Phase Space Diffusion and Low Temperature Aging

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Phase Space Diffusion and Low Temperature Aging

Alain Barrat, M. Mézard

To cite this version:

Alain Barrat, M. Mézard. Phase Space Diffusion and Low Temperature Aging. Journal de Physique

I, EDP Sciences, 1995, 5 (8), pp.941-947. �10.1051/jp1:1995174�. �jpa-00247118�

Classification Physics Abstracts

075.50 02.50 05.40

Short Communication

Phase Space Dilllusion and Low Temperature Aging

A. Barrat and M. Mézard

Laboratoire de Physique

Théorique

de l'Ecole Normale

Supérieure(* ),

24 _rue Lhomond, 75231 Paris Cedex 05, France

(Received

18 April 1995, ^received in final form 22 May 1995, accepted 29 May

1995)

Abstract. We study the dynamical evolution of _a system with _a phase _space consisting of

configurations

with random energies. The dynamics _{we use} is of Glauber type. It allows for _some dynamical evolution and aging even at very low temperatures, through the search of

configurations with lower energies. This simple model provides _an example of _{a new} type of mechanism for the aging elfect.

Many

efforts _are

being

devoted _to the

study

of the

out-of-equilibrium dynamics

of

spin-glasses,

in order to understand several

experimental findings,

such _as

aging

and the

joint

existence of

a new

dynamics

and _memory in temperature

cyding

experiments

iii. Roughly speakiug,

the

aging

^{effect is observed} in the

decay

of the thermoremaueut

magnetizatiou (TRM),

where this

decay

is found _to

depend

_on the _age of the system

(the

time it has spent in the low

temperature phase)

in all the accessible time

regimes.

It

implies

that the behaviour of the system

depends

on ail its

history,

and that

non-stationary dyuamics persists

for

arbitrary long

time-scales. Iii

a first approximation, ^trie

decay

which is observed

experimentally

_or

numerically

is often well described

by

trie fact that the TRM measured after _a duration _time

r after

turning

off the

magnetic

field

depends

_on the _age t~v,

roughly

Iike _a function of

r/t~v.

Various

approaches

bave been used to try to understand this

effect, including

_some

phenomenological

studies based _on

droplet

_or domain

growth

_[2], numerical simulations [3], ^{mean-field models}

[4-6],

diffusion _on tree like structures

iii,

and models based _on the existence of traps ⁱⁿ

phase

_space.

In this note _we shall elaborate

along

the lines of this last

approach.

It deals

directly

with the structure of the

phase

_space, made of _many metastables states,

figuring

the

configurations

of _a spin system. ^Such a

picture

has been put ^forward [8], ^and

generalized

_[9],

presenting

the

phase

space as a random _energy

landscape

made of traps, with _a broad distribution of

trapping

times:

the

energies

_are the low

lying

_ones of _a random _energy model

(MM) [loi,

_so

they

have _an

exponential probability distribution,

and the

trapping times, given by

_an Arrhenius

law,

have _a

power-law

distribution with infinite _mean. The dimension of the space is infinite

(equivalently,

(*) Unité _propre du CNRS, associée à l'Ecole Normale Supérieure et à l'Université de Paris Sud

© Les Editions de Physique 1995

942 JOURNAL DE PHYSIQUE I N°8

all the traps _are

connected).

In this

frame,

the diffusion is anomalous

[8,9, iii,

and

aging

is present. ^One virtue of this

approach

has been to

point

out a

simple

"kinematic"

ingredient

for

aging:

the distribution of

trapping

times itself does not evolve _m time, but if this distribution _is broad

ii-e-,

it is _a

Lévy

Iaw with _an infinite _mean

trapping time),

the

probability

to be at time t~v in a trap ^{of lifetime} r

depends

_on the number of visited traps ^{and thus} on t~v, ^which induces

aging.

^{Besides this kinematic}

effect,

there

might

well exist

another, "dynamical",

_source ^of

aging, namely

^the

explicit

^{evolution with} time of the distribution of

trapping

times. We shall

provide

hereafter such _an

example

of _a

dynamical aging

_process.

