It includes unique material on separatrix chaos, small nonlinearity chaos, fractional kinetics, and discussions on maxwells. The influence of quantization on the onset of chaos in hamiltonian systems. This type of billiard has long been known as a prototype chaotic dynamical system. Pdf zeta function zeros, powers of primes, and quantum chaos. Sections 4 demonstrate how quantum chaos can be detected and quanti ed using circuit complexity while section 5 discusses the otoc and its relation to the results obtained from the circuit complexity. The connections to quantum chaos and semiclassical physics are discussed. Quantum chaotic tunneling in graphene systems with electronelectron interactions.
The topics treated include a detailed exploration of the quantum aspects of nonlinear dynamics, quantum criteria to distinguish regular and irregular motion, antiunitary symmetries generalized time reversal, random matrix theory and a thorough account of the quantum. For an energy conserved hamiltonian system, the volume of a subspace in its phase space is conserved as time elapses. This is an important topic for discussion, because general relativity in its hamiltonian formulation is a constrained system, and there is strong evidence that it exhibits chaotic features. Therefore, there can be augmentation and compression for the subspace, or its dynamics can be chaotic. Modeling, synchronization, and fpga implementation of. Origins of chaos in nonlinear oscillatory hamiltonian. This book aims to familiarise the reader with the essential properties of the chaotic dynamics of hamiltonian systems. Kolmogorov entropy as a common basis for discussion, the onset of chaotic motion in classical mechanical and quantum mechanical systems is. We have already seen that properties such as singularity structure of the solutions and stability of particular solutions have a definite role in the dynamics of the. This introduction to the theory of hamiltonian chaos outlines the main results in the field, and goes on to consider implications for quantum mechanics. Classical hamiltonian systems and stochastic mechanics. Furthermore, chaos in hamiltonian systems is at the heart of such fundamental questions as the foundations of statistical mechanics and the stability of the solar system. Guozhen wu, in nonlinearity and chaos in molecular vibrations, 2005. Introduction the properties of a system that are responsible for the regular or chaotic behaviour are not known clearly.
If you want to understand the ideas underlying an ergodic behavior in a large system you have to turn your attention to quantum mechanics and to the lieb and simon theorem. The connection of the quantum chaos qc with the reflection symmetry of the qd, disappearance of the qc in the tunnel coupled chaotic qd, electron localization and transition. Hamiltonian chaos and the standard map poincare section and twist maps. The term quantum chaos designates a body of knowledge which has been established in an attempt to understand the implications of classical chaos for quantum systems.
Zeta function zeros, powers of primes, and quantum chaos. Poincarebirkhoff theorem and destruction of rational tori. Hamiltonian chaos ii department of physics university of rhode. In this paper, fivedimension euler equations are constructed by integrating two of subeuler equations, which are contributory to the exploration of higherdimensional systems. Hamiltonian system whose quan tum sp ectrum is the primes. Thus, in chapter 11 we shall treat the connection between chaos in hamiltonian systems.
Any novice can master chaosbook part i geometry of chaos andor online course part 1 indeed, any scientist, engineer or mathematician would pro. Chaos, dirac observables and constraint quantization. Chaos poses technical challenges to constrained hamiltonian systems. Upon addressing the properties of hamiltonian systems we will then explore how chaos arises. The transition to chaos, in conservative classical systems. Hamiltonian systems an overview sciencedirect topics. Chaos and quantization cambridge monographs on mathematical physics on. These systems can be studied in both hamiltonian mechanics and dynamical systems.
Example 1 conservation of the total energy for hamiltonian systems 1 the hamiltonian function hp,q is a. In open hamiltonian systems, quantum chaotic scattering 3343 has been investigated extensively. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics hamiltonian mechanics was first formulated by william rowan hamilton in 1833, starting from lagrangian mechanics, a previous reformulation of classical mechanics introduced by joseph louis. Hamiltonian systems and chaos overview liz laneharvard, melissa swager abstract.
