Semiclassical limit and quantum chaos

HAL Id: jpa-00246723

https://hal.archives-ouvertes.fr/jpa-00246723

Submitted on 1 Jan 1993

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.

P. Lecheminant

To cite this version:

P. Lecheminant. Semiclassical limit and quantum chaos. Journal de Physique I, EDP Sciences, 1993,

3 (2), pp.299-309. �10.1051/jp1:1993131�. �jpa-00246723�

J. Phys. I France 3

(1993)

299- 309 _FEBRUARY 1993, PAGE 299

Classification Physics Abstracts

03.65N 03.80 05.45

Sendclassical findt and quantum ^chaos

P. Lecheminant

Laboratoire de Physique Thdorique des

Liquides,(°)

Universitd Pierre et Marie Curie, 4 place Jussieu 75252 Paris Cedex 05 France

(Received

20 May 1992, accepted 25 June 1992)

Rdsumd. Dans cet article, _nous prdsentons le domaine

sur lequel R. Rammal travaillait dans les demiers moments de _sa vie le chaos quantique. Nous dtudions numdriquement le

comportement de plusieurs distributions pour des bfllards _{avec ou sans} champ magndtique. Nous trouvons des lois exponentielles _pour la distribution des longueurs des trajectoires, _pour celle de la surface balayde _par la particule et ainsi que pour la distribution du nombre de r4flections

sur les parois du billard. Ces rdsultats confortent l'hypothbse _que la signature de la diffusion classiquement chaotique dans le domaine quantique est l'apparition de fluctuations de la matrice S

(on

de conductance _pour des conducteurs

ballistiques)

dans la limite semiclassique.

Abstract. In this _paper

we present the field _on which R. Rammal _was working ⁱⁿ the last

moments of his life _: quantum ^chaos. The behavior of various distributions is investigated _nu-

merically for different planar billiards in _presence of _a magnetic field _or not. We find exponential laws for the distributions of the trajectory lengths, of the algebraic _areas, and of the number of boundary reflections. These results support ^the conjecture that the signature of the classical chaotic scattering in the quantum description is the appearance of fluctuations of the S-matrix

(or

conductance for ballistic

conductors)

in the semiclassical limit.

1. Introduction.

The

discovery

of chaotic behavior in deterministic

dynamical

systems has had _a

major

effect

in _{many areas} of

physics iii.

Chaos _{means an}

exponential sensitivity

to initial conditions in _a

bounded

region

which leads to

stretching

and

folding

in classical phase _space.

Quantum

chaos has been defined _as the

study

of quantum systems, ^{whose classical} counterparts

display

chaotic

behavior,

in the semiclassical limit

(h

_-

0)

_[2]. It has stimulated many efforts _to understand

more

deeply

the connection between classical and quantum ^theories

ill.

For

integrable

systelus, this

correspondence

is made

through

the Einstein-Brillouin-Keller

quantization

conditions for

(°) Unit6 de Recherche Assoc14e _au CNRS.(URA 76s)

classical invariant

tori[3].

Gutzwiller and

Balian-Bloch,

have

developed

_a semiclassical

theory,

the swcalled

periodic-orbit theory,

to understand quantum systems ^{whose classical behavior} is

chaotic[4].

In this

theory,

the link between classical and quantum ^mechanics stems from

classical

periodic

^orbits and _can be written

symbolically

_{[3] as}

~(quantal energies)n_o~- ~j _{(classical periodic orbits). (1)}

Gutzwiller

Ill

has noticed that this formula _agrees with the

Selberg

Trace formula for the motion of _a

particle

on a compact surface of constant

negative

curvature. This semiclassical

approach

allows the

study

of quantum spectra ^{for chaotic} systems ^{where the absence of invariant tori} rules out the Einstein-Brillouin-Keller

quantization.

As _a consequence, there is _now strong evidence that the statistical

properties

of the semiclassical spectrum ^{of bound} systems ^with a

chaotic classical limit _{are very} similar to those of random matrix ensembles in nuclear

physics

[5].

Classically,

chaotic systems ^{have levels with} a strong ^local

repulsion

described

by

_an

appropriate

member of the random matrix

theory. By using

the

periodic-orbit theory, Berry

has found that _some

particular

statistics

(spectral rigidity

_or the number

variance), characterizing

the fluctuations of excited energy

levels,

_can be described in terms of three

universality

classes [6].

