exhibiting SOC. This mechanism _is shown to apply to models of sandpiles, earthquakes, depinning, fractal growth and forest tires, which have been proposed _as exarnples of SOC.

1. Introduction

Following

_a number of

early insightful

studies

[1-6],

the past ^{decade has witnessed} a clear

acknowledgement

that _many natural

phenomena

must be described

by

_power law statistics.

Correspondingly,

_an intense

activity

has

developed

in order to understand the

origin

of these

ubiquitous

_power law tails. This has led in

particular

_to the concept

of'self-organized criticality'

(* ^{CNRS URA190.}

© Les Editions de Physique 1995

(SOC)

_{[7, 8],}

according

to which certain

dynamically

driven

spatially

extended systems ^evolve

spontaneously

^towards a critical

globally stationary dynamical

state with _no characteristic time

or

length

scales-

The fundamental idea

underlying

SOC is

that,

unlike

phase

transitions in

equilibrium

statis- tical

physics,

the critical state is reached without the need of

fine-tuning

_a control parameter, 1-e- the critical state is _an attractor of the

dynamics.

To illustrate the basic ideas of

SOC,

Bak

and co-workers used _a cellular automaton

inspired by

the creation of avalanches in _a

pile

of Sand. In these

models,

"sand" is added

grain by grain

_on a lattice until _a local

slope

becomes

unstable,

and _an avalanche is initiated. In this way, the

pile

reaches _a

stationary

critical state, characterized

by

_a critical

slope,

in which additional

grains

^of sand will fall off the

pile

via avalanches of ail sizes from

grain

to lattice

scale,

distributed in size and lifetime

according

to a power law.

In spite of _a

large

theoretical

effort,

the

general

conditions under which _a

physical

system exhibits SOC _are still

largely

unknown. Some facts have however been established:

. Some systems

qualify

_as SOC if their

large

scale evolution

obeys

_a diffusion

equation (possibly

non-linear but with _no characteristic time

scale)

which satisfies _a

global

_con-

servation law [9,

loi.

. There exists _a Mass of systems which exhibit diffusion-like _response but do not

obey

_a

global

conservation law and nevertheless _seem to exhibit SOC

il1-14].

In these _cases, the

underlying

mechanism for SOC is still trot clear _even if there _seems to exist _a

deep

rela-

tionship

with the

problem

^of

synchronization

of

coupled

threshold oscillators of relaxation

occurring

in _some domain of the parameter _space

[8,15-18].

. More

generally,

a feedback mechanism must operate

which,

from the perspective ^{of usual} critical

phenomena,

descnbes the action of the order parameter onto the control _parame- ter [19] ^and attracts the

dynamics

to _a critical state. This then suggests _a mechanism for

transforming

usual "unstable critical

phase

transitions" into SOC [19]. ^{We call unstable} those critical

phenomena

which _are not

self-organized.

In summary, the present theoretical

understanding

of SOC is rather

fragmented

with _no real

unifying

perspective. Our

goal

here _is to attempt to present _a

general

theoretical framework for

SOC,

based _on the

recognition

that it is

nothing

but the

expression,

"unfolded" in _a

suitable parameter _space, ^of an

underlying genuine

unstable critical point. ^This

correspondence provides,

what _we believe _is, the fundamental mechanism for SOC and _more information _on the relevant critical exponents _con ^{be obtained} within this framework.

2. The Nature of

Self-Organized Criticality

2.1. GENERAL MECHANISM. The _essence of _our

approach

_con be summarized in _a few _sen-

tences. Consider _a "standard" unstable critical

phase transition,

such _as the

Ising ferromagnet

or,

analogously,

bond

percolation.

Here, _a spin, _up or

down,

_is

assigned

to each site with _an

exchange coupling

constant J.

Furthermore,

_one defines _{two nearest}

neighbour

sites to be _con- nected with _a

probability

_p

= 1

e~~~/~B~'

_{if both have}

spin

_up. For _zero externat

field,

this defines _a critical temperature T~ _or

bond-density

_p~ below which the order parameter _mû, ^the

magnetization

or the

probability

of _an infinite

cluster,

is _zero and behaves _{as mû oc} (T~

T)P

above. The transition is further characterized

by

_a

diverging

correlation

length (

_oc

(T T~)~"

and

susceptibility

_{x oc}

(T T~)~~

_as T~ ^is

approached,

hence

quantifying

the

spatial

fluctua- tions of the order parameter.

