Mean field dynamics of random manifolds

HAL Id: jpa-00246423

https://hal.archives-ouvertes.fr/jpa-00246423

Submitted on 1 Jan 1991

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

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

Mean field dynamics of random manifolds

T. Vilgis

To cite this version:

T. Vilgis. Mean field dynamics of random manifolds. Journal de Physique I, EDP Sciences, 1991, 1

(10), pp.1389-1394. �10.1051/jp1:1991215�. �jpa-00246423�

f Phys. Ifrance 1

(1991)

1389-1394 OmOBRE1991, PAGE 1389

Classification

PhysicsAbstracts

05.40-61.40

Sham Communication

Mean field dynandcs of random manifolds TA~vflgb

Max-Planck-Institut for

Polymerforschung,

Postfach 3148, D-65W Maim,

Germany

(Received19 June 1991,

accepted

b1

final fern 15Ju~y1991)

Abstract. tile _mean field

dynamics

of manifolds in _a

quenched

random potential is discussed by

means of the

Martin-Siggia-Rose (MSR)

method. In a self-consistent way we obtain for the dynamic exponent _z ^the value _z

=

(~

~

where D is the dimension of the manifold and ₇ the noise charac- +

7)

teristics of the

potential.

lllis

implies

immediately for the

wandering

exponent ( ₌ ^{~ ~}

,

i.e. that

2(1

+

7)

obtained by hierarchical replica _symmetry

breaking.

tile general scaling law _z

=

j

_{( is suggested.}

Moreover, _we find _as in the replica theory two, different

regimes

for the wandering exponent as a

function of the noise correlation function.

1 Iutloductiou.

Polymers

in

quenched

random media have been studied

extensively by

various methods

[I].

This

problem

has its relevance in many di%erent fields such _as

travelling

waves [2], ^surface

growth

[3],

randomly

stirred fluids [4, 5] ^and

spin glasses

_[2]. The

problem

can be unified and

generalhed

to the behaviour of manifolds in

quenched

random

potentials

_[6,7~. The

problem

of the manifold in

a

quenched

random media can be formulated

by

the Hamiltonian [6,_7~

H ₌

/ _d~

s

( ~)~~~

^~

+

/ _{d~s V(R(s), s) (1)}

~~~ s~

where D is the dimension of the manifold. R is _an N dimensional vector. For D

= I and N

= 3

the Edwards Hamiltonian of the usual random directed

polymer

is recovered. _s h the _contour variable in this case and

R(s)

the chain vector. We _assume

(to

_compare to M62ard's and Parisi's

results)

a random

potential

with two cases. First _we consider the disorder to be correlated

by

(V(r,

_S)

V(r', S'))

=

A(r r')~~~~'~ 6(S S') (2)

JOURNAL DE PHYSIQUE I T t, At to, OCTOBRE twt 55

1390 JOURNAL DE PI IYSIQUE I N° lo

and

secondly

a local disorder

(V(r,

_S)

V(r', _S'))

= A

6(r r')

_6(S

S') (3)

The _mean value of the

potenthl (V(r, s))=0

in both _cases.

M62ard and Parisi _[7~ have

analysed

the stathtical

properties

of

(I)

in detail

by

use of _a vari- ational method where the variational

parameter (which depends

on the

replica indices)

shows

replica symmetry breaking (RSB). By

use of hierarchical RSB

they

find for the

wandering

_expo-

nent (, which measures the transverse fluctuations

I(R(s) R(s'))~)

+~

(s _s')~< (4)

~"~ (1 ~)

~~~

which is

actually

the value obtained

by

a

simple FloTy

argument [8]. ^The purpose ^{of this note is} to

derive

(5)

in _a self consistent mean field way

using dynamical methods,

which allow to avoid

repli-

cas [9]

(despite

the

beauty

of the

replica theory). Moreover,

_we calculate the

dynamic

_{exponent z}

which _measures the time

dependence

of the _transverse fluctuations of the _same value of _s, I.e.

I(R(s, t) R(s, t'))~)

~

(t t')~ ₍₆₎

and propose a

(connectivity) scaling

low for

(

and _z.

2. The

equation

of motion and the

dynamic

functional.

The

appropriate dynamical

formulation for the

problem

_can be done

by

the aid of the Martin-

Siggia-Rose theory fIASR) [10].

^We cannot use conventional

polymer dynamics,

since _we have _to average over the random

potential.

