Dynamic scaling and non exponential relaxations in the presence of disorder. Application to spin glasses

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Dynamic scaling and non exponential relaxations in the presence of disorder. Application to spin glasses

B. Castaing, J. Souletie

To cite this version:

B. Castaing, J. Souletie. Dynamic scaling and non exponential relaxations in the presence of dis- order. Application to spin glasses. Journal de Physique I, EDP Sciences, 1991, 1 (3), pp.403-414.

�10.1051/jp1:1991142�. �jpa-00246340�

Classification

Physics

Abstracts

75.40G 75.50K 75.60N

Dynamic scaling and

_non

exponential relaxations in the presence of disorder. Application to spin glasses

B.

Castaing

and J. Souletie

Centre de Recherches sur les Trds Basses

Tempdratures,

C.N.R.S., BP166 X, 38042 Grenoble Cedex, France

(Received 24

September

1990, revised 21 November1990,

accepted

5

December1990)

Rksumk. En supposant

qu'un

_processus

hidrarchique

_gouveme l'dvolution d'un niveau au

suivant dans _un processus de renorrnalisation de Kadanoff, on trouve que la

rdponse dynamique

en

prbsence

de ddsordre _est ddcrite _par des distributions

log-normales

des temps ^de relaxation. Il y a deux limites naturelles _au moddle

qui justifient,

l'une

l'hypothdse

du

scaling

_»

dynamique,

l'autre celle de la

dynamique

activ6e. Nous

justifions

_{avec ces} bases le succds des lois habituelles

(exponentielle

dtirde _ou loi

Cole-Cole)

_avec

lesquelles

on rend compte ^des relaxations _non

exponentielles qui

sont observdes _en fonction du temps ou de la

frdquence

dans les verres de

spins.

Nous donnons des

expressions

_pour

reprdsenter

la variation _avec la

tempbrature

des

paramdtres principaux qui

contrblent _ces lois. Nous proposons aussi _pour ddcrire le passage du

rdgime ergodique

au

rdgime

_non

ergodique,

_un moddle

qui

_conserve la continuitd de 3 In T/3(1/7~ au

point

de

gel.

Nous discutons

quelques expbriences

dans (es _verres de

spins

mais

nos conclusions devraient Etre valables pour

n'importe quelle

transition continue.

Abstract.

Assuming

that _a hierarchical

procedure

_govems the evolution from _one to the next step ^{of Kadanoflls} renormalization in the _presence of _some disorder _one finds that the

dynamic

response is described

by

a

log-normal

distribution of the relaxation times. There _are two natural limits to the model which restitute either the

dynamic scaling

_or the activated

dynamic hypotheses.

This

provides

the basis for a

simple justification

to the _success of the usual fits by _a stretched

exponential

_or

by

_a Cole-Cole law which _are

commonly

made to the

non-exponential

responses observed in the time _or in the

frequency

domains in

spin glasses.

We

provide expressions

for the temperature

dependences

of the parameters ^which control these laws. We also propose a model for

freezing

which is based _on the idea that a In _Tla (^I In should be continuous at the

freezing point.

We discuss available data in

spin glasses

but _our conclusions should be _very general.

1. Introduction.

The

non-exponential

relaxations which _are observed in

glasses

and

spin glasses

_are often attributed to the effect of _a

large

distribution of relaxation times. The detailed

shape

of this distribution

could,

in

principle,

be deduced from _a fit _to the

experiments.

A stretched

exponential [1-5]

M(t)

=

Mo

_exp

i- (t/Tj

)fl

~"j (1)

is often used to fit the

experiments

in the time domain such _as the relaxation of the

magnetization

of _a

spin glass,

_or that of the stress in _a

glass

_{or a}

polymer.

In the

frequency domain,

_many authors

rely

_on the Cole-Cole

equation [6-1Ii

x(W)

=

x'(W ) ix"(W)

=

(~ _~j~~~ ⁽²⁾

to fit their measurements of the

complex susceptibility (or modulus).

A

problem

arises

though

because the distributions which would

justify

these two

phenomenological equations

differ from each other. In the

following

_we will

justify

_a

log-normal

distribution in the framework of

a model which combines Kadanofrs renormalization with _some disorder. This distribution also fits well the data

[12], except

at short

times,

and _we shall show that it

gives

_a

justification

for the _success of

equations (I)

and

(2)

in _some range. Predictions _can be made for the

dependence

of the In

r variance in the

ergodic regime

and _we propose an

argument

^which would

permit

to extend the discussion to the

non-ergodic regime,

when the relaxation time

becomes

longer

than the age of the system.

