Replica field theory for composite media

HAL Id: jpa-00246861

https://hal.archives-ouvertes.fr/jpa-00246861

Submitted on 1 Jan 1993

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Replica field theory for composite media

M. Barthélémy, Henri Orland

To cite this version:

M. Barthélémy, Henri Orland. Replica field theory for composite media. Journal de Physique I, EDP

Sciences, 1993, 3 (11), pp.2171-2177. �10.1051/jp1:1993239�. �jpa-00246861�

Classification

Physics

^Abstracts

02.50-43.50D-77.00

Replica field theory for composite media

M.

Barth414my(~>*)

^{and H.}

Orland(~>*)

(~)

CEA-Centre d'6tudes de

Limeil-Valenton,

Service Th60rie et Etudes

Nouvelles,

94195

ViUeneuve-St-Georges Cedex,

France

(~)

CEA-Centre d'6tudes de

Saclay,

Service de

Physique Th60rique,

91191 Gif-sur-Yvette

Cedex,

France

(Received

28 June 1993, accepted 21 July

1993)

Abstract. In this paper, we use the

replica

trick in order to compute the effective per-

mittivity

of _a medium where the local

permittivity

is _a random

binary

variable. A Gaussian variational _treatment leads

us to self-consistent equations which _are solved and

yield

_a

replica diagonal

solution

(no replica

symmetry

breaking).

We obtain

an effective-medium formula _pre-

viously

derived by _a cumulant

expansion.

This formula is

satisfactory

from _many

points

of views

(Hashin

and Shtrikman

bounds,

low

density

expansion,

etc) showing

that the

replica

method does _not violate _some basic

principles.

1 Introduction.

One

speaks

about effective medium when _{one can} describe _a

heterogeneous

material _as

quasi- homogeneous (on

observation

length

much

greater

^the

typical

size of

inhomogeneities).

Determining

the effective-medium

properties

^{of disordered materials}

(such

_as

composite, suspensions,

etc. is _a very difficult

problem,

_even if _one knows

exactly

the

microscopic

_prop-

erties of the medium. From _a theoretical

point

of

view,

this

problem

is very

general

since it

amounts to

determining

the _average of _a

propagator,

a very common

problem

in the

physics

of

disordered

systems.

^This

problem

is also of

great importance

in

Engineering

Science because of its

large

number of

practical applications.

We will concentrate here _on the

problem

of the effective

permittivity

of _a

binary mixture,

but all the results

presented

here hold for other

physical quantities

such _as electrical and thermal

conductivity,

diffusion _constant and

magnetic permeability.

There is _a

large

number of effective-medium results available for this

problem (see

the reviews

ill

and [2] dud Refs.

therein).

Beside these

approximate theories,

there exist exact bounds which should be satisfied

(*)

Also at:

Groupe

de

Physique Statistique,

Universit6 de

Cergy-Pontoise,

47-49 Av. des

Genottes,

BP. 8428, 95806

Cergy-Pontoise Cedex,

France

2172 JOURNAL DE

PHYSIQUE

I N°11

by

all effective

quantities (Hashin

and Shtrikman

[3],

see also

[4]).

^In recent years, there have also been numerical studies about this

problem _{[5, 6].}

We propose here _{a new}

approach

based _on the _use of the

replica

trick. It _was first introduced within the framework of

spin glasses

_[7] where it _was used in order _to

compute

^the average of the free _energy. The

validity

of this

approach

relies _on the existence of _an

analytic

continuation when the number of

replicas

_goes to _zero.

Unfortunately,

in most cases, it is out of reach to

demonstrate the existence of such _a continuation and _we will not address this

problem

in this paper.

The

replica

method used

together

with _a variational method _was introduced

by

Muthukumar and Edwards [8]

(see

also

[9-11]). Here,

_{we use a} Gaussian ansatz and solve the self-consistent

equations

derived from the variational

principle.

In

part

² we introduce the model. In

part 3,

we introduce the

replica

trick in order to obtain

an effective hamiltonian

coupling

the

replicas.

To

study

this

Hamiltonian,

_{we use a} variational ansatz which

yields

self-consistent

equations.

^{We discuss} the solution of these

equations

^and conclude in

part

^4.

