Post on 19-Jan-2016
Pavel Stránský1,2
3rd Chaotic Modeling and Simulation International Conference, Chania, Greece, 2010
3rd January 2010
1Institute of Particle and Nuclear PhycicsFaculty of Mathematics and PhysicsCharles University in Prague, Czech Republic
MANIFESTATION OF CHAOS IN COLLECTIVE MODELS OF
NUCLEI
2Instituto de Ciencias NuclearesUniversidad Nacional Autonoma de México
Collaborators:
Michal Macek1, Pavel Cejnar1
Alejandro Frank2, Ruben Fossion2, Emmanuel Landa2
1. Model- Geometric Collective Model of nuclei (GCM) (restricted to pure vibrations)
2. Classical chaos in GCM- Measures of regularity- Geometrical method
3. Quantum chaos in GCM- Short-range correlations and Brody parameter- Peres lattices- Long-range correlations and 1/f noise- Comparison of classical and quantum dynamics
MANIFESTATION OF CHAOS IN COLLECTIVE MODELS OF
NUCLEI
1. Geometrical Collective Model of nuclei
(restricted to pure vibrations)
T…Kinetic term V…Potential
Hamiltonian
Neglect higher order terms
neglect
Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)
Corresponding tensor of momenta
1. Geometric Collective Model of nuclei
Surface of homogeneous nuclear matter:
Quadrupole deformations = 2
G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)
4 external parameters
T…Kinetic term V…Potential
Hamiltonian
Neglect higher order terms
neglect
Quadrupole tensor of collective coordinates (2 shape parameters, 3 Euler angles)
Corresponding tensor of momenta
1. Geometric Collective Model of nuclei
Surface of homogeneous nuclear matter:
Quadrupole deformation = 2
Scaling properties
4 external parametersAdjusting 3 independent scalesenergy
(Hamiltonian)
1 “shape” parameter
size (deformation)
time
1 “classicality” parametersets absolute density of quantum spectrum (irrelevant in classical case)
G. Gneuss, U. Mosel, W. Greiner, Phys. Lett. 30B, 397 (1969)
Principal Axes System (PAS)
Shape variables:
1. Geometric Collective Model of nuclei
Shape-phase structure
Deformed shape Spherical shape
VV
B
A
C=1
Nonrotating case J = 0!
(a) 5D system restricted to 2D (true geometric model
of nuclei)
(b) 2D system
2 physically important quantization options(with the same classical limit):
Classical dynamics– Hamilton equations of motion
• oportunity to test Bohigas conjecture for different quantization schemes
Quantization– Diagonalization in oscillator basis
Principal Axes System
1. Geometric Collective Model of nuclei
2. Classical chaos in GCM
Fraction of regularity
REGULAR area
CHAOTIC area
freg=0.611
vx
x
2. Classical chaos in GCM
A = -1, C = K = 1B = 0.445
Measure of classical chaos
Poincaré section
Different definitons & comparison
Surface of the chosen Poincaré section
regular
totalnumber of
trajectories (with random initial conditions)
control parameter
E = 0
Statistical measure
2. Classical chaos in GCM
Complete map of classical chaos in GCM IntegrabilityIntegrability
Veins ofVeins of regularityregularity
chaotichaoticc
regularegularr
control parameter
““ Arc
of
Arc
of
regula
rity
”re
gula
rity
”
Global minimum and saddle pointRegion of phase transition
Sh
ap
e-p
hase
Sh
ap
e-p
hase
tr
ansi
tion
transi
tion
2. Classical chaos in GCM
Geometrical method
L. Horwitz et al., Phys. Rev. Lett. 98, 234301 (2007)
Hamiltonian in flat Eucleidian space with potential:
Hamiltonian of free particle in curved space:
Conformal metric
Application of methods of Riemannian geometry
inside kinematically accesible area induce nonstability.
