Spectral Rigidity in the Large Modal Overlap Regime: Beyond the Ericson-Schroeder Hypothesis

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Spectral Rigidity in the Large Modal Overlap Regime:

Beyond the Ericson-Schroeder Hypothesis

Olivier Legrand, Fabrice Mortessagne, Didier Sornette

To cite this version:

Olivier Legrand, Fabrice Mortessagne, Didier Sornette. Spectral Rigidity in the Large Modal Overlap

Regime: Beyond the Ericson-Schroeder Hypothesis. Journal de Physique I, EDP Sciences, 1995, 5

(8), pp.1003-1010. �10.1051/jp1:1995179�. �jpa-00247111�

Classification Physics Abstracts

03.65Sq 05.45+b

Spectral Rigidity in the Large Modal Overlap Regime: Beyond

the Ericson-Schroeder Hypothesis

Olivier

Legrand,

Fabrice

Mortessagne

and Didier Sornette

Laboratoire de Physique de la Matière

Coudeusée(*),

Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France

and

X-M, Parc-Club 28, _rue Jean Rostand, 91893 Orsay Cedex, France

(Jleceived

27 March 1995, accepted 4 May

1995)

Abstract. Spectral correlations for the total _cross section iii chaotic scatteriug _or, alterna-

tively, for the _resonance density fuuctiou in chaotic reverberant _{rooms, are} studied within the frame of the random matrix theory. This framework allows _{us to} develop _a theory of spectral

correlations in the large modal overlap regime which goes beyond trie Ericson- Schrceder result of Lorentzian autocorrelation functions. Spectral rigidity _is showu to lead to _a different autocor-

relation function which is universal _in the limit of large _resonance overlap. Numerical evidence for this signature of spectral rigidity is given withiu the frame of _a 2-dimeusioual chaotic billiard model of _a reverberant _room, for which level repulsion and spectral rigidity _are knowu to be well

described by the Gaussian Orthogonal Ensemble

iii trie absence of absorption.

Introduction

Since the works of Ericson in the field of nuclear reactions

Iii

and of Scuroeder for _room acoustics [2] ⁱⁿ tue sixties, ii is

only recently

tuat tuere

appeared

_a renewal of interest in tue statistical

properties

of eituer _wave

scattering turougu

a cuaotic

region [3-5]

_or in tue

frequency

response function of _a reverberant _roorn wuicu

displays

cuaotic _ray

dynarnics

[6]. ^{In nudear} reactions, at

uigu energies,

_many melastic cuannels _open

leading

to _an

overlapping

of _rnany

resonances associated to tue interrnediate _states of the

cornpound

nucleus. In tuis

situation,

peaks

occur in tue _cross section wuose inverse _rnean widtu r~~

corresponds

to tue average life titre of tue

compound

nucleus. The fluctuations of the

partial

cross sections bave been assumed

by

Ericson to be dorninated

by

tue Gaussian randorn cuaracter of uncorrelated

partial

widtus

connecting

^tue

impur

or output states with the interrnediate _ores. This led him _to establish the Lorentzian _covanance of the _cross section which is due to the

exponential decay

of the

cornpound

at the rate r.

Using

the _sanie basic arguments, Schroeder established

independently

the _same Lorentzian

"frequency

autocorrelation function" for the

frequency

_response function

(*) CNRS URA 190

© Les Editions de Physique 1995

1004 JOURNAL DE PHYSIQUE I N°8

between two points ⁱⁿ a reverberant _room where the sourd _energy

exponentially decays

due to

absorption.

Both assumed that trie total widths

(or decay rates)

of the _resonances do trot fluctuate much around trie _average width r and this _cari be

justified by

trie fact that these total widths _rnay be viewed _as

cornposed

of _many

partial

widths. Bonn also obtain results where trie

specific

statistical

properties

of trie

positions

of trie _{resonances are} irrelevant.

Ii is trie aim of trie present _paper to show that trie _resonance fluctuations described

by

random-rnatrix

theory (RMT) (e.g.,

Gaussian

orthogonal

ensemble

(GOE)

for the spectra ^of Hamiltonian systerns with tirae-reversai

symmetry)

_{may prove} to be of chief

importance

when

evaluating

_energy correlations for total _cross sections _in chaotic

scattering

_or,

analogously,

fre-

quency correlations for

properly spatially-averaged

_response functions in _wave

problems

witu

absorption.

