Quantum Chaos: an Introduction Via Chains of Interacting Spins-1/2
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From Deterministic Dynamics to Thermodynamic Laws II: Fourier's
FROM DETERMINISTIC DYNAMICS TO THERMODYNAMIC LAWS II: FOURIER'S LAW AND MESOSCOPIC LIMIT EQUATION YAO LI Abstract. This paper considers the mesoscopic limit of a stochastic energy ex- change model that is numerically derived from deterministic dynamics. The law of large numbers and the central limit theorems are proved. We show that the limit of the stochastic energy exchange model is a discrete heat equation that satisfies Fourier's law. In addition, when the system size (number of particles) is large, the stochastic energy exchange is approximated by a stochastic differential equation, called the mesoscopic limit equation. 1. Introduction Fourier's law is an empirical law describing the relationship between the thermal conductivity and the temperature profile. In 1822, Fourier concluded that \the heat flux resulting from thermal conduction is proportional to the magnitude of the tem- perature gradient and opposite to it in sign" [14]. The well-known heat equation is derived based on Fourier's law. However, the rigorous derivation of Fourier's law from microscopic Hamiltonian mechanics remains to be a challenge to mathemati- cians and physicist [3]. This challenge mainly comes from our limited mathematical understanding to nonequilibrium statistical mechanics. After the foundations of sta- tistical mechanics were established by Boltzmann, Gibbs, and Maxwell more than a century ago, many things about nonequilibrium steady state (NESS) remains un- clear, especially the dependency of key quantities on the system size N. There have been several studies that aim to derive Fourier's law from the first principle. A large class of models [29, 31, 12, 11, 10] use anharmonic chains to de- scribe heat conduction in insulating crystals. -
Ergodic Theory Plays a Key Role in Multiple Fields Steven Ashley Science Writer
CORE CONCEPTS Core Concept: Ergodic theory plays a key role in multiple fields Steven Ashley Science Writer Statistical mechanics is a powerful set of professor Tom Ward, reached a key milestone mathematical tools that uses probability the- in the early 1930s when American mathema- ory to bridge the enormous gap between the tician George D. Birkhoff and Austrian-Hun- unknowable behaviors of individual atoms garian (and later, American) mathematician and molecules and those of large aggregate sys- and physicist John von Neumann separately tems of them—a volume of gas, for example. reconsidered and reformulated Boltzmann’ser- Fundamental to statistical mechanics is godic hypothesis, leading to the pointwise and ergodic theory, which offers a mathematical mean ergodic theories, respectively (see ref. 1). means to study the long-term average behavior These results consider a dynamical sys- of complex systems, such as the behavior of tem—whetheranidealgasorothercomplex molecules in a gas or the interactions of vi- systems—in which some transformation func- brating atoms in a crystal. The landmark con- tion maps the phase state of the system into cepts and methods of ergodic theory continue its state one unit of time later. “Given a mea- to play an important role in statistical mechan- sure-preserving system, a probability space ics, physics, mathematics, and other fields. that is acted on by the transformation in Ergodicity was first introduced by the a way that models physical conservation laws, Austrian physicist Ludwig Boltzmann laid Austrian physicist Ludwig Boltzmann in the what properties might it have?” asks Ward, 1870s, following on the originator of statisti- who is managing editor of the journal Ergodic the foundation for modern-day ergodic the- cal mechanics, physicist James Clark Max- Theory and Dynamical Systems.Themeasure ory. -
AN INTRODUCTION to DYNAMICAL BILLIARDS Contents 1
AN INTRODUCTION TO DYNAMICAL BILLIARDS SUN WOO PARK 2 Abstract. Some billiard tables in R contain crucial references to dynamical systems but can be analyzed with Euclidean geometry. In this expository paper, we will analyze billiard trajectories in circles, circular rings, and ellipses as well as relate their charactersitics to ergodic theory and dynamical systems. Contents 1. Background 1 1.1. Recurrence 1 1.2. Invariance and Ergodicity 2 1.3. Rotation 3 2. Dynamical Billiards 4 2.1. Circle 5 2.2. Circular Ring 7 2.3. Ellipse 9 2.4. Completely Integrable 14 Acknowledgments 15 References 15 Dynamical billiards exhibits crucial characteristics related to dynamical systems. Some billiard tables in R2 can be understood with Euclidean geometry. In this ex- pository paper, we will analyze some of the billiard tables in R2, specifically circles, circular rings, and ellipses. In the first section we will present some preliminary background. In the second section we will analyze billiard trajectories in the afore- mentioned billiard tables and relate their characteristics with dynamical systems. We will also briefly discuss the notion of completely integrable billiard mappings and Birkhoff's conjecture. 1. Background (This section follows Chapter 1 and 2 of Chernov [1] and Chapter 3 and 4 of Rudin [2]) In this section, we define basic concepts in measure theory and ergodic theory. We will focus on probability measures, related theorems, and recurrent sets on certain maps. The definitions of probability measures and σ-algebra are in Chapter 1 of Chernov [1]. 1.1. Recurrence. Definition 1.1. Let (X,A,µ) and (Y ,B,υ) be measure spaces. -
