Towards an experimental determination of the number of metastable states in spin-glasses?

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Towards an experimental determination of the number of metastable states in spin-glasses?

J. Bouchaud, E. Vincent, J. Hammann

To cite this version:

J. Bouchaud, E. Vincent, J. Hammann. Towards an experimental determination of the number of metastable states in spin-glasses?. Journal de Physique I, EDP Sciences, 1994, 4 (1), pp.139-146.

�10.1051/jp1:1994127�. �jpa-00246886�

Classification Physics Abstracts

75.50L 05.40

Towards

_an

exper1nlental determination of the nunlber of nletastable _states in spin-glasses?

J-P-

Bouchaud(~,~),

E.

Vincent(~)

and J.

Hammann(~)

(~) TCM Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, U-K-

(~) Service de Physique de l'Etat Condensé, Orme des Merisiers, CEA-Saclay, 91191 Gif

s/Yvette

cedex, France

(Received

25 August 1993, accepted 15

October1993)

Abstract. We argue that trie systematic deviation from perfect aging

(which

_means that

trie relaxation function only depend _on the ratio

£,

_{where tw}

is trie waiting

time)

observed in spin-glasses _can be accounted for if trie

numberlf

_{available metastable states}

is finite. A

significant fraction of these metastable states will be visited after _an "ergodic" time which _we estimate from trie experiments to be ci 10~ seconds. This corresponds ta _m 10~~-10~~ dilferent

metastable states per independent subsystem.

1. Introduction.

The

experimental investigations

of the

dynamics

of the

spin-glass phase

have revealed that the observed properties are

evolving

with the time

elapsed

from the

quench:

this _is the

aging

phenomenon

[1, 2]. A well known

example

is the relaxation of the thermc-remanent

magneti-

sation

(TRM):

^after

field-cooling

the

sample

from above

Tg,

the

longer

the

waiting

^time tw before the field is

cut-off,

^the slower the

subsequent magnetisation decay,

_as if the

spin-glass

had become "stilfer" with time. Similar elfects _are observed in the mechanical

properties

of

polymer glasses

[3, 4].

Aging experiments

also olfer _a

unique

_way of

"scanning"

the

phase

space ^structure of

complex

systems [5], ^and a detailed

understanding

of these

experiments

will

probably

oifer _a

key

for this

long

debated

problem

_[6-9].

Recently,

_one of _us

proposed

_{[10] a}

phenomenological theory

of

aging

^based on the assumption ^that energy barriers _are distributed

exponentially,

as

suggested by

_mean ^field solutions of the

spin-glass problem.

The

probability

to find the system in _a

given

metastable state with lifetime _T is found to be

eztremely

broad in the

spin-glass phase.

As usual with broad

distributions,

_one is

only

sensitive to the few

largest

events encountered

(see

_e-g.

[Il]).

In the present _case, ^the

dynamics

is

entirely

controlled

by

the

longest "trap"

encountered after time tw, which is of order tw itself.

Only

metastable states with lifetime of the order of tw have _a finite

probability

to be

observed;

in

particular

140 JOURNAL DE PHYSIQUE I N°1

-~ 0 6

j

t

~

(se~)

*5 _{0 14}

ÎÎÎ

m 31o~

é

T

j ((Î

à 0 12 j

~ ~

lC

~ ~ ^o t.Ot~+t

~j

H

oJà _oo~

~ Ag Mn 2 6z

~

't'"' _' T=9K=0 87 Tg

oJ

E _0.06

10-5 io-4 io-3 io-2 io-i ioo 101 10?

t / tw

_

0 16

1

~§ ^{0 14}

~

§

ù 12

àÎ ~

$

_{0 lu}

1

7J

0 08 Aq.Mn2.6z

1

1=9K=0 87 Tg

fi

~ÎÎ-5

io-4 io-3 jo-2 io-1 10Ù loi '02

t / tw

Fig.

l.

a)

TRM relaxation in

Ag:Mn

2.6% ai 9 K, _{as a} function of

log(

), for tw= 300, 1000, iw

3000, 10000 and 30000 seconds. The systematic shift towards faster relaxation

as tw increases _is clearly

seen. Insert: sketch of the experimental TRM procedure.

b)

Same data _{as in} figure la, but with the

"interrupted aging" correction, equation

(3),

expanded to second order _in

~i~.

