The Cavity Approach for Metastable Glassy States Near Random First Order Phase Transitions

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The Cavity Approach for Metastable Glassy States Near Random First Order Phase Transitions

T. Kirkpatrick, D. Thirumalai

To cite this version:

T. Kirkpatrick, D. Thirumalai. The Cavity Approach for Metastable Glassy States Near Random First Order Phase Transitions. Journal de Physique I, EDP Sciences, 1995, 5 (7), pp.777-786.

�10.1051/jp1:1995168�. �jpa-00247102�

J. Phys. I France 5

(1995)

777-786 JULY 1995, PAGE 777

Classification Physics Abstracts

75.10N 64.60M 64.70P

The Cavity Approach for Metastable Glassy States Near Random First Order Phase llYansitions

T.R.

Kirkpatrick

and D. Thirumalai

Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, U-S-A-

(Received

23 January1995, received in final form 5 April 1995, accepted 7 April

1995)

Abstract. Ii is shown that trie cavity method developed to describe equilibrium states _in trie mean-field Ising spin glass model _con be generalized to describe metastable stores _in trie mean-field Potts glass. Trie crucial input is trie existence of _an extensive number of statistically

similar but incongruent metastable states with

a weighted Gaussian free energy distribution. It is

argued that the inherent quenched randomness _m trie Potts glass Hamiltonian _is net important for most of the arguments and that this approach provides insight into recent work _on trie

structural glass problem.

1. Introduction

In ibis _{paper we} show how trie

Onsager (reaction

field

approach)

_or

cavity

method

developed iii

to describe

equilibrium

states in trie mean-field

Ising

_spin

glass

model [2] con be modified to describe metastable

glassy

states near random first-order

phase

transitions. For concreteness we

focus _on trie mean-field p-state ^Potts

glass (PG)

model

(with

_p

>4)

where trie metastable states have been well characterized

by

orner methods

(3,4]. ^However,

we stress that trie ideas behind this

approach

_are rather

generic

and in

particular they

do non

depend

in _a sensitive way on trie fact that trie PG Hamiltonian bas

quenched

randomness built into it. As _a consequence, we

argue that trie ideas

presented

here _con be used to understand _some work [Si on trie structural

glass problem

which bas _no

quenched

randomness.

Trie

general

type ^of system ^and situation _we bave _m mmd is trie

following.

First, trie system is above its

equilibrium (ideal) glass

transition temperature ^which we denote

by TK,

_or

by

trie

Kauzmann temperature

(in general

TK _may be ai _zero

temperature). Second,

trie

incipient phase

transition is most

naturally

described

by

_a

probability

_measure for _a random order parameter ^with zero mean. Trie usual observable order parameter is trie average of _a

squared

random variable referred to

commonly

_as trie Edwards-Anderson order parameter, _q (2]. ^Ai

TK,

_q

jumps discontinuously

from _{zero to a} finite value at _a random first-order

phase

transition.

For ail temperatures greater thon

TK,

but less thon _a temperature denoted

by

TA, there exists

locally

stable

glassy

metastable states. In mean-field models these _{states are}

infinitely long- lived,

whereas in

short-ranged

models

they

_can

presumably

be

theoretically

well-defined

by neglecting droplet

fluctuations _or

activated, A,

_transport. For trie systems we consider, there

Q Les Editions de Physique 1995

exists _an extensive number of

statistically

similar but

incongruent

states (6] metastable states below TA and ii is these states that _are considered in this paper. Because of trie extensive

solution

degeneracy,

these states _are

expected

to control trie

dynamical

behavior of trie

glass

for ail T _< TA

13,4, ii.

In trie above

description: (1)

An extensive number of solutions _occurs when the

logarithm

of trie number of states scales with the system size,

(2)

Two states are

statistically

similar when

they

have the _same value of _q,

(3)

States _are

incongruent

when the

overlap

function

(Eq. (3.6))

is _zero.

