Decay of long-ranlye field fluctuations induced by random structures: a unified spectral approach

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random structures: a unified spectral approach

Didier Sornette

To cite this version:

Didier Sornette. Decay of long-ranlye field fluctuations induced by random structures: a uni- fied spectral approach. Journal de Physique I, EDP Sciences, 1993, 3 (11), pp.2161-2170.

�10.1051/jp1:1993238�. �jpa-00246860�

J. Phys. I IFance 3

(1993)

2161-2170 NOVEMBER1993, PAGE 2161

Classification Physics Abstracts

02.90 41.10D 47.15 61.70

Decay of long-range field fluctuations induced by random structures:

_a

unified spectral approach

Didier Sornette

Laboratoire de Physique de la Matibre

Condens6e(*),

Universit6 de Nice-Sophia Antipolis,

B-P- 71, Parc Valrose, 06108 Nice Cedex 2, France

(Received

14 June 1993, revised 9 August 1993, accepted 23 August

1993)

Abstract The problem of the determination of the powerlaw decay of the standard deviation

a~

+~ z~" of the fluctuations of the field generated by _a random _array of elements

(multipoles,

ensemble of dislocations,

etc.)

_{as a} function of the distance _z from the _array is reduced to

the determination of two quantities:

I)

the spectral _power of the disorder in the low k limit

and 2) the structure of the Green function, _{as a} function of wavenumber and distance, for _a

periodic _array of the constituting elements. We thus

recover straightforwardly all results known

previously and derive

new ones for _more general constitutive elements. The general expression of the decay exponent is found to be

o = 3 fl + 2(b

c),

where fl characterizes the self-affine

structure of the disorder (fl ₌ 2 _: strong disorder and fl

= 0 _: weak

disorder)

and the exponents b and _{c are} the exponents ^{of the} algebraic powerlaw corrections, in wavenumber and distance

respectively, to the dominating exponential decay of the Green function for _a periodic _array

of the constituting elements. The proposed spectral method solves automatically the generally

difficult problem of renormalization and screening in arbitrary random structures.

1 Introduction.

"Self-screening"

of

periodic

assemblies of elements whose individual influence is

long-range

is well-known. Consider for instance _a

periodic

column _or

plane

of electric

dipoles.

Even

though

the field of each individual

dipole decays

_as I

/(distance)~,

the total field

decays exponentially.

This is due to the almost exact cancellation between the

angular

structure of the

dipole

fields.

This effect is also well-known for

periodic

_arrays of dislocations

iii

and for

periodic boundary

conditions [2].

Recently,

_a number of _papers have addressed the

problem

of the robustness of this cancellation in presence of disorder

[3-7].

The

highlight

of these works is _as follows: for weak disorder around an average

periodicity,

the ensemble average field still

decays

_exponen-

tially.

However, the standard deviation a~

or second moment of the

field,

which

quantifies

the

(*) CNRS URA190

amplitude

of its

fluctuations, decays only algebraically

as

(distance)~~

a ~-< b2

>1/2 z-3/2, ₍₁₎

where _< b~ _> is the _{mean square}

amplitude

of the fluctuations. This slow _power law

decay (I)

of the field fluctuations is due to the breakdown of the almost exact cancellation of the _sources at all

multipole

orders. An alternative view point _[5] is that Fourier components ^of

arbitrary large wavelengths

ⁱⁿ the _power spectrum of the _{source array appear} and _are

responsible

for

the slow

decay

of the field fluctuations.

Since small random fluctuations around _a

periodic

modulation _are

always

present ⁱⁿ na-

ture, ^the slow

decay

_as

(distance)~~/~

of the

typical

field fluctuations _must be

ubiquitous

in

nature. For instance, this has been

proposed

to be the _case for dislocations in Ti-Y-O

alloys,

in order to

explain

the

experimental

observations of attraction and dissolution of

precipitates by migrating grain

boundaries [8]. ^{In electro- and}

magneto-rheological

fluids [9], chains _are found to interact _more

strongly

than

predicted

from _a naive

prediction

based _{on a}

regular periodic

necklace structure. A small

polydispersity

of colloid size is

enough

to

produce

_some

disorder around the

periodic

structure. As _a consequence, the above mechanism _{must appear} and

give

rise to stronger

long-range

interactions.