In _a series of _papers in collaboration with Bellissard [7], ^{R. Rammal} has derived _a semiclassical

approach

of the _energy levels for Bloch electrons in _a

magnetic

^{field. Ever}

since,

R. Rarrtmal has been _very attracted

by

the methods

developed

for

studying

quantum

chaos,

with the idea

that this

flourishing

field could

bring

_new

perspectives

in _{many areas} of

physics.

For

instance, according

to

him,

quantum ^{chaos could} have _some connections with

strongly

correlated systems,

especially

the

high

temperature

superconductors

and the quantum Hall effect [8].

Despite

_a

great

^deal of effort and _{some progress, new} methods _are still

lacking

for

a better

understanding

of these

phenomena.

In _my

opinion,

R. Rammal

thought

that _near

the supra

region,

where there _{are many} excited energy

levels,

the

study

of spectra and levels statistics should

provide

_new

insight

into the

high

temperature

superconductivity.

Another

example

of this

implication

is the Wannier-Azbel-Hofstadter

problem, namely

the

properties

of _a quantum ^fractal [9] : the energy spectrum _{as a} function of

magnetic

field has _a Cantor set structure. Since these Cantor sets may be

interpreted

_as chaotic

repellers

_or

semiattractors,

the

investigation

of R. Rammal in relation with quantum chaos would have

been, undoubtedly,

fruitful.

Besides,

he studied the quantum

scattering

of _a

particle

in lD and 2D

quasicrystal

[10] structures. He found _energy levels with _a local attraction instead of _a usual

repulsion

for

chaotic systems

ii Ii.

This work has had consequences in

magnetism (Kondo problem) ill]

but also for the

study

of conductance fluctuations

ill].

The aim of this _paper is to present ^the field in which R. Rammal _was

working

in the last moments of his life

but,

above

all,

I would like to thank him and pay him tribute for

introducing

_me in

physics

research _on quantum chaos.

2. Chadtic

scattering.

Recent studies of quantum ^{chaos have}

paid

attention to unbounded

scattering problems

_[12].

These _appear in _a

large variety

of fields such _as inelastic

scattering

in atomic and molecular systems or ballistic conductors. The

signature

of chaos in these _open systems is

quite

subtle since the

dynamics

is not confined to _a finite volume of

phase

_space

thus,

the

folding

is

by

no means obvious. In the most _common view, _a system

displays

classical chaotic

scattering

if

the variation of the parameters

specifying

the outcome of the collision

(such

_as the

scattering

angle)

occurs on all scales of the

incoming

parameters. The

scattering

S-matrix is the relevant

N°2 SEMICLASSICAL LIMIT AND QUANTUM CHAOS 301

concept ^for a

complete

_quantum

description

of the process. It is _a

unitary

_operator, due to flux

conservation,

and is

symmetric

for systems with time-reversal syrrtmetry. ^The crucial

point

for

understanding

the

implication

of classical chaotic

scattering

in the quantum domain is the semiclassical

approximation

for the S-matrix [13]. ^{Bliimel and}

Smilansky

_[14] have shown that

a manifestation of quantum chaos in

scattering

is the _presence of fluctuations of the S matrix

equal

to those

predicted by

random matrix

theory.

More

precisely,

the semiclassical derivation of the _energy correlation is

can,(e)

_=<

sj~,(E)sun,(E

_{+ e)} >=

can'(.°), ₍₂₎

1-

j

where

Cnn,(e)

is

averaged

_{over an energy} domain AE which is

specified

below. This

equation

was derived

through

two main steps

Let

ST

be the set of initial conditions such that _a

particle spends

_a time

exceeding

T in the interaction

region.

The _measure p of this set is then

P(ST)

_*

exP(-7T), (3)

where ₇ is the classical _{escape rate.} In the T _{- oo}

limit, ST

is _a fractal set and _{7 is} connected to the

Lyapunov

exponent

I,

which

gives

_an estimate of the

exponential stretching along

the

unstable

direction, by

7 =

1(1 d), (4)

where d is the Hausdorff dimension of the fractal [12].