Propagation of Rays in 2D and 3D Waveguides: a Stability Analysis with Lyapunov and Reversibility Fast Indicators G
Propagation of rays in 2D and 3D waveguides: a stability analysis with Lyapunov and Reversibility fast indicators G. Gradoni,1, a) F. Panichi,2, b) and G. Turchetti3, c) 1)School of Mathematical Sciences and Department of Electrical and Electronic Engineering, University of Nottingham, United Kingdomd) 2)Department of Physics and Astronomy , University of Bologna, Italy 3)Department of Physics and Astronomy, University of Bologna, Italy. INDAM National Group of Mathematical Physics, Italy (Dated: 13 January 2021) Propagation of rays in 2D and 3D corrugated waveguides is performed in the general framework of stability indicators. The analysis of stability is based on the Lyapunov and Reversibility error. It is found that the error growth follows a power law for regular orbits and an exponential law for chaotic orbits. A relation with the Shannon channel capacity is devised and an approximate scaling law found for the capacity increase with the corrugation depth. We investigate the propagation of a ray in a 2D wave guide whose boundaries are two parallel horizontal lines, with a periodic corrugation on the upper line. The reflection point abscissa on the lower line and the ray horizontal I. INTRODUCTION velocity component after reflection are the phase space coordinates and the map connecting two consecutive re- The equivalence between geometrical optics and mechanics flections is symplectic. The dynamic behaviour is illus- was established in a variational form by the principles of Fer- trated by the phase portraits which show that the regions mat and Maupertuis. If a ray propagates in a uniform medium of chaotic motion increase with the corrugation amplitude. -
© 2020 Alexandra Q. Nilles DESIGNING BOUNDARY INTERACTIONS for SIMPLE MOBILE ROBOTS
© 2020 Alexandra Q. Nilles DESIGNING BOUNDARY INTERACTIONS FOR SIMPLE MOBILE ROBOTS BY ALEXANDRA Q. NILLES DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate College of the University of Illinois at Urbana-Champaign, 2020 Urbana, Illinois Doctoral Committee: Professor Steven M. LaValle, Chair Professor Nancy M. Amato Professor Sayan Mitra Professor Todd D. Murphey, Northwestern University Abstract Mobile robots are becoming increasingly common for applications such as logistics and delivery. While most research for mobile robots focuses on generating collision-free paths, however, an environment may be so crowded with obstacles that allowing contact with environment boundaries makes our robot more efficient or our plans more robust. The robot may be so small or in a remote environment such that traditional sensing and communication is impossible, and contact with boundaries can help reduce uncertainty in the robot's state while navigating. These novel scenarios call for novel system designs, and novel system design tools. To address this gap, this thesis presents a general approach to modelling and planning over interactions between a robot and boundaries of its environment, and presents prototypes or simulations of such systems for solving high-level tasks such as object manipulation. One major contribution of this thesis is the derivation of necessary and sufficient conditions of stable, periodic trajectories for \bouncing robots," a particular model of point robots that move in straight lines between boundary interactions. Another major contribution is the description and implementation of an exact geometric planner for bouncing robots. We demonstrate the planner on traditional trajectory generation from start to goal states, as well as how to specify and generate stable periodic trajectories. -
(2020) Physics-Enhanced Neural Networks Learn Order and Chaos
PHYSICAL REVIEW E 101, 062207 (2020) Physics-enhanced neural networks learn order and chaos Anshul Choudhary ,1 John F. Lindner ,1,2,* Elliott G. Holliday,1 Scott T. Miller ,1 Sudeshna Sinha,1,3 and William L. Ditto 1 1Nonlinear Artificial Intelligence Laboratory, Physics Department, North Carolina State University, Raleigh, North Carolina 27607, USA 2Physics Department, The College of Wooster, Wooster, Ohio 44691, USA 3Indian Institute of Science Education and Research Mohali, Knowledge City, SAS Nagar, Sector 81, Manauli PO 140 306, Punjab, India (Received 26 November 2019; revised manuscript received 22 May 2020; accepted 24 May 2020; published 18 June 2020) Artificial neural networks are universal function approximators. They can forecast dynamics, but they may need impractically many neurons to do so, especially if the dynamics is chaotic. We use neural networks that incorporate Hamiltonian dynamics to efficiently learn phase space orbits even as nonlinear systems transition from order to chaos. We demonstrate Hamiltonian neural networks on a widely used dynamics benchmark, the Hénon-Heiles potential, and on nonperturbative dynamical billiards. We introspect to elucidate the Hamiltonian neural network forecasting. DOI: 10.1103/PhysRevE.101.062207 I. INTRODUCTION dynamical systems. But from stormy weather to swirling galaxies, natural dynamics is far richer and more challeng- Newton wrote, “My brain never hurt more than in my ing. In this article, we exploit the Hamiltonian structure of studies of the moon (and Earth and Sun)” [1]. Unsurprising conservative systems to provide neural networks with the sentiment, as the seemingly simple three-body problem is in- physics intelligence needed to learn the mix of order and trinsically intractable and practically unpredictable. -