Negative eigenvalues of the matrix
2. Classical chaos in GCM
Geometrical criterion= Convex-Concave transition
Global minimum and saddle point
Region of phase transition
Geometrical method- gives good estimation of regularity-chaos transition
2. Classical chaos in GCM
y
x
(d)
(c)
(b)
(a)
(b)
(c)
(d)
(a)
1. Classical chaos in GCM
Geometrical method
Geometrical criterion= Convex-Concave transition
Global minimum and saddle point
Region of phase transition
- gives good estimation of regularity-chaos transition
3. Quantum chaos in GCM
Spectral statistics
GOE
P(s)
s
Poisson
CHAOTIC systemREGULAR system
Nearest-neighbor spacing distribution
Bohigas conjecture (O. Bohigas, M. J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984), 1)
Brodydistributionparameter
- Tool to test classical-quantum correspondence
- Measure of chaoticity of quantum systems- Artificial interpolation between Poisson and GOE distribution
3. Quantum chaos in GCM
Peres lattices Quantum system:
A. Peres, Phys. Rev. Lett. 53 (1984), 1711
Infinite number of of integrals of motion can be constructed (time-averaged operators P):
nonintegrable
E
<P>
regular
E
Integrable
<P>
chaoticregular
B = 0 B = 0.445
Lattice: energy Ei versus value of
lattice always ordered for any operator P
partly ordered, partly disordered
3. Quantum chaos in GCM
Principal Axes System
Nonrotating case J = 0!
(a) 5D system restricted to 2D (true geometric model
of nuclei)
(b) 2D system
IndependentPeres operators in
GCM
H’
L22DL2
5D
Hamiltonian of GCM
3. Quantum chaos in GCM
Increasing perturbation
E
Nonintegrable perturbation
<L2>
B = 0 B = 0.005
<H’>
Integrable Empire of chaos
Small perturbation affects only localized part of the lattice
B = 0.05 B = 0.24
Remnants ofregularity
3. Quantum chaos in GCM
Island of high regularity B = 0.62
<L2>
2D
<VB>
5D
(different quantizations)
E
• – vibrations resonance
3. Quantum chaos in GCM
Zoom into sea of levels
Dependence on the classicality parameter
E
<L2>
3. Quantum chaos in GCM
Selected squared wave functions:
E
Peres operators & Wavefunctions
<L2>
<VB>
2D
Peres invariant classically
Poincaré sectionE = 0.2
3. Quantum chaos in GCM
Classical and quantum measure - comparison Classical
measure
Quantum measure (Brody)
B = 0.24 B = 1.09
3. Quantum chaos in GCM
1/f noise
Power spectrum
2. Quantum chaos in GCM
A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002)
CHAOTIC system = 1 = 2
- Direct comparison of
REGULAR system
= 2
= 1
1 = 0
2
3
4
n = 0 k
k
- Fourier transformation of the time series
Integrable case: = 2 expected
3.0 - 1.92x
6.0 - 1.93x
Shortest periodic classical orbit
Universal region
(averaged over 4 successive sets of 8192 levels, starting from level 8000)
(512 successive sets of 64 levels)
2.0 - 1.94x
log<S>
log f
1/f noise
3. Quantum chaos in GCM
Mixed dynamics A = 0.25
reg
ula
rity
freg
- 11 -
E
Calculation of :Each point –
averaging over 32 successive sets of
64 levels in an energy window
1/f noise
3. Quantum chaos in GCM
Summary
1. Geometrical Collective Model of nuclei • Complex behavior encoded in simple equations• Possibility of studying manifestations of both
classical and quantum chaos and their relation
2. Peres lattices• Allow visualising quantum chaos• Capable of distinguishing between chaotic
and regular parts of the spectra• Freedom in choosing Peres operator