2. The model _: definition.

We assume that _a d-dimensional

microscopically isotropic

medium is

composed

of local scalar

permittivities e(r)

at

point

r which _are random

independent (from

site to

site)

variables dis- tributed

according

to _a

binary

law:

P(E)

₌

Pb

_{(E Ei +}

(I P)b

_{(E E2)}

The _mean value of

e(r)

will be denoted

by

_eo. Note that the _average medium is

isotropic

and

homogeneous.

We define the effective

permittivity

e* of the medium

through

the

following

relation:

(E(T)E(T)I

₌

E*(E(T)I (1)

where E is the electric field and _{r a} d-dimensional _vector and the brackets denote the

quenched averaged

_over

e(r).

We want here _to solve the Maxwell

equation

satisfied

by

the electric field:

div(e(r)E(r))

= p~~~

where _p~~~ creates the external field

applied

to the

sample.

Writing e(r)

_{= eo} +

be(r)

where be is the

fluctuating part off,

_{one can}

easily

show that the variations of the electric field _are

governed by

the

following equation _[12]:

E, (ri)

" Eo>

(ri)

₊

~j / _{dr2G,j (r12)}

_be

(r2) Ej (r2) (2)

~

~

where

G~

is the

dipolar

_tensor and where the

quantity Eo depends only

_on the

boundary

conditions and is assumed to be uniform. Let _us note that the model has to be considered _as the continuous limit of _a discretized _one in order _to avoid short-distance

singularities.

There

exists _an

underlying

cut-off

(denoted by a)

which is of order of the size of the

inhomogeneities.

We will _use the Fourier transform of

G~(r) given by:

~

_(~)

_~

~l~j

(~)

lJ ~ ~2

the value of which is

equal

to

~~

at k

= 0.

deo

Averaging equation (2)

_over the disorder

(the averaged

^fields do not

depend

_on

space),

_one

obtains:

~~~

~

l

+~i

^~~~

where be*

= e* _eo.

Before

averaging, equation (2)

_can be written in the

following

_way:

E,(r)

=

dr' M,j(r, r')Eoj(r') (5)

(we

_{sum over}

repeated indices)

where the kernel M is

given

_as the inverse of the operator:

Mj~(r, ^r')

₌

b,jb(r r') G,j(r r')be(r') ₍₆₎

After

averaging, equation (5)

reads:

(E~)

_m

/ _{dr'im,~ jr,}

r'))j ^{Eo~ (7)}

From obvious translational invariance of the

averaged medium, (M) depends only

_{on r} r' and is

diagonal

in I indices. We _are thus interested in the k

= 0 limit of the Fourier transform of

(M)

im~-o (M~ (k)j

₌

~i~, (8)

1 + m

3.

Replicas

and the variational treatment.

3. I REPLICA TRICK. We _can write M

using auxiliary

scalar fields:

M~(r, ri~

₌

' IIil~>i >i(T)>i(T') e-1'

~~~/

L,>

',~~) Mi~i~>~~>4>(~~)

j IIivjj

^e-

^I f

_d~d~'

L,,

4,(~)

Mji(~,~,)i,(~,) (9)

In order _to

compute

the _average of this

quantity,

_we write the denominator _as the limit when

n goes ^to zero of:

j

_{~ p}

_{~ ~~} ^{l f}

^d~d~'

^L,,

^4,

^(~)Mj~

(~,~')4>

~')j

^{~ ~}

i i

We thus introduce _n

replicas

of the fields

ii

which will be denoted

by ii (a

₌

I,

...,

n).

We first consider _{n as an}

integer

and then take the limit _n

going

to _zero at the end of the calculation.

We _{can now} write M _as:

Mzj(T, T')

"

/ nz,aD#1(~) ~f

^~

x e-

i J

d~d~'

L,,,~

Ii (~) lb,> 6(~-~')-G,>

(~-/)6~(~')l '1(~'> lo)

where the limit _{n -} 0 is

implicitly

taken and where

41

=

(#], if,

,

#/)

2174 JOURNAL DE

PHYSIQUE

I N°11

It is _now easy ^to average the

operator

M _over the disorder. We obtain the

following

result:

M~j

(T,

T'))

^"

/ IIi,aD#I(T) ~f~ ^~~~"

~&)

where the effective Han~iltonian is:

~in

₌

I /

_dT

_{~j (#I(T))~} ^(~ _A~ / _{dTdT' ~j II}

(T)G>j

(T

T')#](T')

i,a i,j,a

(lib)

