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Transcript of Pavel Stránský Complexity and multidiscipline: new approaches to health 18 April 2012 I NTERPLAY...
Pavel Stránský
Complexity and multidiscipline: new approaches to health
18 April 2012
INTERPLAY BETWEEN REGULARITY AND CHAOS IN SIMPLE PHYSICAL
SYSTEMSEuropean Centre for theoretical studies in
nuclear physics and related areas
Trento, Italy
Physics of the 1st kind: CODINGComplex behaviour → Simple equations
Physics of the 2nd kind: DECODINGSimple equations → Complex behaviour
1. Classical physics 2. Quantum physics
Hamiltonian
It describes (for example):Motion of a star around a galactic
centre, assuming the motion is restricted to a plane
(Hénon-Heiles model)
Collective motion of an atomic nucleus(Bohr model)
1. Classical physics
y
x
Trajectories
1. Classical chaos
(solutions of the equations of motion)
y
x
vx
vx
Section at
y = 0
x
ordered case – “circles”
chaotic case – “fog”
(hypersensitivity of the motion on the initial
conditions)
We plot a point every time when a trajectory crosses a given line (y = 0)
Trajectories
1. Classical chaos
Coexistence of quasiperiodic (ordered) and chaotic types of motion
Poincaré sections
(solutions of the equations of motion)
y
x
vx
vx
Section at
y = 0
x
ordered case – “circles”
chaotic case – “fog”
(hypersensitivity of the motion on the initial
conditions)
Trajectories
1. Classical chaos
Poincaré sections
Phase space4D space comprising coordinates and
velocities
(solutions of the equations of motion)
We plot a point every time when a trajectory crosses a given line (y = 0)
Coexistence of quasiperiodic (ordered) and chaotic types of motion
REGULAR area
CHAOTIC area
freg=0.611 x
vx
Fraction of regularity
Measure of classical chaos
Surface of the section covered with regular trajectories
Total kinematically accessible surface of the section
1. Classical chaos
Complete map of classical chaosTotally regular limitsTotally regular limits
Veins ofVeins of regularityregularity
chaotichaoticc
regularegularr
control parameter
Phase
tra
nsi
tion
Phase
tra
nsi
tion
1. Classical chaos
Highly complex behaviour encoded in a simple
equation
P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 046202
2. Quantum Physics
Discrete energy spectrum
2. Quantum chaos
Spectral density:
smooth part
given by the volume of the classical phase space
oscillating part
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)
Wigner
P(s)
s
Poisson
CHAOTIC systemREGULAR system
Brodydistributionparameter
- Measure of chaoticity of quantum systems- Artificial interpolation between Poisson and GOE distribution
Spectral statistics
Nearest-neighbor spacing distribution
2. Quantum chaos
Schrödinger equation:(for wave function)
Helmholtz equation:(for intensity of el. field)
Quantum chaos - examples
2. Quantum chaos
They are also extensively studied
experimentally
Billiards
Riemann 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 function
Quantum chaos - applications
2. Quantum chaos
GOE
Correlation matrix of the human EEG signal
P. Šeba, Phys. Rev. Lett. 91 (2003), 198104
Quantum chaos - applications
2. Quantum chaos
1/f noise
Power spectrum
A. Relaño et al., Phys. Rev. Lett. 89, 244102 (2002)E. Faleiro et al., Phys. Rev. Lett. 93, 244101 (2004)
CHAOTIC system = 1 = 2
Direct comparison of 3 measures of chaos
REGULAR system
= 2
= 1
1 = 0
2
3
4
n = 0k
k
- Fourier transformation of the time series constructed from energy levels fluctuations
J. M. G. Gómez et al., Phys. Rev. Lett. 94, 084101 (2005)
Ubiquitous in the nature (many time signals or space characteristics of complex systems have 1/f power spectrum)
2. Quantum chaos
Peres lattices
A. Peres, Phys. Rev. Lett. 53, 1711 (1984)
A tool for visualising quantum chaos (an analogue of Poincaré sections)
nonintegrable
E
<P>
regular
E
Integrable
<P>
chaoticregular
B = 0 B = 0.445
Lattice:
lattice always ordered for any operator P
partly ordered, partly disordered
2. Quantum chaos
energy Ei versus the mean value of a (nearly) arbitrary operator P
Increasing perturbation
E
Peres lattices in GCM
<L2>
B = 0 B = 0.005
<H’>
Integrable Empire of chaos
Small perturbation affects only a localized part of the lattice
B = 0.05 B = 0.24
Remnants ofregularity
Peres lattices for two different operators
(The place of strong level interaction)
P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), 066201
2. Quantum chaos
Narrow band due to ergodicity
Zoom into the sea of levels
Dependence on the classicality parameter
E
<L2>
Dependence of the Brody parameter on energy
2. Quantum chaos
Classical and quantum measures - comparison Classical
measure
Quantum measure (Brody)
B = 0.24 B = 1.09
2. Quantum chaos
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
2. Quantum chaos
Appendix. sin exp x
Fourier basis
…
Signal
Fourier transform
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)
Appendix
1. Interplay of many basic stationary modes
2. sin exp x
Features:• A very simple analytical prescription• An Intrinsic Mode Function (one single frequency at any time)
Appendix
SummaryThank you
for your attention
http://www-ucjf.troja.mff.cuni.cz/~geometric
http://www.pavelstransky.cz
Enjoy the last slide!
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