Suppose

_now that it turns out to be natural for the system ^under

consideration

that,

instead of

controlling T,

the

"operator"

controls the order parameter _mû and furthermore takes the

limiting

_case of

fixing

it to a

positive

but

arbitrary

small value. The condition _{mû -} o+ _is

equivalent

to T _-

Tj. (Specifically,

the _scenano above _{comes more} natural in the _context of _a "zero

strength"

^infinite

cluster,

where

only

_one bond needs _to be

broken,

thon in that of _a

ferromagnet.)

In other

words,

the system is at the critical value of the unstable critical

point

and must therefore exhibit fluctuations _at ail scales in its response.

This is

nothing

but the hallmark of the

underlying

unstable critical point. As will be shown

explicitly

in the

following examples,

this _scenario

applies

most

naturally

to

out-of-equilibrium

driven systems.

More

precisely,

_we shall _argue that systems

exhibiting

SOC present a

genuine

critical tran- sition when forced

by

_a suitable control parameter, ^{often in the form of} a

generalized

force

(torque

for

sandpile models,

stress for

earthquake models,

force for

depinning systems). Then,

SOC _{appears as soon as one} controls _or drives the system via the order parameter ^{of the critical}

transition

(it

tums out

that,

in these systems, the order parameter is also the

conjugate

of the control parameter in the _sense of mechanical Hamilton-Jacobi

equations).

The order parameter hence takes in

general

the form of _a

velocity

_or flux. The control of the order parameter, a

flux,

is natural in

out-of-equilibrium

_systems. The condition that the

driving

_is at M _- o+

illuminates the

special

role

played by

the constraint of _{a veTy} slow

dTiuing

Tate _common to ail systems

exhibiting SOC,

_as

being

the exact condition to control the order parameter at o+

which _ensures

positioning

at the _exact critical value of the control parameter.

We shall _now illustrate and

develop

this

general idea, by

_a ^detailed discussion of rive _ex-

amples:

the

sandpiles, earthquake models, pinned-depinned

lines _or

Charge-Density-Waves models,

fractal

growth

_processes and forest tires.

2.2. THE SANDPILE

2.2.1. Model of _an Unstable Transition. Let _us consider trie cellular _automaton

sandpile

model _[7] and put ^{it in} a geometry

mspired

from experiments [20],

namely

that of _a

rotating cylinder.

Trie

cylinder

axis is horizontal and is trie _{same as} trie rotation axis. Trie

cylinder

is

partially

^filled with "sand"

presenting

an

initially

^flat horizontal interface below trie axis.

Suppose

that trie _axis of trie

cylinder

_is held to _a fixed frame

by

_a torsion

spring

_on which _one

can exert a controlled torque ^{T. If T} = 0, trie

cylinder

takes trie

position

such that trie surface of trie sand is

horizontal,

1-e-, trie rotation

angle

9

= o. If _one starts to exert a

non-vanishing T,

trie

cylinder

rotates up ^to an

angle

9 such that trie torque ^exerted

by

trie tilted

sandpile

balances

exactly

trie

applied

T.

Increasing T,

_one

finally

reaches _a cntical value T~ at which trie

sandpile

reaches its

slope

_9~ of

instability corresponding

to trie

triggering

of sand flow J,

whose

magnitude

increases with T _> T~.

We witness _a cntical

sliding

transition, from _a state of _repose

(J

= 0 for T < T~) to an active

sliding

state

(J

_> o for T _> T~)

corresponding

to an average

steady

rotation of trie

cylinder

at _{a non-zero} average

angular velocity (d9/dt).

This rotation _occurs, since

mcreasing

T, ^trie

slope

and therefore torque exerted

by

trie sand _{can no}

longer

increase and balance trie

applied

torque. ^{One thus} enters a

dynamical re#me

in which trie avalanches

overlap

and construct a

non-zero

fluctuating

flow.