-Ad dro log

I ₊ Ae%

f

~~

£,,>,a

~~~~°'~"

~~°~~'~~~~~'

where A is _a cut-off in the Fourier _space of order I

la.

In the

following,

_we will take A

equal

to

unity (this just

renormalizes the

permittivities).

A is

equal

to

be2 bei

and ₌

(I p)/p.

As usual in the

replica method,

the

averaging

_over the disorder introduces _a

coupling

between different

replicas.

In order to

study

this

complicated Hamiltonian,

_we will _{use a} variational

method.

3. 2 VARIATIONAL METHOD. For the sake of

simplicity

we shall _{assume a}

replica symmetric

solution. We address the

problem

of

replica symmetry breaking

at the end of this section. We therefore _{assume a}

diagonal

Gaussian ansatz in

replica

_space, but _we shall make _no further

assumption concerning

other indices:

Zo

"

/ _fl,aD#]e~ ^{I f}

^~~~~

£,,>,a ~~~~~i> ~~~~~'~f~~'~ (12)

As mentioned

previously,

_{we are} interested in the k

= 0 limit of the Fourier transform of

K;j (r).

The variational _energy is:

F[K]

= const +

(7in)o

+

Fo (13)

where

)o

^denotes an average

using Zo

and where

Fo

₌

-log Zo.

The variational free _{energy can} be written _as:

~

~ / ~~~d

~

~°~ ~~~~

_~

~ / ~~~d _~~~~~

~~

_~~~

/ ~~~d _~~~~~~~~~

/ _dro

~~~~

(log I

+

Ae% f ~~'£,,>,a ~~~°~~"~~°~~'~~~~~'~j

o

where fl is the volume of the

sample.

The calculation of this last term

(denoted by

I in the

following)

is the main

difficulty

here.

Expanding

the

logarithm

in _powers of and

averaging

over

Zo,

leads to

compute

terms like:

lfii ^j

^d~'

^£

^41(m)G,,

(m-~')4i(~"

e '"'"

0

where is _an

integer running

^{from I} to

infinity.

One _can

easily

_see that this term is

equal

to:

] Tr

log(6,,6(~-~'>-16 £~

K,k(~-m'Gk,

(m-~'))

e

We _can

expand

the

exponential because,

_{as n goes} to zero, we

only

need the term

proportional

to _n. Thus _{we can} write I _as:

Note

that

the

trace has

here

to be understood as the race

over

I

ndices

and r

Expanding the

logarithm

in

this

pression, one can

easily see _that it is of

To so

that

the

for ^I

:

_)i+i -

- j

2 ~_ l

d~

^ia

/ dr(K(r)G(r) j~ (16)

the variational free _energy

(per

unit volume and _per

replica)

is thus:

~ / ~~~d

_~

~°~ ~~~~

~

/ ~~~d

_~

_~~~~

~~~l / $

~

G(~)~(~) ₍₁₇₎

+1 ^l~~~

>~

Tr1°~ biJ

iA

/ I ^KG)iJ)

We _now have to minimize this free _energy. This

yields

the

following

variational

equation:

b)k)

~

~J~

_~

_~~

_~~~

~"

_~~~

d

~~~ i

(18)

~~

l

~~

^~

^{~~~ ~}

_~~

_@

_~~

mj

which _can be rewritten _as:

K[~

₌

b~ (GA)ij(k) (19)

where

~" ~~~~~

~ ~

(~

~~~~ ~~

l

Al

/@KG~

^~~~~

v

From the last

equation,

it is _seen that A is

independent

of k. Thus _{we can use} the limit for k

going

to 0 in

equation (19)

which

together

with

(8) yields:

A~

=

be*b~ (21)

Introducing

the

projector P,j

₌

kikj /k~,

it is easy ^to invert

K~~

_in

equation (19).

One obtains:

K,j

#

b,j ~[ _P,j

E

2176 JOURNAL DE

PHYSIQUE

I N°11

and therefore

/ ^{~~~d~~~~~ ~*}

~~

Using

=

/~

_due~"~

x o

in

equation (20),

the self-consistent

equation

for e* _can be written _as

co

~-u6/d~°

~~~ _{~~~ ~} ~~ ~~~ ^~

l +

Ae~"~/d~°

^~~~~

which _can be recast in _{a more}

symmetric

form _as

1

j~ ~~/d~°j

~~

/

~~

(~~/d~°j

^~~~~

(where

the brackets denote

again

the average over the distribution of

e). Equation (23)

is the ma"n result of _{our paper.}

We have tried to break the

replica symmetry by using

the usual

one-step replica symmetry

ansatz

[13, 14]