Quantum Chaos in Rydberg Atoms In.Strong Fields by Hong Jiao
Experimental and Theoretical Aspects of Quantum Chaos in Rydberg Atoms in.Strong Fields by Hong Jiao B.S., University of California, Berkeley (1987) M.S., California Institute of Technology (1989) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1996 @ Massachusetts Institute of Technology 1996. All rights reserved. Signature of Author. Department of Physics U- ge December 4, 1995 Certified by.. Daniel Kleppner Lester Wolfe Professor of Physics Thesis Supervisor Accepted by. ;AssAGiUS. rrS INSTITU It George F. Koster OF TECHNOLOGY Professor of Physics FEB 1411996 Chairman, Departmental Committee on Graduate Students LIBRARIES 8 Experimental and Theoretical Aspects of Quantum Chaos in Rydberg Atoms in Strong Fields by Hong Jiao Submitted to the Department of Physics on December 4, 1995, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We describe experimental and theoretical studies of the connection between quantum and classical dynamics centered on the Rydberg atom in strong fields, a disorderly system. Primary emphasis is on systems with three degrees of freedom and also the continuum behavior of systems with two degrees of freedom. Topics include theoret- ical studies of classical chaotic ionization, experimental observation of bifurcations of classical periodic orbits in Rydberg atoms in parallel electric and magnetic fields, analysis of classical ionization and semiclassical recurrence spectra of the diamagnetic Rydberg atom in the positive energy region, and a statistical analysis of quantum manifestation of electric field induced chaos in Rydberg atoms in crossed electric and magnetic fields. -
The Emergence of Chaos in Quantum Mechanics
S S symmetry Article The Emergence of Chaos in Quantum Mechanics Emilio Fiordilino Dipartimento di Fisica e Chimica—Emilio Segrè, Università degli Studi di Palermo, Via Archirafi 36, 90123 Palermo, Italy; emilio.fi[email protected] Received: 4 April 2020; Accepted: 7 May 2020; Published: 8 May 2020 Abstract: Nonlinearity in Quantum Mechanics may have extrinsic or intrinsic origins and is a liable route to a chaotic behaviour that can be of difficult observations. In this paper, we propose two forms of nonlinear Hamiltonian, which explicitly depend upon the phase of the wave function and produce chaotic behaviour. To speed up the slow manifestation of chaotic effects, a resonant laser field assisting the time evolution of the systems causes cumulative effects that might be revealed, at least in principle. The nonlinear Schrödinger equation is solved within the two-state approximation; the solution displays features with characteristics similar to those found in chaotic Classical Mechanics: sensitivity on the initial state, dense power spectrum, irregular filling of the Poincaré map and exponential separation of the trajectories of the Bloch vector s in the Bloch sphere. Keywords: nonlinear Schrödinger equation; chaos; high order harmonic generation 1. Introduction The theory of chaos gained the role of a new paradigm of Science for explaining a large variety of phenomena by introducing the concept of unpredictability within the perimeter of Classical Physics. Chaos occurs when the trajectory of a particle is sensitive to the initial conditions and is produced by a nonlinear equation of motion [1]. As examples we quote the equation for the Duffing oscillator (1918) 3 mx¨ + gx˙ + rx = F cos(wLt) (1) or its elaborated form 3 mx¨ + gx˙ + kx + rx = F cos(wLt) (2) that are among the most studied equation of mathematical physics [1,2] and, as far as we know, cannot be analytically solved. -
Signatures of Quantum Mechanics in Chaotic Systems
entropy Article Signatures of Quantum Mechanics in Chaotic Systems Kevin M. Short 1,* and Matthew A. Morena 2 1 Integrated Applied Mathematics Program, Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA 2 Department of Mathematics, Christopher Newport University, Newport News, VA 23606, USA; [email protected] * Correspondence: [email protected] Received: 6 May 2019; Accepted: 19 June 2019; Published: 22 June 2019 Abstract: We examine the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system’s set of cupolets, which are highly-accurate stabilizations of its unstable periodic orbits. Our discussion is motivated by the bound or entangled states that we have recently detected between interacting chaotic systems, wherein pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external controls. This state is known as chaotic entanglement as it has been shown to exhibit several properties consistent with quantum entanglement. For instance, should the interaction be disturbed, the chaotic entanglement would then be broken. In this paper, we further describe chaotic entanglement and go on to address the capacity for chaotic systems to exhibit other characteristics that are conventionally associated with quantum mechanics, namely analogs to wave function collapse, various entropy definitions, the superposition of states, and the measurement problem. In doing so, we argue that these characteristics need not be regarded exclusively as quantum mechanical. We also discuss several characteristics of quantum systems that are not fully compatible with chaotic entanglement and that make quantum entanglement unique. -