This critical transition _is characterized

by

trie _way trie average flow _increases from _zero above T~

(J

~J

(T T~)P),

as well _as

by

trie

spatial

and

temporal

correlations in trie local burst of

non-zero J below T~. Trie maximum size of these bursts when

increasing

T

by

_a small amount

below T~

(or by mserting

a small

perturbation

such _{as a} local

grain-hole addition)

allows _one to define trie correlation

length f~

Note

that,

from trie constraint of

grain

conservation, trie sand flux J and trie

cylinder angular velocity (d9/dt)

_are

simply proportional

and describe

physically

trie _saine ordered _response of trie system to trie

increasing

control parameter T

(J

~J

(d6/dt)

₌ o for T < T~ and J

~J

(d9/dt)

> o for T >

T~).

^For our purpose, it is _now

more

illuminating

to

speak

of trie order parameter ⁱⁿ terms of

(d9/dt).

Trie above critical

sliding

transition _occurs when

controlling

trie torque T.

However,

_suppose

that _one controls trie

angular velocity d9/dt

and

impose d9/dt

=

o+,

1-e- _a

vanishingly

^small

positive

^value. In trie

language

of

mechanics,

this

corresponds

to _an

interchange

of trie rote of trie

conjugate

^variables T and

d9/dt (defined by

trie fact that their

product gives

trie mechanical

power of trie

system).

It is then dear that T

adjusts

to its critical value T~ and trie response of trie

sandpile

will j1÷st be ^that doc1÷mented

previo1÷sly

_[7], in terms of _power law distribution of avalanches.

Let _us

briefly

conclude this section of trie

sandpile paradigrn by describing

how _a

specific

model for trie critical

sliding instability

_can be

implemented, knowing

trie initial rules of trie

sandpile

cellular automaton. A

given

sand

configuration

is charactenzed

by

trie set of column

heights

and

slopes,

trie total torque

just being

trie _{sum over} ail sites of trie torque ^exerted

by

each column with respect to trie

cylinder

axis of rotation.

Then,

_a

given grain configuration corresponds

to a

global

torque ^T

(it

is dear that _many

configurations

have the _same T

similarly

to trie

finding

that many micrc-states

correspond

to trie SOC macrc-state, as found in _a

study

of Abelian

sandpile

cellular automata [22]).

Trier,

_a small increment of T

corresponds

to _a

global

increase of trie local

slopes possibly leading

to local

instabilities,

_1-e-, avalanches.

Starting

from _a small initial

T,

the avalanche

regime

is

only

transient. When _one reaches and overpasses T~, ^the continuous flow _appears and the order parameter

("sand" flow)

becomes

positive,

the

avalanches

being

order parameter fluctuations.

Thus,

the _size of the characteristic avalanches is trie correlation

length.

Note

that,

in _contrast to _a

previous

discussion of the correction

between SOC and standard critical

phenomena

_{[21], our} framework introduces

naturally

trie current _as trie order parameter. Trie

slope

is

just

a

dynamical

variable which

adjusts

itself _as

a function of trie externat torque

(control parameter).

2.2.2.

Scaling

Laws. Viewed _{as a} cntical

transition,

_{one con} derme trie

corresponding

_power

laws:

J

~J

(T T~)P (1)

for trie _current flow

(order parameter),

x ~ jT~

Tj-~

₍₂₁

for trie

susceptibility

_or trie response function _to (1.e. current flow induced

by)

_a small pertur- bation _very close _to trie critical transition [21] ^and

f

~ lTc

Tl~" (3)

for trie correlation

length

_given

by

trie linear _size of trie domain which is sensitive to a local

perturbation.

Note that _{we assume} here that _rl and _{v are} trie _{same on} both sides of T~.

In

general,

_one should expect ^these three exponents to be

interdependent

and

obey

_a

scaling

relation

expressing

that trie local flux induced

by

_a

perturbation

is _a function of trie

suscepti- bility

and of trie volume of trie correlated domain. From

here,

_{we cari} infer trie

properties

of

SOC upon trie system, 1.e., when

driving

it with J _-

0+,

trie system reacts

by

"avalanches"

distributed in _size

according

to _a power law

P(s)

~J

s~~~+") (4)

The maximum avalanche size is related to

( by

_s~utooE

~

(~

_{where D is trie} fractal dimension of trie avalanches.