Similarly,

the

problem

of the intermediate

wavelength magnetic

anomalies measured at various

heights

above the _{ocean crust} has

recently

been

proposed

to arise from the random structure of

geological polarity

reversals [10]. ^Another

interesting

domain of

application

is the

problem

of sedimentation of suspensions ^and _perme-

ability

of porous media. For

instance,

_one is interested in

characterizing

the sedimentation of

a cloud of

heavy particles

in _a fluid

II Ii.

The fluctuations of

particle speed

around the _average

speed

of the

sedimenting

cloud involves

taking

into account

correctly

the renormalization and

screening

^effects of all other

particles

_{on a}

given

test

particle,

_a

problem

which is still unsolved in

general.

Note however that the formalism

presented

in this _paper allows _one to obtain the

velocity

fluctuations _once the

configuration

of the

particles

is known

(through

the

spectral

power of the

disorder,

_see

below).

The

problem

of the renormalization and

screening

effects of all

particles

_{on a}

given

test

particle

is

automatically

solved

by decomposing

the random

configuration

of

particles

_{as a}

superposition

of

periodic

components

(Fourier theorem),

since

screening

at all orders is

completely

described for each

periodic configuration.

A

simple

_ex-

ample

similar to the electrostatic _cases considered below consists in the flow of _an irrotational

inviscid

low-Reynold

number fluid

through

_{an array} of holes in _a

plate,

_{or an array} of

spheres

or other elements.

Many

studies have addressed the determination of the flow in such

periodic

structures,

notably

in connection with the

development

of the

theory

of _porous media

[12].

For

our purpose,

simply

consider the _case of holes which _are small in

comparison

with their distance A.

Then,

_one has to solve the

Laplace equation

for the

velocity potential

# with the additional

boundary

condition

3#/3z (z=o= Vob(x nA) (dipole sources),

which is

equivalent

to _one of the electrostatic

probleqsjjtudied

below. For the _more

general

_case,

knowledge

of the

velocity

potential

field for _a

periidic

_array of elements

given by

the methods

previously developed

_[12]

allows _one in

principle

to get ^the characteristic of the fluctuations of the fluid

velocity

for _an

arbitrary quenched

random

configuration, using

the

spectral decomposition

described below.

Very recently,

the

question

of the

validity

of the law

(I)

in the presence of strong ^disorder has been addressed [13], ^{with the conclusion that it is modified into}

, ~

~-l/2 (~)

in the limit of _very

large

^{disorder. The}

question

arises about the

universality

of these results

(are

there

only

two

classes?)

and the

dependence

of the

decay

of

a(z)

for various types of

N°11 LONG-RANGE FIELD FLUCTUATIONS 2163

constituting

elements:

dislocations, dipoles, quadrupoles

_{or more}

complex

entities. In the _pre- vious

studies,

three different mathematical methods have been used: _a

perturbation approach

in

conjunction

with the Poisson summation formula [4, 5, 7], ^{brute force} calculations of the standard deviation of the field

expressed

_{as an} infinite series [3,

6,

_13] and _a

spectral

method [5]. ^{All of them} are consistent in the weak disorder

regime

and confirm the

validity

of

(I).

In the strong ^disorder

regime,

the first method in terms of the Poisson summation formula does not hold anymore since it relies _{upon a}

perturbative development

of the disorder around the

perfect periodic

structure. The other two methods _are in

principle applicable

to

arbitrary

situations, the

spectral

method

being

however _more

general

and

powerful

_{as we} wish _now to

demonstrate.

In this _{paper, we} first summarize

briefly

how

expression (I)

is obtained and

specify

the

meaning

of "weak" disorder.

Then,

_we present the very

general

form of the solution of _a

general problem

within the

spectral

method framework. This allows _us to _recover

simply

all

previous

results

including

the result

(2)

for the

large

disorder

regime

and furthermore _to derive

a new series of results for various types ^of constitutive elements. We find that the value of the exponent ^{of the} power law

controlling

the

decay

^{of the} standard deviation a~ of the

field,

which thus determines the

"universality class",

is controlled

by

two

properties: I)

the _power

spectrum ^{of the disorder} at low wavenumber and

2)

the structure of the constitutive elements.

2. Derivation of

expression (I)

and

meaning

of "weak disorder"

Consider the

simplest problem, illustrating

the

general

question addressed in this _paper, of _a

Laplacian

field V

obeying Laplace's equation (i7~V

=

0)

ⁱⁿ a semi-infinite medium bounded

by

_a frontier

38,

with

imposed boundary

values _{or sources} at 38. We take 38 to be the

plane (0x, 0y).

