Learning objectives
The course includes the study of various systems that have complex behaviors, with the goal of finding the phenomenological laws governing the overall behavior of such systems.
Various theoretical models and techniques, both analytical and numerical will be discussed in the field of physics, biology, computer science and economics.
Given the interdisciplinary nature the course is recommended for all addresses.
Prerequisites
First elementary courses in classical and quantum mechanics
Course unit content
The fields of nonlinear dynamics and complexity have grown very much over the last decades and are becoming more and more relevant in different disciplines. We presents a concise introduction to the field of complexity and chaos, suitable for graduate students of all branches of physcis. We introduce the concepts of Hamiltonian dynamical systems' theory combining in a comprehensive way classical and quantum mechanical description. Motivations of the respective subjects and many examples from different fields in physics ease the understanding.
In the classical part, we review Hamilton-Jacobi theory as a fundamental basis for perturbational approaches (secular perturbation and KAM theory) and the definition of regularity and chaos (stable and unstable fixed points, Poincare-Birkhoff, Lyapunov exponents). The quantum part summerizes aspects of semiclassical theory (Wigner functions, Weyl symbols) as well as random matrix theory which is best suited to define chaos and complexity on a quantum level.
Full programme
The fields of nonlinear dynamics and complexity have grown very much over the last decades and are becoming more and more relevant in different disciplines. We presents a concise introduction to the field of complexity and chaos, suitable for graduate students of all branches of physcis. We introduce the concepts of Hamiltonian dynamical systems' theory combining in a comprehensive way classical and quantum mechanical description. Motivations of the respective subjects and many examples from different fields in physics ease the understanding.
In the classical part, we review Hamilton-Jacobi theory as a fundamental basis for perturbational approaches (secular perturbation and KAM theory) and the definition of regularity and chaos (stable and unstable fixed points, Poincare-Birkhoff, Lyapunov exponents). The quantum part summerizes aspects of semiclassical theory (Wigner functions, Weyl symbols) as well as random matrix theory which is best suited to define chaos and complexity on a quantum level.
Bibliography
- Lecture manuscript
- S. Wimberger, Nonlinear Dynamics and Quantum Chaos: An Introduction (Springer, Heidelberg, 2014)
- F. Scheck, Mechanics: From Newton’s Laws to Deterministic Chaos (Springer, Heidelberg, 2007)
- V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer Verlag, New York, 1989)
- J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley Publishing Company, Reading, MA, 1994)
- P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge UK, 1998)
- Online book, P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, G. Vattay, Chaos: Classical and Quantum (Niels Bohr Institute, Copenhagen, 2012) at www.chaosbook.org
- M.L. Mehta, Random matrices (Elsevier, Amsterdam, 2004)
Teaching methods
Class lectures
Assessment methods and criteria
Oral exam on the contents of the lecture course.
Other information
Oral exam