In

particular,

in tue _case of moderate _or

large

_resonance

overlap,

the Lorentzian

shape

turns to be _{a poor}

approximation

for tue _energy autocorrelation function of tue _men- tioned

quantities.

A somewuat related

approach

_was initiated

by Smflansky

wuo tried to put

on a firm basis tue relevance of RMT in quantum ^cuaotic

scattering by using Dyson's

circular

orthogonal

ensemble for the

scattering

matrix [3].

1. A

Simple

Model for the

Spectral

Fluctuations in the

Large

Modal

Overlap Regime

Following

the

simplifying assumptions

made

by

Ericson

iii,

we wnte the total _cross section for transitions from _a

given

in-state _as:

a~°~(E)

_~K

(1

Re

S), (1)

wuere S

=

51°)

₊

i~j ^'~?~~

,

denoting by Ej

= Re

Ej ir/2

tue

complex energies

of the E

~

J j

intermediate states and assuming that the _resonance widths _are ail

equal.

Tue

amplitudes (jj

₍₂ of tue _resonances suould _vary from _resonance to _resonance but also from _a

given

in-state to another _one.

Tuerefore,

in tue

following,

_we consider _{an average} of a~°~ _over in-states tuai

we still denote

by

a~°~ and wuich reads:

°~~~ _" ~~ ~~~~ _~

~Î

(E

Re

ÎÎ2

+

r2/4'

^~~~

This _expression is in close relation to tue

Wigner-Eisenbud

titre

delay r(E)

_{or resonance}

density [7,8]

defined _as:

T(E)

= -1 Tr

St ~~, ₍₃₎

ôE and wuich has the

following pote decomposition:

~~~~ ~

(E

Re

ÎÎ2

+ r2

/4

^~~~

A

formally equivalent problem

is met when

studying

tue real part of tue sc-called input acoustic

impedance used,

for instance, in tue field of reverberation _room acoustics. Tuis will be tue

puysical problern

_we address

turougu

_a numerical model described in tue

following

section.

In tue

following,

we assume tuai tue

rigidity

of tue

complex compound (or

of tue _rever- berant

room)

spectrum

(for

the real part ^{of the}

energies)

is

fairly

well described

by

the GOE

statistics. Wuat _are tuen tue

implications

of tue

corresponding spectral

fluctuations _on tue

autocorrelation function of tue

fluctuating

part of tue _resonance

density

<

T~(E

+

e)r~(E)

_>

where

r~

= r- < r > ?

Denoting by

D the local _mean

spacing

between

adjacent

resonances,

< r > is then

equal

to

1ID. Now, using

^the normalised variables _x

=

E/D,

_xj

=

Ej ID

and j =

r/D,

and

defining

the normalised _resonance

density

function

n(x)

₌

Dr(E),

_{one con}

write the _power spectrum ^of n~

= n -1 _as the Fourier transform of its autocorrelation function

Riz):

m

cos

ÀXR(x)dx

=

e~fl~' il b(À)], là)

Î

wuere

b(À)

is tue twc-level form factor defined _as the Fourier transform of the twc-level duster function

11 lx)

_[10]

(see

tue

Appendix).

For tue

GOE,

tue two-level form factor reads:

1

~

+

)ln(1+ À/7r)

£ 27r