The semi-infinite medium extends from _z

= 0 to +cc. An alternate distribution of

sources

(+)

and

(-)

of

strength

~ So _are

arranged spatially

in the

plane (0x, 0y).

We first

study

a two-dimensional version of the

problem

and shall _return to the three-dimensional _case later. The set of _sources

(+)

and

(-)

_are assumed to be

spatially

disordered around _{an average}

periodic

modulation:

(+) xl

='l~ ₊

d$ (3a)

(-) xi

=

(2n

₊

1)A/2

+

bk (3b)

where

d$

and

b$

_are random variables which _are

independent

from site to site, have _zero average ^<

b)

_>= 0 and have _a variance <

b)b)

_>= 0 if _n

#

n' and _=< b~ _> for _n

= n'. is

the average

period

of the modulation. We _assume furthermore that

V(x,

_{z -}

+cc)

_- 0. The

problem

is thus to solve

+co

i7~V

= -47r So

b(z) ~j

_[b

_{(x xi)}

_b

_(x

_xi

_)] (4)

n=-co

where b is Dirac's function. Note that this formulation

incorporates

different

boundary

condi- tions, ^{for instance the} cases when the distribution of _sources is

replaced by

the

knowledge

of the field V at the

boundary

38. This mathematical

formulation,

which has the

advantage

of

simplicity, already

describes various

problems

of

gravity, magnetic,

temperature ^and resistivity

anomalies and will constitute the backbone of _our derivation.

When disorder is absent

(b)

= 0 for all

n),

_we

replace

the _set of _sources

by

the

boundary

condition

V(x, z)

₌

>So cos(27rx/A)

and _use the method of

separation

of variables

V(x, z)

₌

Z(z)cos(27rx/A)

in order to

satisfy

the

boundary

condition. Substitution in

i7~V

= 0 then

yields d~Z/dz~ _(47r~/A~)

Z ₌ 0 whose solution

obeying V(x,

_{z -}

+cc)

_- 0 is

V(x, z)

=

>So

V

cos(27rx/A)e~~"

^~/~

(5)

The field disturbance introduced

by

the

boundary

condition

(I) decays exponentially

with

depth

in _a distance

proportional

to the horizontal

wavelength

A.

When disorder is present, we use the Green function method. In

2D,

the solution of

equation i7~V

= -47r b

(x xo)

b

(z zo)

with

V(x,

_{z -}

+cc)

_- 0 is [15]

G(~,

_Z)

"

~L°g ((~ ^~0)~

^{+ (Z}

20)~j (6)

Using

this

expression

for the Green

function,

the

general

solution of

equation (4)

reads

formally

+co +co

-V(x, z) /So

₌

~ _{f~ (n}

₊

_{b$ IA)} ~j _{f~ (n}

₊

bill) (7)

n=-co n=-co

where

f+(t)

=

Log [(x tA)~

+ z~]

(8a)

f~(t)

=

Log [(x (2t +1)A/2)~

+ z~]

(8b)

Note that for

b)

= 0 for all _n, the series _can be summed _up

using

the

theory

of

analytical

functions

[Ref.

[15] ^{Tome II}

(1963)

_p.

_1236]

and

yields -V(x, z)/So

=

(Log[tan(27r(x+iz)/A)](~

which reduces to

V(x, z)

= >So V

cos(27rx/A)e~~"

^~/~ for

large z(»

A).

In order to estimate

V(x, z) given by

equation

(7),

_we

develop

each term

f~ (n

₊

b) IA)

_up

to second order in

b) _IA,

_since the disorder is assumed _to be small:

i~ _(n

₊

bill)

=

i~(n)

₊

jbj _IA) di~(n)/dn

₊

_(1/2) jbj _/A)~ d~f~(n)/dn~

₊

(9)

The _{average <}

V(x, z)

> over the disorder reduces to

<

V(x, z)

>

/So

"

f f(n)

+

(1/2) <

(b/A)~

>

f

d~

(n)/dn~j

+..

