Chaotic diffusion in nonlinear Hamiltonian systems

This work investigates diffusion in nonlinear Hamiltonian systems. The diffusion, more precisely subdiffusion, in such systems is induced by the intrinsic chaotic behavior of trajectories and thus is called chaotic diffusion''. Its properties are studied on the example of one- or two-dimensional lattices of harmonic or nonlinear oscillators with nearest neighbor couplings. The fundamental observation is the spreading of energy for localized initial conditions. Methods of quantifying this spreading behavior are presented, including a new quantity called excitation time. This new quantity allows for a more precise analysis of the spreading than traditional methods. Furthermore, the nonlinear diffusion equation is introduced as a phenomenologic description of the spreading process and a number of predictions on the density dependence of the spreading are drawn from this equation. Two mathematical techniques for analyzing nonlinear Hamiltonian systems are introduced. The first one is based on a scaling analysis of the Hamiltonian equations and the results are related to similar scaling properties of the NDE. From this relation, exact spreading predictions are deduced. Secondly, the microscopic dynamics at the edge of spreading states are thoroughly analyzed, which again suggests a scaling behavior that can be related to the NDE. Such a microscopic treatment of chaotically spreading states in nonlinear Hamiltonian systems has not been done before and the results present a new technique of connecting microscopic dynamics with macroscopic descriptions like the nonlinear diffusion equation. All theoretical results are supported by heavy numerical simulations, partly obtained on one of Europe's fastest supercomputers located in Bologna, Italy. In the end, the highly interesting case of harmonic oscillators with random frequencies and nonlinear coupling is studied, which resembles to some extent the famous Discrete Anderson Nonlinear Schroedinger Equation. For this model, a deviation from the widely believed power-law spreading is observed in numerical experiments. Some ideas on a theoretical explanation for this deviation are presented, but a conclusive theory could not be found due to the complicated phase space structure in this case. Nevertheless, it is hoped that the techniques and results presented in this work will help to eventually understand this controversely discussed case as well.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Potsdam

Year

2012

Author dc:creator

• Mulansky, Mario

Contributors dc:contributor

• Pikovskij, Arkadij

• Chaos
• Diffusion
• Thermalisierung
• Energieausbreitung
• thermalization
• energy spreading

Statement dc:rights

• Creative Commons - Namensnennung, Nicht kommerziell, Weitergabe zu gleichen Bedingungen 3.0 Deutschland

Repository record source_url

https://publishup.uni-potsdam.de/frontdoor/index/index/docId/6122

OAI identifier oai:identifier

oai:kobv.de-opus4-uni-potsdam:6122