Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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188
P Manneville
rat
es
P24
=
RT2->T4
and
P42
=
RT4->T2,
which
imm
ed
iately leads
to
(1)
We
next
assume
that
the
two
stat
es
represent
local
minima
of
some po-
tential
V,
V
2
= V(2)
and
V
4
= V( 4),
and
that
the
transitions
are
noise-
activated
processes,
th
e
states
being
separated
by
a barri
er
to
which a po-
tential
Vb
= V(B) is associated,
Vb
- V
2
and
Vb
- V
4
being
the
potential
barriers
to
overcome
when
the
system
goes from
state
T2
to
T4
an
d
T4
to
T2,
with
an
Arrh
enius law in
the
form
(2)
where
W is
the
noise
amplitude.
Parameters
in (1)
can
be
estimated
consis-
tently
by (i) considering
the
steady
state
-!ft
P
2
==
0 (long
after
the
transient
have decayed):
P
2
P42
V4
- = - = -
exp(-(V2
- V
4
)/W)
P
4
P24
V2
and
(ii)
making
the
assumption
that
the
noise affecting variable X follows
from a law
of
large
numbers,
that
is
Wex
1/(n
d
)1
/2
=
1/n
2
,
wh
ere
n is
the
lin
ea
r size of
th
e
network
and
d = 4 is
the
dimension
of
space. S
ett
ing
just
W = 1
/n
2
fixes
the
potential
scale.
This
assumption
is in line
with
early
observations
[2]
and
can
be
checked
by
fitting
log(P
2
/
P
4
)
against
a law lin
ear
in n
2
,
i.e.
10g(Pd
P
4
)
=
a(r)n2
+
b(T)
with
a(r)
= V
4
-
V
2
and
b(r) = log(v4/v2). All
these
assumptions
have
been
validated
by
measuring
P
2
and
P
4
as
the
fractions
of
time
spent
in
states
T2
and
T4
during
very long simulations
(tens
of
millions of
it
e
rations)
at
given
r (here r = rmax) for a series
of
values of n (here n = 13, 16,
19,21,23
,25).
Within
that
framework,
it
is
not
difficult
to
und
e
rstand
the
differences
betw
een
the
bifurcation
diagrams
according
to
the
co
rresponding
protocols
and
values
of
n by varying r in
order
to
find which
state
is
most
likely
to
be
visited. Res
ults
for sev
era
l values
of
n
are
displayed
in
Fig
.
2,
in which
it
is
seen
that
the
amplitude
ratio
gets
larger
as
n increases.
Extrapolating
for
n large, one
obtains
that
P
4
> P
2
for for
rinf
'::::'
3.917 < r <
rsup
'::::'
3.997.
Wh
en
n is small,
the
amplitude
of
the
noise is sufficient
to
trigger
transitions
between
T2
and
T4,
but
when
n is
larg
e,
dep
en
ding
on
initial conditions,
the
system
may
remain
stuck
in
on
e
or
th
e
other
state
for
arbitrarily
long
tim
es
since noise becomes
too
small. Regime
T4
is
strong
ly
pref
erre
d for T ~ 3.95
and
regime
T2
for r < 3.90. For r
~
rinf
or
r
~
Tsup
the
two regimes
are
equ
ally probable.
They
are
both
observed
when
n is small
and
which
on
e is
observed
when
n is
larg
e
strongly
dep
en
ds
on
the
experimental
protocol
, i.e.
wh
et
her
or
not
one
has
been
able
to
construct
initial conditions
depending
in
one
or
the
other
basins
of
attraction.
As
an
example, for n = 54 uniform
di
str
ibutions
of
initial
states
Xj
are
in
the
basin
of
attraction
of regime
T2
Sub-Critical Transitions
in
Coupled Map Lattices
189
-l
0
-2
-4
- 6
-8
3.
88
3.9
3.92 3.94 3.96 3.98 4
Fig.
2.