It is assumed that ₇ is

independent

of _n and

n',

and its variations with _{energy are}

neglected

in the interval AE. This small _energy domain AE has to be chosen such that

emq ^{< AE <} e~j,

(5)

where _e~j and emq denote

,

respectively,

the classical and quantum _energy scales.

It follows from

equation(2)

that the fluctuations of the _cross

section,

for _a

given transition,

_are described

by

_a Lorentzian

:

<

'nn'(El'nn'(E

+ £) >"

~

~~~'~fi~/ ⁽⁶⁾

~

(j)

This law _was obtained

by

Ericson [15] ⁱⁿ

high-energy

nuclear collisions. This should have consequences in the transport

properties

of small

phase-coherent

conductors since

they

_are

completely

determined in terms of the S-matrix for

independent

^electrons at the Fermi _energy.

Accordingly,

2D ballistic conductors may exhibit resistance fluctuations due to the

underlying

classical

scattering.

In these

samples,

where the

typical

features _are much smaller than the

elastic,

_or

inelastic, scattering length,

electrons behave like _{waves or} billiard balls. At very low temperatures

(<

4

K)

and at low

magnetic field,

_a

variety

of

magnetoresistance

anomalies

appears in such

junctions.

It _was shown [16] ^{that the Kubo formula for the} conductance is

~2

~ h ~~~~~~~' ~~~

where _i is the transmission matrix and i+ _{denotes its}

adjoint.

In this _case, the _energy correlation function is derived from

equation(2)

_[17]

C(Ak)

_=<

6g(k

+

Ak)6g(k)

_>=

~(~~

(8)

+

())~

~

A

~

~

~

^R

R

~

~

A

~

~ ~

~2~ ~~~

~ ~

Fig.I.

The different billiards studied in this work,

a)

The open-stadium system,

b)

^The elbow system, c) ^The cross system.

d)

The Bunimovich stadium,

e)

The four-disk structure,

f)

General billiard whose _corners verify the

following

law

: xP+yP

= cst.

Jalabert et al.

ii?]

also obtained _a semiclassical derivation of the

magnetic-field

correlation function

C(AB)

_=<

6g(B

+

AB)6g(B)

_>=

~j~(

,

(9) (1+ (~)~)~

o

with lo _" ^~~

(flux quantum);

_a is

obtained,

like _7, from the classical chaotic

scattering.

The

e

main

assumption

for the

validity

of

equation(9)

is that the distribution of _areas is

given by

the

following

law _:

N(Q)

₌

N(0) exP(-alai), (lo)

where Q denotes the

algebraic

_area enclosed

by

the electron's orbit.

Therefore,

the character- istic scales of the conductance

fluctuations,

_as well _as those of the

S-matrix,

_are ^related to the

underlying

classical chaotic

scattering.

3. Nwnerical results.

To check the

validity

of the above

conclusion, namely

the connection between the characteristic scales of fluctuations and the classical parameters a and ₇

(see Eqs.(8, 9)),

we have calculated these parameters ^{defined in}

equations(3)

and

(10)

for the billiards

depicted

in

figure

I. We choose several types of billiards to _measure the effect of symmetry ^reduction on a and _7. For instance, the last billiard in

figure

I illustrates the influence of the

rounding

of the _corners

(~P+yP

=

cst).

We

study

^first the oval billiard in order to test _our

approach,

but also for its

generic

behavior.

Indeed,

this billiard leads to motion for which the

phase

_space is formed

by

chaotic _areas

intricately

mixed with _areas covered with invariant _curves. On the contrary,

No2 SEMICLASSICAI, LIMIT AND QUANTUAI CIiAOS ₃₀₃

1

,.

,"'~_

,_= ~, ~'_

.~

j'~.(

0.B I-1-"' "'II ,' ." ) ^'' .'',, '."'' .j