The ideas behind the cavity

approach

_{are as} follows.

Imagine

_an

N-spin (or N-partiale)

system ^{and add} a new one to it. One then _{tries to} determine the statistical mechanics of the

(N +1)

_spm system

by studying

the _response of the

N-spin

system to the additional

spin.

In

general,

_{one can}

imagine

two types of

important

fluctuations that act _on the added spin. First, _even if there is _a

single

unique state

(as opposed

to

configuration)

of the N and N

+1-spin

system, there will be _a

fluctuating magnetic

field felt

by

the additional spin due to thermal fluctuations _in the

N-spin

system.

Secondly,

if the order parameter

description

is

intrinsically probabiistic,

_as it is in

glasses,

then to obtain observables _one must average

over the randomness. Here ii is

important

to

distinguish

between _two

possible

_sources of randomness. There is the randomness _m spin

glasses

due to the presence of

quenched

disorder

m the

microscopic

Hamiltoman. Even if there is _a unique macroscopic state for _a

given

set of random bonds the local site

magnetization

fluctuates due to this randomness. Trie second type of randomness is _more novel and does arise _even in

regular Hamiltonians,

1-e-, ⁱⁿ those systems where

quenched

interactions _are not

explicitly

present. ^{If there} are many

macroscopic

states that _are

statistically

similar and bave

nearly equal

free

energies

then _an externat field _cannot be used to

pick

oui _a

particular

state and _one is forced to average over ail these _states. In previous work _on trie structural

glass problem

_{(Si it is} trie second type ^{of randomness that} is shown to be important. Note that this type ^{of randomness} can m

principle

exist _even if trie

microscopic

interactions _are not random.

Subsequent

to Dur initial work _on model Hamiltonians

describing

trie

liquid

to

glass

transition numerous papers bave

recently

aflirmed trie earlier conclusion that

glassy dynamics

_can result in systems without

quenched

randomness

[8-12].

In this

paper, we

provide

_very

general

arguments ^{that this conclusion} is

intimately

tied _to trie notion that _an intrinsic randomness results due _to the _presence of _a

large

number of

statistically

similar macroscopic states. In the _case of spin

glasses

without reflection symmetry ^{these arise}

m the temperature range TK < T < TA where TK ^and TA are the characteristic temperatures introduced earlier.

Explicit

calculations reveal that ii is

precisely

in this temperature _range that there _are

large

numbers of

macroscopically

well-defined metastable states.

Conceptually,

the situation _{we are}

considering

here dilfers from the

problem

considered before

by

Mézard et ai.

iii

for the

Ising

_spin

glass

model because: 1) ^the states we consider do not have _an ultrametric

topology, 2)

^for our case, two states

typically

dilfer in free _energy

by

_an amount

+~O(N~/~)

and the

weighted

distribution of free _{energy states} is Gaussian rather

than

exponential

_[4].

Nevertheless,

_we shall _see that the calculational

methodology developed by

Mézard et ai.

iii

_can be

applied

with

appropriate

modifications.

Finally,

_we point out that the states we consider _are

completely

missed in the usual statistical mechanical

approach

to spin

glasses

_[2j.

Physically

this _is because the natural order parameter ^{to this}

phase

transition

is the Edwards-Anderson parameter ^{which is} a

dynarnical

quantity. ^The plan of this _paper is

as follows. In Section 2 _we rederive the TAP equations for the PG using the cavity method and obtain _an expression for the site

magnetization

m terms of _a random

magnetic

field

acting

on that site. In Section 3 we denve the

probability

_measure ^for the random

magnetic

^field

and obtain the self-consistent

equation

of state for the Edwards-Anderson order parameter.

This result _agrees with that obtained from

approaches [3,4j.

In Section 4 _we conclude with _a discussion.

