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