Preferred
states
at
given r
as
obtained
from
the
ra
tio
P
2/
P
4
.
n = 13: blue;
n = 14: green; n = 15:
red
; n = 16,
cyan
.
for r < 3.9290
and
of regime
T4
heyond. on
the
oth
er
hand
if
the
initial
sta
te
is a 1'2 solution
at
T
~
3.88 it renmins in
the
T2
attraction
hasin for a
ll
values
of
l'
> 3.88
whether
r is increased
ad
i
abat
ically as in sweep-up si
mu
l
at
ions
or
abrupt
ly (in fact it has vanishingly small probability
to
turn
to
T4).
The
ca..'le
n =
27
close
to
r = 4
ca
n be discussed in the
same
way,
thus
expla
ining
the
presence
of
the
sm
all interval
of
T2
regime,
the
w
idth
of \vhich depends
OIl
how slowly r is decreased in sweep-down simulations.
The
data
can
be expl
oited
further
to
give a = V
4
-
V
2
and
b = log(v4/v2)
as functions of
r.
In
fact,
the
distributions of times
spent
in
regimes
T2
and
T4
are
avail
ab
le for a whole
set
of
va
lues of
n.and
r
and
not
on
ly
the
ratio
Pd
P
4.
These waiting times
turn
out
to
be
exponentially
distribu
ted
(F
ig. 3)
and
from
the
dec
ay
rat
es of
th
e
distribution
s, one could in principle determine
the
transition
rates
V2
and
V4
and
the value
of
the
potential
barriers
Vb
- V
2
and
Vb
-
V4"
but
this
post-treatment
remains
to
be done.
To conclude, we have shown
that
coupled logistic
maps
OIl
a 4-dimension
al
hyper-cubic l
at
tice display ordered macroscopic
beh
avio
ur
thou
gh no expla-
nation
from first
prin
ciples is avail
ab
le yet. Local chaos implies t
hat
the
standard
bifurcation framework has
to
be
made
stochast
ic,
and
that
the
dis-
continuous
trans
itions observed as
the
control
parameter
is varied
are
empir-
ica
ll
y well accounted for in
terms
of a
master
eq
uation
wr
i
tten
for a two-level
system
and
an
accompanying
activ
ated
transition
process
with
noise inten-
sity con
tro
lled
by
the
size of
the
system.
Better
und
erstan
di
ng
of
the
glo
bal
dynamics of such systems could also shed some light on
the
statistics of
tran-
sients observed
in
discontinuous
turbu
l
ent
-turbul
ent
trans
itions for which
sma
ll
sca
le turbulence would play
the
role of local chaos
and
large
sca
le flow
pattern
s would correspond
to
the
different regimes
obtaine
d
[6].
190
P.
Manneville
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
Fig.
3.
Quench
expe
riment
s from regime
1'4
at
Ti
= 3.98 down
to
various values
of
r'f
were
performed
with n = 27.
From
200
to
400
transients
were
exam
ined,
the
l
ength
of
a
transient
was
determined
from
the
sudden
decrease
of
the
quantity
X
4k
+
2
-
X
4
(k+l),
finite in regime
1'4
and
statistica
ll
y zero in regime 1'2.
As
understood
from
the
semilog
plot
of
the
fraction
of
systems
still in regime 1'4
at
time
t as a
function
of
t itself,
the
distribution
of
lifetimes displays
an
expo
nential
tail
in
the
form
pet'
> t)
ex
exp(
-tiT),
where
7-
1
varies
rap
idly with r from a small value for
rf
= 3.905
(upper
curve with black dots) down
to
3.895.
Somewhat
above
rr
= 3.92
regime
1'4
is essentially
sustained,
which corresponds
to
7 -
1
=
o.
The
author
acknowledges
the
participation
of
L.A.
Zepeda
Nunez
to
th
e early de-
velop
ment
s
of
this
study
within
the
framework
of
his second year
study
programme
at
Ecole
Poly
technique.
References
1.K. Kaneko. The Theory and Application
of
Coupled Map Lattices.
Wil
ey, New
York, 1993.
2.H.
Chate
and
P.
Mannev
ille. Collective behaviors
in
spatially
extended
systems
with
local
interactions
and
synchronous
updating.
Prog. Them'. Phys. , 87:1- 60,
1992.
3.H.
Chate,
A. Lema'itre,
Ph.
Marcq,
and
P. l'vlannevill
e.
