DOCSLIB.ORG

Black Holes and the Butterfly Effect A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Black Holes and the Butterfly Effect A BLACK HOLES AND THE BUTTERFLY EFFECT A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Douglas Stanford July 2014 © 2014 by Douglas Stanford. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/xc388jb9020 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Leonard Susskind, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Patrick Hayden I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Stephen Shenker Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract What happens if you perturb a small part of a large system, and then you wait a while? If the system is chaotic, one expects the butterfly effect to push the state far from its original trajectory. I will present an analysis of this phenomenon in the setting of a strongly interacting quantum gauge theory, using the tools of gauge-gravity duality. The original state corresponds to a black hole geometry, and the perturbation is represented by a particle falling through the horizon. As time passes, the boost of the particle grows exponentially, creating a shock wave that implements the butterfly effect. Building on this framework, I will relate and explore the dynamics of chaos and the region behind the horizon of a black hole. This thesis is based on the papers [1] and [2], written with Stephen Shenker. It should not be cited without also referencing those papers. iv Acknowledgments I am deeply grateful to both Lenny Susskind and Steve Shenker for sharing some of their intuitions, tools, and enthusiasm for physics. I started learning from Lenny during my first class as a college freshman, and I can't see the top yet. I am also grateful to my other collaborators, including Daniel Harlow, Patrick Hayden, Nima Laskhari, Dan Roberts, Ahmed Almheiri, Don Marolf, Joe Polchinski, and Jamie Sully. My work was supported by the Stanford Institute for Theoretical Physics, by the NSF Graduate Research Fellowship Program, and by a Graduate Fellowship from the Kavli Institute for Theoretical Physics. v Contents Abstract iv Acknowledgments v 1 Introduction 1 2 Black holes and the butterfly effect 4 2.1 Introduction . 4 2.2 A qubit model . 7 2.3 A holographic model . 9 2.3.1 Unperturbed BTZ . 9 2.3.2 BTZ shock waves . 10 2.3.3 Geodesics . 12 2.3.4 Mutual information . 13 2.3.5 Correlation functions . 15 2.4 String and Planck scale effects . 16 2.5 Discussion . 18 2.6 Appendix A: Haar scrambling . 21 2.7 Appendix B: Geometrical generalizations . 23 2.7.1 Higher dimensions . 23 2.7.2 Solutions with localized sources . 24 3 Multiple Shocks 25 3.1 Introduction . 25 3.2 Wormholes built from shock waves . 28 3.2.1 One shock . 28 3.2.2 Two shocks . 29 3.2.3 Many shocks . 32 3.3 Ensembles . 36 vi 3.4 Discussion . 40 3.5 Appendix A: Recursion relations for many shock waves . 41 3.6 Appendix B: Vaidya matching conditions . 42 Bibliography 44 vii List of Figures 2.1 Mutual information (upper, blue) and spin-spin correlation function (lower, red) in 0 the perturbed state jΨ i, as a function of the time of the perturbation tw. The delay is a propagation effect; if the perturbation at site five is sufficiently recent, sites one and two are unaffected. 8 2.2 The Kruskal diagram (center) and Penrose diagram (right) for the BTZ geometry. 10 2.3 The Kruskal and Penrose diagrams for the geometry with a shock wave from the left, represented by the double line. The dashed v = 0 andv ~ = 0 horizons miss by an amount α........................................... 12 2.4 In the unperturbed BTZ geometry (left), a smooth horizon requires the black mode on the left to be highly entangled with the blue mode on the right. By contrast, in the shock wave geometry (right) the black and blue modes are far apart and unentangled. Instead, the black mode is entangled with the green mode coming out of the white hole. The arguments of [65] suggest that the green mode may be complicated in the CFT.............................................. 20 3.1 The geometry dual to Eq. (3.4One shockequation.3.2.4) consists of a perturbation that emerges from the past horizon and falls through the future horizon (left). If t1 is sufficiently early, the boost relative to the t = 0 slice generates backreaction in that frame (right). Note that the horizons no longer meet. 29 3.2 The dual to a two-W state is constructed from the one-W state by adding a pertur- bation near the boundary at time t2 and then evolving forwards and backwards. 