1
Yu.L.Bolotin, I.V.Krivoshei, Sov.J. of Nucl.Phys.(Yad. Fiz). 42, 53
(1985)
DYNAMICAL CHAOS AND NUCLEAR FISSION
Theme de travail:
DYNAMICAL CHAOS
AND NUCLEAR FISSION
REGULAR AND CHAOTIC CLASSIC AND QUANTUM
DYNAMICS IN (2D) MULTI-WELL POTENTIALS
YU.L.Bolotin, NSC KhFTI,
Kharkov, Ukraine
2
Hamiltonian system with multi-well potential energy
surface (PES) represents a realistic model, describing
the dynamics of transition between different equilibrium
states, including such important cases as chemical and
nuclear reactions, nuclear fission, string landscape and
phase transitions).
One-well potential – rare exception
Multi-well – common case
Such system represents an important object, both for the
study of classic chaos and quantum manifestations of
classical stochasticity.
3
Research of any nonlinear system (in the context of chaos) includes the
following steps
• 1. Investigation of the classical phase space, detection of
chaotic regimes
• 2. Analytical estimation of the critical energy transition to
chaos.
• 3. Test for quantum manifestation of classical stochasticity
• 4. Action of chaos on concrete physical effects.
The basic subject of the current report is to realize the
outlined program for two-dimensional multi-well
Hamiltonian systems (of course, only in part)
4
I. Classical dynamics
• SPECIFICS OF CLASSICAL
DYNAMICS IN MULTI-WELL
POTENTIALS —MIXED STATE
5
Quadrupole oscillations of nuclei
deformation potential of the surface quadrupole
oscillations was built by Mosel , Greiner (1968)
U  a0 , a2    Cmn  a  2a
2
0

2 m
2
a  6a  a
n
0
2
2

2 n
0
m ,n
a0 , a2 is coordinate of nuclear surface


R  ,   R0 1  a0Y2,0  ,   a2 Y2,2  ,   Y2,2  ,  
6
Restricting to the member of the fourth degree
and performing simple transformation we obtain
H
p p
2
x
2
x  2a2 ;
2
y

b2 
 U  x, y;W   ;
ac 

y  a0 ; a  2C10 ; b  3C01; c  C20
1
1 3
2
2
2
2
2 2
U ( x, y;W ) 
x  y   x y  y  x  y 

2W
3
W  16  one  well potential
W  16  multi  well potential
U ( x, y)  C3v symmetric potential
7
Why is such potential chosen?
NUCLEAR THEORY
Authorities love order, but very don’t love chaos.
My bosses this dislike briefly formulated as follows:
Chaos could be studied only by those, who have
nothing to do.
We felt himself as partisan.
In this distant time we lived in era Henon-Heiles potential.
U ( x, y ) 
1 2
1 3
2
2
x

y

x
y

y;


2
3
  x2  y 2 
2
I remembered that this potential saw somewhere (W.Greiner book).
We proudly went out from an underground and
chaotic parasite  (1987)  chaotic nuclear physisist
NUCLEAR THEORY AND NONLINEAR DYNAMICS
n6
The surfaces of potential energy of Krypton
isotopes.
9
Full list of «Our» potentials
2
2
px  p y
H
 U ( x, y );
2
1
1 3
 2
2
2
2
2 2
U QO 
x  y    x y  y   x  y  ;