~~~~

~

(À/7r

₊

1j

^{~ ~ ~}

~~~

~

27r ^~ À/7r ^~

The evaluation of the autocorrelation

Riz)

is

easily

achieved in the

limiting

case of _{a very}

large

_j

value,

which amounts to consider the

large

modal

overlap regime.

^{In the}

simple

model

we

analyse, only

this

limiting regime

makes _{sense since} the

approximate

constancy ^of the

decay

parameter is assumed. This

simplifying assumption

is

commonly

used in the acoustic literature [2, 6,9] ^{since it is} claimed to be _true in

practice

in the _case of _a diffuse sourd field [2]. ^We postpone a tentative

justification

of this

assumption

within the frame of tue

model reverberant room discussed in the

following

section.

Due to the

exponential

factor in tue r-h-s of equation

là), only

the

leading

beuavior of 1-

b(À)

~- for small values of is to be retained. Note

that,

in tuis

regime,

the _same

leading

^beuavior is obtained for otuer Gaussian ensembles

(GUE

_or

GSE)

_or

superpositions

^of

independent

spectra [10] ^{but not in tue Poisson statistics}

(whicu

is

conjectured

to

prevail

for spectra ^of

dassically regular systems)

for wuich Y2

lx)

= 0.

Tuus,

_in tue limit of

large

_resonance

overlap,

no otuer information but

spectral rigidity

suould be revealed

by

tue autocorrelation

whicu is found to be related to tue derivative of tue Lorentzian:

~j~~

~

i ^lx/1)~

~

t

1-

lé/r)~

~ ~~~~ _~~~

ir~i~ ₍₁

₊

(x/1)~)~

ir~r~

(1

+

(é/r)~)~

It suould be stressed

again

tuat

R(x)

would be _a Lorentzian in tue _case of _a Poisson sequence

of _resonances witu _no level

repulsion.

Expression (7)

^{of tue} autocorrelation coincides witu tue

limiting

form

given by

Eckhardt in

tue context of _a semidassical

analysis

of _a

scattering problem

wuere the width r is _a dassical

escape rate [8]. ^{A similar form bas also been derived}

by

Suusuin and Wardlaw for _a

particular scattering

^off _a ^{surface of} constant

negative

curvature

il Ii.

In contrast, our derivation makes

no assumption about _an equivalent ^classical

problem

but addresses _a

general dissipative

_wave

problem.

As _an extension of tue tueories of Ericson and Schroeder it relies _on the

validity

of _a

universal statistical feature of the spectrum,

namely

tue

rigidity.

2. The Numerical Model

Here _we

provide

numerical support ^{for tue above}

predictions

within tue fratrie of _a 2D model of _a reverberant _room,

namely

a cuaotic billiard where the _wave equation is solved

subject

to

absorbing boundary

conditions. We _{use a} method based _on ray

trajectories,

introduced

1006 JOURNAL DE PHYSIQUE I N°8

previously [12,13],

to calculate tue

temporal

_response to _a brief

puise

_{in a} chaotic billiard

shaped

2D _room in which the _{pressure p} satisfies the standard _wave

equation

_[12]:

jP(x>

~2

^t)

+

V~P(x> t)

₌

-F(x> t), (8) togetuer

witu tue uniform

absorbing boundary

condition

Î

^"

~~T'

l~~

where

à/ôn

denotes tue inward normal

gradient

at the

boundary

whicu

plays

tue rote of _an

absorbing

watt witu _a

specific

admittance

fl.

Dur construction is based _{on a} summation _over families of _rays wuicli connect a source point to _a limite _size

region

around _{a measure}

point

so tuat tue response is evaluated _{on a}

coarse-grained

scale wuich amounts to a

spatial

and

temporal averaging

at

high frequencies. Accounting

for the

absorption

is achieved

by reducing

the

amplitude

of each contribution

by

_a factor

(cos fl) Ii

_cos +

fl)

for each reflection at tue

watt, being

the

angle

of incidence witu respect to the normal.

The time response in such _a model _{room is}

computed

for _an

isotropic point

_source which is _a Dirac

puise

_in time. Tue enclosure used is _a

dissymmetrized

Sinai billiard in wuicu the

dynamics

of tue _ray

trajectories

is cuaotic in tue strongest sense. It is also well known

(see

for instance Ref.

[loi

tuat tue

uigu-frequency

_spectrum of _a membrane with tue

suape

of _a

cuaotic billiard exuibits statistical features

je-g-,

level

repulsion

_or

spectral rigidity)

wuich _are very well described

by

tue GOE.

By

Fourier

transforming

the

decaying

_response, we calculated tue

frequency

autocorrelation function in _a

frequency

_range wuere tue _mean

spacing

between

resonances is much smaller tuan the

decay

rate. Two types of response were considered. The

first type ^of response

corresponds

to a measure

point

wuicu is different from trie _source location.

For _a

point

_source at y and _{a measure} point at x, trie Fourier transform of trie _response

lits

real part ⁱⁿ

fact),

_or tue transmission

function, approximately

reads [9]:

T(X,

_{y( ld) OE}

~j9~3(X)9~3(Y)j~

~ )2 +

r2/~'

^~~~~

~

~~

In the last expression, q7j is tue

amplitude

of tue j~~ resonance mode of tue _wave

equation subject

to tue

boundary

conditions

given by

equation

(9).

At tuis stage, we show uow, _in tue

large

modal

overlap regime,

tue assumption of constant

decay

parameter _may ^be

justified.

Tue argument _goes as follows. In _our

model, absorption

is

uniformly

distributed _on tue

boundary.

At

uigu frequencies,

tue

decay

pararneter ^{is obtained}

turougu

_{an average} of tue

squared

_{pressure on} tue

boundary

wuicu

essentially

is

equivalent

to _a mufti-channel

dissipative

_process

involving

_a

large

number N of

independent

cuannels

scaling

_as

P/À IF being

tue perimeter ^of tue 2D _room and tue

wavelengtu).

If _{we assume}

a Gaussian distribution for tue pressure

(deduced

from _an

ergodic conjecture

due to

Berry,

see [13] ^{and references}

tuerein)

and tuat tue _average

partial decay

rates for each cuannel _are

equal,

1-e-,

(rj)

₌

IF) IN,

tuen _{a X~} distribution with N

degrees

of freedom is obtained:

~'~~~~~~ ~ÎÎ ^(Î~Î/2)~~~ ÎÎÎ Î'

