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Symmetry vs. Chaos * in nuclear collective dynamics signatures & consequences

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Symmetry vs. Chaos *

in nuclear collective dynamics

signatures & consequences

Pavel Cejnar

Institute of Particle and Nuclear Physics

Faculty of Mathematics and Physics

Charles University, Prague, Czech Republic

* According to Empedocles (cca.490-430 BC), the real world, Cosmos, is an interference of Sphairos, an exquisite world of perfect order originating in symmetry, and Chaos, a world of complete disorder which results from a lack of symmetry.

Kazimierz 2010

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Symmetry

- Dynamical symmetry
- symmetry of a particular system with respect to a dynamical group – a higher group than the one following from the invariance requirements
- invariant symmetry with respect to the dynamical group can be broken, but a number of motion integrals remains preserved
- dynamical symmetry integrability
- perfect order

- Invariant symmetry
- commutation of Hamiltonian with generators of a certain group (group of invariant symmetry)
- conservation laws

Breakdown of dynamical symmetry

Signatures of symmetry do not vanish instantly:

Quasi dynamical symmetry (Rowe… 1988…)

The actual dynamical symmetry of the system is broken but the system partly behaves as if the symmetry were effectively preserved.

Partial dynamical symmetry (Leviatan… 1992…)

All conserved quantum numbers preserved for a part of states or a part of conserved quantum numbers preserved for all states.

- Breakdown of integrability
- Regular character of classical motions is preserved for some orbits: KAM theorem (Kolmogorov, Arnold, Moser 1954-63)
- “quasi integrability”
- “partial regularity”

All kinds of dynamical symmetry relevant in physics of nuclear collective motions. These motions are therefore mostly thought to be regular...

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Chaos

- Quantum chaos
- no genuinely quantum definition of chaos (linearity & quasi periodicity of quantum mechanics)
- chaos studied in connection with classical limit
- Bohigas conjecture (1984): Chaos on quantum level affects statistical properties of discrete energy spectra. Chaotic systems yield spectral correlations consistent with Gaussian random matrix model.

- Classical chaos
- exponential sensitivity to initial conditions (“butterfly wing effect”)
- practical loss of predictability
- quasi ergodic trajectories in the phase space

- Nuclei show neat signatures of quantum chaos!
- data from neutron and proton resonances: Bohigas, Haq, Pandey (1983)
- ensemble of low-energy levels: Von Egidyet al. (1987)

(Wigner)

Nearest-neighbor spacing

Brody distribution interpolates between

Poisson (ω=0) … order

Wigner(ω=1) … chaos

ω=0.62

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Origin of chaos in atomic nuclei

Many-body dynamics

Complex interactions of all particles in the nucleus

too difficult => two complementary simplifications:

?

1) Single-particle dynamics

Nucleonic motions in deformed nuclear potentials

- Arvieu, Brut, Carbonell, Touchard: PRA 35, 2389 (1987)
- Rozmej, Arvieu: NPA 545, C497 (1992)
- Heiss, Nazmitdinov, Radu: PRL 72, 2351 (1994); PRL 73, 1235 (1994); PRC 52, 3032 (1995)

2) Collective dynamics

Nuclear vibrations and rotations

- a) Interacting Boson Model (IBM)Iachello, Arima1975
- Alhassid, Whelan, Novoselsky:PRL 65, 2971 (1990); PRC 43, 2637 (1991); PRC 45, 1677 (1992); PRL 67, 816 (1991); NPA 556, 42 (1993)
- Paar, Vorkapic, Dieperink:PLB 205, 7 (1988); PRC 41, 2397 (1990), PRL 69, 2184 (1992)
- Mizusaki et al.:PLB 269, 6 (1991)
- Canetta, Maino:PLB 483, 55 (2000)
- Cejnar, Jolie, Macek, Casten, Dobeš, Stránský:PLB 420, 241 (1998); PRE 58, 387 (1998); PRL 93, 132501 (2004); PRC 75, 064318 (2007), PRC 80, 014319(2009),PRC 82, 014308 (2010), PRL 105, 072503(2010)
- b) Geometric Collective Model (GCM) Bohr1952
- Cejnar, Stránský, Kurian, Hruška:PRL 93, 102502 (2004); PRC 74, 014306 (2006); PRE 79, 046202 (2009), PRE 79, 066201(2009), JP Conf.Ser. 239, 012002(2010)

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A. Bohr 1952

Gneusset al. 1969

Geometric Collective Model

quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D )

…corresponding tensor of momenta

Hamiltonian

neglect …

Angular momentum → 0

- effectively 2D system

y

x

Shape variables

Principal Axes System

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A. Bohr 1952

Gneuss et al. 1969

Geometric Collective Model

quadrupole tensor of collective coordinates ( 2 shape + 3 Euler angles = 5D )