(10)

n=-co n=-co

with

f(n)

=

f+(n) f~(n) (ii)

using

<

b) _IA

_{>= 0, and <}

b)b$

_>= 0 if _n

#

^n'and <

(b$ /A)~

>=<

(bj /A)~

>+<

(b/A)~

>

Using

Poisson's summation rule:

+m +c~ +m

~j _f(n)

=

~j / f(t) e~"~~dt, (12)

n=-co k=-co ^~"

one can write

equation (10)

_as

<

V(x, z)

_>

/So

"

/

_dt

_[f(t)+

<

(b/A)~

> d~

f/dt~

_{+.. +}

~~

+2

f _j~"

ji(1)+

<

(~/>)2

>

d21/d12

+..

cos(2~rki)

di.

(13)

~=i ^-co

N°11 LONG-RANGE FIELD FLUCTUATIONS ₂₁₆₅

Averaging

the

Laplacian

field _amounts

essentially

to

recovering

the

periodic

_case with _a small

perturbation _proportional

to the second _moment

(< (b/A)~

>«

l)

of the disorder. For

f(t) given by equation (II

with

(8),

the first

integral

in the r-h-s- of

equation (13)

is

identically

zero since

f+(t)

and

f~(t)

cancel

exactly.

The

leading

behavior of

V(x, z)

is thus

given by

the first

(k

=

I)

term in equation

(13)

which _can be shown, ^after some tedious

calculations,

to recover

expression (5).

The fact that the k

= 0 contribution vanishes in this _case is the mathematical translation of the mutual

screening

at all

multipole

orders of the field

produced

by

each _source in the

periodic

_case.

Thus, averaging

the

Laplacian

field would _seem to

imply

that the effect of fluctuations is

negligible.

In

fact,

it is

quantified by

the second moment bV~ of

V(x, z)

defined

by

bV~

+

j< _{jV(x, z)j~}

_{> <}

_{V(x, z)} >~j /Sj

_=<

(b/A)~

>

f id I(n)/dnj~

₊

(14)

The k

= 0 term in Poisson's summation rule

(12) applied

to the _sum in the r-h-s- of equation

(14)

is _now

proportional

to

(< [V(x, z)]~

> <

V(x, z) >~) /S(

_=<

(d/A)~

>

/

id

f(t)/dt]~

_{dt +..}

₍₁₅₎

which cannot be

equal

to zero since d

f(t) /dt

is _{non zero.} The remarkable fact is that this _term

produces

_a

long-range powerlaw decay

_{as a} function of _z.

Using

the substitution T

= At _x,

the

integral

in the r-h-s- of equation

(15)

reads

/ _{id f(t) /dt]~}

_dt

=

/

_dT

_{[T/ (T~}

+

z~)] [(T

+

A/2) / ((T

+

A/2)~

₊

z~)] )~ (16)

which

yields

dv/so

_a

j< jv(z, z)j2

> <

v(z, z) >2j~/~ /so

= c _<

(d/>)2 ^>1/2 (z/>)-3/2

₊ o

jz-2)

(17)

where C is _a numerical factor of order

unity. Expression (17)

exhibits the announced slow

algebraic decay

of the

Laplacian

field fluctuations induced

by

the disorder. It is remarkable that _any

non-vanishing

disorder _as weak _{as can} be generates the "universal"

z~3/~ decay

of

the _r-m-s- fluctuations of the

Laplacian

field. The

strength

of the disorder affects

only

the

prefactor

of the _power law

decay.

3. General

spectral

formulation and solution of the random _source problem.

We now consider _a

general problem,

electric, elastic _or other, characterized

by

the value of the Green function

G(x x', z) giving

the field at

position (x, z)

created

by

_a unit element

placed

at

(x', 0). Here,

the x-coordinates denotes the

position

of _{a source}

transversally

to the axis _z.

The Green function

G(x x', z) fully

characterizes the

given problem.

The element _can be _a

single

_source, a

dipole,

_a

multipole,

_a dislocation... For instance, for the electrostatic

problem

in which the

potential

as well _as the field

obeys Laplace's equation (i7~V

₌

0)

,

G(x x', z)

=

-Log [(x x')~

+ z~] ^{for the}

potential

in 2D and minus its

gradient

for the electric

field,

if the elements _are

monopoles.

The distribution of element sources in the

plane

_{z =} 0 is described

by

the function

p(x').

We do not need here _to

specify

if it is

periodic, weakly

^random _or otherwise.

Then,

the formal solution of the field is

given by

V(x, z)

=

p(x') G(x x', z)

^dx'.