The

organization

of this paper is _as follows: in section 2 _we compare the stretched

exponential

and the Cole-Cole behaviors. We show that both laws _are consistent with _a

log-

normal distribution in _a wide _range of times and _we relate the

exponents

a and

p

^{with the In} r

variance A. In sections 3 and 4 _we discuss _a hierarchical model that Palmer _et al. have used but

in _a different way

[13].

This model _can be considered _{as a}

dynamical counterpart

to

Kadanows renormalization _{near a} transition when the coherence

length f diverges

^and we

show that it has two limits which _are consistent either with the usual

dynamic scaling hypothesis

_or with the activated

dynamics hypothesis.

In the presence of disorder the model leads

naturally

to

log-normal distributions,

_so that

by using

the results of section 2 _{we can}

justify

the stretched

exponential

behavior _{as was}

already

remarked

by

Montroll and Bendler

[14]

or the Cole-Cole behavior. We deduce the temperature

dependence

of

a and

p

^{from the}

knowledge

of the

temperature dependence

of the In

r variance. Our

approach

to

freezing

is discussed in section 5.

All what we say is very

general

and should

apply

to any transition. But most crucial

experimental

tests which _we know _concem

spin-glass physics.

With this in mind _we will

mainly speak

of the

magnetization

_or of the

susceptibility

but the

transposition

_can

easily

be made to

other

quantities.

2. Similitudes and diwerences lietween laws.

In this section _we discuss the relations between the

log

normal distribution and the

distributions which

correspond

to the usual fits with

equations (I)

and

(2).

With _a distribution of relaxation times

P(In r),

the effect of _a field variation is written

M(t)

=

Mo

_exp

(- t/T)

P

(in

_T

)

d in _T

(3)

exp

(-

_t

IT represents

a very

sharp

variation _{on a}

log

scale and it is

possible

to consider it _{as a} cut-off with any distribution of

significant

width in In _r. It follows that

[15]

M(t) lw

=

Mo

P

(In

r

)

d In _r

In (t/rj)

SO that

fl))

=

P

(ill (t/Ti))

The _same argument ^leads to the well known

ar/2

rule introduced

by Lundgren

et al.

[16]

for the

experiments

in

frequency

x"(w

=

] ()(()(

=

]

P

(-

in

(wri)).

With _a Gaussian distribution

j

In~ TIT1)

~ ~~~ ~

A

fi

^~~~ _{2 A}~ ^' ~~~

we have within the cut-off

approximation

dM(t) ^In~ (t/Tj)

in

(-

~ ~~ ⁼ A

~

(5)

With

M(t) given by equation (I),

_we would obtain instead

in

(- fl(~)

= in

(limo)

₊

p

in

(t/Tj

_exp

(fl

in

(t/Tj )

For

p

In

(t/rj)

_«

I,

it is

possible

to

expand exp(p

In

(t/rj))

as

I ₊

fl

in

(t/Ti)

+

(fl ~/2) in~ (t/Tj

+ so that

in

(- fljj

=

iIn (limo) Ii (p12) in~ (t/Ti)

₊

(6)

which _can be identified with

equation (5) provided p

=

I

IA, _except

for corrections of order

(P

in

(t/Ti))~.

As for the Cole-Cole

equation,

it

corresponds

to the distribution

[17]

_:

~ _~~

(l/2

_ar sin

(a

_ar

~ ~ ~

cash

(a

in

(T/Tj))

_{+ cos}

(oar )

tan

(~z~r/2)

a In

(r/rj)

2 _ar 2 _cos

(oar/2) j2j

so that

tan

(oar/2) ^"~ _in~

(T/ri). (7)

In

[P(In

_r

)]

= in

2 _~ 4

cos~ (oar/2)

Again,

_{one recovers}

equation (5) provided

I

^~

/

cos

(oar/2)

^~

~

_~~~

The two

equations (I)

and

(2)

therefore follow from distributions which differ from each other and from _a

gaussian by

terms of order

(p

In

(t/ri )3

^{and above}

(see Fig. I).

Due _to this and to the cut-off

approximation,

the relations

(8) correspond

to the infinite A limit. However the other limit A

- 0

obviously corresponds

to

p

= a =

I. As the

equation (7) directly

compares the

distributions,

_{we can} consider that the

correspondence ^A~~=

a

/(/

cos

(

_am

/2))

is valid in the whole range of A. As for

p,

we show in the

appendix

that _a formula

P

=

i ₊ A

~)-

^~'~

(9)

gives

satisfaction in both limits and would remain _a

good approximation

in the whole range.