2W
3 

1 4
2
2
2
U D5  x  xy  2 y  x ;
4
so called umbilic catastrophe D5
1 6 1 4
3 2
2
U D7  x  x  xy  x  2 y 2
6
2
8
so called umbilic catastrophe D7
We worked also and with other potentials, but nothing substantially new (as
compared to these) did not discover there
10
motionis finite only for E  Esaddle
D5
motionis always finite
QO, W  18
11
D7
catastrophe
motionis finite only for E  Esaddle
12
What is the mixed state?
Yu.L.Bolotin, V.Yu.Gonchar, E.V.Inopin
Chaos and catastrophes in quadrupole oscillations of
nuclei, Yad.Fiz. 45, 350, 1987 (20 anniversary
)
One-well case – Poincare section
py
py
y
(nothing unusual!)
As the energy increase
the gradual transition
from the regular motion
to chaotic one is observed.
y
E  Esaddle
E  0.75Ecr
py
py
y
E  0.25 Ecr
y
E  1.25Ecr
13
py
E  2 Esaddle
y
py
E  Esaddle
y
py
py
py
E  Ecr
y
py
E  Ecr
y
Change of the character of
motion in left and right local
minima is essentially
different!
It means that in this case
so-called
MIXED STATE
may be observed:
at one and the same
energy in different local
minima various dynamical
regimes (regular or chaotic)
are realized
14
Mixed state is common property of
multi-well potentials
D5
D7 QO
15
Why the dynamical behavior is so
unlike in the different local minima:
why in some local minima chaos
begins below the saddle energy, but in
others only above.
If we want answer this question, we
must use different criteria of chaos.
It is a very complicated problem, separate
question, and we do not have time for the
detailed discussion.
If there will be time at last, we will discuss
some details.
16
We used:
1. Negative curvature
criterion (Toda)
2. Geometrical approach
(Pettini et al.)
3. Overlap of nonlinear
resonances (Chirikov)
4. Destruction of stochastic
layer (Delande et al). ….
and many others
17
Result:
we can find critical energy of
transition to, but, we can’t
forecast specificity of behavior
in arbitrary local minimum using
only geometrical terms (for
example, number of saddle,
negative curvature etc.)
18
Regular-Chaos-Regular transition
R-C-R transition is a possible only for the system with
localized domain of instability (negative Gaussian curvature
or overlap of nonlinear resonances)
QO potential
74
R2
C
R1
Kr
R2
R1
K<0
C
The part of phase space S% with chaotic
trajectories as a function of the energy
19
R-C-R TRANSITION IN MULTI-WELL POTENTIAL
E  ES
E  2 ES
E  10 ES
E  280 ES
E  3000 ES
E  4000 ES
20
p2
H ( p, x , t ) 
 Ax n  Fx cos t
2m
I k
Reason of the additional
C-R transition: new
intersection point
Yu.L.Bolotin, V.Yu.Gonchar,
M.Ya.Granovsky, Physica D 86 (1995)
R-C-R transition in a periodically driven
anharmonic oscillator
21
One comment
Stochastization of quadrupole nuclear oscillations is confirmed by the
direct observation of chaotic regimes at simulation of reaction with
heavy ions.
Umar et al. (1985)
TDHF calculation head-on collisions:
He4  C14 ; C12  C12 (0 ); He4  Ne20
M LI (t )   d 3rr LYLM ( ,  )  I ( r , t )
  p ( r , t )   n ( r , t ); I  0
 I (r, t )  
  p ( r , t )   n ( r , t ); I  1
Poincare section for isoscalar quadrupole mode in
Mg 24
22
II. Quantum chaos
Quantum manifestation of classical
stochasticity in mixed state.
(comparison of one-well and multi-well)
23
SPECTRAL METHOD
M.D.Feit, J.A.Fleck, A.Steiger (1982)
1.Calculation of quasiclassical part of the
spectrum for multi-well systems requires
appropriate numerical methods.
2. Matrix diagonalization method (MDM) is
attractive only for one-well potential.
In particular, the diagonalization of the QO Hamiltonian
with W > 16 in the harmonic oscillator basis requires so
large number of the basis functions that go beyond the
limits of the our computation power.
The spectral method is an attractive alternative to MDM
24
 h2

    U ( x, y )  n ( x, y )  En n ( x, y )
 2

The main instrument of spectral method is correlation function
P(t )   dxdy 0* ( x, y ) ( x, y, t )
The solution can be accurately generated with the help of the split operator
method
 h 2 t2 
 ( x, y, t  t )  exp  i
 exp  itU ( x, y ) 
4 

 h 2 t2 
3
exp  i

(
x
,
y
,
t
)



t

4


 
25
 ( x, y, t ) can be expressed as
 ( x, y, t )   an n ( x, y )exp  iEnt / h 
n
P (t )   an exp  iEnt / h 
2
n
T
1
2
P ( E )   dt exp  iEnt / h  P (t ) w(t )  an  T  E  En 
T0
n
T
1
 T ( E )   dtw(t )exp  iEt / h 
T0
if eigenvalues are known :
T
1
 n ( x, y )   dt ( x, y, t ) w(t )exp  iEnt / h 
T0
26
P( E ) for Hamiltonian of quadrupole oscillation (W  18)
27
ANALITICAL METHODS
For simplicity, we only will name analytical methods which we used
(and plan to use) for description of the mixed state.
1. A.Auerbach and S.Kivelson (1985): The path decomposition expansion
Path integral technique which allows to break configuration space into
disjoint regions and express dynamics of full system in term of its parts
2. Kazuo Takatsuka ,Hiroshi Ushiyama, Atsuko Inoue-Ushiyama (1998)
Tunneling paths in multi-dimensional semiclassical dynamics
28
Now we have methods of investigations both classical and quantum
chaos, but ….. Do we have a research object?
Chinese legend
Once upon a time there lived Dzhu,
Who learned to kill off dragons
And gave up all he had
To master art like that.
Chaos vs. regularity
Eternal battle
Three whole years it took,
But, alas, never came up that chance
To present skill and form.
So he took on himself teaching
others the art of slaying dragons.
The last two lines belong
R. Thom