.- _{,., ;} [ _{.i,, =.}

:> :

I "'., ,". .I ."'1

"_

$'

0. 6 ', ,-.' .,' ,t [

",~'

_{[=:i." )( ',' ,,." [.}

..,

0 4 ^" .~j.'.)_~. ^~,j

.. )~"

~ '/[[?

.~ .,;,',_. ;,

j"~ ^)"l,

0.2 ^{~' "'} ..'

.,_,,

; ',,.,l' "~, ~~l' .;( ~

'),'. ,'ll.

'I" '.1" .'

P 0 _{..;.. ..:-'}

_I

,j'

I,"

."

-0.2 '"

~"

;~;:" ^~ ,f..' ," "~l'~

"" ):

.. ,'[ ',' ,')

j'

-0.4 "/. "_' (" I.,I' ~. ''',

", (_ fl'

'.' :,' ...[/..

_,~: _.

.' "..

.j, j. ,', ' ", ') .[

.. ','

-0. 6 '~,""

;_ '~^.. [

-0. 8

["'

^''

"'"

'

,: .C.[-

= .- .= ^""

-1

0 1 2 3

~ 4 5 6 7

Fig.2.

The phase _space struture of the oval bdliard

(obtained

by _a deformation of _a

circle);

S and P denote, respectively, the _arc length and the tangential momemtum 10' _bounces

are required.

1

o 5

,

~

~ /

.l' '~

l'

___~.

'~' 'i'",,,

Y 0

, ' / ,,--~'~~

',

^"

' ''j

", , ' '

, ^'

, , ^/

, '

, ; ,'

-0 5

', 1'

,, ' .'

'~~^L -I'

, I

j,

?'

,

i '

-i -o.5 o o.5 1

x

Fig.3.

Typical classical trajectory in the four-disk structure.

12

~',

10

',

~ ',

, ,

~ ',

8

, ,

~ ',

,

, b,

~" ~ 6

',

, b,

, ,

, bs

4

, ,

, ,

o, ,

,

, o

,

2

',

,

, o

,

, ',

, ,

),~~~

0

0 20 40 60 80 100 120

L

FigA.

The distribution of trajectory lengths for the elbow system. ^A = 0.5 _m, R

= 0.478

m

(values

bom Doron et al. 's experiments [19]). We find ₇

" 0.0956 m~~

12

~ ,,

10

~,

,~,

',

*,

8

'h~

, ,

» ',

'Q

LnN 6 ',

~,

'~

, ',

4 b

',

, ,

o, ',o

,

2 ',

~N ',

~,

, , 0

0 5 10 15 20 25 30 35

L

Fig.5.

The distribution of trajectory

lengths

for the four-disk _structure. A

= 0.5 _m, R

= i _{m. 7}

= o.385 m~~

N°2 SEMICLASSICAL ^{LIMIT AND} QUANTUM CHAOS 305

11

',

~°

'b,

, h~

,

~ 'b

', lK

8

',

~ ',

~,

7

',

L»N

~,

, b,

6

',

d~ ',

,

5

°',

~o',

,

~

4

', bs

,~l~

3

',o

,

',~~

2

0 100 200 300 400 500 600 700 800 900

L

Fig.6.

The distribution of trajectory lengths for the four-disk structure in _a magnetic field. A

=

0.5 _m, R

= i _m; for _a cyclotron radius Rc ₌ 3 m, 7 " 0.01 m~~

12

10

~

? ',

/ «

4 ,

' b

8 '~ 'b

~d ^'

~f ~,

,d ~,

~f ~l

Ln N 6

/ '~

A~ ,

~ ^~

'

~i

j ,

~' b,

4 _' Q

6 ,

,' ~

~~ ,~

,6 ',

, §

2 'O

~ o

/ '

1 '

' ' / 0

-1000 -800 -600 -400 -200 0 200 400 600 800

S

Fig.?.

The distribution of effective _areas for the

open-stadium

system. A ₌ 4 _m, R

= 4 _m, W

=

I _m

(width

between the two

probes).

_{txi =} 0.0099 m~~

(for

Q _< 0), ₀₂ " 0.016 m~~

(for

Q > 0).

12

io

~K

~,

~~S

8 ~,

~i~

~,~

~°~

L~iN 6 ~'o~

~,~

,~

~, )'o~

4 ~,

~,

i~,

,°~S°

2

"~°

', o

~, °

~,

',,

o

0 100 200 300 400 500 600 700 800 900

Nr

Fig.8.

The distribution of the number of boundary reflections for the open-stadium structure A

=

4 _m, R

= 4 _m, W

= i _m

(width

between the _two

probes)

Rc

= 5 m; 71 = 0.072. This distribution is defined by _:

P(N)

=

P(0) exp(-71N)

for the Bunimovich stadium [18], ^{almost all orbits}

explore

the

phase

_space

uniformly (ergodic billiard)

whereas for

ellipses

the motion is

entirely

confined _to invariant _curves

(integrable

bil-

liard). Figure

2

emphasizes

the richness of the orbital _structure for oval billiard

(10~ bounces).

We simulate the

injection

of _a

large

number

(typically

_10~) of

particles

towards the

scattering region

and follow their

trajectories (straight

lines _{or arcs} of circle

depending

_on the _presence of

a