Non
-trivial
collective
behavior in
extens
ively-chaotic
dynamical
systems:
an
update.
Physica
A,
224:447- 457, 1996.
4.A.
LemaItre
and
H. Ch
ate
.
Renormalisation
group
for
strongly
co
upl
ed
maps. J.
Stat. Phys., 96:915 -962, 1999.
5.N.G.
Van
Kampen.
Stochastic Proce8ses
in
Physics and ChemistTY. No
rth-
Holland,
Amsterdam,
1983.
6.
F.
Ravelet, L. Marie,
A.
Ch
iffaudel,
and
F. Daviaud.
Multistability
and
memory
effects in a highly
turbulent
flow:
experimental
evidence for a global bifurca-
tion.
Phys. Rev. Lett., 93:164501.1- 4, 2004.
191
Quantum
Scattering
and
Transport
in
Classically
Chaotic
Cavities:
An
Overview
of
Old
and
New
Results
Pier
A.
MeHol,
Victor
A.
Gopar2,
and
J.
A.
Mendez-Bermudez
3
1
Instituto
de
Ffsica
Universidad
Nacional
Autonoma
de
Mexico
01000
Mexico
Distrito
Federal,
Mexico
(e-mail:
mello@fisica.unam.mx)
2
Departamento
de
Ffsica
Teorica
and
Instituto
de
Biocomputacion
y
Ffsica
de
Sistemas
Complejos
U
niversidad
de
Zaragoza
Pedro
Cerbuna
12, 50009
Zaragoza,
Spain
(e-mail:
gopar@unizar.es
)
3
Instituto
de
Ffsica
Universidad
Autonoma
de
Puebla
Apdo.
Postal
J-48,
Puebla
72570,
Mexico
(e-mail:
antonio.ifuap@gmail.com)
Abstract:
We
develop
a
statistical
theory
that
describes
quantum-mechanical
scattering
of
a
particle
by
a
cavity
when
the
geometry
is
such
that
the
classical
dynamics
is
chaotic.
This
picture
is
relevant
to
a
variety
of
physical
systems,
rang-
ing
from
atomic
nuclei
to
mesoscopic
systems
and
microwave
cavities;
the
main
application
to
be
discussed
in
this
contribution
is
the
electronic
transport
through
mesoscopic
ballistic
structures
or
quantum
dots.
The
theory
describes
the
regime
in
which
there
are
two
distinct
time
scales,
associated
with
a
prompt
and
an
equili-
brated
response,
and
is
cast
in
terms
of
the
matrix
of
scattering
amplitudes
S.
We
construct
the
ensemble
of
S
matrices
using
a
maximum-entropy
approach
which
incorporates
the
requirements
of
flux
conservation,
causality
and
ergodicity,
and
the
system-specific
average
of
S
which
quantifies
the
effect
of
prompt
processes.
The
resulting
ensemble,
known
as
Poisson
's
kernel,
is
meant
to
describe
those
situ-
ations
in
which
any
other
information
is
irrelevant.
The
results
of
this
formulation
have
been
compared
with
the
numerical
solution
of
the
Schri:idinger
equation
for
cavities
in
which
the
assumptions
of
the
theory
hold.
The
model
has
a
remarkable
predictive
power:
it
describes
statistical
properties
of
the
quantum
conductance
of
quantum
dots,
like
its
average,
its
fluctuations,
and
its
full
distribution
in
several
cases.
We
also
discuss
situations
that
have
been
found
recently,
in
which
the
notion
of
stationarity
and
ergodicity
is
not
fulfilled,
and
yet
Poisson's
kernel
gives a
good
description
of
the
data.
At
the
present
moment
we
are
unable
to
give
an
explana-
tion
of
this
fact.
Keywords:
quantum
chaotic
scattering,
statistical
S
matrix,
mesoscopic
physics,
information
theory,
maximum
entropy.
192
P A. Mello, V
A.
COPGI;
Gild
1.
A. Mendez-Bemnidez
1
Introduction
The
problem
of
coherent
multiple
scattering
of waves
has
long
been
of
great
int
ere
st
in physics
and,
in
particular,
in optics.
