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12th CRETE REGIONAL MEETING �IN�STRING THEORY��ORTHODOX ACADEMY OF CRETE 4-10 JULY 2022 ����CHAOTIC LATTICE FIELD THEORIES��EMMANUEL FLORATOS�� PHYSICS DEPT- UoA & INPP-DEMOKRITOS

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PLAN OF THE SEMINAR

  • Introduction to Chaotic Field Theories

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  • Our Motivation

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  • The simplest chaotic unit: The Arnold’s cat map

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  • Symplectic Coupling of n- Fibonacci sequences

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  • D-Dimensional Arnold’s cat map Lattices, Tunable Locality

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  • Exact results for D=1 periodic Lattice,All Orbits,

Liapunov spectra, KS Entropy

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  • Conclusions ,Open Questions

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TO APPEAR SOON IN THE ARXHIVES

Arnol’d cat map lattices

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Minos Axenides(INPP DEMOKRITUS), Emmanuel Floratos (Physics UoA)and Stam Nicolis(UoTours)

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ABSTRACT

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We construct Arnol’d cat map lattice field theories (ACML) with linear symplectic

interactions, of tunable locality in one or higher dimensions.

They provide examples of lattice field theories for interacting many-body deterministically chaotic oscillators.

We study the classical spatio-temporal chaotic properties of these systems by using standard benchmarks for probing deterministic chaos of dynamical systems,

namely the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing.

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In the case of closed chains, with translational invariant couplings of tunable locality,we determine analytically the Liapunov spectra and the K-S entropy, as functions of the strength and the range of the interactions.

The Kolmogorov–Sinai entropy is found to scale as the volume of the system.

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We provide methods to determine the spectrum of the periods of the unstable periodic

orbits of the dynamical system and we observe that it exhibits a strong dependence on

the strength and the range of the interaction

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The above construction provides, exactly solvable, multidimensional chaotic dynamical

systems with tunable Kolmogorov -Sinai entropy and mixing times.

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  • Quantum cat map dynamics on AdS2

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Minos Axenides , Emmanuel Floratos , Stam Nicolis

Eur.Phys.J.C.(2018)78:412

arXiv:1608.07845

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  • The arithmetic geometry of AdS2 and its continuum limit

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Minos Axenides,Emmanuel Floratos

Stam Nicolis

SIGMA 17 (2021), 004

arXiv:1908.06691

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  • Arithmetic Circuits for Multilevel Qudits� Based on Quantum Fourier Transform�� Archimedes Pavlidis,Emmanuel Floratos�� Phys.Rev.A 103,032417(2021)� arXiv:1707.08834

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INTRODUCTION TO CHAOTIC LATTICE FIELD THEORIES

  • CHAOTIC LATTICE FIELD THEORIES ARE FIELD THEORIES OF INTERACTING CHAOTIC UNITS ,ONE AT EACH SITE OF A LATTICE ,WITH LOCAL OR NONLOCAL INTERACTIONS

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  • CHAOTIC UNITS ARE DYNAMICAL SYSTEMS WHICH LIVE ON A COMPACT PHASE SPACE AND WHICH ARE ERGODIC AND MIXING

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  • AS SUCH THEY ARE APPROPRIATE TO DESCRIBE NEW MANY BODY DYNAMICAL SYSTEMS WITH FASTER THERMALIZATION PROPERTIES AND FASTER MIXING OF INFORMATION

THAN THE CONVENTIONAL FIELD THEORIES

WHICH DESCRIBE HARMONIC OSCILLATORS WITH LOCAL OR NONLOCAL

INTERACTIONS.

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  • CHAOTIC FIELD THEORIES APPEARED IN THE 80’s UNDER THE NAME

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OF COUPLED MAP LATTICES (KANEKO 1984) TO STUDY THE

UNIVERSALITY CLASSES OF SPATIOTEMPORAL CHAOTIC BEHAVIOUR OF

COUPLED CHAOTIC SYSTEMS

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  • CRUTCHFIELD, JAMES,P. AND KANEKO, KUNIHIKO (1987). PHENOMENOLOGY OF SPATIOTEMPORAL CHAOS (IN DIRECTIONS IN CHAOS, ED HAO BAI LIN) WORLD SCIENTIFIC, SINGAPORE.

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  • IN THE 90’s AND RECENTLY 2018-2020 THEY WERE PROPOSED BY CVITANOVIC TO

STUDY THE COHERENT STRUCTURES OF WEAK TURBULENCE

USING AS CHAOTIC UNITS LINEAR HYPERBOLIC MAPS ON

A TOROIDAL PHASE SPACE

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  • CHAOTIC FIELD THEORIES EXHIBIT A RICH SPECTRUM OF SPATIOTEMPORAL PATTERNS AND RECENTLY HAVE BEEN USED TO STUDY A UNIVERSALITY CLASS OF SPECIAL STATES OF THESE SYSTEMS

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  • THE CHIMAIRA STATES

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  • A SUBSET OF OSCILLATORS HAVE A COHERENT DYNAMICAL BEHAVIOUR

WHILE ALL THE OTHERS ARE RANDOM

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  • APPLICATIONS TO BOSE LOCALIZATION IN RANDOM MATERIALS
  • MEMORY EFFECTS
  • COGNITIVE PROCESSES

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�ARNOLD’S CAT MAP LATTICES��WORK TO APPEAR SOON IN ARCHIVES IN COLLABORATION WITH ��MINOS AXENIDES (INPP DEMOKRITUS) AND STAM NICOLIS (TOURS UNIV)�

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  • OUR MOTIVATION IS THE STUDY OF FAST THERMALIZATION PROPERTIES OF CLOSED CLASSICAL AND QUANTUM SYSTEMS -

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  • SREDNICKI 1994- EIGENSTATE THERMALIZATION HYPOTHESIS (ETH)

IF A CLOSED QUANTUM SYSTEM IS SUFFICIENTLY RANDOM (CHAOTIC)

ANY OF ITS SUBSYSTEMS WILL THERMALIZE EVENTUALLY TO THE THERMAL

DENSITY MATRIX WITH TEMPERATURE DEFINED BY THE TOTAL ENERGY OF THE

SYSTEM.

  • WE ARE INTERESTED FOR APPLICATIONS TO THE DYNAMICS OF BLACK HOLE HORIZONS
  • -FAST SCRAMBLING : HAYDEN PRESKILL (2007) -SEKINO SUSSKIND(2008)
  • NO-CLONING OF QUANTUM INFO (INSIDE-OUTSIDE BH OBSERVERS)

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  • OUR IDEA

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  • TO ACHIVE THE FASTEST POSSIBLE THERMALIZATION

WE CHOOSE THE CHAOTIC UNITS OF THE CHAOTIC CLOSED AND FINITE

SYSTEM TO BE MUST BE MAXIMALY CHAOTIC AND MIXING

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  • IN SYK MODELS FERMIONS(REPULSIVE DENSITY -DENSITY 4 FERMION

NON LOCAL RANDOM INTERACTION)

ALL INGREDIENTS ARE PUT BY HAND

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  • ACM1

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  • ACM2

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  • ACM3

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  • ACM4

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  • ACM5

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  • ACM6

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  • ACM7

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  • ACM8

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  • ACM9

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  • ACM10

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  • ACM11

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  • ACM12

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  • ACM13

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LIAPUNOV SPECTRA –MODE NUMBER��CONSTANT,INCREASING,DECREASING COUPLINGS

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SORTED LIAPUNOV SPECTRA ��CONSTANT,INCREASING,DECREASIN COUPLINGS

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THANK YOU!