magnetic

^field _or

not)

to

determine,

for

instance,

the

trajectory length

before it scatters out.

Then,

_we

adjust

_an

exponential

law _on the

histogram describing

the

appropriate

distribution

so as to evaluate the value of the classical parameter. ^Because

elasticity

of collision

implies

conservation of energy and hence of

speed,

_we normalize the

speed

of the

scattering particle

to _one.

Figure

3 shows _a

typical trajectory

for the four-disk system

(18 bounces).

When _a

magnetic

field B

(constant, uniform,

and

perpendicular

to the

plane

of the

billiard)

is

applied,

a

trajectory

is _a succession of _arcs of circle with reflections _on the walls. The radius of the circles is

given by (Larmor radius)

R =

§, _(II)

q

where _{m,q, u are,} the _mass, the

charge,

and the

velocity

of the

particle, respectively.

p

We _now

report

our main results. For the distribution of

trajectory lengths,

_we

obtain,

_on the whole,

exponential

laws. 'Ve

find,

for instance, _{7 ~M} 0.096 m~~

(see Fig. 4)

for the elbow

structure in agreement ^{with Doron} et al. [19]. We also observe that

equation(3)

is still valid in the _presence of _a

magnetic

field and

that,

in

general,

_{7 seems} to be reduced

(see

the results for the four-disk structure in

Figs.

5,

6).

We calculated the distribution of elTective _areas

by

N°2 SEMICLASSICAL LIMIT AND QUANTUM CHAOS 307

11

~~

"'b

', h~

9 ^, ,

b ',

8

h~

, ,

~ ',

L,,N ^~

~,

,

,o~,

~

'b ',

, o~

5

',

,o',

~

4

',

,

~ ,

,

3

~s

,

~%l

,

'f~~~~

2

0 100 200 300 400 500 600 700 800 900

Nr

Fig-g-

The distribution of the number of boundary reflections for the elbow system A

= 0.5 _m, R

= 0.478 _{m; 71}

= 0.016.

evaluating

the oriented _area enclosed

by

the electron

path

:

Q ₌

f(dr

^A

^{r),k, (12)}

where k is _a unit _vector

along

B. We assumed that it is _a

good approximation

for

typical

_open

orbits. We obtain that Q has _an

approximately symmetric exponential

law for the open stadium structure and _very weak fields

(see Fig. 7).

If R is less than the

typical

linear dimension of the

junction,

the

predictions

of

equation(10)

are not satisfied. We have also calculated this

distribution for _a

particle displaying

_a Brownian motion. We find the

following

law _:

P(S, t)

=

) ^~

~ ,

(13)

cosh

(fl§)

where i is the number of steps _necessary for the

particle

to return near the

origin

of the

movement. This result is

explicitly

known for _{a wave}

propagating

in _a random medium in two

dimensions where D is the diffusion _constant [20].

Finally,

_we mention the distribution of the number of

boundary

reflections.

Figures

8, 9 show that the distribution follows _an

exponential

law for chaotic billiards whether _a

magnetic

field is

applied

_or not.

Unfortunately,

_we do not know of _any

quantity

for which the characteristic scale of its fluctuations is connected to the

reflection number in the semiclassical limit.

4. Conclusion and

perspectives.

In this

work,

_we have verified that the fluctuations of the S-matrix _or of conductance are related in the semiclassical limit to the

underlying

classical chaotic

scattering.

In

particular,

_we have

checked

numerically

that the

exponential

laws _are

accurately reproduced

for the distribution of

trajectory lengths (or areas).

We stress that the

expression

for the

magnetic-field