There
are
a
great
many
wave-scattering problems,
appearing
in
various fields of physics, where
the
int
er
ference
pattern
du
e
to
multiple
scattering
is so complex,
that
a
chang
e in
som
e
ext
ern
al
paramet
er changes
it
completely
and,
as
a consequence, only
a
statistical
treatment
is feasible
and,
perhaps
, meaningful [1,2].
In
th
e
problems
to
be
discussed here, complexity in wave
scattering
de-
rives from
the
chaotic
nature
of
th
e
underlying
classical dynamics.
Our
discussion will find
applications
to
problems like;
i)
electronic
transport
in
microstructures
called ballistic
quantum
dots,
and
ii)
transport
of
electro-
mag
netic
waves,
or
other
classical waves (like elastic waves),
through
cavities
with
a chaotic classical dynamics.
In
particular,
we shall
study
the
sta
tistical
fluctuations
of
transmission
and
reflection
of
waves, which
are
of
considerable
interest
in mesoscopic physics.
The
ideas involved in
our
discussion
hav
e a
great
generality.
Let
us re-
call
that,
historically, nuclear physics - a
complicated
many-body
problem-
has
offered very go
od
examples
of
complex
quantum-mechanical
scattering.
Here,
the
typical
dimensions
are
on
the
scale
of
th
e fm
(1
fm =
1O-13
cm
).
The
statistical
theory
of
nuclear reactions
ha
s
been
of
great
interest
for
many
years, in
those
cases where,
due
to
the
presence
of
many
resonances,
the
cross
section is so
complicated
as
a function of energy
that
its de
tailed
structure
is of
littl
e
interest
and
a
statistical
treatments
is
then
called for. Most re-
markably, one finds similar
statistical
properties
in
the
quantum-mechanical
scattering
of "simple" one-particle
systems
-lik
e a
particle
scatte
ring from a
cavity-
whose classical
dynamics
is chaotic.
The
typical dimensions
of
the
ballistic
quantum
dots
th
at were
mentioned
above is 1
lun,
while
those
of mi-
crowave cavities is of
the
order
of 0.5 meters.
Thus
the
size
of
the
se
systems
sp
ans
;::::0
14
orders
of
magnitud
e!
The
purpose
of
this
contribution
is
to
review
past
and
recent work in
which various ideas
that
were originally developed in
the
framework
of
nucl
ear
physics, like
the
nuclear
optical
model
an
d
th
e
statistical
th
eory of nuclear re-
actions
[3],
hav
e
been
used
to
give a unifi
ed
treatment
of
quantum-mechanical
sc
at
tering
in
simple
one-particl
e
systems
in
which
the
corresponding
classical
dyn
amics is chaotic,
and
of
microwave
propagation
through
similar cavities
(see Ref. [1]
and
references con
tai
ned
therein).
The
most
remarkable
feature
that
we shall
encounter
is
th
e
statistical
regu
larit
y
of
the
results, in
the
sense
that
they
will
be
expressible in
terms
of
a few relevant physical
parameters,
the
rem
aining
details
being
just
"scaf-
foldings" .
This
feature
will be
captured
within
a framework
that
we
shall
call
the
"max
imum-entropy
approach"
.
In
th
e
next
section
we
pr
ese
nt
some gene
ral
physical ideas re
lated
to
scattering
of microwaves
by
cavities
an
d
of
e
lectron
s
by
ballistic
quantum
dots.
The
formulation
of
the
scattering
problem
is given in Sec. 3, where
we
Quantum Scattering and Transport in Classically Chaotic Cavities
193
employ,
to
be
specific,
the
language
of
electron
scattering.
The
notion
of
do-
ing
statistics
with
the
scattering
matrix
and
the
resulting
maximum-entropy
model
are
reviewed
in
Secs. 4
and
5, respectively.
In
Sec. 6 we
compare
the
predictions
of
our
model
with
a
number
of
computer
simulations.
We
also
present
a
critical
discussion
of
the
results,
based
on
some
recent
findings
on
the the
validity
conditions
of
our
statistical
model.
The
conclusions
of
this
presentation
are
the
subject
of
Sec.
7.
2
Microwave
cavities
and
ballistic
quantum
dots
Consider
the
system
shown
schematically
in
Fig.