The

equation

of motion of the _vector

R(s)

is

given by

the

Langevin equation

~~~~

^{~~ ~}

6is)

~~~' _~~ ~~~

where ro b the kinetic coefficient

(rp'

_{b a}

_friction)

_and

_{f(s, t)}

_b

a Gaussian noise with a corre-

lation function

~~~~' ~~~~~" _~'~~

/0

^{~~~ ~'~ ~~~ ~'~ ~~~}

The

corresponding

MSR functional b

given by

Z

(it, _H)

₌

/

_D

_{k(s, t)}

_D

_{R(s, t)}

_e~

^I

^~~

ⁱ ~~~~°'~~~°'~~~~°'~~~°'~~l

x

x

e-C((his,

t), Rl~,

t)))

j

(9)

where

£

((k, R))

₌

/ _d~s /

_dt

(I#i(s, ^t) () ⁷⁾⁾ _{R(s, t)+}

o t

+ik(s, _t)T7Rjs,

t)

V(R(s, t)s) )k~(s, _t)) (10)

o

N° lo MEAN HELD DYNAMICS OF RANDOM MANIFOLDS 1391

J b _an

appropriate

Jacobian which _ensures the normalisation of

Z,

I.e.

Z(0, 0)

₌ and the response ^and correlation function is

given by simple

derivatives

G = I

(k _R)

=

i~~ (~ ^(10a)

C =

(R R)

=

~~ (~ (10b)

The Jacobian is

actually

J ₌ exp

/ _d~

s dt

~

/(~

~~

which

depends

_on the

quenched

ran- s,

dom

potential.

This has in

principle

to be taken into account. The Jacobian

plays

a technical role

ensuring causality

if

perturbation approximations

are studied. Since _{we use a} self consbtent

approximation

we do not need to consider the influence of the Jacobian. It _can be shown

(since

it does not

couple

to a hatted field

Ii(s, _t)

_{that it does not influence the} correlation function C within our

approximation.

Since Z is

correctly

^normalized we can

perform

the dborder average

immediately

on Z without

introducing replicas

and _we obtain the effective

Lagrangian

for the

potential

correlation function

(2)

£eg

₌

£o ) / _d~s /

_dt

_{k~(s, t)}

o

A

/ _d~s /

_dt

/

_dt'

_{Ii(s, t) (R(s, t) R(s,} t'))~~'Ii(s, _{t') (II)}

where

(R(s, t) R(s, t'))~2'

₊

((R(s, t) R(s, t'))2) ^~'.

If _we had taken the Jacobian like account we would have obtained _an additional term in £eg ₊ A

/ _d~

s dt dt'

(R(s, t) R(s, t))~~'~~)

But for the

approximation

used in thb paper ^{it turns}

out to be irrelevant.

This

dynamical

functional cannot be handled further

exactly

and _we have to use

approximate

methods.

3. Self consistent mean field calculation of the

dynamic

and

wandering

_exponents.

The first step to handle such _a functional is to

replace (R(s, t) R(s, t'))-2' _by

its average

#(t t')

₌

(((R(s, t) R(s, t'))~~') ₍₁₂₎

since thin correlation function

depends only

_on one variable

s we can assume that

#(t) depends

only

on time differences. This is

exactly

true for

polymers.

^We

get

^{then the first}

general

result that the effective noise is _no

longer

white, Le. upon ^{Fourier transform the} new nobe is

given by

a

correlation function

(f(q, w) f(q', w'))

=

()

+

(W)A)

6(q

+

q')

_6(W + W')

(13)

where _q is the Fourier

conjugate

to s. The correlation function

(10b)

b

given by

2

C(q, w)

₌

~~°

^{+ A}

~~"~ (14)

~

~ ~4

~

~ ~4

r2 r2

1392 JOURNAL DE PHYSIQUE I N° lo

Itis the second term in the correlation function which will lead to new effects which comes from the dborder. The first term

simply provides

Rouse type

dynamics

well known in

polymer physics [I Ii.

Setting

A

=

0,

I.e. _no dborder _we find from the first term the classical exponents ^{for manifolds} without excluded volume interaction

lo

₌ ^~

~

~

~~~~~

2 D

(lsb)

zo =

~

where for D

= I

~linear polymer)

^the classical exponents are recovered.

The _new

exponents

are derived from the second term of

(14).

lb obtain the

dynamic

and

wandering exponent simultaneously

_{we assume} for

(12)

a

scaling

law

1(~)

~

i-2~z (16)

I,e.

(R(s, t) R(s, t')

~- it

t'j~

and calculate _z self

consbtently

from the correlation function

equation (14).

Therefore _we fourier transform

equation (16).

Since we are

only

interested in the

asymptotic behaviour,

I.e. in the

exponents

z and

(

we

ignore

in the

following

numerical factors.

Thus

4(W)

_~- ^W~'~~~

(16a)

and we determine _z self

consistently

from the correlation function

(14)

in the limit A »

)

o

((R(s, t) R(s, 0))~)