correla~

tion function is

only

^valid when the Larmor radius is much

larger

than the device dimensions

(namely,

at low

magnetic field).

Due to Rammal's

death,

_many of _our results have not been

exploited.

For instance _we have _not studied in _a

systematic

_way the influence of the _symme- try ^reduction on the classical parameters.

Besides,

the fluctuations of the S-matrix under _a

magnetic

field _{are very} relevant

since,

for _a time-reversal

breaking

system, _one ^{should observe}

a transition between members of the random matrix theory.

Finally,

the dilTerent correla- tion functions for _our billiards could be calculated

directly by

the recursive Green's function method.

Therefore,

this threefold

comparison

between

classical, semiclassical,

and quantum

theory

should

provide

the main characteristic features of the classical chaotic

scattering

in the

quantum ^domain.

Acknowledgements.

I would like to thank the Centre de Recherches _sur les Tr+s Basses

Tempdratures (CRTBT)

where this work _was

performed.

I

gratefully acknowledge

J.C.

Anglbs d'Auriac,

B.

Dougot,

T.

Dombre,

M. Azzouz and Y. Brechet for _many useful discussions. I

specially

^thank J-M-

Maillard,

D. Delande, G.

Tarjus,

P. Viot, B. Bemu, C.

Lhuillier,

P. Azaria and H. Azzouz for their

help.

References

[1] M-C- Gutzwiller, Chaos in Classical and Quantum Mechanics,

Springer Verlag,

New York, 1990.

[2] M-V- Berry, Proc. Roy. Soc. London, Ser. A 413, 183

(1987).

[3] M-V- Berry, Semiclassical Mechanics of Regular and Irregular Motion in Chaotic Behaviour of Deterministic Systems, Les Houches session _no. 36, G. Iooss

,

R-H-G Helleman, and R. Stora, eds, North-Holland, 1983.

[4] ^M-C- Gutzwiller, J. Math. Phys. 8, 1979

(1967);

lo, 1004

(1969);

11, 1791

(1970);

12, 343

(1971);

R. Balian and C. Bloch, Ann. Phys.

(N.Y)

69, 76

(1972),

and 85, 514

(1974).

is]

O. Bohigas, Random Matrix Theories and Chaotic Dynamics in Chaos and Quantum Physics, Les Houches session _no. 52, M-J Giannoni, A. Voros, and J. Zinn-Justin, eds, North-Holland, 1991.

[6] M-V- Berry, Proc. Roy. Soc. London, Ser. A 400, 229

(1985).

[7] ^R. Rammal and J. Bellissard, J. Phys.

(Paris)

51, 1803

(1990);

51, 2153

(1990);

R. Rammal, J.

Bellissard and A. Barelli, ibid. 51, 2167

(1990).

[8] Strongly correlated systems, Les Houches, ^R. Rammal, B. Douqot, and J. Zinu-Justiu, eds, 1991,

to be published.

[9] ^B. Douqot and P-C-E- Stamp, Phys. Rev. Lent. 66, 2503

(1991).

[10] ^D. Schechtman, I. Blech, D. Gratias, J-W- Cahn, Phys. Rev. Lett. 53, 1951

(1984).

iii]

R. Rammal and Y-Y-

Wang,

unpublished.

[12] ^U. Smilansky, The Classical and Quantum Theory of Chaotic Scattering in Chaos and Quantum Physics, Les Houches session _no. 52, M-J Giannoni, A. Voros, and J. Zinn-Justin, eds, North-

Holland, 1991.

[13] ^W-H- Miller, Adv. Chem. Phys. 30, 79

(1975).

[14] ^{R. Blfimel and U.}

Smilansky,

Phys. Rev. Lent. 60, 477

(1988);

64, 241

(1990).

N°2 SEMICLASSICAL LIMIT AND QUANTUM CHAOS 309

[is]

T. Ericson, Phys. Rev. Lent. 5, 430

(1960).

[16] ^{D-S- Fisher and P-A-} Lee, Phys. Rev. B. 23, 6851

(1981).

[17] ^R-A- Jalabert, H-U- Baranger, and A-D- Stone, Phys. Rev. Lent. 65, 2442

(1990).

[18] ^{G. Benettin and J-M} Strelcyn, Phys. Rev. A, 17, 773

(1978).

[19] ^E. Doron, U. Smilansky, and A. Frenkel, Phys. Rev. Lent. 65, 3072

(1990).

[20] ^J-P- Bouchaud, J. Phys.

(Paris)

_{1, 985}

(1991).