30 3.3 As t2 shifts earlier, the time at which the original shock reaches the boundary shifts later, eventually moving onto the singularity (right). 32 3.4 The thermofield double and the first six multi-W states are drawn. In each case, the next geometry is obtained from the previous by adding a shock either from the top left or bottom left corner. The gray regions are sensitive to the details of a collision, but the white regions are not. Using the time-folded bulk of [64], these states can be combined as different sheets of an \accordion" geometry. 33 viii 3.5 A geodesic passes across a portion of the wormhole. It intersects the null boundaries of the central regions halfway across their width. 34 3.6 The large-α four-W geometry is shown. Notice that the post-collision regions are small and isolated near the singularities. The Kruskal diagram at the bottom emphasizes the kinkiness of the geometry. 35 3.7 The wormhole created from a large number of weak shocks (top) becomes a smooth geometry in the α ! 0 limit (bottom). 36 3.8 Bob falls in from the boundary at tB = 0 and experiences a mild interaction with the stress energy supporting the solution. If he jumped in at a much earlier time tB ∼ −t∗, he would experience a dramatic interaction. 39 3.9 A candidate for the geometrical dual to a typical state? . 39 1 3.10 The size of the S at the vertices is labeled rn, and the R parameter of the BTZ geometry forming each plaquette is labeled Rn...................... 41 ix Chapter 1 Introduction Consider a warm bath of water. Or, better, consider a warm bath of water in a physicist's thought laboratory, where we are free to run time backwards and forwards, and we are able to record the precise positions of all of the water molecules. Suppose that we are particularly interested in one of the 1028-odd molecules in the tub. We will refer to this molecule as A. As time passes, A is buffeted by the other molecules, and, under their influence, it traces out a rather random path through space. What determines the shape of this path? In fact, the detailed shape is sensitively dependent on the initial conditions for every other molecule in the system. To reveal this dependence, let us consider the following experiment. First, starting with the configuration at t = 0, we run the system backwards in time for an interval tw. Second, we add a new particle W at some location in the bath. Third, we evolve the system forwards in time again to t = 0. What do we get? In particular, where does A end up? The answer depends on the time tw and the distance between A and W . If tw is short enough, or if the distance is large enough, causality ensures that adding W won't have time to affect A, so the forwards evolution will cancel out the backwards evolution, and A will end up where it started. On the other hand, if tw is large enough, one expects that A will be buffeted by particles that themselves were buffeted by particles that themselves were buffeted... by a particle that was buffeted by W . Under this circumstance, the new momentum of A at t = 0 will be randomly re-oriented with respect to its original t = 0 momentum. This phenomenon is one manifestation of the “butterfly effect:” changing the state of a single particle at time −tw changes the states of all particles a short while later. This effect is best understood in the context of classical mechanics, but a very similar phenomenon is expected for many quantum systems. The trouble is that it is difficult to analyze, almost by definition. Think about the bath of water: to calculate what happens to particle A, we would have to be able to accurately study the fine-grained dynamics of 1028 water molecules. To solve the problem quantum 28 mechanically, we'd have to evolve forward a system of 1010 equations.