JOURNAL DE PHYSIQUE T i, M 3, MARS 1991 19

p Cole

l

stretched

h_

h

",

~-k _{-2 0 2 k}

Ln(T/Til

Fig.

I. The

logarithmic

^derivative

fl~~~

^{of a stretched}

exponential M(t)

=

Moexp(- (i/Tj)fl)

is

n t

compared

with _a Cole-Cole distribution P (In _T

= sin _{a ~T}

/

[2

josh

^{a In}

^t/Tj

^{) + cos a ~T and with a}

gaussian distribution P (In _T) _exp (- In~ (t/Tj )/2

A~)/(

2 _~T A) in _a typical case where

a/(/

cos

(a~T/2))

= 1/A

= p. With these conditions the curvatures of the three _{curves are} matched around their maximum. In the

particular

_case which is considered _a

= p

= I/A

=

1/2.

We compare in the

figure

I the Gaussian distribution and the Cole-Cole distribution with the

logarithmic

derivative of _a

stretched-exponential

in _a

typical

_case

(p l/2).

The three

curves have been matched _to have the _{same curvature} around their maximum

(a

=

p

=

I/A).

For the

largest

times the usual

opinion

favors a behavior

governed by

_a

single largest

relaxation time _»

although

there is _no decisive evidence that _we know about this

point.

The Cole-Cole distribution

gives

the slowest

decay,

like _a power law. This feature turns out to be _an

advantage

at short times

(high frequencies)

where _X

"(w

is well fitted

by

_{a power}

law

t"~".

_But

a

(T),

which is derived in this range

although

it does show _some

pathologies

associated with the

transition,

is not

simply

related with the

position

of the maximum _or with any other

plausible

determination of the

divergence

of the coherence

length [7, 12, 18].

The

logarithmic

derivative of _a stretched

exponential

can

only

be identified with the

corresponding

distribution in the limit where the cut-off

assumption

is

valid,

but it reflects the main features of this distribution it is

asymmetric showing

_a fast cut-off at

long

times and _{a power} law at

short times. Nevertheless several authors have

pointed

out that _a

specific

_power law correction to the stretched

exponential

is necessary ^to fit the relaxation data _at short times

[3, 4].

With the law which

they

_propose

M(t)

t~"

exp[- (t/ri)~(~~]

the

high frequency

features _are related to the _t~ " part ^{of the relaxation and could be of} a different

physical origin

than the

part

^{that the stretched}

exponential

describes. With _a similar power law correction the

log-normal

distribution fits the data _as well _as the stretched

exponential.

We would then recommend to discuss relaxation data in both the time and the

frequency

domains in _terms of the

log-normal approximation

which has several

advantages

it _can be

justified by simple

models _as shown in the

following

sections

its main parameters

(the

mean

position Inrj

and the

log

variance A) are both

measurable and

predictable

_on the whole

temperature

_{range ;}

it _can be related to the usual Cole-Cole and stretched

exponential

fits and we will _use this fact to deduce the variation of the parameters ^which govern these

phenomenological

laws.

3. The model.

In the Kadanofrs renormalization _process, at the

stage

n+

I,

the

(n+ )-clusters

_are

themselves

composed

of

n-clusters,

in _a self-similar way. The model _we shall _{use assumes} that

a

(n

₊ I

)-cluster

_can

change

its state

only

if p~ of the n-clusters sit in _a

given position.

These p~ n-clusters thus _{act as}

keys

for the

(n

₊ I

)-cluster [13].

Let _us call A~ a

typical

_energy difference

(measured

in

K)

between the two states of _{a n-} cluster. Two _{extreme cases are} then to be considered

I)

_{A~ is} much smaller than T. Then the two states have

equal probabilities.

The

typical

time r~~ i between two

changes

of the

(n

₊ I

)-cluster

state is

r~ _~j ⁼

2'~

r~ =

2~'~

To

(10)

Then,

due to the

disorder,

_some noise exists _on the value of _p~. Whatever is its

distribution,

its average value

Hand

its variance

(

_{p ~) are}

independent

of _n because of the

self-similarity

of the renormalization

stages.

Thus In

(r/ro)

=

(2p~)

In 2 is

Gaussian,

with _an average value In

(ri/ro)

=

NH

In 2

(11)

and _a variance

A^~

= N

(

_p~)

^ln~

2

(12)

in the

large

N

limit,

where N is the number of stages

[14].