In the trap

model,

the

probability

of

hopping

from _one

configuration1

to another

config-

uration

j, W~-j, depends only

_on the energy of the site 1: the

energies

_{are m} fact _{seen as} energy

barriers,

which _are then uncorrelated. The model _can be solved in full details: the

equilibrium

correlation

decays

in time _{as a} power law II 2]; ^{however this}

equilibrium

correlation

is _never observed

(for

_an infinite

system)

because of _an

aging

effect due _{to a} broda distribution

of

trapping

^times [8]. ^The

equilibrium dynamics

of the REM has been studied in other

models, corresponding

to some factorized choices of the transition

probabilities W~-j [13,14].

These choices allow for _a solution of the master

equation,

and the

equilibrium

correlation function

displays

various

forms, induding

power law and stretched

exponential

relaxation. Hereafter _we shall consider _{a case} where the transition

probability

is of Glauber type:

W~-j

is not

factorized,

and the _energy barriers _{are now} correlated. This _{seems a}

priori

_{a more} refined _way to define _a

dynamics

for the REM. Furthermore this _case allows for the existence of _a

dynamical

evolution

even at _zero temperature,

through

the search of

configurations

with lower

energies. During

this evolution it becomes _more and _more dillicult for the system to find _a lower

configuration,

which results in _a slow down of the

dynamics.

We shall show

through

_an

explicit

solution at zero temperature ^{that this mechanism}

gives

rise to an

aging

^effect in which the system never

reaches

equilibrium.

This

aging

does not take its

origin

_m

energetic barriers,

but rather in

some kind of

entropic barrier, namely

the low

probability

of

finding

_a favourable direction in

phase

_space. Another

example

of

aging

due _to entropic ^{barrier has been}

proposed recently by

Ritort

lis].

The model _we consider _is defined _as follows: The system can be _{m any} of N

configurations

=

1,..,N.

The

configuration

has _{an energy} E~. The

energies

_are

independent

random variables with distribution

P(E).

The

probability

of

hopping

^from one

configuration

to another

can be defined in several _ways, the

only

_a

priori

constraint

being

the detailed balance:

w

~-pE,

_w

~-pE, j~)

~-J J-~

For

example,

for the trap

model,

where the lifetime of

configuration1

_{is r~}

=

exp(-flE~),

the transition rates are

W~-j

₌

~~~(~~

^Here we consider _a transition

probability depending

on both Ei and

Ej, given by

the Glauber

dynamics:

~~~ ÎÎ

I +

xp(flÎEj

Ei)

^~~~

As mentioned

before,

this system is quite different from the trap ^model

(see Fig. i).

For instance, at zero temperature, the

jump

to _a lower state is allowed in _our

model,

while it is

impossible

to jump out of _a trap.

In this

study,

_we will be interested in

computing

the law of diffusion

(the

number of _con-

figurations

reached at time

t),

the evolution of the _mean energy with time, the

probability,

given

_a time t, to be in _a

configuration

of lifetime _r

(we

shall define the lifetime _r

precisely

below),

which _we will note

pt(r),

and the two-times correlation function

C(t~v

+ t, t~v). ^This

a) b)

Fig. 1.

a)

trap model;

b)"step"

model. Remember that the connectivity _is infinite, _so that such _a picture _can ^be misleading!

last

quantity

is defined _as the _mean

overlap

^between the

positions

of the system at times tw and _{t~v + t:} the

overlap

is

simply

either i if the system is in the _same

configuration,

_or o if it has moved.

Assuming

_an

exponential decay

out of the

configurations,

the correlation function

is related _{to pi}

(r) through:

Citqv

+ t,tqv) =

/~

_dr

pt~

ir)e~~/~ j3)

Before turmng ^{to the} exact solution of the _master

equation

at zero temperature, it is useful _to start with _a discussion of the

trapping

time distributions. When the system ^{is in}

configuration

1, with _energy

E~,

^the

probability

^of _{going away per} unit time is

vs

lE~)

₌

£ _W~-j

₁₄₎

For

large

N and

using

the definition

(2)

of the transition

probability,

_one gets at _zero temper-

ature:

E,

PslE~)

"

/

_dE'

_PIE')

₁₅₎

-co

The

"trapping

time" _r~ is defined _as It

depends only

_on the energy of the

configuration,

ps(E~

through

the relation:

Î ÎÎ~ ^~~~~

^~~~

We deduce

that, regardless of P(E),

the _a

priori

distribution of the lifetimes is:

po(r)àr

=

[à(r _{i) (1)}

As in the trap

model,

this is _a broad distribution with _a

divergent

_mean ^lifetime

(although

here it is

just margmally divergent,

for _any

P(E)).

