Pavel Stránský Complexity and multidiscipline: new approaches to health 18 April 2012 I NTERPLAY BETWEEN REGULARITY AND CHAOS IN SIMPLE PHYSICAL SYSTEMS European Centre for theoretical studies in nuclear physics and related areas Trento, Italy

Physics of the 1 st kind: CODING Complex behaviour → Simple equations Physics of the 2 nd kind: DECODING Simple 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 vxvx vxvx 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 vxvx vxvx 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 space 4D 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 f reg =0.611 x vxvx 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 chaos Totally regular limits Veins of regularity chaotic regular control parameter Phase transition 1. Classical chaos Highly complex behaviour encoded in a simple equation P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009),

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, (2011)

Wigner P(s)P(s) s Poisson CHAOTIC system REGULAR system Brody distribution parameter  - 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), Quantum chaos - applications 2. Quantum chaos

1/f noise Power spectrum A. Relaño et al., Phys. Rev. Lett. 89, (2002) E. Faleiro et al., Phys. Rev. Lett. 93, (2004) CHAOTIC system  = 1  = 2 Direct comparison of 3 measures of chaos REGULAR system  = 2  = 1  1 = 0 22 33 44  n = 0 kk k - Fourier transformation of the time series constructed from energy levels fluctuations J. M. G. Gómez et al., Phys. Rev. Lett. 94, (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 regular E Integrable chaotic regular B = 0 B = Lattice: lattice always ordered for any operator P partly ordered, partly disordered 2. Quantum chaos energy E i versus the mean value of a (nearly) arbitrary operator P

Increasing perturbation E Peres lattices in GCM <L2><L2> B = 0 B = Integrable Empire of chaos Small perturbation affects only a localized part of the lattice B = 0.05 B = 0.24 Remnants of regularity Peres lattices for two different operators (The place of strong level interaction) P. Stránský, P. Hruška, P. Cejnar, Phys. Rev. E 79 (2009), Quantum chaos Narrow band due to ergodicity

Zoom into the sea of levels Dependence on the classicality parameter E Dependence of the Brody parameter on energy 2. Quantum chaos

Classical and quantum measures - comparison Classical measure Quantum measure (Brody) B = 0.24 B = Quantum chaos

Mixed dynamics A = 0.25 regularity f reg   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

Summary Thank you for your attention 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