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

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REGULAR AND CHAOTIC CLASSIC AND QUANTUM DYNAMICS IN (2D) MULTI-WELL POTENTIALS

2

YU.L.Bolotin, NSC KhFTI, Kharkov, Ukraine

3

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.

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)

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5

I. Classical dynamics

• SPECIFICS OF CLASSICAL DYNAMICS IN MULTI-WELL POTENTIALS —MIXED STATE

6

2 2 2 20 2 0 2 0 2 0

,

0 2

0 0 2,0 2 2,2 2, 2

, (1968)

, 2 6

,

, 1 , , ,

m nnmn

m n

deformation potential of the surface quadrupole

oscillations was built by Mosel Greiner

U a a C a a a a a

a a is coordinate of nuclear surface

R R a Y a Y Y

Quadrupole oscillations of nuclei

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2 2 2

2 0 10 01 20

22 2 2 3 2 2

, ; ;2

2 ; ; 2 ; 3 ;

1 1( , ; )

2 316

16

x y

Restricting to the member of the fourth degree

and performing simple transformation we obtain

p p bH U x y W

ac

x a y a a C b C c C

U x y W x y x y y x yW

W one well potential

W m

ulti well potential

3( , ) vU x y C symmetric potential

Why is such potential chosen?

NUCLEARTHEORY

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.

22 2 2 3 2 21 1( , ) ;

2 3U x y x y x y y x y

I remembered that this potential saw somewhere (W.Greiner book). We proudly went out from an underground and

NUCLEAR THEORY AND NONLINEAR DYNAMICS

(1987)chaotic parasite chaoticnuclear physisist

9

The surfaces of potential energy of Krypton isotopes.

6n

10

5

7

2 2

22 2 2 3 2 2

4 2 2 2

5

6 4 2 2 2

7

( , );21 1

;2 3

12 ;

4

1 1 32

6 2 8

x y

QO

D

D

p pH U x y

U x y x y y x yW

U x xy y x

so called umbilic catastropheD

U x x xy x y

so called umbilic catastropheD

Full list of «Our» potentials

We worked also and with other potentials, but nothing substantially new (as compared to these) did not discover there

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5D , 18QO W

saddlemotionis finite only for E E motionis always finite

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7D catastrophe

saddlemotionis finite only for E E

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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 )

0.75 crE E

1.25 crE E

saddleE E

0.25 crE E

(nothing unusual!)As the energy increasethe gradual transition from the regular motionto chaotic one is observed.

One-well case – Poincare sectionyp

y

yp

yp yp

y

y y

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crE E

crE E

saddleE E

2 saddleE EChange 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

yp

yp

yp

yp

y

ypyp

y

y

y

15

Mixed state is common property of multi-well potentials

5D 7D QO

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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.

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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

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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.)

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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

K<0

1R

1RC C

2R2R

The part of phase space S% with chaotic trajectories as a function of the energy

74Kr

SE E 2 SE E

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R-C-R TRANSITION IN MULTI-WELL POTENTIAL

280 SE E 3000 SE E 4000 SE E

10 SE E

21

2

( , , ) cos2

npH p x t Ax Fx t

m

kI

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

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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:

4 14 12 12 4 20; (0 );He C C C He Ne

3( ) ( , ) ( , )

( , ) ( , ); 0( , )

( , ) ( , ); 1

LLI LM I

p n

Ip n

M t d rr Y t

t t It

t t I

r

r rr

r r

Poincare section for isoscalar quadrupole mode in 24Mg

One comment

II. Quantum chaos

Quantum manifestation of classical stochasticity in mixed state.

(comparison of one-well and multi-well)

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SPECTRAL METHODM.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

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2

( , ) ( , ) ( , )2 n n nU x y x y E x y

h

The main instrument of spectral method is correlation function

*0( ) ( , ) ( , , )P t dxdy x y x y t

The solution can be accurately generated with the help of the split operator method

2 2

2 23

( , , ) exp exp ( , )4

exp ( , , )4

tx y t t i i tU x y

ti x y t t

h

h

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2

2

0

0

0

( , , )

( , , ) ( , )exp /

( ) exp /

1( ) exp / ( ) ( )

1( ) ( )exp /

:

1( , ) ( , , ) ( )exp /

n n nn

n nn

T

n n T nn

T

T

T

n n

x y t can be expressed as

x y t a x y iE t

P t a iE t

P E dt iE t P t w t a E ET

E dtw t iEtT

if eigenvalues are known

x y dt x y t w t iE tT

h

h

h

h

h

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( ) ( 18)P E for Hamiltonian of quadrupole oscillation W

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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

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Now we have methods of investigations both classical and quantum chaos, but ….. Do we have a research object?

Once upon a time there lived Dzhu,Who learned to kill off dragons

And gave up all he hadTo master art like that.

Three whole years it took,But, alas, never came up that chance

To present skill and form.

So he took on himself teachingothers the art of slaying dragons.