(18)

This

expression

consists of _a convolution between the distribution of _sources and the Green function which

depends

_on the

depth

_z of the _source

layer

beneath the observation

plane.

Convolution in the space domain becomes _a

multiplication

in the Fourier domain, _so that _we have

v(k,z)

_m

p(k) G(k,z), (19)

where k is the wavenumber in the

x-plane.

Note that the Fourier transform

G(k,z)

of the Green function is

nothing

but the field V created

by

_a

periodic

_array of elements with

periods 27r/k~

and

27r/ky

^{in the} x and _y directions

respectively.

For suitable

elements,

such _as

dipoles,

dislocations,

etc.,

G(k, z) decays exponentially

with _{z as} [2,5]

(see

also Sect.

2) G(k,z)

r~

e~~~, (20)

where k

=

(k(,

thus

recovering

the

general "self-Screening"

property of

periodic

systems of

dipoles

_or dislocations discussed above.

By

inverse Fourier

transform, expression (19)

trans- forms into

V(x,z)

=

/ p(k) G(k,z)

e~~~ dk.

(21)

The ensemble average field and the second moment of the field _are

given respectively by

<

V(x, z)

>=

/

<

p(k)

_>

G(k, z)

e~~~

dk, (22)

and

a~

_+<

[V(x, z)]~

> <

V(x,z) ^>~= / /

<

p(k) p*(k')

_>

G(k, z) G*(k', z) e~~~~~'~~dkdk', (23)

where the

symbol

* stands for the

complex conjugate.

Consideration of these _two equations allows _us to _recover

previous

^results

rapidly.

In

periodic

systems _or weak disordered system,

<

p(k)

>= po d

(k ko),

which

together

with

(20) yields

the

exponential decay

<

V(x, z)

>r~

e~~°~e~~°~,

characteristic of the

"self-creening"

property. Note that _we exclude here the _case where the

spatial

_average

yields

a non-zero contribution at k

= 0,

implying

_a

non-vanishing

source

density

in the continuous limit. Our discussion

applies only

to the contributions at

finite wavevectors, the k ₌ 0 contribution

being easily

taken into account in the continuous limit. For weak disorder such that the power spectrum <

p(k)p(k')

> of the distribution

p(x')

is

given by

<

p(k) p(k')

>r~

k~b(k k')

_{[5, 14], we} obtain a~

r~

z~3 _{at distances}

z »

kp~,

which _recovers

(I).

Note that the _z

dependence

of the field and of its fluctuations at

large

distances _z is controlled

by

the behavior of these

integrals (21-23)

in the low k domain. As _a consequence, we can

neglect

the

x-dependence

in these

integrals,

since the

exponential

^{e~~~ in} equation

(21)

becomes unity for k « x~~.

Anyway,

the

x-dependence

is washed _out

by

the ensemble

averaging yielding ,2 since,

in

general,

<

p(k) p(k')

>r~

b(k k')

due to the property ^of _average translation

invariance

(average

^uniform

spatially

distributed

disorder)

in _an infinite system.

N°11 LONG-RANGE FIELD FLUCTUATIONS 2167

These considerations allow _us to

give

the

general

rules

controlling

^the

decay

of the standard deviation a~ of the field fluctuations. If _a

given physical problem

is such that the _power spectrum ^{of the disorder} is

given by

<

p(k) p(k')

_>m

kab(k k'), ₍₂₄₎

and the field created

by

_a

periodic

_array of elements of _wavevector k is

given by

G(k, z)

_cs

k~z~e~~~, (25)

where _a, b and _{c are} in

general positive

but

possibly negative

exponents, ^{then from} expression

(23)

we

immediately

get ^from power

counting

a~

r~

z~°, (26)

(27)

where

~ = i + a +

2(b

_C).

Several comments _are in order.

First,

expression

(27)

shows that the

universality

class of the standard deviation

decay problem

is

entirely

determined

by

three exponents _a, b and _c,

defined

by I)

the

spectral

content of the disorder at low wavenumber

given by expression (24)

which

gives

the value of exponent "a" and

2)

the structure of the constitutive elements and the

nature of the

physical problem,

which determines the kernel structure

given by equation (25)

and which

gives

the exponents "b" and "c". It is clear from

expression (25)

_or

(20)

that the characteristic

decay length

of _a

spectral

_component of wavenumber k is of the order of k~~

Thus,

the

powerlaw decay

of a~

can be traced back to the existence of _very low wavenumbers in the power spectrum of the disorder. Disorder destroys the exact cancellation of all

multipoles

and introduces _a continuous spectrum down to k _- 0.