Example 2 conservation of the total linear and angular momentum we consider a system of nparticles interacting pairwise with potential forces depending on the distances of. Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. In this paper we will give an overview of hamiltonian systems with specific examples, including the classical pendulum example. Our aim is to study the change in quantum chaos as a is varied and compare the classical and quantum. Origins of chaos in nonlinear oscillatory hamiltonian systems a thesis presented to the division of mathematics and natural sciences reed college in partial ful llment of the requirements for the degree bachelor of arts mateo r. In section 6, we discuss a manybody system quantum eld theory where the inverted oscillator arises. The study of nonlinear dynamics, and in particular of chaotic systems, is one of the fastest growing and most. Although the hamiltonian method generally has no advantage over and in fact is invariably much more cumbersome than the lagrangian method when it comes to standard mechanics problems involving a small number of particles, its superiority becomes evident when dealing with systems at the opposite ends of the spectrum.
The sensitivity to initial conditions is important to chaos but does not itself di erentiate. In quantum mechanics, a hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system this addition is the total energy of the system in most of the cases under analysis. Chaos and quantum chaos in cosmological models core. Relativistic quantum chaos an emergent interdisciplinary field. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. Chaos and ergodicity in hamiltonian systems stack exchange. Introduction to quantum chaos denis ullmo1 and steven tomsovic2 1lptms, univ parissud, cnrs umr 8626, 91405 orsay cedex, france 2department of physics and astronomy, washington state university, pullman, wa 991642814 usa july 17, 2014 keywords. The influence of quantization on the onset of chaos in. According to this, the function exhibits quantum chaos with a broken timereversal symmetry. The study of nonlinear dynamics, and in particular of chaotic systems, is one of the fastest growing and most productive areas in physics and. A hamiltonian system is a dynamical system governed by hamiltons equations. For example, in closed chaotic hamiltonian systems, the basic phenomena that have been and continue to be studied include energy levelspacing statistics319 and quantum scarring.
The study of nonlinear dynamics, and particularly, chaotic systems, is one of the fastest developing areas in physics and applied mathematics. In addition, hamiltonian mechanics and its structure are reflected in quantum mechanics. This by now classic text provides an excellent introduction and survey to the continuously expanding field of quantum chaos. Quantum chaotic tunneling in graphene systems with. On the quantization procedure in classical mechanics and. Given a collection of observations of such a hamiltonian system over time, we extract phase space coordinates and a hamiltonian function of them that acts as the generator of the system dynamics. A possible new technique to investigate classical integrability and define quantum chaos. What is the relationship between quantum mechanics and classical chaos. Standard map as time sections of kicked oscillator link with quantum chaos. The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in. Quantum mechanics postulates that we can calculate the result of any. The influence of quantization on the onset of chaos in hamiltonian. The corresponding quantum system, on the other hand.
The primary question that quantum chaos seeks to answer is. Lieberman, applied mathematical sciences vol38, springer verlag, new york, 1982. The kolmogorov entropy interpretation ronnie kosloffa and stuart a. Constrained hamiltonian systems courses in canonical gravity yaser tavakoli december 15, 2014 1 introduction in canonical formulation of general relativity, geometry of spacetime is given in terms of elds on spatial slices, whose geometry is encoded by a three metric hab, presenting the con guration variables. The relationship between the defined kolmogorov entropy, previous interpretations of kamlike onset of chaos in quantum mechanical systems, and the role played by preparation and observation of a system in influencing the intramolecular dynamics, are discussed.
The topics include a brief introduction to classical hamiltonian chaos, a detailed exploration of the quantum aspects of nonlinear dynamics, quantum criteria used to distinguish regular and irregular motion, and antiunitary generalized time reversal and unitary symmetries. Similarly, gue is said to model classically chaotic systems possessing no timereversal symmetry. Hamiltonian mechanics is an equivalent but more abstract reformulation of classical mechanic theory. Pdf chaos, dirac observables and constraint quantization. Conservative chaotic systems have potentials in engineering application because of their superiority over the dissipative systems in terms of ergodicity and integer dimension. Write the equations of motion in poisson bracket form. This introduction to the theory of hamiltonian chaos considers its implications for quantum mechanics as well.
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