We have a chaotic dragon and even can present some trophy
29
O.Bohigas, M.Giannoni, C.Shmit: (1983)
Hypothesis of the universal fluctuations of energy spectra
regular system (in classical limit )  level clusterization
p ( s ) : exp(  s)
chaotic system (in classical limit )  level repulsion
p ( s ) : s exp(  s 2 )
Fluctuation properties of QO spectra
W  13
Rigid lines are Poisson’s prediction
Dashed lines are GOA prediction
Qualitative agreement with Bohigas hypothesis
30
Fluctuations of energy spectrum in mixed state
E  Ecr
E  Ecr
E  Ecr
E  Ecr
E  Ecr
Rigid lines are Poisson and Wigner prediction; dashed lines – fitting by Berry-RobnikBogomollny distribution (interpolation between Poisson and Wigner distribution)
A priori FNNSD  weighted superposition Poisson and Wigner
In that case we deal not with statistics of mixture of two spectral
series with different NNSD, but with statistics of levels that none
of them belongs to well-defined statistics.
Statistical properties of such systems were not studied at all up to
now, though namely such systems correspond to common situation.
31
Evolution of shell structure in the process
R-C-R transition and in mixed state
Very old problem (W.Swiateski, S.Bjornholm): how
one could reconcile the liquid drop model of the
nucleus (short means free path) with the gas-like
shell model?
To account for such contradiction investigation of
shell effect destruction in the process R-C-R
transition plays the key role
More exact formulation:
How do shell dissolve with deviation from regularity,
or, conversely,
How do incipient shell effects emerge as the system is
approached to an integrable situation?
32
We used nonscale version of the Hamiltonian QO
Classical prompting
H
Interesting
px2  p 2y
2
a 2
1 3
 2
2
2
2 2
 x  y   b x y  y   cx  y 
2
3 

W  13
W  3.9
In the interval 0<W<4 for all energies the motion remains
regular (in this interval K>0)
33
The destruction of shell structure can be
traced, using analog of thermodynamic
entropy
Sk    C
2
k
NLj
ln C
2
k
NLj
N ,L , j
k 
k
C
 NLj NLj
N ,L , j
R2
Regular domain:
change of entropy correlates
with the transition from shell to
shell
C
Chaotic domain:
1. quasiperiodic dependence of
entropy from energy is violated;
R1
2. Monotone growth on average
towards a plateau corresponding
to entropy of random sequence.
34
We obtain this result for QO potential, but it is
general result
Regularity-chaos transition in
any potential is always
accompanied destruction of
shell structure
35
Quantum chaos and
1/ f
noise
Relano et al. 2002: the energy spectrum fluctuations
of quantum systems can be formally considered as a
discrete time series. The power spectrum behavior of
such a signal are characterized by
1/ f noise
Spectral fluctuations described by
n
n
i 1
i 1
 n    si  s    wi
Power spectrum of a discrete time series
S (k )   k
2
1
; k 
N
 2 kn 
n  n exp   N 
36
Example of chaotic system is nucleus at high excitation energy
1
S (k ) : 
k
  1.11  0.03  Na 34 
  1.06  0.03  Mg 24 
The average power spectrum of the  n function for Mg
34
(sd shell) and Na (very exotic nucleus) using 25 sets from 256
levels for high level density region. The plots are displaced to avoid
overlapping.
24
37

Power spectrum of the n function for GDE (Poisson) energy
levels compared to GOE,GUE, GSE (Relano et al. 2002)
38
 : 1.14
 : 1.87
39
Signature of quantum chaos in wave
function structure
In analysis of QMCS in the energy spectra the main role
was given to statistical characteristic: quantum chaos was
treated as property of a group of states
In contrast, the choice of a stationary wave function as a basic
object of investigation relates quantum chaos to an individual
state!
Evolution of wave function during R-C-R transition can be studied
with help:
1. Distribution on basis.
2. Probability density.
3. Structure of nodal lines.
40
Degree of distribution of wave function
k 