Load more

Recommended publications
  • It from Qubit Simons Collaboration on Quantum Fields, Gravity, and Information
    It from Qubit Simons Collaboration on Quantum Fields, Gravity, and Information GOALS Developments over the past ten years have shown that major advances in our understanding of quantum gravity, quantum field theory, and other aspects of fundamental physics can be achieved by bringing to bear insights and techniques from quantum information theory. Nonetheless, fundamental physics and quantum information theory remain distinct disciplines and communities, separated by significant barriers to communication and collaboration. Funded by a grant from the Simons Foundation, It from Qubit is a large-scale effort by some of ​ ​ ​ ​ the leading researchers in both communities to foster communication, education, and collaboration between them, thereby advancing both fields and ultimately solving some of the deepest problems in physics. The overarching scientific questions motivating the Collaboration include: ● Does spacetime emerge from entanglement? ● Do black holes have interiors? Does the universe exist outside our horizon? ● What is the information-theoretic structure of quantum field theories? ● Can quantum computers simulate all physical phenomena? ● How does quantum information flow in time? MEMBERSHIP It from Qubit is led by 16 Principal Investigators from 15 institutions in 6 countries: ​ ● Patrick Hayden, Director (Stanford University) ​ ● Matthew Headrick, Deputy Director (Brandeis University) ​ ● Scott Aaronson (MIT) ​ ● Dorit Aharonov (Hebrew University) ​ ● Vijay Balasubramanian (University of Pennsylvania) ​ ● Horacio Casini (Bariloche
    [Show full text]
  • Chaos Theory
    By Nadha CHAOS THEORY What is Chaos Theory? . It is a field of study within applied mathematics . It studies the behavior of dynamical systems that are highly sensitive to initial conditions . It deals with nonlinear systems . It is commonly referred to as the Butterfly Effect What is a nonlinear system? . In mathematics, a nonlinear system is a system which is not linear . It is a system which does not satisfy the superposition principle, or whose output is not directly proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. the SUPERPOSITION PRINCIPLE states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). So a nonlinear system does not satisfy this principal. There are 3 criteria that a chaotic system satisfies 1) sensitive dependence on initial conditions 2) topological mixing 3) periodic orbits are dense 1) Sensitive dependence on initial conditions . This is popularly known as the "butterfly effect” . It got its name because of the title of a paper given by Edward Lorenz titled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? . The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena.
    [Show full text]
  • Spacetime Geometry from Graviton Condensation: a New Perspective on Black Holes
    Spacetime Geometry from Graviton Condensation: A new Perspective on Black Holes Sophia Zielinski née Müller München 2015 Spacetime Geometry from Graviton Condensation: A new Perspective on Black Holes Sophia Zielinski née Müller Dissertation an der Fakultät für Physik der Ludwig–Maximilians–Universität München vorgelegt von Sophia Zielinski geb. Müller aus Stuttgart München, den 18. Dezember 2015 Erstgutachter: Prof. Dr. Stefan Hofmann Zweitgutachter: Prof. Dr. Georgi Dvali Tag der mündlichen Prüfung: 13. April 2016 Contents Zusammenfassung ix Abstract xi Introduction 1 Naturalness problems . .1 The hierarchy problem . .1 The strong CP problem . .2 The cosmological constant problem . .3 Problems of gravity ... .3 ... in the UV . .4 ... in the IR and in general . .5 Outline . .7 I The classical description of spacetime geometry 9 1 The problem of singularities 11 1.1 Singularities in GR vs. other gauge theories . 11 1.2 Defining spacetime singularities . 12 1.3 On the singularity theorems . 13 1.3.1 Energy conditions and the Raychaudhuri equation . 13 1.3.2 Causality conditions . 15 1.3.3 Initial and boundary conditions . 16 1.3.4 Outlining the proof of the Hawking-Penrose theorem . 16 1.3.5 Discussion on the Hawking-Penrose theorem . 17 1.4 Limitations of singularity forecasts . 17 2 Towards a quantum theoretical probing of classical black holes 19 2.1 Defining quantum mechanical singularities . 19 2.1.1 Checking for quantum mechanical singularities in an example spacetime . 21 2.2 Extending the singularity analysis to quantum field theory . 22 2.2.1 Schrödinger representation of quantum field theory . 23 2.2.2 Quantum field probes of black hole singularities .