~~~~

Tuus _as N increases

(at uigu frequencies)

tue variance decreases:

ai

₊

_{(F~l iF)~}

"

(iF)~

"

jif)~. ⁱ¹²⁾

0.8

0.6

~,,,,,,,~~~

0.4

""'~"'~~,~~~~~~

0.2

o

-0.2

0 0.5 1.5 2

E

Fig. 1. The normalized autocorrelatiou

Jl(é)/R(0)

of the frequeucy _respouse calculated in _a two- dimensioual model of _a reverberaut _room with tue suape of _a Sinai billiard is suown

(squares)

in tue

case of two uncorrelated _source aud measurement locations. The value of tue decay rate r m 1.17

(deduced

from _an exponeutial fit of tue temporal decay

response)

is substituted in tue Lorentzian

il

+

(e/r)~]~~

which is plotted

(dashed fine)

for comparison. Frequency averaging is obtaiued in _a higu frequency ^{wiudow whose width is arouud} 50 decay rates. The corresponding frequency _range is in trie

large modal overlap regime _since the estimated value of trie local _mean spaciug D betweeu adjacent

levels is of trie order of 0.015.

An estimate for

jr)

_can be obtained

turougu

Sabine's formula

(ri

=

Po/7rS (S being

tue _area of tue 2D

room)

wuere _o is tue

equivalent absorption

coefficient [13] ^{tue realistic values of}

whicu

being

of tue order

of,

_or smaller tuan,

unity. Using

tue

leading

term of

Weyl's

law at

high frequencies,

tue _mean

spacing

D is

approximately /S. Tuerefore,

tue

large

modal

overlap regime (ri ID

»

immediately implies

that _or be mucu smaller tuan

(ri

thus

ensuring

^the

approximate

constancy of the

decay

parameter.

Here it suould be noted tuat tuere exist two _sources of fluctuations for expression

(loi.

One is associated witu the random cuaracter of tue mode

amplitudes

_q7j, the otuer is _re- lated to tue fluctuations of the spectrum

(real

_parts of tue

eigenfrequencies).

If

only

tue randomness of tue

products

_q7j

(x)q7j (y)

is considered _{one recovers} tue theories of Ericson and Schroeder.