For

simplicity (p~)

stands for

I (p ip ) )~l

ii) If,

_on the contrary, _{A~ is}

larger

than

T,

the

keys

will have in

general

_a lower

probability

to be in the

good position

than in the other _one.

Introducing lI~

~ j which is the _sum of the various _A~ of the different

keys

_r~~i

= r~ exp

~~~ lI~, thus, represents

the

typical

T

barrier between the two states of _a

n-cluster,

whose

typical

_energy difference is

A~.

Again

the

self-similarity implies

that

W~)

=

k

(A~)

and

Wj)

₌ ^k'

(Aj)

with k and k'

independent

of _n. For

simplicity (lFj)

stands for

((H§ (ll~) )~).

^Then

H§~

j will be

Gaussian,

for

large enough

_n. We have

(W~~j)

=

(p/k)(lI~) (13)

and

((wn+1)~)

"

(i~/k')((wn)~) (14)

~~~~ _'

(p/kl'~ j

_Wo>

in

(Ti/To) (15)

= yn

and

~

(H/k')~ ((%)~)

~

"

~2

^~~~

4. The

neighborhood

of the transition.

Close to a transition when the coherence

length f diverges

_we

have,

_as a result of Kadanows renormalization

~ in

((la)

(17)

inn

where b is the renormalization step

(sometimes

called

rescaling length)

and _{a an} atomic distance.

Furthermore,

for T~ T~, we have from the static

scaling hypothesis

that

ila

=

(i Tc/Tl-

^v

(18)

As _a result _rj and A

diverge

and the exponents _a ^and

p

in

equations (I)

and

(2)

cancel. The way

they

do _so is different in the two limits

I)

and

iii.

In _case

I), eliminating

N between

equations (11)

and

(17)