1.
It
consists
of
a
cavity
connected
to
the
external
world
by
two waveguides, ideally
of
infinite
length.
__
bt,;)
...
----
a(2)
n
Fig.
1.
The
2D
cavity
referred
to
in
the
text.
The
cavity
is
connected
to
the
outside
via
two
waveguides,
each
supporting
N
running
modes.
The
arrows
inside
the
waveguides
indicate
incoming
or
outgoing
waves.
In
waveguide
I = 1, 2
the
wave
amplitudes
are
indicated
as
a~),
b~),
respectively,
where
n = 1,
...
,
N.
We
may
think
of
performing
a m'icmwave
experiment
with
such
a device
[4].
A wave is
sent
in
from
one
of
the
waveguides,
the
result
being
a reflected
wave
in
the
same
waveguide
and
a
transmitted
wave
in
the
other.
One
given
point
inside
the
system
receives
the
contributions
from
many
multiply
scattered waves
that
have
bounced
from
the
inner
surface
of
the
cavity, giving
rise
to
the
complex
interference
pattern
mentioned
in
the
Introduction.
An
alternative
interpretation
of
the
same
situation
is
that
the
outgoing
wave
has
suffered
the
effect
of
many
resonances
that
are
present
inside
the
cavity.
The
net
result
is
an
interference
pattern
which
shows
an
appreciable
sensitivity
to
changes
in
external
parameters:
for
example,
a
small
variation
in
the
shape
of
the
cavity
may
change
the
pattern
drastically.
In
experiments
on
electmnic transport
[5]
performed
with
ballistic mi-
crostructures,
or
quantum
dots,
one
finds, for sufficiently low
temperatures
and
for
spatial
dimensions
on
the
order
of
llLm
or
less,
that
the
phase
coher-
ence
length
11>
exceeds
the
system
dimensions;
thus
the
phase
of
the
single-
electron
wave
function
-
in
an
independent-electron
approximation-
remains
194
P.
A.
Melia,
V.
A. Gopar,
and
1.
A. Mendez-Bermudez
coherent
across
the
system
of interest.
Under
these conditions,
these
systems
are
called mesoscopic [6-
8].
The
elastic
mean
free
path
lei
also exceeds
the
system
dim
ensions:
impurity
scattering
can
thus
be
neglected, so
that
only
scattering
from
the
boundaries
of
the
system
is
important.
In
these
sys-
tems
the
dot
acts
as
a
resonant
cavity
and
the
leads as
electron
waveguides.
Experimentall
y,
an
electric
current
is
established
through
the
leads
that
con-
nect
the
cavity
to
the
outside
,
the
potential
difference is
measured
and
the
conductance
G is
then
extracted.
Assume
that
the
microstructure
is placed
between
two reservoirs
(at
different chemical
potentials)
shaped
as
expand-
ing
horns
with
negligible reflection
back
to
the
microstructure.
In a
picture
that
takes
into
account
the
electron-electron
interaction
in
the
random-phase
approximation
(RPA),
the
two-terminal
conductance
at
zero
temperature
-
understood
as
the
ratio
of
the
current
to
the
potential
difference
betw
een
the
two reservoirs- is
then
giv
en
by
Landauer's
formula [10,9,
1]
e
2
G=2-T
h '
where
the
total
transmission
T is given by
(1)
(2)
t being
the
matrix
of
transmission
amplitud
es
tab
associated
with
the
res
ulting
self-consiste
nt
single-electron
problem
at
th
e Fermi energy
tF,
with
a
and
b
denoting
final
and
initial channels.
The
factor
of
2 in Eq. (1) is
due
to
the
two-fold
spin
degeneracy in
the
absence
of
spin-orbit
scattering
that
we
shall assume here.
In
this
"scattering
approach"
to
elec
tronic
transport,
pioneered by
Landauer
[10],
the
problem
is
thus
reduced
to
understanding
the
quantum-mechanical
single-electron
scattering
by
th
e
cavity
under
study.
When
th
e
system
is
imm
ersed in
an
e
xternal
magnetic
field B
and
the
latter
is varied,
or
the
Fermi e
nergy
tF
or
the
shape
of
the
cavity
are
varied,
the
relative
phase
of
the
various
partial
waves changes
and
so
do
the
interfer-
ence
pattern
and
the
conductance.