Then,

trie flow caused

by

_a local

perturbation

below T~ is

simply

trie _average

size of

avalanches,

_x

~' ÎÎ~~~°~~

sP(s)ds

~J

s)j$~, _yielding

_p

= 1

rl/(vD).

This

expression

bas been derived

previously

and checked with numerical simulations [21] ^and

yields

_{p ct} o.l in 2D and _{p ci} 0 in 3D. In

general,

trie avalanches _are compact

(D

=

d,

where d is trie _space

dimension), showing

that p can be determined from trie

properties

of trie critical transition

(rl

and

v).

Trie determmation of trie

dynamical

exponent _z, ^defined

by

trie

scaling

of _an avalanche duration t _~J

(~,

involves trie exponent

fl

in

equation (1).

In trie

simplest picture

where trie

dynamics

is

dilfusive,

trie sand flux _is

proportional

to trie diffusion coefficient which is itself

proportional

to

(~/t.

This

expression

_assumes that there is _no renormalization of trie

microscopic

transport coefficient entenng trie definition of trie diffusion coefficient and leads to z = 2+

pu.

This last

expression

is uncertain in

general

since trie

simple

diffusion approximation is in

question.

2.2.3. Non.Conservative

Sandpile

Models. Our framework

provides

_a

simple

and natural

explanation

for trie observation

by

_[11] that non-conservative

sandpile

models _may be SOC.

Trie authors present ^{their model} as a toy model for

earthquakes.

Trie local variable is trie total force exerted _{on a} site. In their initial

model,

trie force _on each site is

increasing

_very

slowly

at a constant rate. When trie force _{on a}

given

site reaches _a

threshold,

it is reset to _zero while

a fraction is redistributed _on trie nearest

neighbors.

Trie non-conservation _stems from trie fact that trie total amount which is distributed is less thon trie initial value.

We _{now assume} that _we work at fixed

global force,

1.e. _we

impose

that trie _{sum over} ail sites

of the local forces is constant.

Increasing

this

global

force to _a

larger

value _{may or may not}

trigger readj1÷stments.

^The point

again

_is ^{that there exists} _a ^{cntical value for the}

global

force above which trie avalanches _never stop,

characterizing

_{a non-zero average}

velocity

_v.

Again, ass1÷ring

v - o+

places

trie system at trie critical

point by adj1÷sting

trie

global

^force to its critical value. It is then dear

why

trie condition of conservation is not crucial for trie _appearance of SOC: SOC is _seen to

rely fundamentally

_on trie existence of _an

underlying sliding

critical

point,

^which does not need

conservation,

as well-documented in st1÷dies of trie critical

dynamics

of1÷nstable second order

phase

transitions [23].

2.3. EARTHQUAKES AND TECTONICS. It lias been

argued by

several authors

[24,

25] ^that trie

earthquake phenomenology

in

geology

_is trie

signature

of SOC. Let _us thus consider _a

model elastic tectonic

plate [26, 27],

^{scaled down} m trie

laboratory

to

perform

_a mechanical

thought experiment, namely

_a shear deformation

imposed

at its border for instance. Trie

simplest

situation is where _a shear force F is

imposed

_on two

opposite

borders

(trie

other two

being free),

_say,

by

_a

spring

set-up.

(One

could

similarly

consider _a

compressive experiment

as done in triaxial tests [28], ^without any

change

in _our discussion. As trie

applied

force F increases, trie

plate (which

_can contain

pre-existing damage

such _as cracks and

faults)

_starts to deform

increasing

trie internai

damage

[29]. ^For

sufliciently

low

F,

alter _some transient

during

which the system ^{deforms and}

adjusts

itself to the

applied force,

trie system becomes static and

nothing happens:

trie strain rate or

velocity

of deformation becomes _zero

(here

_we

neglect

_any

additional _{creep or} ductile

behavior).

As F increases, ^trie transient becomes

longer

and

longer

since

larger

and

larger plastic-like

deformations will

develop

within trie

plate.