we obtain

~~~~°

(~/~)~

with _z

=

lL In 2

in b ~~~

This is

interesting,

_as it

justifies (with

_or without

disorder)

the

dynamic scaling assumption,

in

an unusual way. Often the critical

slowing

down is

justified by

a diffusion process,

r =

f~/D _(where

D is a diffusion

constant)

which would lead to _z

= 2.

Here,

_we have the

possibility

of

larger

values for _{z, as} is indeed observed. In reference

[19]

the _same result In _r

~ z In

f

is obtained for finite

Ising

fractal clusters.

The behavior of is

given by equations (12)

and

(17).

We have

Very

close _to the

transition,

where is

large,

the exponents a and

p

thus _vary like

(in ((la) )-

~'~.

In terms of

temperature,

we can write

fl

=

(1 T~/T~-zv (21)

and ^~ is

proportional

to In

(I T~/Tl.

Experimentally,

_a and

p

_are observed to decrease when

approaching

^the transition and

p

saturates to a value of the order of

1/3.

Our model

predicts

that

p

cancels _at

T~, ^{however the}

huge

increase of the measurement time

for(ids

^the observation at

equilibrium

very close to T~, ^which results in _a virtual limit for ^~

=

~~

~~ ~

ln

t~~~/ro)

hence for

p

and lL

a. This formula _{can account} for the observations

(Fig. 2).

But _our limit is _non universal and

time-dependent by

contrast with another

description,

based _on diffusion in the

hypercube

of

spin configurations,

which makes a

point

that the limit should be

1/3 [23].

We have thus here _a criterion which should

permit

to decide between these two

approaches.

In _case

iii,

the behavior of _ri is

given by equations (15)

and

(18)

Tj

if~)

_~j~o)

_f

In z

=

exp(N

In

(p/k)

=

(22)

~0 T T _a

with

In

(p/k)

z =

In b

Using equation (18),

close _to T~, this is

equivalent

to the activated

dynamics

law

[20]

rj = To exp

1°

with _«

= zv and 0

o ⁼

T) ll§) (23)

(

T

~

T~)"

equilibrium response

j

^{z0 = 8/8}

prachcal law T limit

~'~~

1~2>

0 0.1 0.2 0.3 0.k

~/~

^0.5

Fig. 2. We _use the

expression

p

(l

₊ A~)- ^"~ which describes

correctly

the

approach

to the two limits

= 0 and A

-

~c,

^{to deduce the variations of the exponent p T~ of a stretched}

exponential

from the variation A^~

=

~~

~~

~zv

_{In (I}

_T~/n

_{of the width of the Gaussian} distribution

predicted

in the

dynamical

scaling

_case. At low temperatures p reaches _an effective limit 0.22

p/(p~)

when zvIn (I

-Tjn~

In

(T/To)

reaches values of the order of 30. We have used _~

~2(~~)

and

zv 8 to illustrate _our

point.

As for A^~ it also behaves _{as a power} of

(la

A^~

=

$~

_exp

_(N

_In

_{(p /k') )}

=

~/~ _~

_~'

(24)

T T a

with

,

In

(

_p

/k')

~ ln b

In contrast with _case

I)

we have not in

general

_a

simple proportionality

between

A^~ and In

(rj fro);

In terms of

temperature,

the

exponents

a and

p

_{are now}

going

to zero as

power laws of

(T- T~).

^Close to the

transition,

for instance

p

₌

po(I TJTI~'~'~

A

special

mention must be made of transitions where T~ =

0,

_as several

examples

have

perhaps

been found

experimentally [11, 18, 5]. Following

_our

approach,

^the

only

difference sits in the modification of

equation (18)

_:

~

= exp

~

(25)

a T

This law _can be obtained from

equation (18) by letting

_{T~ go} to _zero. As

f

should be _an

analytical

function of

T~/T

at all

temperatures

T~ T~, ^it

implies

that

(la

l ₊

vT~/T

_+...

where

vT~

₌ 0 is finite. With this constraint

equation (18)

generates

equation (25)

when 7~ ^{cancels. In} case

I),

for

instance,

the consequence is that _ri behaves in _a

purely

activated

way

[21]

_:

ri zo

= exp

(26)

To T

and

I/A

hence _a

(or p)

_are

proportional

to

fi _{(Fig. 2).}

The

figure

3 condenses several of the above

predictions.

The

equilibrium

result at T~ T~ for

x"

or

f

vs. ln t

(in

the cut~off

approximation)

is _a Gaussian centered at n t

In(r/ro)

=

NH

In 2 whose

amplitude

increases like

$

_{and which tends}

to _a 3 function _at infinite

temperature.

^{In the} case

I) NH

ln2

diverges

like _zv In

(I T~/Tl

which tends to

zv

T~/T generating

Arrhenius

dynamics

when T~ cancels.

~n

j~

~ x p^Ln2 _~°°~

~o j~

t iTobsl ~a

Tobs>Tc equilibrium

la

~rJ

fini~e

~ ,

°~

~

Q

(

~ i

/~

o i/Tgi/T~ i/T~bs.

Fig. 3.- At

equilibrium

P(Inr) hence X" _or ^~~ are Gaussian functions of In t centered on

d In _t In

(Tj(T)/To)

= N~ In 2

= zv In (I

Tjn

for T~ finite. The

requirement

of

analycity

for T> T~

implies

that _zv T~^is finite _so that Arrhenius

dynamic

is obtained in the 2~ = 0 limit. Each tangent to the In _{T vs.}

I/Tcurve

describes _a

particular

frozen _state which _can be labelled

by

the renormalization step

N(T~)

at the contact

point.

An

aging

t~ from B to C is thus related to an evolution from A _to D of the cooling rate. The drift of the response time _t~~~ with the

waiting

time t~ ^{in the} frozen state _ceases when t~b~= T(T) if

T>T~

and is endless at

T~T~ (see

insert). For _a given observation window (In

(T/ro)~25

to

30) high

levels of renormalization _can be reached

only

_{near T~.} If

zvT~>

TIn

(r/To) [T~ Tj4]

_{we stay} in the level N

=

0 where the situation is

entirely

determined

by

the state of disorder of the

particular sample

which is considered and TIn t

scaling

is

obeyed.

5.

Freezing.

At _some

temperature, depending

_on the rate of

cooling,

the

system

^{will «freeze». The}

question

which _we address in this

paragraph

is how the relaxation times evolve after

freezing.

We shall discuss it in _a

diagram representing

In _{r vs.}

I/T (Fig. 3).

At

equilibrium,

In _r is

diverging

at I

In.

We shall

present arguments

for the

freezing

to _occur without

jump

in the derivative of In _r.

Indeed _{we may} write In

r as

~~

=

ASA