3. Methods of Riemannian geometry• Approximate location of the onset of
chaoticity in classical systems
4. 1/f Noise• Effective method to introduce measure of
chaos using long-range correlations in quantum spectra
5. Other models studied• Interacting boson model, Double
pendulum
Thank you for your attention
http://www-ucjf.troja.mff.cuni.cz/~geometric
~stransky
This is the last slide
Appendix. Double pendulum
3. Chaos in IBM
Angular momenta
Quantization:
Peres operators:
Ambiguous procedure
(noncommuting in the kintetic term)
Hamiltonian
Double pendulum
freg - Double pendulum
(a) E = 1
(b) E = 5
(c) E = 14
(c)
(a)
(b)
Double pendulum - results
= l = = 1
Double pendulum in ISS • No gravity• Integrable case• m = l = 1
Libration
Rotationin
distinguishing different classes of motion
Peres lattices
Double pendulum - results
Introducing gravity = 0 = 1
Chaotic band
Double pendulum - results
Classical-Quantum Correspondence
(a) E = 1
(b) E = 5
(c) E = 14
(c)
(a)(b)
Harm
onic
ap
pro
xim
ati
on
Em
pir
e o
f ch
aos
Pre
vale
nce
of
rota
tions
reg
ula
rity
freg
- 1
1 -
IBM Hamiltonian
3 different dynamical symmetries
U(5)SU(3)
O(6)
0 0
1
Casten triangle
a – scaling parameter
Invariant of O(5) (seniority)
3. Chaos in IBM
3 different dynamical symmetries
U(5)SU(3)
O(6)
0 0
1
Casten triangle
Invariant of O(5) (seniority)
a – scaling parameter
3 different Peres
operators
3. Chaos in IBM
IBM Hamiltonian
Regular lattices in integrable case
3ˆ.ˆ SUQQ
dn̂v
- even the operators non-commuting with Casimirs of U(5) create regular lattices !
40
-40
-2020
10
30 -10
-30
0
-40
-20
-10
-30
0
0
3ˆ.ˆ SUQQ
6ˆ.ˆ OQQ
dn̂
v
L = 0
commuting non-commuting
U(5)
limit
N = 40
3. Chaos in IBM
Different invariants
= 0.5
N = 40
U(5)
SU(3)
O(5)
Arc of regularityArc of regularity
classical regularity
3. Chaos in IBM
Different invariants
= 0.5
N = 40
U(5)
SU(3)
O(5)
Arc of regularityArc of regularity
classical regularity
3. Chaos in IBM
GOE<L2>
Application: Rotational bands
dn̂
N = 30L = 0
η = 0.5, χ= -1.04 (arc of regularity)
3ˆ.ˆ SUQQdn̂
3. Chaos in IBM
N = 30L = 0,2
η = 0.5, χ= -1.04 (arc of regularity)
3ˆ.ˆ SUQQdn̂
3. Chaos in IBM
Application: Rotational bands
Application: Rotational bands
N = 30L = 0,2,4
η = 0.5, χ= -1.04 (arc of regularity)
3ˆ.ˆ SUQQdn̂
3. Chaos in IBM
3ˆ.ˆ SUQQ
N = 30L = 0,2,4,6
η = 0.5, χ= -1.04 (arc of regularity)
dn̂
3. Chaos in IBM
Application: Rotational bands
How to distinguish quasiperiodic and unstable trajectories
numerically?1. Lyapunov
exponent
Divergence of two neighboring trajectories
2. SALI (Smaller Alignment Index)
• fast convergence towards zero for chaotic trajectories
Ch. Skokos, J. Phys. A: Math. Gen 34 (2001), 10029; 37 (2004), 6269
• two divergencies
1. Classical chaos in GCM
Wave functions<L2>
E
<VB>
Probability densities
regular regularchaotic
2. Quantum chaos in GCM
Wave functions and Peres lattice
convex → concave (regular → chaotic)
E
E
OT
Probability density of wave
functions
Peres lattice
B = 1.09
2. Quantum chaos in GCM
Long-range correlations
• number variace
• 3 („spectral rigidity“)
• Short-range correlations – nearest neighbor spacing distribution
Only 1 realization of the ensemble in GCM – averaging impossibleChaoticity of the system changes with energy – nontrivial dependence on both L and E
2. Quantum chaos in GCM