Indeed,

for

spatially

uncorrelated

points

_x and _y, tue

frequency

autocorrelation function

(Tito

+

e)T(w))

is

expected

to be of the Lorentzian type

Il

+

(e/r)~]~~,

due _to the fact that the fluctuations of

Tito)

_are ^dorninated

by

those of the mode

amplitudes,

at least for

sulliciently

^{small values} of _e. In

Figure

1 this

prediction

_is

partially

validated

by comparing

it with trie autocorrelation function obtained

numerically

for _a value of trie

decay

rate r deduced

directly

from the

early exponential decay

of trie time

dependent

_response. On the contrary when the response is measured at the location of trie source,

expression (loi

reduces to the _so-

1008 JOURNAL DE PHYSIQUE I N°8

0.2 0

-0.2

0 0.5 _{1.5 2}

E

Fig. 2. Same _{as in}

Figure

1 wuen measuring tue _response at tue location of tue _source. For tue sake of comparison, tue prediction of formula (7) is suown

(dasued-dotted fine),

_as well _as tue Loreutzian

(dashed fine),

both with r _m 1.17. The agreement of the uumerical result with

our prediction witu

no

adjustable pararneter _is ^excellent.

called input

impedance

of tue _source

~~~'"~

_"

~

~~~~~(w wÎ~+ _r2/4'

^~~~~

Wuen

averaging

^this

quantity

_{over a} number of different _source

locations,

_one is left with _a

quantity

which

essentially

is the _resonance

density

function

given

in

equation (4)

~

" ~~

Î

(w

w~~+

r2

/4'

^~~~~

In trie last expression trie overbar denotes the average over the _source locations.

Tuerefore,

tue autocorrelation of tue

fluctuating

part ^of

equation (14)

suould bear tue mark of tue

spectral rigidity

_as

predicted

in tue

preceding

section. Indeed tuis is wuat _we have

found,

as shown in

Figure

2 where both the Lorentzian and _our

prediction given by expression I?i

are

cornpared

to the

numerically

obtained autocorrelation. The Lorentzian cannot account for the

shape

and width of the autocorrelation in this _case whereas _our formula lits trie numerical data very well.

Again,

it should be reminded

that,

for the norrnalized autocorrelation

Rie) /R(0),

there is _no

adjustable

parameter left since the value of r is deterrnined

separately by

_a direct measurement. In both

figures,

tue

frequency averaging

was

performed

in the _same

frequency

window _w E

[95,

150] wuere tue _mean spacing D between

adjacent

_resonances is of tue order

0.015 _as estimated

by using

^tue

leading

term of

Weyl's

formula for tue _mean

density

of _resc-

nances. Tue

absorption

_we used in botu _cases leads to _a characteristic

decay

rate r _m I.Ii.

For values of trie

frequency

mismatch _e

larger

^than _a few widths

(not

shown in trie

figures),

the

frequency

autocorrelation functions _we observe

clearly display

non-universal oscillations

Ii.e.,

not

predicted by assuming spectral rigidity

as

given by RMT).

Tuese suould be related to trie existence of non-universal features of trie form

factor,

_as

emphasized by Berry _[14],

associated with trie suortest

periodic

orbits of tue billiard. Furtuer

investigation

is needed to account for the non-universal

long

_range behavior of the correlations

probably through

semidassical

calculations

involving

tue least unstable

periodic

orbits.

Conclusion

We bave

provided simple

arguments wuicu show uow

spectral rigidity

in

complex

spectra _cari be revealed in

decaying

processes even in tue

regime

wuere tue _{resonances are not} resolved. Dur

approacu

is based _{on a}

simple

model

already

used

by

Ericson and Scuroeder but indudes _one

more essential

ingredient, namely

tuat tue _resonances bave real parts ^{tuat should be described}

by

tue random matrix

tueory. Using

tue

twc-point

correlation function of the Gaussian _ensem- bles of random matrices, _we ^suowed tuat the autocorrelation function of the _resonance

density

suould be different from tue standard Lorentzian and establisued its

expression

in the _case of

large

modal

overlap.

We then

presented

numerical support for _our

predictions by analysing

the

spectral

content of the time response of _a model 2D reverberant _room. We _surmise that this

signature

^of

spectral rigidity

in

dissipative complex

_{wave systems} has _some close relation with

tue recent

findings

of Weaver

concerning

weak Anderson localization and enuanced couerent

backscattering

in reverberation _rooms in the time domain

Ils].