+

~~~

= ln

~

+

~~~

where

AFA, ASA

^and

T T _To T

AUA

are the differences in free energy, entropy ^and

potential

_energy between the critical state and the activated state _» which limits the evolution towards _a final

equilibrium. (From

_now

on, for

simplicity,

_we will refer to

AFA

_as

F, AU~

_as U and

AS~

_as

sl.

In the framework of the

theory

of the

quasi~equilibrium [24]

which supposes that _an

equilibrium

is created and maintained between the initial and the intermediate state, the final and the initial state _are

separated by

_a barrier

height

whose value U

= 3

(F/Tl/3 (1/7~

is

precisely

the

slope

of In

r vs.

I

IT. By assuming

the

continuity

of the derivative _{we assume} the

continuity

of the

potential

barrier

U( T~. Freezing

then would appear as a continuous transition _», which

corresponds

to the idea that the

dynamical properties just

after

freezing

_are identical to what

they

_are

just

before. On the other

hand,

after

freezing,

the value T

= T~ ^has no further

significance

for the system. ^The characteristic _{range on} which U will vary in this frozen situation is thus of the order of T itself. In the

neighborhood

of T~, therefore the evolution of In

r after

freezing

will coincide with the

tangent

to the

equilibrium

_curve.

The consequences of these

speculations

_are illustrated _on the

figure

3. If _a

system

^{is cooled} in _a time to the

equilibrium

_response describes the situation down to T~ such that

r(T~)

₌ to. ^{If the} temperature is further decreased in

essentially

the _same time _our model

assumes that the characteristic _response times at different temperatures ^{for the} same frozen

state characterized

by T~(t)

are related to each other and to the

equilibrium

relaxation time at T~

by

_an Arrhenius law _:

r = To exp

~~~~ ~~~~

exp

~~~~

=

r(

_exp

~~~~

(27)

Tg

^{T T}

which

corresponds

_to the tangent at T~ ^{to the}

equilibrium

In

(r(T~/ro)

_vs. I

IT

_curve. Thus

dM/d

In t has its maximum at In t~ where this

tangent

intersects the I

IT

line.

Conversely

if _we wait _an additional time t~ at the

temperature

T the system ^evolves towards _{a new} frozen state characterized

by

a new

T(

^{whose relaxation time t'=}

r(T() corresponds

to the _new age t~+ t~ of the

sample

at the

temperature

T. The maximum of the response function

dM/d

In t will _now sit at the

correspondent

of the _{new r} which is t~ + t~ ^{at the}

annealing

temperature and _can be deduced of

equation (27)

for any other

temperature.

^This

equation

defines _{an energy}

U(T~)

^which

diverges

at T~ ^{where the} apparent

r(

cancels. In the _case

I)

where

FIT

zv In

(I T~/Tl

_we have

respectively

u(T~)

₌

zvT~/(i T~/j)

T

S(T~)

₌ ^In

^(rilTo)

= zv

In (I T~/lj)

₊

~

~~

g C

It

provides

a

simple justification

for effective

r(

values which would be difficult to

accept

otherwise. For

example, Lundgren

et al. found that an

annealing

time t~ ^of 103 _s

produces

at 0.91 T~ a shift

by

¹⁰³ _s. The _same shift at the _same temperature

requires

_an

annealing

time of 104 _{s at} 0.9 T~. ^The

interpretation

with _an Arrhenius law would

yield r(

=

10~~°

_{s !!} which

would

correspond

to I

TjT~

~

0.07 with _zv 8 and _To 10~ ^~~ _s.

Suppose

_now that _we cool the

sample by

successive steps, _say : we cool at

Tj

and wait _a time ti ^{and then} cool _{at a} lower T2 ^where we wait _t~, There will be two distributions of lower

amplitude

_one at In

(t2)

which

corresponds

to the last _move and _one at In

(t(

₊

t~)

which

corresponds

to the first _move

(t(

is the

correspondent

at T~ ^of a time tj at

Tjj,

If In

(ti

₊

t2)

In

(t2)

is smaller than the width of the distribution then the whole process will be _{seen as a}

single

_move with _a distorted distribution.

There _{are many} other

signs

that _our

picture

is

good.

For

example,

in

glasses,

_{a cross over} is often observed in the behaviour of In t _vs. I

IT

without _a

change

of the derivative and which could be

interpreted

_as

freezing [22].

In the

figure

3 it is _now

possible

_to label different _areas of the N=Inn _vs.

I/T plane by

the abcissa of the

corresponding Inr~

value

I.e.,

the value N of the renormalization level which has been reached. This

picture

shows that the

only

_way to reach

high

levels of renormalization in

laboratory

times is to sit in the

vicinity

of the transition

temperature

_T~, In

particular

the

right

hand side of the tangent, at the

origin,

to the In

(r/ro)

_vs. I

IT

curve

belongs

to the N

= 0 order of renormalization, It is reached when

In(r/ro)~zvT~/T.