This
sensitivity
of
G
to
small changes in
parameters
through
quantum
interference is called conductance fluctuations.
The
exp
e
riments
on
quantum
dots
given in Ref.
[5]
re
port
cavities in
the
shape
of
a
stadium,
for which
the
single-ele
ctron
classical
dynamics
would
be
chaotic,
as
we
ll
as
experimental
ensembles
of
shapes
.
The
average of
the
conductance,
its fluctuations
and
its full
distribution
were
obtained
over such
ensembles.
In
what
follows we shall
phrase
the
probl
em in
the
language
of
electronic
scattering;
spin will
be
ignored, so
that
we
shall d
ea
l
with
a scalar wave
equation
. However,
our
formalism is applicable
to
scattering
of
microwaves,
or
other
classical waves, in
situations
wh
ere
the
scalar wave
equation
is
a
r
ea
sonable
approximation.
Quantum Scattering and Transport in Classically Chaotic Cavities
195
3
The
scattering
problem
In
our application
to
mesoscopic physics we consider a
system
of
noninteract-
ing "spinless" electrons
and
study
the
scattering
of
an
electron
at
the
Fermi
energy
EF
by a 2D
microstructure,
connected
to
the
outside
by two l
ea
ds
of
width
W (see Fig. 1).
Very generally, a
quantum-mechanical
scattering
problem is described by
its
scattering
matrix,
or S
matrix,
that
we
now define.
The
confinement
of
the
electron in
the
transverse direction in
the
leads gives rise
to
the
so called
transverse modes,
or
channels.
If
the
incident Fermi
momentum
kF satisfies
N<kFW/7r<N+1,
(3)
there
are
N
transmitting
or
running
modes (open channels)
in
each
of
the
two leads.
The
wavefunction in lead l (l =
1,2
for
the
left
and
right leads,
respectively) is
written
as
the
N-dim
ensional vector
(4)
the
n-
th
component
(n =
1,
...
,
N)
being a linear combination of plane waves
with
constant
flux (giv
en
by
h),
i.e.,
Tn
being
the
electron mass.
In
(5), k
n
is
the
"longitudinal"
momentum
in
channel
n, given by
(6)
The
2N
-dimensional S
matrix
relates
the
incoming
to
the
outgoing ampli-
tudes
as
(7)
where
a,ll),
a(2),
b(1) , b(2)
are
N-dimensional
vectors.
The
matrix
S
has
the
structure
(
r
t')
S = t
r'
,
(8)
where r, t
are
the
N x N reflection
and
transmission matrices for particles from
the
left
and
r',
t'
for those from
the
right: this
thus
defines
the
transmission
matrix
t used earlier in Eq. (2).
The
requirement
of
flux conservation
(Fe)
implies
unitarity
of
the
S ma-
trix
[1],
i.e.,
sst =
I.
(9)
This
is
the
only requirem
ent
in
the
absence
of
other
sym
metrie
s.
This
is
the
so-called
unitary
case, also
denoted
by
(3
= 2 in
the
literature
on
random-
matrix
theory. For a time-reversal-invariant (TRI)
problem
(as is
the
case
196
P.
A. Melia,
V.
A. Gopar,
and
1.
A. Mendez-Bermudez
in
the
absence
of
a
magnetic
field)
and
no
spin,
the
S
matrix,
besides
being
unitary,
is
symmetric
[1]:
(10)
This
is
the
orthogonal case, also
denoted
by
fJ
=
l.
The
symplectic case
(fJ
=
4) arises
in
the
presence
of
half-integral
spin
and
time-reversal
invariance,
and
in
the
absence
of
other
symmetries,
but
will
not
be
considered
here.
4
Doing
statistics
with
the
scattering
matrix
In
numerical
simulations
of
quantum
scattering
by
2D
cavities
with
a
chaotic
classical
dynamics
one
finds
that
the
S
matrix
and
hence
the
various
trans-
mission
and
reflection coefficients
fluctuate
considerably
as a
function
of
the
incident
energy
E (or
incident
momentum
k),
because
of
the
resonances
oc-
curring
inside
the
cavity
[1].
