Transcript ppt - Pavel Stránský

INTERPLAY BETWEEN REGULARITY AND
CHAOS IN SIMPLE PHYSICAL SYSTEMS
Pavel Stránský
European Centre for
theoretical studies in nuclear
physics and related areas
Trento, Italy
Complexity and multidiscipline: new approaches to health
18 April 2012
Physics of the 1st kind: CODING
Complex behaviour → Simple equations
Physics of the 2nd kind: DECODING
Simple equations → Complex behaviour
Hamiltonian
It describes (for example):
Motion of a star around a galactic centre,
assuming the motion is restricted to a plane
(Hénon-Heiles model)
1. Classical physics
Collective motion of an atomic nucleus
(Bohr model)
2. Quantum physics
1. Classical physics
1. Classical chaos
Trajectories
(solutions of the
equations of motion)
y
x
1. Classical chaos
Trajectories
We plot a point every time
when a trajectory crosses a
given line (y = 0)
(solutions of the
equations of motion)
vx
y
x
vx
chaotic case – “fog”
(hypersensitivity of the motion
on the initial conditions)
Section at
y=0
ordered case – “circles”
x
Poincaré sections
Coexistence of quasiperiodic (ordered) and chaotic types of motion
1. Classical chaos
Trajectories
We plot a point every time
when a trajectory crosses a
given line (y = 0)
(solutions of the
equations of motion)
vx
y
Phase space
x
4D space comprising coordinates and velocities
vx
chaotic case – “fog”
(hypersensitivity of the motion
on the initial conditions)
Section at
y=0
ordered case – “circles”
x
Poincaré sections
Coexistence of quasiperiodic (ordered) and chaotic types of motion
1. Classical chaos
Fraction of regularity
Measure of classical chaos
Surface of the section covered
with regular trajectories
Total kinematically
accessible surface of the
section
vx
REGULAR area
CHAOTIC area
freg=0.611
x
1. Classical chaos
Complete map of classical chaos
chaotic
Phase transition
Totally regular limits
Veins of
regularity
regular
control
parameter
Highly complex behaviour
encoded in a simple equation
P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202
2. Quantum Physics
2. Quantum chaos
Discrete energy spectrum
Spectral density:
smooth part
oscillating part
given by the volume of
the classical phase space
Gutzwiller formula
(the sum of all classical periodic
trajectories and their repetitions)
The oscillating part of the spectral density can give relevant
information about quantum chaos (related to the classical trajectories)
Unfolding:
A transformation of the spectrum
that removes the smooth part of the level density
Note: Improved unfolding procedure using the Empirical Mode Decomposition method in: I. Morales et al., Phys. Rev. E 84, 016203 (2011)
2. Quantum chaos
Spectral statistics
Nearestneighbor
spacing
distribution
P(s)
Poisson
s
REGULAR system
Brody
Wigner
CHAOTIC system
distribution
parameter w
- Artificial interpolation between Poisson and GOE distribution
- Measure of chaoticity of quantum systems
2. Quantum chaos
Quantum chaos - examples
Billiards
They are also
extensively studied
experimentally
Schrödinger equation:
(for wave function)
Helmholtz equation:
(for intensity of el. field)
2. Quantum chaos
Quantum chaos - applications
Riemann z
function:
Prime numbers
Riemann hypothesis:
All points z(s)=0 in the complex plane lie on the line s=½+iy
(except trivial zeros on the real exis s=–2,–4,–6,…)
GUE
Zeros
of z function
2. Quantum chaos
Quantum chaos - applications
GOE
Correlation matrix
of the human EEG signal
P. Šeba, Phys. Rev. Lett. 91 (2003), 198104
2. Quantum chaos
Ubiquitous in the nature (many time signals or space
characteristics of complex systems have 1/f power spectrum)
1/f noise
- Fourier transformation of the time series
constructed from energy levels fluctuations
dn = 0
dk
d4
Power spectrum
d3
k
d2
d1 = 0
a=2
a=2
a=1
CHAOTIC system REGULAR system
Direct comparison of
3 measures of chaos
A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002)
E. Faleiro et al., Phys. Rev. Lett. 93, 244101 (2004)
a=1
J. M. G. Gómez et al., Phys. Rev. Lett. 94, 084101 (2005)
2. Quantum chaos
Peres lattices
A tool for visualising quantum chaos (an analogue of Poincaré sections)
Lattice: energy Ei versus the mean value of a (nearly) arbitrary
operator P
Integrable
lattice always ordered
for any operator P
nonintegrable
B=0
partly ordered,
partly disordered
B = 0.445
<P>
<P>
regular
E
E
regular
chaotic
A. Peres, Phys. Rev. Lett. 53, 1711 (1984)
2. Quantum chaos
Peres lattices in GCM
Small perturbation affects only a localized part of the lattice
(The place of strong level interaction)
B=0
B = 0.005
B = 0.05
B = 0.24
Narrow band due
to ergodicity
<L2>
Peres lattices for two different operators
Remnants of
regularity
<H’>
E
Integrable
Increasing perturbation
Empire of chaos
P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 066201
2. Quantum chaos
Dependence on the classicality parameter
<L2>
Zoom into the sea of levels
E
Dependence of the Brody
parameter on energy
2. Quantum chaos
Classical and quantum
measures - comparison
B = 0.24 Classical measure
B = 1.09
Quantum measure (Brody)
2. Quantum chaos
1/f noise
Mixed dynamics A = 0.25
regularity
a-1
Calculation of a:
Each point –
averaging over 32
successive sets of 64
levels in an energy
window
1-w
freg
E
Appendix. sin exp x
Appendix
Fourier transform
Fourier basis
Signal
Fourier transform calculates an “overlap”
between the signal and a given basis
How to construct a signal with the 1/f noise
power spectrum? (reverse engineering)
1. Interplay of many
basic stationary modes
…
Appendix
2. sin exp x
Features:
• A very simple analytical prescription
• An Intrinsic Mode Function
(one single frequency at any time)
Enjoy the last slide!
Summary
Thank you for
your attention
1. Simple toy models can serve as a theoretical
laboratory useful to understand and master
complex behaviour.
2. Methods of classical and quantum chaos can be
applied to study more sophisticated models or to
analyze signals that even originate in different
sciences
http://www-ucjf.troja.mff.cuni.cz/~geometric
http://www.pavelstransky.cz