_The

error bar _{is an}

ierg

estimate of the systematic uncertainty _in the vertical _axis.

the

microscopic

time scale

totally disappears

from the

problem [loi. Hence, quite naturally,

ail

dynamical phenomena

indeed take

place

on the scale of tw.

Simple

arguments,

building

_upon

these

ideas,

show that say in thermoremanent

magnetisation

relaxation experiments the

magnetisation decays

_as

fi _),

where t is the time after the switch off of the

magnetic field,

tw

tw is the

"waiting

^time"

during

which trie field _{is on,} and

f(it)

is _a function which behaves _{as a}

"stretched

exponent1al"

for small _it, and _{as a power} law for

large

_it

[10,

12]. ^This

prediction

_is

in very

good

agreement ^with experiments _over

nearly

rive time decades for _a

given

tw. However,

curves for different tw fait _to

collapse

onto the _{same curve} when

plotted

_versus trie variable

(see Fig. la):

there _{is a} systematic

deviation,

which suggests that trie relaxation of older tw

systems ^is "faster" thon it should be

(although

of _course, when

plotted

_versits t, trie older trie system, trie slower trie

relaxation).

Trie _sonne effect is observed both in metallic and

insulating

spin-glasses

[2], ^{but also} on trie viscoelastic

properties

of

polymeric glasses

[3]. ^{In order} to find

a

unique

"Master" _curve, trie authors of _[2]

proposed

to take into account this

systematic

^shift

by introducing

_a relaxation time distribution of trie form _p ^~

,

where _{/t is} around

ii

+

tw)"

0.9. This allows _one to rescale ail data

quite satisfactonly

s

a

unct~n

of _a reduced time variable which is

equalto )

_{in trie t < tw limit.}

_(A

_similar

_procedure

was in fact

proposed by

Struik

[3]).

^From a fundamentalw

point

^of view this

procedure

bas _an

intriguing

consequence:

on dimensional

grounds,

trie

scaling

^variable should

really

be written _as

~~ ~

Trie value

T -"tw

of trie time scale T* _con be extracted from trie

experiments

and is

extremely large

_{[2]: m} ^10~°

secondsl This is

surprising

since

Orly

two time scales _{are a}

priori

available: trie

microscopic

time scale _To OE 10~~~ seconds

(which,

_as

explained above,

should

disappear

in trie

spin-glass phase)

and trie

experimental waiting

time tw.

Trie aim of this letter is to

give

_a

physical meaning

to T* and _{to propose a new way} to estimate it within trie framework of trie

thecry developed

in

[loi.

We _argue that trie

systematic

deviation from _a

simple scaling

_con be

explained provided

_one takes into accoitnt the

jinite

nitmber

tw

of

available metastable stores in real

sarnples

made of

"grains"

of finite size. In

short,

trie idea is trie

following:

a finite size system ^will

eventually

find trie

"deepest trop"

_in its

phase

_space,

which

corresponds

to trie

equilibrium

state

[loi.

This will take _a

long

but finite time

terg,

^when

trie

waiting

time tw exceeds this

"ergodic"

time t~rg,

aging

is

"interrupted"

because trie

phase

space bas been

faithfully probed. Beyond

this time

scale,

conventional

stationary dynamics

resume. A comparison of this

eqitilibriitm dynamics

with trie _numerous available results [fil,

like e-g- a-c-

susceptibility experiments

at net toc low

frequency,

is however

beyond

trie _scope of trie present

simple

model.

It is reasonable _{to assume} that trie

experimental sarnples

_are collections of

independent subsystems,

^{each of which}

evolving

within _{its own}

complicated phase

_space. There will thus be _a distribution of

ergodic

times _some shorter and _some

longer.

Trie

systematic

deviation

from _a

perfect scaling

is trie consequence of trie

part1al

"death" of trie system, 1-e- that tw

aging

is

interrupted

in _a certain fraction of

subsystems.

This is what _was mimicked

by

trie

previous

~ ~

scaling proposed

in [2]. Here _we show how _one may

quantify

this

effect,

T* -"tw

which allows _{us to extract} from trie

spin-glass

data _a

typical ergodic

time of trie system which

is in tum related to trie number of available metastable states. In

fact,

this

"interrupted aging"

phenomenon

bas been observed in 2-D

spin-glasses

[13], ⁱⁿ

polymer glasses

_[14] and

possibly

in

glassy Charge Density

Wave systems [15]. Ail these systems ^however reveal important

qualitative

and

quantitative

differences. For

example,

trie

ergodic

time scale in CDW _appears to be much shorter thon in

spin-glasses.