With In

(r/ro)~30

and zv~8 this

yields T<T~/4,

No universal behaviour is

expected

in this _{case as} the

problem

is

entirely

determined

by

the disorder in the

particular sample

which is considered, For _a

given sample

there is however _an

equivalence

of the effect of the

temperature

^{and of the}

logarithm

of time which follows from the Arrhenius law and leads to the TIn _t

scaling

which has been described in detail in reference

[15].

6. Conclusion.

Assuming

that _a hierarchical

procedure

controls the

evolution,

and

introducing

_some

disorder,

the

dynamic

_response should be described with a

log~normal

distribution of

relaxation times centered around In _rj, We have shown the

following

consequences.

At

equilibrium

at T _~ T~ ^{either the}

dynamic scaling relation,

I.e.

Ti/To

=

(fla)~

₌

(l TJT~~~

or an

activated~dynamic

law _occurs

naturally

in this frame.

The

non~exponential

character of the relaxation is the consequence of disorder and renormalization in _our model. It is not the result of _an ad hoc

assumption

_on the evolution of the hierarchical criterion like in

[14]

_{or on} the distribution of cluster sizes like in all fractal

[3, 25, 26]

_or

droplets

models

[27]

nor on the time evolution of _any domain size

[28].

In the

non~equilibrium regime,

times _are renormalized

along

_an Arrhenius law. The argument ^{of the} tangent

explains

the

strongly

non

physical attempt frequencies

which _are measured _near T~.

Our model fails to describe the power laws which _are observed at short times in

spin glasses.

In other words if the solution _can be written

M(t)

t~ ^~ exp

[- (t/r)~

we do not

explain

the t~" part.

By

the way we would rather write this

equation

t~"erf

[In (r/rj)]

where

erf

(x)

is the _error function.

It turns out that

log

normal distributions are also very successful in

describing

_some aspects of turbulence

[29].

A natural

explanation

is that

long

_range turbulences must be

dissipated

at

short range

[30]

_so that _a

hierarchy

of processes is

implied

which is not unlike the _one which govems cluster renormalization in the model which _we

adopt [13].

It is not clear that there _are any further

deeper implications

of this

parallel.

Acknowledgments.

The authors would like _to thank M.

Papoular

and J. Hamman.

Appendix.

In order to go farther than the cut-off

approximation

let _us start from the

equation (4). Then,

with _u

= In

(t/rj),

v ₌ In

(r/rj),

we can write

h(u)=fl=Mo f(u-v)p(v)dv (Ai)

where

f

=

~

[exp(- t/r)]

₌ ^e~ exp

(- e~). Taking

the Fourier transform of

equation (Al)

du

yields

_:

3C(1i) =Mow(1i) «(1i)

where

ar(~J )

=

~

exp

(-

^~

^~'~

^~ is the Fourier transform of P and _q~ that of

f

fi

2

Approximating f

with _a Gaussian _:

f(U)=e~~eXp ~2

2

(I

₊ A~) ~J~

gives 3Cl'J)

" C eXP

~

~2

and

h~U)