There exists _a critical

plasticity

threshold F~ at which trie

plate

becomes

globally "plastic",

in trie _sense that it starts to flow with _{a non-zero} strain rate

de/dt

under fixed F. As F increases above F~,

trie shear strain rate

de/dt

increases. In models and in

laboratory experiments,

this

plastic

transition from _a brittle to a ductile behavior is critical in trie usual _sense. F is trie control parameter and

de/dt qualifies

at trie order parameter

(de /dt

= o for F _< F~ and de

/dt

_> o for

F _>

F~).

Instead of

controlling

trie force exerted _on trie system, ^[et us

apply

_a constant

(very small)

shear _rate at trie border of trie

plate.

We thus _recover trie "natural"

boundary

condition for

earthquakes

^and

plate

tectonic deformations

(the typical

relative

plate velocity

is of order 1

cm

/year compared

to the fast rupture

velocity during

_an

earthquake

of order 2 km

/s).

Such

a situation has been studied in many works

showing

the existence of

SOC,

both in the

shape

of _power law

earthquake

_size distribution and in the fractal fault geometry ^selected

by

the

earthquake dynamics.

It _now becomes dear that this "natural" condition is

nothing

but

driving

^the

plate

at the critical

point

F

= F~

by controlling

the order parameter

de/dt

to a very small value, thus

ensuring

the critical properties of the systems. Note that SOC will appear if the

corresponding

transition is critical. This may not be the _case for _some

specific

models such _as the

spring-block

models [30].

2.4. DEPINNING TRANSITIONS. Consider _an elastic fine

lying

within _a random system of

pinning impurities

_or

asperities

^and

pulled by

a force

density

_per unit

length

_or at its two ends in _a direction

perpendicular

to its average direction. This _can be _a stretched _or directed

polymer

with electric

charges

at its ends which _are submitted to an electric field

[31, 32],

a vortex line in _a

superconductor

of type ^{II with} a

superficial

electric current which in the

presence of _a

magnetic

field will create a Lorentz force _on the two ends of the vortex line at the

sample

borders [33], an interface created

by

_a fluid

displacing

another fluid in _a porous

medium [34] or a

magnetic

domain wall in the presence of

quenched

disorder. The electric _or

magnetic

field

(the

field E from _now

on) plays

the role of the control parameter. For _a

given (small)

E and _some initial

conditions,

the

overdamped dynamics

will _ensure the relaxation of the line in _a well-defined line

configuration.

When mcreasing ^E from _some small value

by

some small increment, the hne may stay ^fixed or may

readjust

itself via _a sequence of localized

sliding

events into another static conformation. As E

increases,

it has been well-documented that there is _a critical value E~ above which the fine

begins

to move. The

point

E ₌ E~ is

a

pinned-depinned

critical

point

_very similar to usual

dynamical

critical

points,

endowed with ail the

corresponding

properties. The order parameter is the _average

velocity

_v of the fine and

scales _{as v}

~J

(E E~)P.

Suppose

_now that instead of

controlling E,

_one drives the line ends _{at a constant}

vanishingly

small

velocity,

1e. _one controls the

conjugate

to the

field, namely

the order parameter.

By

the

same arguments as

above,

the fine is then

automatically

at its critical

pinned-depinned

tran- sition

point.

As a consequence,

long-range spatial

correlations and

large

fluctuations _appear, reflected _in the distribution of burst-like

sliding

events

occurring along

the line.

One _can extract the "avalanche" distribution from numeric simulations _{on an}

overdamped

elastic

string

in _a random medium

pulled by

_a force _{per unit}

length

_[31].

Using

trie fact that trie

typical

transverse fluctuation _{over a} scale _{z <}

( along

trie hne _is of order _z

implies

that trie characteristic

jump

_size for _a

portion

of the line of

length

z is

proportional

to z.

Then,

the average ^transverse motion _{over a} correlation

length

is

proportional

to

J/ _zP(z)dz

~J

(~~",

where

P(z)

~J

z~~~+"1

_{is the} distribution of

sliding

events in the

corresponding

SOC

problem.

It _seems reasonable to _assume that this average transverse motion _{occurs over a} time scale

proportional

to

( (it

is easy ^{to correct} for _any other

scaling),

which leads to _an average

velocity just

above threshold

scaling

_as J

~J

(E E~)P

_~J (~",

#ving

the

prediction

_p

=

plu

_ci o.25

using

the results

fl

_ct o.25 and _{v m} 1 [31].