N°7 THE CAVITY APPROACH FOR METASTABLE GLASSY STATES 779

2. Thermal Fluctuation in _a

Single

^State:

Tap Equations

for trie PG

In this section _{we use} the cavity

approach

to derive the TAP

equations

for the PG. The derivation in _a

generalization

of that

given

elsewhere _[2] for the mean-field

Ising

_spin

glass.

We repeat ^it here because _some of the results _are needed in the next section where fluctuations due to solution

degeneracy

_are taken into account.

The mean-field _or infinite

ranged

PG Hamiltonian _is

(3,13],

p-1 l~

"

~ ~ _{~çS,asja. (2.1)}

~~~

i, j

(Î ~

j)

The PG has _p components m a

(p-1)-dimensional

vector space. The bonds

(J,j )

_are

randomly

distributed

according

to,

P(J,j)

=

~

~ exp

~~j~~j

,

(2.2)

2~rJ

~ ~

2J

with N the number of lattice sites. To _{suppress a}

ferromagnetic

transition _one must

properly

fix the average of

J,j.

We avoid this

problem by setting

the

ferromagnetic

order parameter,

N

Ma =

~jm,a

= 0.

(2.3)

~

i=J

equal

to _zero. The S,a in

equation (2.1)

_are Potts variables chosen from the set

(e(),

é

=

1, 2,.. _p,

given

in reference [14]. ^{Some useful identities} _are,

~j

P

ele(

"

Pôab, (2.4a)

1=1

~j

p-1

e(e$'

=

pâli»

1.

(2.4b)

a=1

~j

P

_e(

= 0.

(2.4c)

1=1

We next add _an additional Potts

spin

at a site 0. The vector

configurational magnetic

field

acting

_on site 0

(the (N +1)~h spin)

due to the

original N-spms

is,

N

hi

=

fl ~j

_Jokska.

_(2.5)

k=1

If the

N-spin

system ^is m the _state

a(N)

then

by

definition the number of

configurations, dN(EN),

of the

N-spm

_system with energy between EN and EN +dEN is,

dN(EN

=

e~(~N)dEN

=

exp[fIEN flF~(NjÎdEN ^(2.6)

The

magnetic

field

given by equation (2.5)

is

specified by

_a

probability distribution, P(h)dh,

equal

to the

probability

that

choosing

at random _any relevant

configuration

_in

a(N)

the value of

ha is in the interval

(ha,

ha+dha

). By

^definition

P(h)

is

independent

of EN which is determined

by J,j

^with 1,

j #

o.

By

trie central limit theorem

P(h)

is Gaussian with parameters,

N

(h()~(Nj

"

fl~jJ0k _(Ska)ce(Nj

k=1

(2.7a)

N

"

à~J0kfilÎÎ~~

" ~Î~~~>

k=1

~~~

(h~

_~

h"~~~)

~

^hC

b

^OE(N)j

b

~

~(N)

~ fl~

~

_~°kJ0£

_jska

lllj~~~) jS~~

^~OE(N)j

k,£=1

~ £b

N

OE(N)

Î ~ ^~~~

'~~ÎÎ~~) (Skb m~)~~)

~~ ~~~

~~~ OE(N)

" Plôab qab] "

Pôab Ii ql'

The second

equality

_in

equation (2.7b) (p

_w

fl~J~)

follows from trie

clustering

property ^of

a pure state,

a(N),

and trie third

equality

is due to linear _response in _a

surgie valley

with

zero net

magnetization, equation (2.3).

In

equation (2.7b),

_{qab is} trie Edwards-Anderson order parameter,

~ab

~

D2~~~~~Î~~~

~~ab> (~.~C)

i=1

which in _a frozen state, with

equation (2.3),

must be

diagonal

in trie vector labels _a, b.

To each

configuration

of trie

N-spin

system there exists p

configurations

of trie

(N +1) spin

system. ^Trie energy of trie

(N +1) spin

system is

EN+i

₌

EN hasoa.

For fixed

EN+i,

h and

So

trie number of

configurations

_is,

(h ha(Nj)

²

dN

(EN+i, h,

So _" exp

-fifre(Nj

+

fIEN+i

+ h

So dEN+iDh. (2.8a)

21L(1

q)

IV

l)

with Db

=

~~~

~~ ~

db.