2. Trie mortel.

In order to be

self-contained,

let _us first recall trie results of reference

[loi.

An

exponent1al

distribution of barrier

heights (suggested by

trie mean-field models of

spin-glasses [16,

7] induces

a

power-law

distribution for trie lifetime of metastable states:

ifilT)

= ~~

fi _lT

_{» To)}

Ii)

where _To is _a

microscopic

time scale (~- 10~~~

s). ~(T)

is _a temperature

dependent

parameter

describing

trie structure of

phase

_space: trie characteristic

free-energy

barriers _are of trie order

142 JOURNAL DE PHYSIQUE I N°1

of ^~

I?i,

^and

might

thus be temperature

dependent,

_x is between 0 and 1 in trie

spir~-glass

~(T)

phase,

and thus, from

equation il),

_or~e finds that the average

exploration

time is infir~ite. The

probability P(T)

to fir~d the system in a metastable state of lifetime _T is

proportional

to T~fi(T)

which,

for _{~ <}

l,

is r~on-normalisable [9, 10,

1?i.

For _a fir~ite observation time tw,

however,

trie distribution is cut-off above _{T t} tw hence

leading

to _a tw

deper~dent

normalisation. Trie

~~-i

resulting P(T)

is then found _to be

independent of

_Toi

P(T)

_cc ^~

(for

_{T <}

tw).

These

only ingredients

_are suilicier~t _to induce

aging

effects [18]. ^For x > 1, _onT~ trie other

bond,

trie _average

exploration

time is finite and cor~ventional

stationary

relaxation _occurs.

There

might

however be

physical

mechanisms which limit trie

possible

^{values of} T. For

exarnple if,

_as discussed

above,

trie total number of metastable states of _a giver~

subsystem

^is

jinite,

=

S,

trie

"ergodic

time" is of trie order of trie

longest

_T drawn from distribution

il):

terg(S)

ce

TOSÎ.

Trie relaxation of trie

magnetisatior~ mit, tw,terg(S))

after _a

given waiting

time tw is found to be, for tw <

terg(S), fi

with

f(it)

= exp

~ it~~~ _for

it < 1, and

tw

il

~

f lit)

_cc it~?II for _{it »} 1

[loi,

_~fI is _a number

quantifying

^trie

tendency

of trie system to torse its

magnetisation

rather thon continue to evolve within

magnetized

states. ~fII is

proportional

to

~fi, with _{an x}

dependent

coefficient

(which

_was, for

simplicity,

^taken to be in

[loi

_see

below).

For

longer

times tw »

t~rg(S), mit,

tw,

t~rg(S))

becomes _a function of This function

t~rg(S)

is in fact _very

nearly

trie _same

f [loi land

trie small difference does net affect trie

following

conclusions:

only

its monotonic character is of

importance). Hence,

trie

magnetisation

^of _a

given subsystem

_can be written in _a compact ^forrn _as:

Îllil,

_ÎW>

Îergis))

" iÎÀQIS)1

l~[~ _~~leml

¹²⁾

with

t*[

₌ tw for t + tw <

t~rg(S)

^and t*I,j =

terg(S)

^for t + tw »

t~rg(S).

In

words, equation (2) simply

_means that _once equilibrium is achieved

ii

_{+ tw »}

terg(S)),

the

magnetisation

is

much smaller than its "short" time

extrapolation, fi ).

tw

Now, assuming

that the number of metastable _states varies from _one

subsystem

to ar~other

according

to a distribution

Pi

^~

(So

is thus _a

"typical"

number of metastable states per

subsystem),

the total

(observed)

o

magnetisation Mit, tw)

is obtained _as trie _{average over} P of

equation (2).

^With these

only assumptions,

_one

finally

finds:

Mit, tw)

~j

^t

~

~

jjt

^{+ tw}

^~j

^j~~

M t to

o _w

erg

where

t(r~

+

TOS)

is trie

"ergodic

time" of trie system, ^and

Fij

is _an

ir~creasir~g

functior~ which

depends

_on trie choice of

mo(S)

and P. Before

taking

_a

specific example,

it should be noted that

equation (3) qualitatively

_agrees with trie

experimental

trend: older systems indeed relax '~faster" when

plotted

_versus

tw

3.