The _same type ^of

reasoning applies

to _a

charged-density-wave (say

_an elastic line

pulled

in

a direction

parallel

to its average

direction)

which is well-known to exhibit _a

depinning

critical transition at some critical value of the

driving

field [35].

Again, driving

the CDW _{at a constant} very small

velocity

creates _an SOC system ^with a power law distribution of

sliding

events.

2.5. FRACTAL GROWTH PRocEssEs. It has been

proposed

that fractal

growth

_processes,

such _as

diffusion-limited-aggregation (DLA),

fluid imbibition _as

exemplified

for instance

by

invasion

percolation,

dendritic

growth,

dielectric

breakdown,

rupture in random

media,

_con-

stitute another Mass of systems

exhibiting

SOC

[36-38].

These

growth problems

_seem however _to

belong

to _a diiferent Mass of

phenomena

thon

sandpile

models since e-g- trie internai

geometrical

structure of _an aggregate is

quenched (and only

its

perimeter

is

active)

whereas trie critical state of _a

sandpile

is

continuously rearranging

under trie action of externat

forcing.

Here,

_we want to

point

out that _a

stronger similarity

_emerges when

using

trie

proposed

framework and trie fractal

self-organization

of these

growth

_{processes con} be linked to trie existence of _an

underlying

critical

point.

We shall illustrate _our ideas _on trie annealed dielectric breakdown model in _a

cylindrical

geometry

(best

suited to define _a

steady

state

regime),

which

is known to be

equivalent

to DLA [39]. ^{Trie base of trie}

cylinder

is fixed at

potential

o and its other end is fixed _{at some non-zero} value V. Trie rate of

growth

from trie base is assumed to be _a stochastic _process controlled

by

_a

probability proportional

to trie electric field _on trie base

growth

surface. Trie DLA model is recovered in trie limit where the

growth

is done

quasi- statically.

This situation

corresponds

to

fixing

trie

growth

rate

(equivalently

trie

particle

flux

m trie DLA

model)

J _- o+.

Let _{us now} consider trie _case where _we

impose

_a constant average

potential gradient -dV/dz,

1.e. electric field

E, along

trie

cylmder

axis.

Furthermore,

[et _{us assume} that dielectric break-

down,

1.e.

growth

_{on a} site, can

only

_occur above _a certain

threshold,

which is _a

quenched

random variable distributed

according

to a

given

distribution. This condition _ensures that trie

growth problem

_now becomes similar to a

pinned-depinned

transition. For those sites above their

threshold,

trie rate of

growth

is

again

assumed to be _a stochastic process controlled

by

a

probability proportional

to trie electric field. If trie

applied

electric field is very

small,

_a few sites will break down and trie

growth

duster will evolve until all sites _are below their threshold.

This is trie bound state.

Increasing

trie

applied

electric field above _a certain threshold E~

(a

function of trie threshold

distribution),

trie system

begms

to grow in _an unbound _way at _a finite rate which increases _as E gets

larger.

Trie finite

growth velocity

is determined

by

the details of the

dynamical

breakdown processes

(which

_we do not describe

here).

Note that _a finite

velocity corresponds

to a limite

particle

flux in trie DLA model. For trie critical transition, trie control

(resp. order)

parameter ^{is trie electric field} or

partiale

concentration

gradient (resp.

trie

growth velocity

or

particle flux).

In trie

re#me

above E~, trie

growing

clusters do not exhibit

self-similarity

at all

scales,

but _a finite correlation

length

_appears ^which is _a

decreasing

function of trie

growth

rate. In trie

language

^{of fluid}

interfaces, pushed

with _an average

velocity

_c, trie

relevant dimensionless number is trie ratio of trie Bemoulli

velocity

_pressure

pv~

to trie

Laplace

pressure

a/b (similar

to a Bond

number),

where _{p is} trie

pushing

fluid

density,

_a is trie surface tension and b trie Hele-Shaw cell thickness _or trie interface radius of curvature.

In summary, the fractal structure of DLA dusters _can be viewed _as trie result of trie quasi- static

regime,

1.e. the control of the order parameter

(growth rate)

of the

corresponding

^critical

transition at _an infinitesimal

(positive)

value.

Similar

mappings

_can be defined for the invasion

percolation problem

and rupture

problems.