Integrating

_over h and

summing

_over So

gives, by definition,

27r

dN(EN+i

_{" exp}

-flF~(

_N+i) +

fIEN+i dEN+1, (2.8b)

with

Fa(

_N+ij trie free _energy of trie

(N

₊

1)-spin

system,

FOE(N+i) =

Fa(N) ~(P ^{l)(1 q)} j1°g~exP _(lll~~~e~j ^(2.8C)

1

To derive trie TAP

equations

for trie PG _{we use} that

equation (2.8a) implies

that trie

probability

that trie

(N +1)

_spin system ^bas a certain value So and field h

acting

_on So _is, ^for fixed

EN+1, j~

~oe(N) )2

~~~'~°~

~~~~

2p(1- q)

^~

~°Î

~~'~~

N°7 THE CAVITY APPROACH FOR METASTABLE GLASSY STATES 781

with K _a normalization constant. From finis

result,

trie average

magnetization

in trie _state

a(N +1)

is,

(S0a)a(N+i~

^OE

m()~~~~

#

~j _{/DhP(h, 50)S0a}

~~~

~jeÎeXPÎh~~~~e~Î (2.10a)

~

i

~jexp[h~(~)e~]

^'

1

and trie _average

magnetic

field

is,

N

(h ~~ _J~~~~~~~~~

~ °~~~~~

~~~

~~

(2.10b)

=

hÎ~~~

₊

pli q)m()~~~~

To relate this _to

previous

TAP work for trie PG _{(3] we use,}

equations (2.4a)

and

(2.10a)

to

obtain,

h[(~l

=

£

e~

log il

⁺

_e(m()~~~~j ^{j2 lia)}

p

~

~

In

deriving equation (2.lia)

trie

identity

_[3]

exp

- £ _log

^~

^~~~~~~~~~

=

£

_exp[e~

^h~(~) _(2.

_Il

_b)

1 i'

i

was used. With

equations (2.10)

and

(2.Il)

_{we recover} trie TAP

equations

used before. Al-

though

non needed in Section 3 trie

underlying

TAP free energy [3] is, _suppress trie

o(N +1) index,

àFTAP

_"

~(l

₊

_m,ae~)

_ÎOg ^{~ ~~~~~}

iiJ,>m,~m~~

ljp~

i~ji

_~~~.

12.12~

1<j

These results reduce _to trie free energy for trie usual SK mortel for _p

= 2.

3. Fluctuations Due to

Many

Solutions: Self-Consistent

Equation

for qab

In this section _we first quote ^trie known result for the number of

statistically

similar but incon- gruent ^metastable states in the PG above the true

equilibrium phase

transition temperature.

We then argue that this result is easy ^to understand and that it is

probably generic

for _a class of

problems. Following

finis, we determine trie statistical properties of trie

magnetic

field which acts _on trie

(N +1)

spin due to trie fact that trie

magnetizations

_are

specified by

a

probability

measure and because there _{are many}

statistically

similar metastable states.

Finally,

_{we use}

these results to derive _a self-consistent

equation

for trie Edwards-Anderson order parameter ^for trie PG. Trie final result is in agreement with trie

equation

of state derived

by

orner methods.