Analysis

of trie

experimental

data.

The

simplest assumption

_is that _mû ar~d P behave

regularly

when their arguments are small.

In this _{case, one}

finds,

for t + tw < t(~~

F(it)

₌ Ait + Bit~ ₊

,

where A is _a constant which

Table I. Parameters ~, nfI, nfII, t(~~ and

"complezity"

fl

for

the

Ag:Mn

2.6$i

sample

[2]. ~, nfI were e~tracted by

jitting

the "Initial" part

of

trie TRM relaxation

(t

<

0.3tw)

ta _a stretched

e~ponential,

while _~fII is obtained

from

the

long

lime part

(t

> 2tw

) of

the rela~ations. Note thon

Q(T

₌ 10 K

)

coitld net be estimated becaitse

of

_oint lack

of precision

in the vertical

positionning of

the rela~ation _citrves. This là _a

problem

close ta Tg ^{since the} magnetisation

faits rapidly

ta

small valites.

Table I: AgbIn: 2.6% (T =10.4 K)

TII<) _{z 11 ~/jj} t~r«(10~ s.) ^Q

8 0.66+0.05 0.083+0.010 0 090+0 003 4+2 12.1+0.9

9 0.65+0 06 0.0124+0.016 0.127+0.003 2+1 11 9+0 9

g-à 0.68+0.06 0 153+0.020 0 lôâ+0.005 0.8+0.3 12.2+0.9

10 0.îâ+0.05 0.221+0.030 0.26â+0.005

Table II. Same _as in table

I,

bitt

for

the

Cdcri.7Ino_354 sample

_[2].

Here,

the initial part

of

the rela~ation _was extended ta t < 0.stw. The valite

of

the parameters

given

here _are

compatible

with those obtained in

reference [loi,

where the additionnai assitmption _~fI +'fII was made.

Table II. Cdcrj 7In0 354 lT =16.î Ii)

TII<) _{x ni iii} t~~~j10 s Q

10 0 76+0 05 0.059+0.008 0 058+0.003 5+2 14.2+0.9

12 0 76+0 0â 0.100+0 01 0 100+0 003 3+1 14.0+0.9

14 0 î3+0 0â 0 200+0 02 0 182+0.010 0 6+0.2 13.0+0.9

con be reabsorbed in the definition of

t$~~ [19]. ^{We show in}

figure

16 the _same data _{as in}

figure

la, but with the

"interrupted aging"

correction, equation

(3).

The

only

free parameters are

the

ergodic

time of the system

t$~~(S)

^{and the value of B}

(the

value of _~ is fixed

by fitting

the

"youngest"

curve see

[19]).

^{We have however}

imposed

trie _same value of B

= -3

ii-e-

the _same functional form for

P)

for ail temperatures and

samples.

Trie

collapse

of trie _curves is quite

good

in ail _cases

(see Fig. lb). Hence,

and this is trie main result of this paper, oint

interritpted aging

scenario là able ta mimic _a " "

scaling

with _{/t <} 1.

tw

The

corresponding

values of the parameters

~,~fi,'fII,t$~~,

^{and of trie}

"complexity"

fl e

x

logio( ~f)

are

given

ⁱⁿ tables I and II, for different temperatures and two different

samples:

Ag:Mn

^2.6$i

(Agmn),

which is metallic and

Cdcri.7Ino.354 (CrIn)

which is _an

insulating

spir~

glass.

It is

interesting

to see that both

samples

lead _to similar conclusions:

a)

The

ergodic

time t$~~ ^is of trie order of106

seconds,

and

thus,

within _our interpretation, trie number of metastable states per

"subsystem"

is found to be _ce 10~~ -10~~

(see

Tabs.

I, II).

This number is however not easy ^to discuss sir~ce _we do not kr~ow

precisely

what these

"subsystems"

are.

They

_may

be,

for

example, magr~etically

disconnected

regions (like grains)

trie size of which

being

determined

by

trie

sample preparation,

and thus is temperature

independent.

It could also be that trie

phase

_space of 3d

spin-glasses

is broken ir~to

mutually

inaccessible

regions ("true" ergodicity breaking).

In this _case, S would denote the number of metastable

states within _one such

ergodic

component. This number would then be _an intrinsic property

of 3d

spin-glasses.