In this respect, it is

interesting

to realize that the

quasi-static

rupture ^models

extensively

studied in the literature [40] are characterized

by

the rupture ^{of elements} one

by

_{one, i-e-}

by controlling

the rate of rupture

(order parameter)

to _a

vanishingly

small value.

Truly dynamical

models of rupture [41] are in this _sense

beyond

the unstable critical

point, separating

the stable finite

damage phase

from the

fully

rupture

phase.

2.6. THE FOREST-FIRE MODEL. The

self-organized

critical forest-tire model introduced

by

Drossel and Schwabel [13, 42] ^{is another}

example

of the

proposed

^mechanism. The model is defined _{on a} d-dimensional

lattice,

where each site is either empty, ^tree or tire. The lattice _is

updated synchronously according

to trie

following

four rules:

1)

A tree will grow on an empty site with

probability

_p;

2)

Fire becomes empty;

3)

Fire

spreads

to _u-n trees and

4)

A tree catches tire

spontaneously

with

probability f.

Trie existence of _a critical

point

in trie limit

f/p

_- o is

expected,

_smce trie average number of _trees

destroyed by

_a

lightning

is

(si

=

f/p)~l (1- pt)/Pt (pt

is trie _mean tree

density) provided f

< p <

1/T(s)

_[42], where

T(s)

is trie _average time it takes to bum _a tree cluster of size _s. This

separation

of time scales is

again nothing

^but

a condition of slow

dTiuing

_common for SOC.models.

Furthermore,

_{pt <} 1 in order for

(si

to

diverge.

This then suggests _pt as trie control parameter for trie critical transition in

agreement

with trie

proposed

framework.

Changing

trie first rule of trie model allows _{one to tune} pt ^to its critical value: Each time _a tree is bumed down _{a new} tree _is put at random thus

keeping

_pt

fixed to _a controlled value. Trie order parameter ^{is then trie}

density

of tires _pf, trie condition

f/p

_- o above

corresponding

to pf =

o+. In trie _case of

f

=

o,

trie forest tire _con

only

propagate on

dynamically

connected tree

clusters,

thus

defining

_a critical tree

density

below which _a forest tire will

finally extinguish

itself and bum forever above. It is then dear that trie

parameter

f plays

trie role of _an external field thus

modifying

trie transition and

explaining

trie

systematic

deviations from _{a pure power} law _seen in simulations

[42, 43].

3.

Concluding

Remarks

1. We bave

proposed

_a

general conceptual

framework for

self-organized criticality,

which consists _m

mapping

^SOC onto unstable critical

points,

controlled

by driving

trie _corre-

sponding

order parameter at an infinitesimal value.

Our present

theory

clarifies and extends trie

previously recognized

mechanism of _a feed- back of trie order parameter on trie control parameter as

being

essential for SOC [19].

Trie

mapping

between SOC and usual critical transitions oifers _a novel but natural route to

study

^further trie

properties

associated with

SOC, namely by characterizing

^{trie critical}

properties

of trie

underlying

critical

point

itself. In _a way, we bave

only displaced

trie search for trie

underlying

mechanism for SOC _to that of trie _appearance of unstable critical points.

However,

_we feel that this is _a

significant improvement

for two _reasons:

1)

some

of trie

underlying

unstable critical

points

_are known and bave been documented _{per se.}

Their

knowledge

_can thus shed _new

light

_on trie SOC models. In

particular,

trie present framework illummates trie

physical meaning

of the slow

driving

_common to all systems

exhibiting

SOC.

2)

Even if the

underlying

critical point ^is _new, ^its

study

_con

probably

be

quite

efficient

by employing

the

large

toolbox

developed

in trie last twenty _{or more}

years in this field.

2.

SupeTcTitical bifuTcations:

Note that _our framework

applies directly

to

supercritical

and

Hopf bifurcations,

when

considering

trie situation where trie order parameter is controlled at trie value o+. _{This situation}

can be modelled

analytically by writing

_a

general (possibly complex) Landau.Ginzburg equation

for trie order parameter

fluctuations,

conditioned

to bave _a

vanishingly

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

3. RenoTmalization gToup and

fized-scale tTansfoTmation: Controlling

trie order parameter J _- o+ _of

an unstable critical

point

does not allow for _a standard renormalization group

procedure. Indeed,

in this situation, there is _no control parameter ^{and trie} critical _expo-

nents cannot be obtained

by

trie standard

procedure

in terms of trie derivatives of trie

renormalization group transformation of trie control parameters under scale transforma- tion. Thus, trie exponents related to trie distance from trie critical point do not exist.