3.1. NUMBER OF METASTABLE STATES. In reference _{[4] ii was} shown that for trie PG trie number of

statistically

similar metastable states in _a

N-spin

_system with free _energy FN above the

equilibrium phase

transition temperature _was extensive [3] and

given by,

dN(FN)

" ~~P

~~N ~~Î~

2àN ~~~ ~~~

~~~

(~'~~

=

exp[fIFN flF(]dFN.

Here

F(

is the canonical free _energy in the

N-spm

system,

ÉN

is the component

averaged

free energy for

N-spins

and

AN

z~J

0(N)

is the variance in the

weighted

free _energy distribution about

ÉN

_Î15]. Note that the last term in the first

equality

in equation

(3.1)

is of relative

magnitude

_zero unless FN

ÉN

z~J

0(N). Restricting

ourselves to states with free _energy close to

ÉN,

_we have the second

equality

in

equation (3.1).

In

general F( # ÉN

if there is _an

extensive solution

degcneracy.

We next

interpret

equation

(3.1)

and argue it is

generic

for _a Mass of

problems.

The systems

we have considered with ramdam first-order

phase

transitions have _an extensive number of metastable _states that _are

statistically

similar but

incongruent.

In terms of

overlap

functions

(cf. below)

these states have _zero

overlap. By

the central limit theorem this

implies

that

the

weighted

distribution of free _energy states must be Gaussian. The

weighted

distribution

is defined

by examining

the

partition

function which is

given by

a sum over ail

statistically

similar states,

~N "

~

_eXPÎ~~~OEÎ

"

/

_d~N

~~j@~ ^{eXPÎ~~~NÎ. (3.~)}

~

N

Equations (3.1)

and

(3.2)

give a

weighted

free _energy distribution

P(FN), P(FN)

z~J

~~~~~~

exp[-fIFNÎ, (3.3)

dFN

with _an average free _energy

ÉN

=

~f~exp[-flf~j /ZN

and _a canonical free _energy

flfjj

=

-logZN.

The

density

of states

given by

~ equation

(3.1)

is the unique

density

consistent with the above stated

requirements.

Finally,

_we note that the cavity

approach

in the _next subsection shows that

Fjj

does non

depend

_on the

ordering

into the metastable

glassy

states. This has been

interpreted

else- where

[3,4, 7j.

The crucial point ^{is that}

dynamical

mean-field

theory predicts freezing

into _a

metastable

glassy

state for T _< TA ^because the extensive solution

degeneracy implies

that with

probability

_one the initial

configuration

of the

N-spm

_system will be _{in a}

glassy

metastable

state. Because nucleation ont of _a metastable state _is

impossible

in _a mean-field model it will

stay m a

particular

metastable state forever. In non-metastable models _one expects

droplet

fluctuations to cause activated transitions to the lower free energy

paramagnetic

state and to other metastable

glassy

states (3j.

3.2. SITE FLUCTUATIONS DUE TO THE PRESENCE OF MANY STATES. We denote the

free _energy states of the

N-spin

system

by

the label _a

= 1, 2,.

,

M and the

corresponding magnetic

^fields

acting

_on the

(N +1)

_spin

by (h~(/~1).

^An _average of _our

observable,

0, over

solutions _is defined

by [lj

1°)r(N)

"

ù É

1°)a(N) ^13.4)

~=i

N°7 THE CAVITY APPROACH FOR METASTABLE GLASSY STATES 783

Note that this is _an

unweighted

_average because _we bave

already

restricted ourselves to states with similar free

energies.

^The solution

averaged magnetic

^field at site 0

is,

N

/~o(N)) ^~ jj

_j

jy~~

/~r(N) j~

~)

~ r~~/~ ^{°~ ~~ r(N) " a}

k=1

By

definition the field

hÎ~~~

fluctuates

only

due _to the random

(Jok).

For _{our case} there _are

many ^states

(M

+~

e/~)

that _are ail

incongruent,

~ÎÎ ~Î ~Îa~$

_{~ ~°~}

~

# ~' (3.~)

~

Denoting

the disorder _average

by

_],

equations (3.5)

and

(3.6) yield,

j/~r(N)j

~ ^~

(3.7) [/~r(Nj/~r(Njj

~ ~ ^~

or

equivalently, hÎ~~~

= 0. With equation

(3.5) this,

in tum, seems to

imply (mka)r(N)

" 0

(self-consistent

arguments

yield ((mka)r(N)(

^z~J

N~~/~).