=

C'~XP ~

~~i

₊ ~ _~~

fit,f

^U

~ 0

~ _l+h~

which

gives

the

correspondence

p

_{~ ~~} ~ ~ ~~~ ^i/~

~~~

We shall further

verify

that _we have the

good

behavior for

p

in the

neighborhood

of

A

= 0. When is small _{we can}

expand fin equation (Al)

_:

f(u

_v ~2

=

f(u)

₊

vf'(u)

₊

jf"(u)

Then

h(u)

=

Mo f(u )

P

(v)

dv ₊

f~~"~

v2 _P

(v)

dv ₊ 2

=

Mo (f(u)

⁺

~f"(u)

₊

Taking

the

logarithm

_:

g

(u)

= In h

(u)

= In

Mo

₊ In

f (u)

₊ In

(1

⁺

§(~)~

~ ²

~,,~~~

~

= in

(Mo)

₊ in

f(u

₊

~ ~ ~~~

As

§)"/

₌ I e~ and

§~~[~

^{= l 3 e~ +}

^e2~

it is _easy to find that the maximum of

g(u)

is for _u

=

(

^~ _{and that}

g"(-~~ ^=p~m

^I

^-A~

2

which agrees with

equation (A2).

References

[I]

STRUIK L. C. E.,

Physical Aging

in

Amorphous

Polyrners and Other Materials

(Elsevier

Scientific

Publishing Company, 1978).

(2] CHAMBERLIN R. V., MAZURKEVICH G, and ORBACH R., Phys. Rev. iett. 52

(1984)

867.

[3] ALBA M., OCIO M. and HAMMAN J.,

Europhys.

Lett. 2 (1986j 45 _;

HAMMAN J., OCIO M. and VINCENT E.,

Workshop

_{on «} Relaxation in

Complex Systems

and Related

Topics

_», Torino, 16~20 October 1989

~Plenum

Press, New

York),

to be

published.

[4] LUNDGREN L., J.

Phys. Colloq.

France 49

(1988)

C8~1001 and papers

quoted

therein.

[5] BILJAKOVIt _{K., LASJAUNIAS} J. C., MONCEAU P. and LEVY F.,

Phys.

Rev. Lett. 62

(1989)

15I2.

JOURNAL DE PHYSIQUEI T i,M 3, MARS [Ml 20

[6] COLE K. S. and COLE R. H., J. Chem. Phys. ⁹

(1941)

341.

[7] SAITO T., SANDBERG C. J. and MIYAKO Y., J. Phys. Soc. Jpn 54

(1985)

231.

[8] HUSER D., VAN DUYNEVELDT A. J., NIEUWENHUYS G. J. and MYDOSH J. A., J.

Phys.

C19

(1986)

3697.

[9] PAULSEN C. C., WILLIAMSON S. J. and MALETTA H.,

Phys.

Rev. Lent. 59

(1987)

128.

[10] DIRKMAAT A. J., NIEUWENHUYS G. J., MYDOSH J. A., KETTLER P. and STEINER M.,

Phys.

Rev.

836

(1987)

352.

[I Ii DEKKER C., ARTS A. F. M., _DE WIJN H. W., VAN DUYNEVELDT A. J. and MYDOSH J. A., Phys.

Rev. Lent. 861

(1988)

1780.

[12] HbCHLI U. T.,

Phys.

Rev. Lent. 48

(1982)

HOCHLI U. T. and MAGLIONE M., J.

Phys.

Cl

(1989)

2241;

MAGLIONE M., BOHMER R., LOIDL A. and HOCHLI U. T.,

Phys.

Rev. 840

(1990)

l1441.

[13] PALMER R. G., STEIN D. L., ABRAHAMS E. and ANDERSON P. W.,

Phys.

Rev. Lett. 53

(1984)

958.

[14] MONTROLL E. W. and BENDLER J. T., J. Stat.

Phys.

34

(1984)

129.

[15] ^OMARI R., PRtJEAN and SOULETIE J., J.

Phys.

France 45

(1984)

1809.

[16] LUNDGREN L., ^SVEDLINH P. and BECKMAN O., J. Magn. Magn. Mater. 25

(1981)

33.

[17] ^{FUOSS R.} and KIRKWOOD J. G., J. Am. Soc. 63

(1941)

385.

[18]

BERTRAND D., REDOULtS J. P., FERRt J., POMMIER J. and SOULETIE J.,

Europhys.

Lent. 5

(1988)

271.

[19] RAMMAL R., J.

Phys.

France 48

(1987)

1837.

[20] VILLAIN J., J. Phys. France 46 (1985) _;

FISHER D. S. and HUSE D. A.,

Phys.

Rev. Lett. 56

(1986)

1601;

BRAY A. J. and MORE M.,

Phys.

Rev. Lett. 58

(1987)

57.

[21] SOULETIE J., J.

Phys.

France 49

(1988)

21II.

[22] SOULETIE J., J.

Phys.

France 51

(1990)

883.

[23] CAMPBELL I. A., FLESSELLES, J. M., JULIEN R. and BOTET P.,

Phys.

Rev. 837

(1988)

3825.

[24] See e.g. BURKE J., The kinetics of

phase

transformations in metals

(Pergamon)

1965.

[25] CONTINENTINO M. A. and MALOzEMOFF A. P.,

Phys.

Rev. 833

(1986)

3591.

[26] CARRt E., PUECH L., PRtJEAN J. J., BEAUVILLAIN P., RENARD J. P.,

Heidelberg Colloquium

_on

Glassy Dynamics

and

Optimization (Springer Verlag,

1987).

[27]

FISHER D. S. and HUSE D. A., Phys. Rev. 838

(1988)

373.

[28]

KOPER G. J. M. and HILHORST H. J., J.

Phys.

France 49

(1988)

429.

[29]

CASTAING B., GAGNE Y. and HOPFINGER E. J.,

Physica

D 46 (1990) 177.

[30] MONIN A. S. and YAGLOM A. M., in Statistical Fluid Mechanics, vol. 2 (M.I.T. _press

1975).