^{where each element} is _a d x d matrix

depending

on k and

,

~ ~~i~(

^'

~(o)

K

~(i)

^'

~

~

~

'

"., _K(i)

K(o) k

~~~ ^~

~ "

where the blocks _are of size _m. The variational

equations

lead to the

unique

solution:

K(o)

~

~(i)

~ ~

and the

remaining equation

for

k

_{is identical to}

_equation (18). Therefore,

_we conclude that there is _no

replica _symmetry breaking

in this

problem.

4. Discussion.

Formula

(23)

_was first obtained

by

Hori

[12] through

a cumulant

expansion.

The calculations involved _are rather tedious and _our method is

simpler.

This solution _was studied in

great

^detail

by

Hori and Yonezawa [15] and we refer the reader to this _paper. We

just

note that it is not very

good

in _one dimension

(where

the exact solution is

known)

but it is

expected

that the variational method

gets

better in

higher

dimensions.

Note that the

duality property

^{in d} ₌ ² is not satisfied

by equation (23).

The

expansion

in powers of I

Id

_agrees with the exact result [16] up ^to the third order included. For d >

2,

the solution verifies the Hashin-Strikhman bounds

(which

is _a crucial test for all effective-medium

theories). Moreover,

at small

concentration,

this effective-medium formula satisfies _a

general condition,

based _on

general principles _[17].

Beside these crucial

criteria,

the solution

predicts

_a

percolation

threshold

given by

_{p~ =}

I

e~~/~ (which

is the mean-field result for

percolation)

and exponents s = t ₌ I

(which

is the usual effective-medium result

ill ).

We end this paper

by noting

that this method _can be extended to the _case of local

anisotropic permittivities

and to the

important problem

of the effective-medium

properties

at finite fre- quency.

Moreover,

this kind of calculation

(replicas

+ variational

treatment)

could be

applied

to _a wide class of other

physical problems,

since it _amounts

essentially

_to

computing

the _aver- age of _a

propagator,

a very common

problem

in the

physics

of disordered

systems (localization,

random

matrices,

etc.

).

Acknowledgements.

One of _us,

MB,

is indebted to Dr. J.-P. Bouchaud for

having suggested

this kind of

approach

to this

problem

and also to Dr. P.-G. Zdrah for _a careful

reading

of the

manuscript

and for

helpful

discussions.

References

ill

Clerc

J-P-,

Giraud

G., Lauder

J-M- and Luck J-M-, Adv. Phys. 39

(1990)

191.

[2] Bergman D-J- and Stroud D., Solid State Phys. 46

(1992)

147.

[3] ^{Hashin Z. and Shtrikman} S., J.

Appl. Phys.

33

(1962)

3125.

[4]

Bergman

D-J-, Ann.

Phys.,

138

(1982)

78.

[5] ^In Chan Kim and

Torquato S.,

J.

Appl. Phys.

69

(1991)

2280.

[6] St61zle S., Enders A. and Nimtz

G.,

J.

Phys.

I IFance 2

(1992)

401.

[7] ^Edwards S-F- and Anderson

P-W-,

J.

Phys.

F 5

(1975)

965.

[8] ^Edwards S-F- and Muthukumar M., J. Chem. Phys. 89

(1988)

2435.

[9] Garel ^{T. and} Orland

H., Europhys.

Lett. 6

(1988)

307.

[10] Shakhnovich E.I. and Gutin

A-M-,

J.

Phys.

A, ²²

(1989)

1647.

ill]

M6zard M. and Parisi

G.,

J. Phys. ^I France 1

(1991)

809.

[12] ^Hori M., J. Math. Phys. 18

(1977)

487.

[13] ^Blandin A., J.

Phys.

C 6

(1978)

1578.

[14] ^Parisi G., Phys. Lett. 73A

(1979)

203.

[15] ^{Hori M. and Yonezawa}

F.,

J.

Phys.

C 10

(1977)

229.

[16] Luck

J-M-, Phys.

Rev. B 43

(1991)

3933.

[17] ^{Landau L-D- and Lifchitz}

E.,

Course of theoretical

physics,

^vol. 8,

Electrodynamics

in continous media

(MIR,

Moscow,

1969).