    [Show full text]
  • Synchronization in Nonlinear Systems and Networks
    Synchronization in Nonlinear Systems and Networks Yuri Maistrenko E-mail: [email protected] you can find me in the room EW 632, Wednesday 13:00-14:00 Lecture 4 - 23.11.2011 1 Chaos actually … is everywhere Web Book CHAOS = BUTTERFLY EFFECT Henri Poincaré (1880) “ It so happens that small differences in the initial state of the system can lead to very large differences in its final state. A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.” Ray Bradbury “A Sound of Thunder “ (1952) THE ESSENCE OF CHAOS • processes deterministic fully determined by initial state • long-term behavior unpredictable butterfly effect PHYSICAL “DEFINITION “ OF CHAOS “To say that a certain system exhibits chaos means that the system obeys deterministic law of evolution but that the outcome is highly sensitive to small uncertainties in the specification of the initial state. In chaotic system any open ball of initial conditions, no matter how small, will in finite time spread over the extent of the entire asymptotically admissible phase space” Predrag Cvitanovich . Appl.Chaos 1992 EXAMPLES OF CHAOTIC SYSTEMS • the solar system (Poincare) • the weather (Lorenz) • turbulence in fluids • population growth • lots and lots of other systems… “HOT” APPLICATIONS • neuronal networks of the brain • genetic networks UNPREDICTIBILITY OF THE WEATHER Edward Lorenz (1963) Difficulties in predicting the weather are not related to the complexity of the Earths’ climate but to CHAOS in the climate equations! Dynamical systems Dynamical system: a system of one or more variables which evolve in time according to a given rule Two types of dynamical systems: • Differential equations: time is continuous (called flow) dx N f (x), t R dt • Difference equations (iterated maps): time is discrete (called cascade) xn1 f (xn ), n 0, 1, 2,..
    [Show full text]
  • Finite-Cutoff JT Gravity and Self-Avoiding Loops Arxiv
    Finite-cutoff JT gravity and self-avoiding loops Douglas Stanford and Zhenbin Yang Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305 Abstract We study quantum JT gravity at finite cutoff using a mapping to the statistical mechanics of a self-avoiding loop in hyperbolic space, with positive pressure and fixed length. The semiclassical limit (small GN ) corresponds to large pressure, and we solve the problem in that limit in three overlapping regimes that apply for different loop sizes. For intermediate loop sizes, a semiclassical effective description is valid, but for very large or very small loops, fluctuations dominate. For large loops, this quantum regime is controlled by the Schwarzian theory. For small loops, the effective description fails altogether, but the problem is controlled using a conjecture from the theory of self-avoiding walks. arXiv:2004.08005v1 [hep-th] 17 Apr 2020 Contents 1 Introduction 2 2 Brief review of JT gravity 4 3 JT gravity and the self-avoiding loop measure 5 3.1 Flat space JT gravity . .5 3.2 Ordinary (hyperbolic) JT gravity . .8 3.3 The small β limit . .8 4 The flat space regime 9 5 The intermediate regime 11 5.1 Classical computations . 11 5.2 One-loop computations . 13 6 The Schwarzian regime 18 7 Discussion 19 7.1 A free particle analogy . 20 7.2 JT gravity as an effective description of JT gravity . 21 A Deriving the density of states from the RGJ formula 23 B Large argument asymptotics of the RGJ formula 24 C Monte Carlo estimation of c2 25 D CGHS model and flat space JT 26 E Large unpressurized loops 27 1 푝 2/3 -2 푝 푝 = ℓ β large pressure effective theory Schwarzian theory flat-space (RGJ) theory 4/3 β = ℓ -3/2 = ℓ4β3/2 푝 = β 푝 푝 = ℓ-2 0 0 0 β 0 β Figure 1: The three approximations used in this paper are valid well inside the respective shaded regions.