A similar result _was used in _our

previous

work _on structural

glasses

_[Si where the

analog

of Jok is non-random and the

only

average is _over the solution

degeneracy.

The solution _average of the _square of the

magnetic

^field at site 0 is,

(h[(/~)h)~~~)

^{= pqab =}

^{pqôab. (3.8)}

r(N)

Because the states are uncorrelated _{we can} take

h]~~~

_{e hia to} be _a Gaussian random variable

with _vanance

given by equation (3.8)

in _an ensemble of

N-spin

states. The number of free

energy ^states with

N-partiales

with field hi

acting

_on the

(N +1) spin

_is,

dN(FN,

hi

" exp

(fIFN flF[ h( _{/2~1q) dFNDhi, (3.9)}

jp 1j

with Dhi

=

(~~)

^{~ dhi The} distribution _m the variables

FN+i

and hi _is

given by,

27r

~

j~ _(FN+i,

_hi

=

/dN(FN, ^h)à[FN+i

^FN

^àF(h)]

~+i

=

Dlli

_exP

fl _FN+i

flfi Ill/2Pq

₊

iog ~llie~ 13.1°)

+i _{iv i)ii} ⁾¹

~

where _we have used equation

(2.8c)

with

h°(/~)

- hi

Integrating equatiou (3.10)

_over hi must

yield,

~~l~N+1)

" ~XP

ÎfI~N+i fIFÎÙ+iÎ dFN+1 (3.lia)

From

equation (3.10)

_we obtain,

~~N+1 ~7c/~

2fl

~

jp i)ji

_q)

fl log /Dhexp ^{[-hÎ/2~1q) ~je~~~~}

1

(3.llb)

" ~Î~

~~~

~~

~°~~

As

already

mentioned, this result

implies

that the canonical free energy does not

depend

_on trie

glassy ordering

characterized

by

_q.

Equation (3.116)

is trie

expected

result for _a

paramagnetic

PG if _one takes into _account that trie J's in

Fjj

~~ ^are normalized to

(N +1)~~,

while those in

Fjj

_are normalized _to N~~.

A self-consistent

equation

for trie Edwards-Anderson order parameter can be obtained

by noting

that equation

(3,10)

_gives trie distribution of hi for fixed

FN+1~

P(h)Dh

= K exp

-

_~~

+

log £

e~i~~

Dhi, (3.12)

~l~~

1

with K

being

_a normalization constant. From equations

(2.10a), (2.

Il and

(3.12)

trie equation of state is,

/Dhima(hi ^)mb(hi

^exp

-

_~~

£ _log

^~

^~~~~~~~~ )j

21Lq P

~ P

~~~ ~~~~

/Dhi

exp

(- ^{~Î £ log}

^~

^~~~~~~~

~~ ~~~~

211q P

~ P

with

ma(hi given by,

hia

_"

~j

_e$

_log

(1 ⁺

e(mb(hi

_)j

(3.13b)

P ~

Equations (3.13)

_are closed

equations

for _q and _{are m} agreement ^{with trie} results from orner static and

dynamical approaches (3,4].

4. Discussion

In this _{paper we} bave

provided

_very

general

arguments ^for trie

origin

of

glassy

behavior in systems ^that

undergo

random first-order transitions. Dur results

emphasize

that trie slow

dy-

namics

resulting

in

glass-like

relaxation in _a

myriad

^of

seemingly

unrelated systems can be understood from _one central concept. This crucial idea _is

that,

whenever _one bas _a froc _en- ergy

landscape

such that there _are extensive numbers of basins of attraction

corresponding

to

(macroscopically) statistically

^{similar but}

incongruent

metastable states, ^{then trie} system _gen-

erates intrinsic randomness because _{one is} forced to average over trie solution

degeneracy.