Chinese legend

The last two lines belong R. Thom

We have a chaotic dragon and even can present some trophy

Chaos vs. regularity

Eternal battle

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O.Bohigas, M.Giannoni, C.Shmit: (1983)

Hypothesis of the universal fluctuations of energy spectra

2

( )

( ) exp( )

( )

( ) exp( )

regular system in classical limit level clusterization

p s s

chaotic system in classical limit level repulsion

p s s s

:

:

Fluctuation properties of QO spectra

13W Rigid lines are Poisson’s prediction

Dashed lines are GOA prediction

Qualitative agreement with Bohigas hypothesis

crE E crE EcrE E

crE E

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Rigid lines are Poisson and Wigner prediction; dashed lines – fitting by Berry-Robnik- Bogomollny distribution (interpolation between Poisson and Wigner distribution)

crE E

Fluctuations of energy spectrum in mixed state

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.

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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?

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We used nonscale version of the Hamiltonian QO

2 2

22 2 2 3 2 21

2 2 3x yp p a

H x y b x y y c x y

13W 3.9W In the interval 0<W<4 for all energies the motion remains regular (in this interval K>0)

Interesting

Classical prompting

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The destruction of shell structure can be traced, using analog of thermodynamic

entropy

2 2

, ,

, ,

lnk kk NLj NLj

N L j

kk NLj

N L j

S C C

C NLj

2R

C

1R

Regular domain:

change of entropy correlates with the transition from shell to shell

Chaotic domain:

1. quasiperiodic dependence of entropy from energy is violated;

2. Monotone growth on average towards a plateau corresponding to entropy of random sequence.

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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

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Quantum chaos and noise1/ f

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

1 1

n n

n i ii i

s s w

Power spectrum of a discrete time series

2 1 2( ) ; expk k n

n

knS k

NN

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The average power spectrum of the function for (sd shell) and (very exotic nucleus) using 25 sets from 256 levels for high level density region. The plots are displaced to avoid overlapping.

n 24Mg34Na

Example of chaotic system is nucleus at high excitation energy

1( )S k

k:

34

24

1.11 0.03

1.06 0.03

Na

Mg

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Power spectrum of the function for GDE (Poisson) energy levels compared to GOE,GUE, GSE (Relano et al. 2002)

n

39

1.87 :1.14 :

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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.

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Yu.L.Bolotin, V.YU.Gonchar….Yud.Fiz. (1995)

, , 1

; ,2Ljk

k NLjN L j

PC NLj NLj NL j N L

Nordholm, Rice (1974) Degree of distribution of wave function arises in the average along with the degree of stochasticity.

Degree of distribution of wave function

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Isolines of probability density2

( , )k x y

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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 integrablenonintegrable, avoided intersection of nodal lines

A.G.Manastra et al. (2003)

Mixed state: QMCS in structure of wave function

QO potential

5QO potential D potential44

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.

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

V.P.Berezovoj, Yu.L.Bolotin, V.A.Cherkaskiy, Phys. Lett A (2004)

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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.

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Optic Billiard

acousto-optic deflectors

laserbeam

verticalhorizontalbilliardplane

KHz 40~)/(1

KH 4 ~/v

2max

0amin

AODaccessscan

scan

Tf

wf

max

scanff max

scanff 10 KHz 100 KHz

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How Do We Observe Chaos in the Wedge ?

Stable trajectoriesdo not “feel” the hole

Chaotic trajectoriesleak through the hole

Cs

450 m m

55 mm 55 m m

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Experiment vs Numerical Simulations

20 25 30 35 40 45 50 550

.4

θ (deg)

3

90o

2

90o

4

90o

.3

.2

.1

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Exponential decay is a common property expected in strongly chaotic systems

(W.Bauer, G.F.Bertch,

1990)

For the chaotic systems exponential decay law

For the nonchaotic systems power decay law

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Numerical experiment on the Sinai billiard

( ) (0)exp( )

,

N t N t

p

Ap the absolute value of momentum

the opening of width

A total coordinate space available

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Once more “mixed state”

crE E crE E: saddleE E 2 saddleE E

5D

Quadrupole

oscillations

At energy higher than saddle energy the phase space structure preserves division on chaotic and regular components

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D5

QO

Decay law for mixed states in the D5 (a) and QO (b) potentials

Solid lines – numerical simulation for E/E(saddle)=1.1,1.5, 2.0.

Dotted and dashed lines – analytical exponential and linear decay law

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The result for different potential are evidently similar and have such characteristic features:

1. Decay law saturates

( )0

( )

( )

" "

ne

ne

N t N

is relative phase valume

of never escaping trajectories

regular trajectories

2 ( ) exp

( )

for t E the decay law has the onential form

as billiard

3. ( )for t E the decay law is linear

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Relative area of stability island

Fraction of non-escaping particles

Rigid correlations

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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.

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1. Quantum decay of the mixed states (current activity)

2. Investigation of dynamical tunneling in 2D multi-well potentials (current activity).

3. Tunneling from super- to normal deformed minima in nuclei (only plan)

Now a few words about our current activity.

Superdeformation in nuclei mixed state

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Chaos Regularity

T.L.Khoo Lecture in Institute of Nuclear Theory (???)

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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

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

Billiard potential (mixed state)

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BBCM

The energy splitting is increased on a lot of orders as chaos increases.

Energy splittingMeasure of chaos

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Dynamical tunneling in QO potential

We plan to realize Bohigas”s billiard problem for multi-well potentials

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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.

Bohigas et al.

What I wanted about, but did not have time to tell

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1. Our analitical results2. Birkhoff-Gustavson normal form (classic and quantum)3. Wave packet dynamics4. Numerical methods (apart “spectral method”)…………………..

Thank you for attention

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grateful acknowledgment to prof. Egle Tomasi for all!