Let _us stress that this is not due to _a

special

attractive property of trie critical

point,

as claimed in [44], ^{but results}

solely

from trie

special driving

conditions which _ensure trie exact

positioning

of trie

dynamics

_on trie unstable critical

point.

In this

situation,

_a

generalized

renormalization

procedure

should reflect the _exact

positioning

_{on a} critical

point,

1-e- should

give

trie

signature

of _an attTactiue critical point. This is indeed boni out in trie renormalization _group

procedure

introduced in [44]. ^{We believe} that

finding

a

general

renormalization

theory

for systems

right

at their critical

point

could

provide

a new class of tools for

general

critical

phenomena.

This could be related to conformai invariance

theory

_[45].

4. Relation with Goldstone modes: Our

approach

allows _{us to}

clarify

trie

proposai

_[46] that SOC stems from trie non-linear

dynamics

of Goldstone

modes,

however

by returning

trie

logic

of Obukhov's argument:

criticality

in SOC is not trie eifect of interaction of Gold-

stone

gapless

modes _as

claimed;

the

gapless

^modes result rather from the

underlying

unstable critical

point,

stabilized

by

the

special driving

condition. Let _us recall

that,

in trie

long-wavelength limit,

Goldstone modes reduce to _a

homogeneous displacement

of trie

sample

_or to _a uniform rotation of trie whole spin system. ^Since _an ^SOC system is

right

at trie critical point ^of a standard unstable cntical

phase

transition characterized

by

_a spontaneous symmetry

breaking,

the avalanches _cari be viewed _as trie Goldstone

fluctuations (1.e.

large

scale

displacements) attempting

to restore trie broken symme- try. In trie _case of _a discrete symmetry

breaking,

^{trie avalanches}

correspond

to

droplet

fluctuations [47].

5. Relation with

singulaT diffusion:

The

singular

diffusion property ^of continuous

equations

obtained

by taking

trie

hydrodynamic

limit of SOC models [48] ^derives

straightforwardly

from _our framework, since it is trie direct

signature

of _precise localization _{at an} unstable critical

point. Thus,

ail theories of SOC in terms of

singular

^diffusion

[48,

49] are

only

^trie expression ^{that trie}

goveming equation

is that of _a system

sitting precisely

at _a critical

point.

Let _us recall

that,

_more

generally, singular

^diffusion _{occurs on} trie

approach

to any

supercritical bifurcation,

trie best known

example being

trie

Rayleigh-Bénard instability.

In this _case, trie order parameter is trie _average convection

velocity

and trie fluctuations

are associated with streaks _or

patches

of _non-zero

velocity occurring

below trie critical

Rayleigh

number R _at which

global

convection starts off. Trie

spatial

diffusion coefficient

D(R) diverges

as

D(R)

~J

(R~ R)~~/2

and reflects trie existence of _very

large velocity

fluctuations. A

scaling

argument _can be written to get this

powerlaw

_[Soi.

Similarly,

trie

singular

^diffusion found in SOC models _can also be derived from similar

scaling

reasoning

[51],

showing

trie _common

ongin

^{of trie}

singular

diffusion.

6. Our

proposed

framework is not in contradiction with trie recent

recognition by

several authors

[8,15-18]

that SOC is

deeply

connected with trie mechanism of

partial synchro-

nization of relaxation oscillators with thresholds. Under _a slow

driving

of trie order parameter, ^{each threshold element}

(for

instance _a column _m trie

sandpile)

taken in isola- tion

undergoes periodic

oscillations of relaxation. The

complete problem corresponds

to descnbe the result of the

coupling

between these oscillations in terms of _a

competition

between

synchronization

and

desynchronization

eifects. In other

words,

the

dynamics

of relaxation oscillators describes the detailed _response of the critical point ^{under the slow}

driving

of the order parameter.

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