_~-

^t~

_~-

/ _d~q / ^~)~ _{~(l coswt) (17)}

and find

immediately

~

4(1~)

~~~~

On the other hand _{we can} calculate the transverse

fluctuations,

I.e. the

wandering exponent

from

1(R(S, t) R(°, t))~)

_~-

^S~

and find

immediately

4 D

2(1

+

7)

^~~~~

which turns out to be consistent with that calculated from the variational

replica

method.

The

connectivity scaling

^{law which is valid for}

(o

and _zo

(see (15))

^holds also for the

quenched

dhorder _case and we find the

scaling

relation

(

₌ 2z

(20)

which seems to be

general

at least in the frame of _{our mean} field considerations.

N° lo MEAN HELD DYNAMICS OF RANDOM MANIFOLDS 1393

4. The _case of local disorder

We consider

finally

the case of

purely

^local

disorder,

I.e. the random

potential

correlation function is

given by

_[3]. M6zard and Parisi related thb _case to a

specific

^{value of} _7, ^I.e.

7 =1+

( ₍₂₁₎

where the codimension

N(=

d

D)

b

just

the dimension of the vector

R(s).

Thin _can be

directly

verified in the framework of _our self consbtent

dynamical

method.

The short range correlated

potential replaces

the disorder term in the effective action £eg

(see Eq. (ll)) by

Foflowing

the _same

analysis

we find

immediately

4

(2

⁺

~)

~

~

~~~

and

4 D

(24) (

_"

/~

2

(2

⁺

~)

which is

exactly

what _we had before with _{7 = +} ^~ 2

5. Discussion.

We have studied the mean field

dynamics

of manifolds in

quenched

random medh. A self _con- sbtent

dynamical

calculation recovered the

wandering exponent

^{of the} transverse

(static)

fluctu- ations which has been

recently

calculated

by

_means of

replica theory

combined with a variational

calculation.

Moreover,

we

provide

the _mean field

exponent

^{for the} transverse

dynamical

^fluctu-

ation _z which b to our

knowledge

a new result. We find the

general scaling

relation

(

₌ 2z at least in our mean field limit. The next

step

^b to go

beyond

the self consbtent calculation of the

exponents

and

provide

a more theoretical basis for _our considerations.

But there b _{even more} in this

simple

self consistent method. Consider

tie

fun correlation function

((R(s, t) R(0, t))~)

with the fun correlation function

(14)

where we

put

ⁱⁿ

#(w)

with the self consbtent _z. We obtain then

4 D

(R(s, i) R(o, ₁₎₎₂

~

cis2-D

_{+ c~s}

^{(1+ 7)} ₍₂₅₎

where _cl and _{c2 are some} constants

depending

on the cut off

length

etc. As

long

as 7 ^<

~ D the second term b in the limit _s

- cc

always larger

and we have

2(1

+

7)

^~~~~

1394 JOURNAL DE PHYSIQUE I N°10

as

given by

^M62ard and Parbi. On the other hand if 7 >

~

the first term in

equation (25)

2 D

wins and _we have

(

₌ ^{~ ~}

(27)

2

which b

exactly

what M62ard and Parhi calculated in [7~

(see Eq. (5.25)

in [7~). ^Therefore we are

able to recover the central results of the beautiful

replica

calcultation _[7~

by

our

simple dynamical

method. The _same limits hold for the

dynamical _exponent

_z, I.e.

z = zo =

~

~

~

7 ^>

) ₍₂₈₎

~

4(1~)

~ ~

2

~D

~~~~

Finafly

we want to criticize ourselves. The self consbtent

dynamical

calculation b not _as rich

as the

replica

calculation. We could not obtain

explicitely

_many states or

sample

fluctuations.

Nevertheless, we believe it b useful to have _an altemative derivation of the

wandering exponent ( and,

_moreover, to

provide

an

explicit expression

for the

dynamical _exponent

_z of the manifold.

Finally

let _us comment on the

validity

of the self consistent

approximation.

Md2ard and Parisi [7~ have shown that their Hartree-Fock

(variational)

method becomes exact in the limit of

large

codimensions N _{- cc.} These authors have transformed the Hamilton%n with

auxilliary

fields to a form where for

large

N the saddle

point

method

(steepest descent)

can be used. Such _an argument ^is

(to

our

knowledge)

not

possible

here. But if _one goes ^{back to}

equation (22)

and uses the average ^{for this}

expression

it becomes

~- A

/ / _{k(s, t)}

~~

,~(~

~~j~~~

k(s, _t')

and it can be _seen that the

dynamical

correlation

drops

very fast for N ^» I. Thus _we may ^conclude that corrections to our mean field

approximation

become less

important.

But this remains to be proven yet.

References

Y.C., Phys. ^{Rev Left.} s7(1987~ 2087.

Phys. Sl

(1988)

817.

G., ZHANG Y.C., Phys. Rev Left. 56

(1986)

889.

FORSmR D., NELSON D.B., ^STEPHAN

MJ.,

Phys. Rev A 16

(1977)

732.

HusE D.A., HENLEY C.L., FISHER D.S., Phys. ^{Rev LetL} SS

(1985)

^2924.

MtzARD M., PARisi G., f Phys. A 23

(1990)

L 1229.

MtZARD M., ^PARISI G., f Phyg. ^{I France 1}

(1991)

809.

HALPIN-HEALEY T, Phys. ^{Rev LetL 62}

(1989)

44~

DE DOMINICIS C., Phys. Rev la

(1978)

4913.

MARnN PC., SIGGIA E.D., ROSE H.A., Phys. Rev AS

(1973)

423.

DOT M., EDWARDS S-E, tile

Theory

of

Polymer Dynamics (Cxford University

Press, 1986).