As _was shown in reference [8], ^{this fact} in itself creates _an

aging

effect. In _{our case} there is _an additional effect because the effective distribution of lifetimes evolves with time. After k

jumps

the system will be in _a lower energy

configuration,

and the

probability Pk(r)

of

having

_a lifetime _r is different from

Po(r).

The

zero temperature

dynamics gives

the recursion relation:

Pk+i(r)

₌

Po(r)

^dr'

r'Pk(r')H(r r') (8)

944 JOURNAL DE PHYSIQUE I N°8

which leads _to:

~

Pk

(r)

=

~')))Î ^{~(T 1) (9)}

The

typical

^lifetime thus increases

exponentially

with k

(it

_means that the diffusion is

loga- rithmic).

This will add _up to the usual eifect of _a

diverging

_mean lifetime in order to induce

aging.

We _now

proceed

to the solution of the master

equation

at zero temperature

using

the

Laplace

transform.

Denoting by p~(t)

the

probability

of

being

on

configuration

at time t, we have

jp~(t)

=

j _{T~ p~(t) (io)}

with

Tq

₌

Wj-~

for

# j,

and

~j _Tq

= 0. The

Laplace

transform jl~(çi) =

/

_dt

p~(t)e~"

~

~

satisfies the

equation:

p~(0)

+

) £~ Ù(Ej E~)jlj(çi) fi~l#)

" i

(ii)

çi + T~

where the lifetime _r~ is defined _as before

by

₌

~j

_{Tj~ and where}

we will take

p~(0)

₌

j.

~ J#~

To solve this

equation

we introduce the

Laplace

transform of the

occupation probabilities

for all

configurations

of _energy E:

f(E,

_{çi) e}

~j

_jJj

_{(çi)à(E Ej (12)}

Using equation (11)

_one denves for

f

the

equation:

fjE, i)

₌

gjE, i) (i

⁺

/°'dE'fjE',1)) ^j13)

where

giE> <)

=

( _L ~)j _Î~~

=

~ )~Î i14)

J TJ T(E)

The

self-consistency

equation

(13)

is

easily

solved and

gives:

RE, #)

=

glE, #)

_exP

/~ _glE',

#)dE') ^ils)

We _{can now use} this solution to compute the

physical quantities

of interest. We start with the

probability pt(r)

to be at time t in _a

configuration

of lifetime _r. Its

Laplace

transform with respect to t is

given by:

Pô(T)

=

£ _ji~(çi)à(r

_r~)

=

P°(T) f(E(T), #)

_{çi +} 1

~

Il

₊

1) 9(E(T), #) (i

₊

çir)2 (16)

This

Laplace

transform _can be inverted and

gives:

Pi(r)

=

~~

~(

~ _~

exp

(-() _{o(r i) (ii)}

We _see that this

expression

decreases _as

t/r~

for _{t « r, as} for _a model of traps for

Po(r)

=

1/r~,

but the

exponential

term makes the

probability

of

being

at time t in _{a con-}

figuration

with lifetime smaller than t very small.

The correlation function _can also be

computed

for

large

t and t~v:

C(t~v + t,t~v) t

~'°

(18)

t~v + t

We _see

immediately

the

t/t~v scaling

of the correlation function. The behaviour lim

C(t~v

+

t-co

t,_t~v = 0, sometimes called "weak

ergodicity breaking"

_[8] shows the existence of _{a non}

equilib-

rium

dynamics

_even at zero temperature, ^{while the behaviour} in the other

limit,

lim

C(t~v

+ t, t~v) = 1, reflects the fact that the

equilibrium dynamics

is frozen _at T

= o.

t~-co

Let _us

emphasize

that ail these results _are

independent

of the distribution

P(E).

The two main

hypotheses

of the derivation _are the fact that the

connectivity

is

infinite,

and the temperature ^{has been taken}

equal

to _zero. These results _can be

partially

extended in the

case where

P(E)

is _an

exponential distribution, P(E)

m~

pexp(pE)Ù(-E).

Such _a distribution has been found in _mean field

spin glass

^models [16], ^{and it is} at trie heart of the trap ^model

de8cription,

since it Ieads _{to a} broad distribution of Iifetimes when

fl

> p.