    [Show full text]
  • Writing the History of Dynamical Systems and Chaos
    Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as
    [Show full text]
  • The Anthropic Principle and Multiple Universe Hypotheses Oren Kreps
    The Anthropic Principle and Multiple Universe Hypotheses Oren Kreps Contents Abstract ........................................................................................................................................... 1 Introduction ..................................................................................................................................... 1 Section 1: The Fine-Tuning Argument and the Anthropic Principle .............................................. 3 The Improbability of a Life-Sustaining Universe ....................................................................... 3 Does God Explain Fine-Tuning? ................................................................................................ 4 The Anthropic Principle .............................................................................................................. 7 The Multiverse Premise ............................................................................................................ 10 Three Classes of Coincidence ................................................................................................... 13 Can The Existence of Sapient Life Justify the Multiverse? ...................................................... 16 How unlikely is fine-tuning? .................................................................................................... 17 Section 2: Multiverse Theories ..................................................................................................... 18 Many universes or all possible
    [Show full text]
  • Tom W B Kibble Frank H Ber
    Classical Mechanics 5th Edition Classical Mechanics 5th Edition Tom W.B. Kibble Frank H. Berkshire Imperial College London Imperial College Press ICP Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Kibble, T. W. B. Classical mechanics / Tom W. B. Kibble, Frank H. Berkshire, -- 5th ed. p. cm. Includes bibliographical references and index. ISBN 1860944248 -- ISBN 1860944353 (pbk). 1. Mechanics, Analytic. I. Berkshire, F. H. (Frank H.). II. Title QA805 .K5 2004 531'.01'515--dc 22 2004044010 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2004 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore. To Anne and Rosie vi Preface This book, based on courses given to physics and applied mathematics stu- dents at Imperial College, deals with the mechanics of particles and rigid bodies.
    [Show full text]
  • The Multiverse: Conjecture, Proof, and Science
    The multiverse: conjecture, proof, and science George Ellis Talk at Nicolai Fest Golm 2012 Does the Multiverse Really Exist ? Scientific American: July 2011 1 The idea The idea of a multiverse -- an ensemble of universes or of universe domains – has received increasing attention in cosmology - separate places [Vilenkin, Linde, Guth] - separate times [Smolin, cyclic universes] - the Everett quantum multi-universe: other branches of the wavefunction [Deutsch] - the cosmic landscape of string theory, imbedded in a chaotic cosmology [Susskind] - totally disjoint [Sciama, Tegmark] 2 Our Cosmic Habitat Martin Rees Rees explores the notion that our universe is just a part of a vast ''multiverse,'' or ensemble of universes, in which most of the other universes are lifeless. What we call the laws of nature would then be no more than local bylaws, imposed in the aftermath of our own Big Bang. In this scenario, our cosmic habitat would be a special, possibly unique universe where the prevailing laws of physics allowed life to emerge. 3 Scientific American May 2003 issue COSMOLOGY “Parallel Universes: Not just a staple of science fiction, other universes are a direct implication of cosmological observations” By Max Tegmark 4 Brian Greene: The Hidden Reality Parallel Universes and The Deep Laws of the Cosmos 5 Varieties of Multiverse Brian Greene (The Hidden Reality) advocates nine different types of multiverse: 1. Invisible parts of our universe 2. Chaotic inflation 3. Brane worlds 4. Cyclic universes 5. Landscape of string theory 6. Branches of the Quantum mechanics wave function 7. Holographic projections 8. Computer simulations 9. All that can exist must exist – “grandest of all multiverses” They can’t all be true! – they conflict with each other.
    [Show full text]
  • Annotated List of References Tobias Keip, I7801986 Presentation Method: Poster
    Personal Inquiry – Annotated list of references Tobias Keip, i7801986 Presentation Method: Poster Poster Section 1: What is Chaos? In this section I am introducing the topic. I am describing different types of chaos and how individual perception affects our sense for chaos or chaotic systems. I am also going to define the terminology. I support my ideas with a lot of examples, like chaos in our daily life, then I am going to do a transition to simple mathematical chaotic systems. Larry Bradley. (2010). Chaos and Fractals. Available: www.stsci.edu/~lbradley/seminar/. Last accessed 13 May 2010. This website delivered me with a very good introduction into the topic as there are a lot of books and interesting web-pages in the “References”-Sektion. Gleick, James. Chaos: Making a New Science. Penguin Books, 1987. The book gave me a very general introduction into the topic. Harald Lesch. (2003-2007). alpha-Centauri . Available: www.br-online.de/br- alpha/alpha-centauri/alpha-centauri-harald-lesch-videothek-ID1207836664586.xml. Last accessed 13. May 2010. A web-page with German video-documentations delivered a lot of vivid examples about chaos for my poster. Poster Section 2: Laplace's Demon and the Butterfly Effect In this part I describe the idea of the so called Laplace's Demon and the theory of cause-and-effect chains. I work with a lot of examples, especially the famous weather forecast example. Also too I introduce the mathematical concept of a dynamic system. Jeremy S. Heyl (August 11, 2008). The Double Pendulum Fractal. British Columbia, Canada.