Appendix

A

Here,

tue normalised variables defined in tue _{text are} used. The normalised _resonance

density

function is written _as:

~~~~ =

~

~~

~~~~'~~~~

~~'" XÎ~Î _~2 /4~°')

^jA_~~

wuere,

for tue sake of

simplicity,

tue

xj's

stand for tue real parts, ^{and the delta}

density

function

z(x)

is introduced:

z(x)

=

~j _à(x

xj

). (A.2)

J

It is then known

tuat,

from the

knowledge

of the correlation functions of

z(x)

which _are deduced from trie n-level distribution functions of the Gaussian

ensemble,

_{one may,} in

principle,

find the correlation functions of the _resonance

density

_n. The twc-level correlation function of _z

reads [16]:

lzlx

+

X)zlx))

=

ôlX) Y21X), lA.3)

where

Y2(x)

is the twc-level duster function of the sequence of

xj's

which have unit

density

on the _average.

For _a

given stationary

random _process

six),

_one dermes the _power spectrum

S(À)

_as:

Sis;

_À)

=

/~~ e"~jsjx

₊

X)sjx))dX. jA.4)

1010 JOURNAL DE PHYSIQUE I N°8

For the

density

function _z, the power spectrum thus reads:

siz-

< _z >;

>)

m i

bi>), jA.s)

where the twc-level form factor b(À) ^{is the} Fourier transform of Y2

[loi

_:

bjÀ)

₌

/~~ e'~~Y2ix)dx. jA.6)

Then trie _power spectrum ^of n reads:

/~°~ ^{e'>x linix}

⁺

^{x)nix)) in)21}

^{dx =}

e-~'>' ii bi>)). IAi)

References

[Il Ericson T., Phys. Reu. Lett. 5

(1960)

430-431; id., Ann. Phys. 23

(1963)

390-414.

[2] Schroeder M.R. and Kuttruff K.H., J. Acoust. Soc. Am. 34

(1962)

76-80; Scuroeder M.R., J.

Acoust. Soc. Am. 34

(1962)

1819-1823.

[3] ^Blümel R. and Smflansky U., Phys. Reu. Lent. 64

(1990)

241-244.

[4] Smilansky U., Chaos and Quantum Puysics, M.-J. Giannom, A. Voros and J. Zinn-Justin, Eds.

(Elsevier,

Amsterdam, 1991) _pp 371-441.

[Si Doron E., Smilansky U. and Frenkel A., Phys. Reu. Lent. 65

(1990)

3072-3075.

[6] Weaver R.L., J. Sound Vib. 130

(1989)

487-491.

[7] ^{Doron E. and} Smilansky U., Phys. Reu. Lent. 68

(1992)

1255-1258.

[8] Eckhardt B., Chaos 3

(1993)

613-618.

[9] Davy J-L-, J. Sound Vib. 77

(1981)

455-479.

[loi

Bohigas O., Chaos and Quantum Physics, M.-J. Giannoni, A. Voros and J. Zinn-Justin, Eds.

(Elsevier,

Amsterdam, 1991) _pp. 87-199.

iii]

Shushin A. and Wardlaw A., J. Phys. A ₂₅

(1992)

1503.

[12] Mortessagne F., Dynamique et interférences géométriques dans les billards chaotiques. Application

à l'acoustique des salles, Thèse de l'Université de Paris 7

(1994).

[13] Mortessagne F., Legrand O. and Sornette D., Chaos 3

(1993)

529-541; Mortessagne F., Legrand

O. and Sornette D.

(prepnnt).

[14] Berry M.V., Chaos and Quantum Physics, M.-J. Giannoni, A. Voros and J. Zinn-Justin, Eds.

(Elsevier,

Amsterdam, 1991) _pp. 251-303.

[15] ^{Weaver R.L. and Burkhardt} J., J. Acoust. Soc. Am. 96

(1994)

3186-3190.

[16] Brody T.A. et ai., Reu. Med. Phys. 53

(1981)

385-479.