Second,

_a

given

value of _a does not need to

correspond

to the _same

physics,

since there _are many combinations of _a, b and _c for _a

given

value of the

decay

exponent _a. For

instance,

the

case a = 3 discussed in sections I and 2 is recovered for _a

= 2 and b _{= c}

= 0, which _was the

case studied in reference [5] for a system of

planar dipoles

ⁱⁿ electrostatics. The _same value

a = 3 is also obtained for _a

planar

lattice of dislocations in the

elasticity problem

[4, 7] for which _a

= 2 and b

= c = I. More

generally,

_we expect ^{that the}

decay

exponent a may be modified either due to _a

cl~ange

of the

spectral

content of the disorder or due to the

multipole

nature of the constitutive element in the _{source array.}

4. Limit of

large

disorder and other _cases.

The strong ^disorder case considered in [13]

corresponds

to

assuming

that each constitutive element

(in

this _{case, a}

dislocation)

_can have _any

position

within the

plane.

This

corresponds

to a

complete

decorrelation of the position of the dislocations. In this case, the power spectrum of the disorder reads

<

p(k) p(k')

>r~ En Em < e~~~" e~~'~m _>,

(28)

where _xz~ is the position of the nth element in the _source

plane (z

=

0).

For uncorrelated positions of the

elements,

all the "interference" _terms in the double _sum such that _n

#

_m

average to _zero since < e~~X» e~~'Xm _>=< e~~X» _> < e'~'Xm _>= 0 if _m

#

_n.

Expression (28)

reduces to

<

p(k) p(k')

_>

r~ En < e~~~~~'~~» _>,

(29)

~ ll ~

~i(k-k')x

_~

_(~~)

due to the

independence

of the

positions

of the elements and the fact that _{we assume} that

they

are

identically spatially

distributed. N is the number of elements _per unit

length.

In _an infinite

domain,

< e~(~~~'~~ _>=

b(k k'), leading

to a value _a

= 0 for the disorder exponent ^{defined in}

equation (24).

As _a consequence, for the _same

physical problem (same

b and _c

exponents),

_we

find that the

decay

exponent a for the standard deviation a~ of the field fluctuations _is reduced

by

2 in the

large

disorder limit, in comparison to the weak disordered _case. In

particular,

for

b _c

= 0

(electrostatic dipoles

_or

dislocations),

_{we recover} the announced result [13]

given by expression (2):

^a~ _r~ ^z~~ Note that the present

spectral

method also allows _us to address

the _more

general

_case of _an

arbitrary

disorder and to

quantify

the _{cross over} from weak to

strong ^disorder.

Mathematically,

this is done

by computing expression (28)

for the _case at hand characterized

by

the

corresponding spatial

distribution of _xz~.

It is

maybe

useful to list a few

problem archetypes

and their

corresponding decay

laws. The first

simplest problem

is that of _a

planar

_array of

monopole

_sources. The _case of _a

periodic

array with

alternating signs yields

the well-known

exponential decay V(x,z)

r~

e~k°z

e~k°~

For _a weak disorder around this

periodic

structure, a = 2 and b

= c = 0 which

yields

_{a =} 3.

For _a strong ^{disorder with} a = 0, _we obtain _a

= I.

Suppose

_now that the

monopoles

_are

grouped

in pairs so as to constitute

dipoles lying parallel

to the

plane

to which

they belong.

Summing

the

dipole

field _{over a}

periodic

_array of

dislocation, using

for instance the Poisson summation formula,

yields again

b _{= c =} 0. The _same result holds for vertical

dipoles

which

are

perpendicular

to the

plane

to which

they belong.

The

decay

exponent o is thus

uniquely given

in terms of the exponent "a" determined from the _nature of the disorder. We

again

recover a = 3 and _a

= I for the weak and strong ^disorder cases,

respectively.

Let _us consider _{now an array} of electric

quadrupoles.