These
resonances
are
moderately
overlapping
for
just
one
open
channel
and
become
more
overlapping
as
more
channels
open
up.
An
example
is
illustrated
in
Fig.
2,
which shows
the
total
transmission
2
1.5
Ll
i
T
0.5
2.5
kWln
3
3.5 4
Fig.
2.
Total
transmission
coefficient
of
Eq.
(2) as a
function
of
the
incident
mo-
mentum,
obtained
by
solving
numerically
the
Schrodinger
equation
for
the
system
indicated
in
the
upper
left
corner
(from Ref. [11]), whose classical
dynamics
has
a
chaotic
nature.
coefficient
T(E)
of
Eq.
(2)
[proportional
to
the
conductance,
by
Landauer's
formula
of
Eq.
(1)],
obtained
by
solving
numerically
the
Schrodinger
equa-
tion
for a
cavity
whose
underlying
classical
dynamics
is chaotic. Since we
are
not
interested
in
the
detailed
structure
of
T(E),
it
is
clear
that
we
are
led
to
a statistical description
of
the
problem,
just
as
was
explained
in
the
Introduction.
Quantum Scattering and Transport
in
Classically Chaotic Cavities
197
As E changes,
5(E)
wanders
on
the
unitarity
sphere,
restricted
by
the
condition
of
symmetry
if
TRI
applies.
The
question
we
shall
address
is:
what
fraction
of
the
"time"
(energy)
do
we find 5 inside
the
"volume
element"
dp,(5)? [dp,(5) is
the
invariant
measure
for 5
matrices,
to
be
defined below.]
We call
dP(5)
this
fraction
and
write
it
as
dP(5)
=
p(5)dp,(5),
(11)
where
p(5)
will
be
called
the
"probability
density".
Once
we
phrase
the
prob-
lem
in
this
language,
we realize
that
we
are
dealing
with
a
mndom
5-matrix
theory.
In
the
next
section
we
shall
introduce
a
model
for
the
probability
density
p(5).
The
concept
of
invariant
measure
dp,(5) for 5
matrices
mentioned
above
is
an
important
one
and
we now briefly
describe
it.
By
definition,
the
invariant
measure
for a given
symmetry
class
does
not
change
under
an
automorphism
of
that
class
of
matrices
onto
itself
[12,13]' i.e.,
(12)
For
unitary
and
symmetric
5
matrices
we
have
5'
= u
o
5ul, (13)
U
o
being
an
arbitrary,
but
fixed,
unitary
matrix.
Eq.
(13)
represents
an
automorphism
of
the
set
of
unitary
symmetric
matrices
onto
itself
and
de-
fines
the
invariant
measure
for
the
orthogonal
((3
= 1)
symmetry
class
of
5
matrices.
We recall
that
this
class is
relevant
for
systems
when
we ignore
spin
and
in
the
presence
of
TRI.
When
TRI
is
broken,
we
deal
with
unitary,
not
necessarily
symmetric,
5
matrices.
The
automorphism
is
then
5'
= U
o
5V
a
,
(14)
U
a
and
Va
being
now
arbitrary
fixed
unitary
matrices.
For
this
unitary,
or
(3
=
2,
symmetry
class
of
5
matrices
the
resulting
measure
is
the
well
known
Haar
measure
of
the
unitary
group
and
its
uniqueness
is well known.
Uniqueness for
the
(3
= 1 class
(and
(3
= 4,
not
discussed here) was
shown
in
Ref.
[12].
The
invariant
measure,
Eq.
(12), defines
the
so-called Circular
(Orthogonal, Unitary)
Ensemble
(COE,
CUE)
of
5
matrices,
for
(3
=
1,2,
respectively.
5
The
maximum-entropy
model
We
shall
assume
that
in
the
scattering
process
by
our
cavities two
distinct
t'/me scales
occur
[1,3,14 -17]: i) a
prompt
response
arising
from
short
paths;
ii)
a delayed
response
arising
from
very
long
paths.
Historically,
this
assumption
was first
made
in
the
nuclear
theory
of
the
compound
nucleus,
where
the