Specializing

in this case, we can first

explain

in _more detail the _zero temperature

dynamics

^{studied above.}

Indeed,

the relation between _energy and

trapping

time _can be

explicited

_{as: r~}

=

e~P~,

and the energy distribution evolves then _as

j_~~k+i

~k

PklE)

₌

~,

exPlPE)°1-E) l19)

The _{mean energy} decreases _as

-k/p

with the number of visited

configurations,

_or

equivalently

as

log t/p (remember

that the diffusion _is

logarithmic).

For finite teniperature, the set of self-consistent

equations

is _more

complicated;

_we write a~ =

exp(flEj),

_so that

=

ai/~ j~'

~'°

i

/l~/~ 120)

and _we obtain

p~

io)

₊

/~° ^{d>e-Àa, fj>,}

^çi)

fl>1#) =

°

~

~

j ¹²¹⁾

flÀ, #)

=

glÀ, #)

+

/~

_d~l

_glÀ

₊

^Î, _{#)fl/1, #) 122)}

~~~'~~

_"

( ( Î)1'

=

j _f~

_d~

^uP/Pe-vu

'

°

çi + ~lp/p

j"

^{~'~ du}

¹²³⁾

~ i + ~~~~

Taking

=

r~/P

_and

writing g(r, çi), f(r,

çi) ^{instead of}

g(r~/P, _çi), f(r~/P, _çi),

_{we obtain} for

f

the

following scaling:

f(T, #)

=

jT~~' ^{h(#T) (24)}

co

with

h(x) behaving

_as

I/x~

for _{x »} I and

/ _dxh(x)

_{finite. After}

some calculations, it _can o

be shown that

pi(r)dr

behaves Iike

~~~

^for I « t _{« r,} and _as ^~~

(~) ^~~~

^for I « r « t,

r r t

and for

fl

» 1.

946 JOURNAL DE PHYSIQUE I N°8

We obtain thus

qualitatively

the _sanie behaviour for

pt(r)dr

_as

before,

and also for the correlation function: the

dynamics

is not modified

by

_a small temperature.

Note that this behaviour holds in the limit of infinite

N;

for _any finite N the system eventu-

ally thermalises,

after _a time

proportional

to N

(for example,

the minimal energy of N _states with

exponentially

distributed

energies

is

-logN/p

_so it takes _a time N to find

it).

For this

model,

_some of _our results _are similar to those of reference [9]: ^the mean energy decreases _as

-log(t),

and at time _t the most

probable configurations

are of lifetime _r

= t.

Nevertheless _{we must}

emphasize

that the mechanism is

totally

diiferent: in _a model of traps, the _{mean energy} decreases because the system visits _more and _more traps, and _so it has _more and _more chances to find

deep

_ones. At each step

Pk(E)

remains the _same: Pk

" Po. The

diffusion is in

tP/~~,

and the energy ^at step ^{k evolves like}

log(k),

because the minimum of k

energies

distributed

according

to the

exponential

distribution is in

log(k).

In _our

model,

_on

the opposite, the energy distribution that the

partide

_sees

evolves,

and _so does the distribution of

trapping

times

(see Eq. (9));

the _energy decreases in fact because it is easier to find _a

configuration

with lower _energy

lit

takes _a time

proportional

_to

exp(-pE))

than to move

by

thermal activation

la

time

exp(-flE)

is

needed).

It _means that the system

spends

less time in

a given

configuration

but, since the energy ^at step ^{k decreases} like -k, the diffusion is much slower

(logarithmic

instead of _a

power~law),

_{so we}

finally

get the _same behaviour for the _mean energy as a function of time. However the main feature is that there exists _a zero-temperature

dynamic,

which is

qualitatively

not modified

by

_a small temperature.

It would of _course be _very

interesting

to be able to

generalize

this

approach beyond

the _case of

an infinite

connectivity.

For instance _{a more} realistic definition of the REM

dynamics

could be to start from the definition of the REM in terms of

spins

with p

(- co) spin

interaction

[10,17],

and _use the transition matrix

resulting

from

single

_spm

fiip dynamics.

This _seems rather

complicated

at the moment.

Acknowledgments

We thank J.P.

Bouchaud,

R. Burioni, L.

Cughandolo,

D. Dean, C. De Dominicis and J. Kurchan for many

helpful

discussions on related topics.

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