A

given quadrupole

_can be constituted

by approaching

two horizontal

dipoles

of

opposite

direction to within _a close

distance,

_say

d,

in the z-vertical direction. The kernel

Gquadrupoie(k, z)

^{for the}

^periodic

array of

quadrupoles

is then

given by

Gquadrupole(k,

Z) =

Gdipole(k,Z

+

d) Gdipole(k,

Z)

* kd

3Gdipole(k,

Z)

/32. (31)

This

expression (31)

shows that b ₌ I and _c

= 0 for the system of

quadrupoles.

Then this

yields

_o

= 5 and _a

= 3 for the weak and strong ^disorder _cases,

respectively. Generalizing

this

reasoning

to

arbitrary multipoles

of order

p(p

= 0 _:

monopole,

_{p =} I _:

dipole,

_{p =} 2

quadrupole,

etc.

),

_we obtain b

=

Max(p

_-1,

0)

and thus _o

= 3 +

2(p I)

and _{a =} 1+

2(p I)

respectively

for the weak and strong disorder _cases, for _p > 1.

5. Conclusion.

The

problem

of the determination of the

decay

law for the standard deviation of the fluctuations of the field

generated by

_a random _array of elements

(multipoles,

ensemble of

dislocations,

etc.

has been reduced to the determination of _two quantities:

I)

the

spectral

_{power [14]} of disorder in the low k limit and

2)

the structure of the Green function

[15],

as a function of wavenumber

and

distance,

for _a

periodic

_array of the

constituting

^elements. In turn, ^this Green function _can be evaluated

easily

from the

knowledge

of the Green function for _a

single

element and from the

use of the Poisson summation formula

allowing

the discrete

periodic

_sum to be transformed into _a continuous

integral

_over the

plane,

_or more

generally

_over the manifold,

containing

the _sources. In

general,

this continuous

integral

has the form of _a Fourier transform which

generically yields

_a spatial

dependence given by equation (25),

controlled

by

_an

exponential

decay

decorated

by algebraic

corrections in powers of k and _z.

N°11 LONG-RANGE FIELD FLUCTUATIONS ₂₁₆₉

This formulation is

completely general

and

quite powerful technically

since it has allowed

us to _recover

straightforwardly

all results known

previously

and _to derive _{new ones} for _more

general constituting

elements. It _can be

easily adapted

_{to any}

physical problem,

_as

long

_as

the Green function for _a

single

_source is known. Its second interest is in the

particularly

clear

physical picture

that it

provides

in order to understand the

origin

of the slow

algebraic

powerlaw

decay

of the field fluctuations. In this

framework,

it stems from the creation

by

disorder of _very low wavenumber components ^which

provide

the

dominating

contribution _to the

decaying

field at

large

distances. In the weak disorder _case, the _appearance of such low wavenumber components in the disorder spectrum results from the

progressive

destruction of the cancellation of the field

occurring

in the

periodic

case.

However,

the

picture

is _not restrictive to this limit and

fully

characterizes the

problem

from the

knowledge

of the disorder spectrum.

An

interesting

consequence is the

analysis

of _a different

c(ass

^of

disorder, exhibiting long

_range

"fractal-like" _power law correlations in space and characterized

by

_a spectrum

b(k k')

k~~~

for small k with 0 <

fl

<

2,

I-e- _a

= 2

fl, yielding

_a

decay

exponent a =

(3 fl)/2

_[5]. To summarize, the

general expression

of the

decay

_{exponent a} for the standard deviation a~ of the fluctuations of the field

generated by

_a random array of elements is a~

r~ z~" with

a = 3

fl

+

2(b c), (32)

where

fl

characterizes the self-affine structure of the disorder and exponents b and _{c are} defined

by expression (25).

The domain of

validity

of

expression (26)

with

(32)

is

typically

for distances

z

larger

than the average

period

in the weak disorder _case around _{an average}

periodicity.

In the strong ^disorder case, it is valid in _an ensemble _{average sense over many} disorder

configurations.

It is

possible

that _a

simple

_space

integral (or

_space

average)

_over the

in-plane

coordinate _x for

a

single

disorder

configuration

is

sufficient, implying

_some kind of

ergodic

property. This is

currently being

checked

by

numerical simulations [10].

Acknowledgements.

Stimulating

conversations with E.

Bouchaud,

J.-P.

Bouchaud,

G. Saada and A-A- Nazarov _are

acknowledged.

I _am

grateful

to D. Stauffer _as the editor for

suggesting

the sedimentation and

permeability problems

and for useful comments.

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ill

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