We however think that numerica1simulations [20]

and/or experiments

_on mesoscopic

spin-glasses

_{[21] are}

really

needed to understand how the

ergodic

lime t$~~ ^is related to the number of

spins

contained in _a

subsystem.

b)

As shown in

figure

2, ^{the number of metastable} states of the

ir~sulating sample (Crin)

tends to increase when the temperature ^is lowered

(it roughly

doubles when trie temperature is

144 JOURNAL DE PHYSIQUE I N°1

15

14

ll ,

l0 ^°

0.5 o-G 0.7 0.8 o-g TIT

Fig.

2. Evolution of the "complexity" f1 % ~logio ^~~~~

^o

with reduced temperature

(

_{for both}

Agmn and CrIn. Note that the complexity does trot _seem to go ^to zero continuously ai Tg.g Insert:

Evolution of the complexity _per spin _versus reduced temperature ^for the ~-spin model, with, from bottom to top: _{p =} 2

(SK model),

3,4,5 and 6 ~p = ce is the Random Energy

Model).

Note that f1 is

discontinuous ai Tg _as soon as p ^> 3. From reference [24].

lowered

by

1

Kelvin). However,

in contrast to trie SK model [22], ^{this nitmber ares} net _seem ta vanish

continitoitsly

ai the

spin-glass

transition

Tg.

^Such a behaviour _was found in "Potts

glas"

models [23] ^{and in models with}

p(> 2)- spin

interactions [16],[24] see

figure

2

(insert).

Trie number of metastable states of trie metallic

sample (Agmn)

_seems to be

roughly

temperature

indeper~dent, although

trie rather

large

_errer bars leave _room for trie _sonne behaviour _{as in}

CrIn.

c)

The parameter x grows towards 1 when T _- Tg

(at

least for

AgMr~,

but other experi-

ments on CrIn showed the _same

tendency [2]).

This behaviour _was also found in _a very clear

way in reference

[25],

^{where the initial}

decay

of the

magnetisation

was also fitted

by

_a stretched

exponer~t1al.

Dur results

(fl(Tg)

^finite and

~(Tg)

m

1)

suggest ^{that the}

spir~-glass

transition

is,

in these

samples,

^much doser to the "Random

Energy

Model" scenario [16], ^than to trie "SK"

scenario where

fl(Tg)

₌ 0 and

~(Tg)

₌ 0 1?i. ^{We however note} that _~

significar~tly departs

from trie REM linear behaviour _{~ =}

~,

which would hold if the _energy barriers _were temperature Tg

independent. Furthermore,

trie REM

"replica

_symmetry

breaking"

scheme is

certainly

net ricin

enough

to accour~t for _more

complicated aging protocoles

_{[Si or} "second

spectrum" experimer~ts

[21].

d) Finally,

_{one can see} that trie

simplifying hypothesis

_{~fI "'fII} is

quite

accurate. In

fact,

_a

doser look at equations

(8-10)

of reference

[loi

reveals that trie ratio ^'~~~ ir~creases with _~

land

'fi

diverges

for _{x -}

1),

^which is consistent with trie

experimental

trend

(see

Tab. 1).

4. Conclusion.

Let _{us summarize our}

findings: quantifying

trie idea that

aging

stops ^{when trie}

exploration

of

phase

space is

completed,

_we bave

proposed

_{a way} to rescale relaxations obtair~ed for different

waiting

times _{on a}

unique

"Master" _curve, ar~d to understand the

physical meaning

of the

phenomenological

^"

scaling.

This allows to extract from the

experimer~ts

_{a very}

long,

tw

temperature

dependent,

time scale t~rg

beyond

which

aging

_ceases and trie nature of relaxation

changes.

This is

interesting

from _two

standpoints:

from _a fundamental point ^of _view, it allows to our

knowledge,

for the first time to estimate the number of metastable states in _a

complex

system, and its temperature

dependence. Perhaps

_more

importantly,

it

allows,

from _a

practical point

^of

view,

to extract from

relatively

short time measurements _an

ergodic

time

beyond

which trie relaxation

changes qualitatively:

_we bave in mind here trie aging

experiments

_on

polymer glasses,

where trie mechanical

properties

_{are very} similar to those discussed here

[3],[4].

^It could be of

importance

to know when these mechanical

properties

wiII

appreciably change.

Trie ideas

developed

here

might perhaps

be useful to discuss

physical agir~g

^{in trie} context of other

(molecular

_or

meta1Iic) glasses.