Chao-Dyn/9402001 7 Feb 94
chao-dyn/9402001 7 Feb 94 DESY ISSN Quantum Chaos January Einsteins Problem of The study of quantum chaos in complex systems constitutes a very fascinating and active branch of presentday physics chemistry and mathematics It is not wellknown however that this eld of research was initiated by a question rst p osed by Einstein during a talk delivered in Berlin on May concerning Quantum Chaos the relation b etween classical and quantum mechanics of strongly chaotic systems This seems historically almost imp ossible since quantum mechanics was not yet invented and the phenomenon of chaos was hardly acknowledged by physicists in While we are celebrating the seventyfth anniversary of our alma mater the Frank Steiner Hamburgische Universitat which was inaugurated on May it is interesting to have a lo ok up on the situation in physics in those days Most I I Institut f urTheoretische Physik UniversitatHamburg physicists will probably characterize that time as the age of the old quantum Lurup er Chaussee D Hamburg Germany theory which started with Planck in and was dominated then by Bohrs ingenious but paradoxical mo del of the atom and the BohrSommerfeld quanti zation rules for simple quantum systems Some will asso ciate those years with Einsteins greatest contribution the creation of general relativity culminating in the generally covariant form of the eld equations of gravitation which were found by Einstein in the year and indep endently by the mathematician Hilb ert at the same time In his talk in May Einstein studied the -
Feynman Quantization
3 FEYNMAN QUANTIZATION An introduction to path-integral techniques Introduction. By Richard Feynman (–), who—after a distinguished undergraduate career at MIT—had come in as a graduate student to Princeton, was deeply involved in a collaborative effort with John Wheeler (his thesis advisor) to shake the foundations of field theory. Though motivated by problems fundamental to quantum field theory, as it was then conceived, their work was entirely classical,1 and it advanced ideas so radicalas to resist all then-existing quantization techniques:2 new insight into the quantization process itself appeared to be called for. So it was that (at a beer party) Feynman asked Herbert Jehle (formerly a student of Schr¨odinger in Berlin, now a visitor at Princeton) whether he had ever encountered a quantum mechanical application of the “Principle of Least Action.” Jehle directed Feynman’s attention to an obscure paper by P. A. M. Dirac3 and to a brief passage in §32 of Dirac’s Principles of Quantum Mechanics 1 John Archibald Wheeler & Richard Phillips Feynman, “Interaction with the absorber as the mechanism of radiation,” Reviews of Modern Physics 17, 157 (1945); “Classical electrodynamics in terms of direct interparticle action,” Reviews of Modern Physics 21, 425 (1949). Those were (respectively) Part III and Part II of a projected series of papers, the other parts of which were never published. 2 See page 128 in J. Gleick, Genius: The Life & Science of Richard Feynman () for a popular account of the historical circumstances. 3 “The Lagrangian in quantum mechanics,” Physicalische Zeitschrift der Sowjetunion 3, 64 (1933). The paper is reprinted in J. -
Ballistic Electron Transport in Graphene Nanodevices and Billiards
Ballistic Electron Transport in Graphene Nanodevices and Billiards Dissertation (Cumulative Thesis) for the award of the degree Doctor rerum naturalium of the Georg-August-Universität Göttingen within the doctoral program Physics of Biological and Complex Systems of the Georg-August University School of Science submitted by George Datseris from Athens, Greece Göttingen 2019 Thesis advisory committee Prof. Dr. Theo Geisel Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Examination board Prof. Dr. Theo Geisel (Reviewer) Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus (Second Reviewer) Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Ulrich Parlitz Biomedical Physics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stefan Kehrein Condensed Matter Theory, Physics Department Georg-August-Universität Göttingen Prof. Dr. Jörg Enderlein Biophysics / Complex Systems, Physics Department Georg-August-Universität Göttingen Date of the oral examination: September 13th, 2019 Contents Abstract 3 1 Introduction 5 1.1 Thesis synopsis and outline.............................. 10 2 Fundamental Concepts 13 2.1 Nonlinear dynamics of antidot superlattices..................... 13 2.2 Lyapunov exponents in billiards............................ 20 2.3 Fundamental properties of graphene......................... 22 2.4 Why quantum?..................................... 31 2.5 Transport simulations and scattering wavefunctions................