    [Show full text]
  • Instructional Experiments on Nonlinear Dynamics & Chaos (And
    Bibliography of instructional experiments on nonlinear dynamics and chaos Page 1 of 20 Colorado Virtual Campus of Physics Mechanics & Nonlinear Dynamics Cluster Nonlinear Dynamics & Chaos Lab Instructional Experiments on Nonlinear Dynamics & Chaos (and some related theory papers) overviews of nonlinear & chaotic dynamics prototypical nonlinear equations and their simulation analysis of data from chaotic systems control of chaos fractals solitons chaos in Hamiltonian/nondissipative systems & Lagrangian chaos in fluid flow quantum chaos nonlinear oscillators, vibrations & strings chaotic electronic circuits coupled systems, mode interaction & synchronization bouncing ball, dripping faucet, kicked rotor & other discrete interval dynamics nonlinear dynamics of the pendulum inverted pendulum swinging Atwood's machine pumping a swing parametric instability instabilities, bifurcations & catastrophes chemical and biological oscillators & reaction/diffusions systems other pattern forming systems & self-organized criticality miscellaneous nonlinear & chaotic systems -overviews of nonlinear & chaotic dynamics To top? Briggs, K. (1987), "Simple experiments in chaotic dynamics," Am. J. Phys. 55 (12), 1083-9. Hilborn, R. C. (2004), "Sea gulls, butterflies, and grasshoppers: a brief history of the butterfly effect in nonlinear dynamics," Am. J. Phys. 72 (4), 425-7. Hilborn, R. C. and N. B. Tufillaro (1997), "Resource Letter: ND-1: nonlinear dynamics," Am. J. Phys. 65 (9), 822-34. Laws, P. W. (2004), "A unit on oscillations, determinism and chaos for introductory physics students," Am. J. Phys. 72 (4), 446-52. Sungar, N., J. P. Sharpe, M. J. Moelter, N. Fleishon, K. Morrison, J. McDill, and R. Schoonover (2001), "A laboratory-based nonlinear dynamics course for science and engineering students," Am. J. Phys. 69 (5), 591-7. http://carbon.cudenver.edu/~rtagg/CVCP/Ctr_dynamics/Lab_nonlinear_dyn/Bibex_nonline..
    [Show full text]
  • Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory
    Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Roberts, Daniel A., and Douglas Stanford. "Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory." Phys. Rev. Lett. 115, 131603 (September 2015). © 2015 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevLett.115.131603 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/98897 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 115, 131603 (2015) PHYSICAL REVIEW LETTERS 25 SEPTEMBER 2015 Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory Daniel A. Roberts* Center for Theoretical Physics and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA † Douglas Stanford School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA (Received 10 March 2015; revised manuscript received 14 May 2015; published 22 September 2015) We study chaotic dynamics in two-dimensional conformal field theory through out-of-time-order thermal correlators of the form hWðtÞVWðtÞVi. We reproduce holographic calculations similar to those of Shenker and Stanford, by studying the large c Virasoro identity conformal block. The contribution of this block to the above correlation function begins to decrease exponentially after a delay of ∼t − β=2π β2E E t β=2π c E ;E à ð Þ log w v, where à is the fast scrambling time ð Þ log and w v are the energy scales of the W;V operators.
    [Show full text]