Acknowledgments.

We wish to express warm

gratitude

towards M.

Ocio,

who gave us access to bis data _on

Agmn,

and for

enlightening

critical discussions. We aise thank K.

Biljakovic,

F. Lefloch and M.

Weissmann for

interesting

conversations.

References

[1] Lundgren L., Svedlindh P., Nordblad P., Beckmann O., Phys. Rev. Lent 51

(1983)

911;

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[2]

a)

Alba M., Ocio M., Hammann J., Europhys. Lent. 2

(1986)

45; J. Phys. Lent. 46

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L-l101;

Alba M., Hammann J., Ocio M., Refregier Ph., J. Appt. Phys. 61

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3683.

b) Vincent E., Hammam J., Ocio M., Recent Progress _in Random Magnets, D.H. Ryan Ed

(World

Scientific Pub. Co. Pte. Ltd, Singapore, 1992).

[3] Struik L.C.E., Physical

Aging

_in Amorphous Polymers and Other Materials

(Elsevier,

_Houston,

1978)

[4] ^Perez J., Physique et Mecanique des Polymères Amorphes

(Technique

et Documentation Lavoisier, Paris, 1992).

[si Hammann J., Lederman M., Ocio M., Orbach R., Vincent E., Physica A 185

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278; Lefloch F., Hammam J., Ocio M., Vincent E., Europhys. Lent. 18

(1992)

647.

[GI Binder K., Young A.P., Rev. Med. Phys. 58

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801.

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Scientific, Singapore

1987)

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1601; Phys. Rev. B 38

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1705.

[Il]

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[12] ^{Some of these} predictions have recently been confirmed by theoretical and numerical calculations:

see [20]

Cughandolo L.F., ^Kurchan J., Phys. Rev. Lent. 71

(1993)

173;

Cugliandolo L.F., Kurchan J. and Ritort F., prepnnt

(lune

93);

Marinan E., Pansi G., prepnnt

(August 93).

146 JOURNAL DE PHYSIQUE I N°1

[13] ^Schins A.G., Dons E-M-, Arts A.F.M., de Wijn H-W-, Vincent E., Leylekian L. and Hammann J., to appear in Phys. Rev. B

(Dec.

_1~~, 1993).

[14] ^Lee A., Mc Kenna G-B-, Polymer 29

(1988)

1812;

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497.

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1902;

Biljakovic K., Lasjaunias J-C-, Monceau P., Phys. Rev. Lent 62

(1989)

1512.

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2613.

[17] ^Another physical situation where _a non-normalisable probability distribution _occurs is presented

in: Bardou F., Bouchaud J-P-, Cohen-Tannouji C., Emile O., Subrecoil laser cooling and Lévy flights, preprint ENS

(July 1993),

submitted to Phys. Rev. Lent.

[18] Simple numerical simulations of models based _on equation (1) indeed confirm that _{aging occurs}

as soon as x < 1 without any further physical ingredient: E. Vincent

(unpublished)

[19] ^Since t$rg turns out to be _ci 10~ seconds,

one may proceed "backwards" _as follows. The "youngest"

relaxation

curve is such that the correction _term in equation (3)is negligible: this allows to identify

the Master _curve

fi

Now, plotting the ratio of -say- the oldest relaxation _curve to the youngest tw

versus the variable

(t

+ tw)~ determines the function

F(~).

In the interval of _~ which is probed,

a fit of the form

F(~)

= ~ + B~~ _is

very satisfactory.

[20] Rieger H., preprint, J. Phys. A 26

(1993)

L615.

[21] ^{Alers G.} B., Weissmann M. B., Isrealoff N. E., Phys. Rev. B 46

(1992)

507;

Weissmann M. B., Isrealoff N.E., Alers G. B., J. Magn. Magn. Mater. l14

(1992)

_87; Weissmann M. B., Rev. Med. Phys.

(July,1993,

to appear).

[22] Bray A. J., Moore M. A., J. Phys. G13

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L469.

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5388;

Kirkpatrick T. R., Wolynes P. G.,Phys. Rev. B 36

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8552.

[24] Rieger H., Phys. Rev. B 46

(1992)

14655.

[25] Hoogerbeets R., ^Luo W. L., Orbach R., Phys. Rev. B 34

(1986)

1719. Note however that in this paper, ~ was interpreted _as 1- _~. See [10] ^for a discussion of this point.