N , L , j 1
k
CNLj
NLj ; NLj 
PLj
2
 NL
 j N , L

Nordholm, Rice
(1974) Degree of
distribution of wave
function arises in the
average along with
the degree of
stochasticity.
Yu.L.Bolotin, V.YU.Gonchar….Yud.Fiz. (1995)
41
Isolines of probability density
 k ( x, y )
42
2
The topography of nodal lines of the stationary wave function.
R.M.Stratt, C.N.Handy, W.N.Miller (1974): system of nodal lines of the
regular wave function is a lattice of quasiorthogonal curves or is similar to
such lattice. The wave function of chaotic states does not have such
representation
separable
nonseparable,
but integrable
nonintegrable,
avoided
intersection of
nodal lines
A.G.Manastra et al. (2003)
43
Mixed state: QMCS in structure of wave function
Usual procedure of search for
QMCS in wave function implies
investigation their structure below
and above critical energy
Problem: necessity to separate
QMCS from modification of wave
functions structure due to trivial
changes in its quantum numbers
QO potential
The main advantage of our approach:
In the mixed state we have possibility to
detect QMCS not for different wave function,
but for different parts of the one and the
same wave function.
QO potential D5 potential
V.P.Berezovoj, Yu.L.Bolotin,
V.A.Cherkaskiy, Phys. Lett A (2004)
44
Decay of the Mixed States
The escape of
trajectories (particles) from
localized regions of phase or configuration space
has been an important topic in dynamics, because
it describes the decay phenomena of metastable
states in many fields of physics, as for example
chemical and nuclear reactions, atomic ionization
and induced nuclear fission.
45
Optic Billiard
acousto-optic deflectors
laser
beam
billiard
plane
f
min
scan
horizontal
 v a / w0 ~ 4 KH
vertical
max
f  f scan
max
f  f scan
10 KHz
100 KHz
max
2
f scan
 1 /(Taccess
  AOD ) ~ 40 KHz
46
How Do We Observe Chaos in the Wedge ?
450 mm
Stable trajectories
do not “feel” the hole
55 mm
Chaotic trajectories
leak through the hole
Cs
47
Experiment vs Numerical Simulations
.4
90 o
4
90 o
3
90 o
2
.3
.2
.1
0
20
25
30
35
40
45
50
55
θ (deg)
48
(W.Bauer,
G.F.Bertch,
1990)
Exponential decay is a common property
expected in strongly chaotic systems
For the chaotic systems  exponential
decay law
For the nonchaotic systems  power
decay law
49
Numerical experiment on the Sinai billiard
N (t )  N (0) exp(  t )
p

A
p  the absolute value of momentum ,
  the opening of width
A  total coordinate  space available
50
Once more “mixed state”
E  Ecr
E : Ecr
E  Esaddle E  2 Esaddle
D5
Quadrupole
oscillations
At energy higher than saddle energy the phase
space structure preserves division on chaotic
and regular components
51
Decay law for mixed states in the D5
(a) and QO (b) potentials
D5
QO
Solid lines –
numerical
simulation for
E/E(saddle)=
1.1,1.5, 2.0.
Dotted and dashed
lines – analytical
exponential and
linear decay law
52
The result for different potential are evidently similar and have such
characteristic features:
1. Decay law saturates
N (t  )   ( ne ) N 0
 ( ne ) is relative phase valume
of " never  escaping " trajectories 
regular trajectories
2
for t   ( E ) the decay law has the exp onential form
(as billiard )
3. for t   ( E ) the decay law is linear
53
Rigid correlations
Relative area of stability island
Fraction of non-escaping
particles
54
Decay of mixed states may find practical
application for extraction of required particle
number from atomic traps.
Changing energy of the particles trapped
inside the “regular” minimum we can
extract from the trap any required
number of particles.
Obtained results may present an interest for
description of induced nuclear fission in the
case of double-humped fission barrier.
55
Now a few words about our current activity.
1. Quantum decay of the mixed states (current
activity)
2. Investigation of dynamical tunneling in 2D multiwell potentials (current activity).
3. Tunneling from super- to normal deformed
minima in nuclei (only plan)
56
Superdeformation in nuclei   mixed state
Chaos
Regula
rity
T.L.Khoo Lecture in Institute of
Nuclear Theory (???)
57
Our aim: to transform
O.Bohigas, D.Boose, R.E. de Carvachlo, V.Marvulle
(BBCM) (1993)
“Quantum tunneling and chaotic dynamics”
to dynamical tunneling in the mixed state
Billiard

potential (mixed state)
BBCM: the tunneling is increased as the transport through
chaotic regions grows.
Why?
The energy splitting of a given doublet is very sensitive to is
position in the energy spectrum as well as to its location in
phase- space
58
BBCM
Measure of chaos
Energy splitting
The energy splitting is increased on a lot of
orders as chaos increases.
59
We plan to realize Bohigas”s billiard
problem for multi-well potentials
Dynamical tunneling in QO potential
60
Bohigas et al.
The energy splitting of a tunneling doublet
(spectral method)
We will do animation
for the splitting levels
as function of chaos
in central minima
QO.
61
What I wanted about, but did not have time to
tell
1. Our analitical results
2. Birkhoff-Gustavson normal form (classic and
quantum)
3. Wave packet dynamics
4. Numerical methods (apart “spectral method”)
…………………..
62
Thank you for attention
grateful acknowledgment to prof. Egle Tomasi for all!
63