This situation _can

naturally

arise _even for Hamiltonians which do non

explicitly

contain

quenched

random interactions. Trie

glassy

behavior results from this

self-generated

randomness. Dur arguments ^bave been buttressed

using

trie Potts

glass

as a concrete

example.

However trie

similarity

of trie results with _our

previous

work

involving

model fluid Hamiltonians _[Si that do not

explicitly

contain any randomness shows that trie

conceptual

aspects of _our work should be

applicable

to _a wide

variety

of systems. We also note that similar argument _can be used to

obtain

analogous

results for trie J~-spin

glass

model

(16j.

N°7 THE CAVITY APPROACH FOR METASTABLE GLASSY STATES 785

These ideas _can be used to understand trie recent novel work

(8-loi by several'groups

who bave discovered that

glassy

behavior _can be observed _even in model Hamiltonians that do _not contain

explicitly

contain

quenched

disorder. In ail these _cases trie systems gel ^frozen into

one of trie

possible

metastable states below _a characteristic temperature. ^{In reference}

iii]

_an

explicit computation

of trie solution

degeneracy

for _one of these niodels bas been

done,

and

a

dynamical phase

transition bas been related to trie _emergence of _an extensive number of

statistically

similar states.

A crucial

point

that is worth

making

is that ail _our results follow from trie Gaussian distri- bution of free

energies.

This is to be contrasted with trie SK

model,

where trie free

energies

are

exponentially

distributed below _a threshold value

[18j.

Thus unlike trie SK

model,

alI trie consequences of trie law of

large

numbers are valid in _{our case.} It is

perhaps

for this _reason that trie behavior _we bave discovered _niay be

generic

for _a

variety

of classical disordered systems.

Furthermore,

it appears natural that these concepts con be used _to

provide

_a framework for

understanding dynamical

behavior in structural

glasses (19j.

It is also important to

emphasize

that trie

major

consequences

resulting

from solution de- generacy do non require trie

replica analysis.

In

fact,

trie

only

relevant order parameter ^{in trie} temperature range TK < T < TA is trie self

overlap

between trie

replicas, narnely

trie

dynarni-

cal Edwards-Anderson order parameter.

This,

_agam, is in contrast with trie much studied SK model in which there is _no

analogue

ofTA

Description

^of trie

spin glass phase

in trie SK model

requires

trie fuit broken

replica

_symmetry treatment.

Further

insight

into trie nature of trie states m trie range TK < T < TA _may be obtained

by mferring

trie barrier

height separating

two states. From

equation (3.1)

it is obvions

(3,7j

that trie free

energies separating

_any states o and

fl typically

dilfer

by @.

_{Trie free energy of}

activation between two genenc metastable states may be estimated _as follows. Elsewhere [20j

we showed that, at least close _to

TK,

trie activation borner àF* scales _as, [20]

àF*

+~

t~~"~. (4.1)

In finite _range

models,

there exists _a natural finite size correlation

length, (

+~

t~". In _a correlation volume, trie number of _spms N scales

(~. Making

these substitutions in

equation (4.1)

_gives

àF*

+~

N~/"~ (4.2)

For random first-order transitions, du ₌ 2

Id,

which

implies

that àF*

+~

@,

_{1-e-, trie} barrier between states is

comparable

to their dilference in free energy. In trie usual SK model àF* is

estimated (21] to scale _as

N~H.

_Trie

same arguments ^{bave been used}

recently

to estimate lime scales for

folding

of

proteins

_[22] which _are

mesoscopic.

Trie _success of finis

application

further

underscores trie

ubiquity

of

complex

_energy

Iandscapes

_{m a} wide vanety ^of systems.

Acknowledgments

This work _was

supported by

trie National Science Foundation under grant ^numbers DMR-92- l7496 and CH&93-07884.

References

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