…corresponding tensor of momenta

Hamiltonian

neglect higher-order terms

neglect …

Shape-phase diagram

prolate

spherical

B

C >0

y

oblate

A

x

Shape variables

Principal Axes System

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

independent scales

energy

time

coordinates

… but a fixed value of Planck constant

Hamiltonian

neglect …

External parameters

Two essential parameters

1) “shape” parameter

2) “classicality” parameter

prolate

spherical

B

C >0

oblate

A

integrability

- path crossing all parabolas
- all equivalent classes of Hamiltonians

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

Classical chaos

map ofthedegreeof chaos

Regular phase space fraction

chaos

order

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

Classical chaos

map ofthedegreeof chaos

change of the shape of the border of the accessible domain in the xy plane

convex

concave

y

y

x

x

Regular phase space fraction

concave

convex

y

y

chaos

order

x

x

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

Classical chaos

J = 0, E = 0, A = −5.05, B = C = K = 1

examples of trajectories and a Poincaré section

Poincaré section

y

x

50,000 passages of 52 randomly chosen trajectories through

the section y=0

x

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

J=0

Quantum chaos

classicality parameter

Two quantization options

(b) 5D system restricted to 2D

(true geometric model of nuclei)

(a) 2D system

The 2 options differ also in the metric (measure) for calculating matrix elements.

► Possibility to test Bohigas conjecture

in different quantization schemes.

with additional constraints

(to avoid quasi-degeneracies due to the symmetry of V)

even/odd

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

Quantum chaos

comparison of classical and quantal measures

freg … classical regular fraction

1−ω … adjunct of Brody parameter

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

Quantum chaos

1) <L2>

2) <H’>

departure from integrable regime

Visualmethod

by A. Peres (1984)

ChoiceofP

1) Quasi-2D angularmomentum

2) Hamiltonian perturbation

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

Quantum chaos

complex mixture of regular and chaotic patterns: ordered vs. chaotic states

B=0.62

1) <L2>

2) <H’>

Visualmethod

by A. Peres (1984)

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Stránský, Cejnar…

2004 ……. 2010

Geometric Collective Model

Quantum chaos

complex mixture of regular and chaotic patterns: ordered vs. chaotic states

B=0.62

1) <L2>

2) <H’>

Visualmethod

by A. Peres (1984)

■

♦

●

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Macek, Cejnar, Jolie… 2007

Interacting Boson Model

More than 1 classical control parameter

But there exist regions of almost full compatibility

- multi-dimensional chaotic map.

with the GCM.

integrable

regime

J = 0, E = 0

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Macek, Dobeš, Cejnar… 2010

Interacting Boson Model

Consequences of regularity for the adiabatic separation of intrinsic and collective motions: rotational bandsexist even at very high excitation energies if the corresponding region of the J =0 spectrum is regular.

intrinsic wave functions for various band members in the SU(3) basis

- Selection of hypothetical bands of rotational states based on the maximal correlation of the intrinsic SU(3) structures.

N = 30

0

0 15 30 J

- Product of 0+-2+ and 0+-4+ correlation coefficients for intrinsic wave functions in the given “band”

- Energy ratio for 4+ and 2+ states in a given “band”

- Classical regular fraction in the respective energy region

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- Conclusions
- Nuclear collective motions exhibit an intricate interplay or regular and chaotic features (despite the presumption that collective = regular).
- Models of collective motions may serve as a laboratory for general studies of chaos (profit from the coexistence of simplicity and complexity).
- Order/Chaos have relevant nuclear-structure consequences (e.g. for the adiabatic separation of collective and intrinsic motions etc.).
- Thanks to PavelStránský(now UNAM Mexico)
- Michal Macek (soonUni Jerusalem)

GCMJ = 0, E = 0

integrable

regime

Appendices

Dependence

on energy

J=0, A=−0.84, B=C=K=1

GCM

GCM

Peres lattices

A visual method to study quantum chaos in 2D systems due to Asher Peres,

PRL 53, 1711 (1984)

► In any system there exists infinite number of integrals of motions:

e.g. time averages of an arbitrary quantity along individual orbits

(note: in fully or partly chaotic systems, these integrals do not allow one to build

action-angle variables since they are strongly nonanalytic)

► Quantum counterparts of such observables can be found:

(1934-2005)

► One can construct a lattice: energy versus value of

► In a fully regular (integrable) system, the lattice is always ordered

(the new integral of motion is constant on tori => it is a function of actions)

► In a chaotic system, the lattice is disordered

► In a mixed regular & chaotic system, the lattice is partly ordered & disordered

regular

mixed

chaotic

2D even

GCM

Relation to the regular fraction

B=1.09

classicality parameter