TY - THES A1 - Mulansky, Mario T1 - Localization properties of nonlinear disordered lattices N2 - In this thesis, the properties of nonlinear disordered one dimensional lattices is investigated. Part I gives an introduction to the phenomenon of Anderson Localization, the Discrete Nonlinear Schroedinger Equation and its properties as well as the generalization of this model by introducing the nonlinear index α. In Part II, the spreading behavior of initially localized states in large, disordered chains due to nonlinearity is studied. Therefore, different methods to measure localization are discussed and the structural entropy as a measure for the peak structure of probability distributions is introduced. Finally, the spreading exponent for several nonlinear indices is determined numerically and compared with analytical approximations. Part III deals with the thermalization in short disordered chains. First, the term thermalization and its application to the system in use is explained. Then, results of numerical simulations on this topic are presented where the focus lies especially on the energy dependence of the thermalization properties. A connection with so-called breathers is drawn. N2 - In dieser Arbeit wird das Verhalten nichtlinearer Ketten mit Zufallspotential untersucht. Teil I enthaelt eine Einfuehrung in das Phaenomen der Anderson Lokalisierung, die Diskrete Nichtlineare Schroedinger Gleichung und ihren Eigenschaften sowie die verwendete Verallgemeinerung des Modells durch Einfuehrung eines Nichtlinearitaets-Indizes α. In Teil II wird das Ausbreitungsverhalten von lokalisierten Zustaenden in langen, ungeordneten Ketten durch die Nichtlinearitaet untersucht. Dazu werden zuerst verschiedene Lokalisierungsmaße besprochen und außerdem die strukturelle Entropie als Messgroeße der Peakstruktur eingefuehrt. Im Anschluss wird der Ausbreitungskoeffzient fuer verschiedene Nichtlinearitaets-Indizes bestimmt und mit analytischen Absch¨tzungen verglichen. Teil III behandelt schließlich die Thermalisierung in kurzen, ungeordneten Ketten. Dabei wird zuerst der Begriff Thermalisierung in dem verwendeten Zusammenhang erklaert. Danach erfolgt eine numerische Analyse von Thermalisierungseigenschaften lokalisierter Anfangszustaende, wobei die Energieabhaengigkeit besondere Beachtung genießt. Eine Verbindung mit sogenannten Breathers wird dargelegt. KW - Anderson KW - Lokalisierung KW - Unordnung KW - Ausbreitung KW - Chaos KW - Anderson KW - Localization KW - Disorder KW - Spreading KW - Chaos Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-31469 ER - TY - JOUR A1 - Mulansky, Mario A1 - Picovsky, Arkady S. T1 - Re-localization due to finite response times in a nonlinear Anderson chain JF - The European physical journal : B, Condensed matter and complex systems N2 - We study a disordered nonlinear Schrodinger equation with an additional relaxation process having a finite response time tau. Without the relaxation term, tau = 0, this model has been widely studied in the past and numerical simulations showed subdiffusive spreading of initially localized excitations. However, recently Caetano et al. [Eur. Phys. J. B 80, 321 (2011)] found that by introducing a response time tau > 0, spreading is suppressed and any initially localized excitation will remain localized. Here, we explain the lack of subdiffusive spreading for tau > 0 by numerically analyzing the energy evolution. We find that in the presence of a relaxation process the energy drifts towards the band edge, which enforces the population of fewer and fewer localized modes and hence leads to re-localization. The explanation presented here relies on former findings by Mulansky et al. [Phys. Rev. E 80, 056212 (2009)] on the energy dependence of thermalized states. Y1 - 2012 U6 - https://doi.org/10.1140/epjb/e2012-21040-5 SN - 1434-6028 VL - 85 IS - 3 PB - Springer CY - New York ER - TY - JOUR A1 - Mulansky, Mario T1 - Scaling of chaos in strongly nonlinear lattices JF - Chaos : an interdisciplinary journal of nonlinear science N2 - Although it is now understood that chaos in complex classical systems is the foundation of thermodynamic behavior, the detailed relations between the microscopic properties of the chaotic dynamics and the macroscopic thermodynamic observations still remain mostly in the dark. In this work, we numerically analyze the probability of chaos in strongly nonlinear Hamiltonian systems and find different scaling properties depending on the nonlinear structure of the model. We argue that these different scaling laws of chaos have definite consequences for the macroscopic diffusive behavior, as chaos is the microscopic mechanism of diffusion. This is compared with previous results on chaotic diffusion [M. Mulansky and A. Pikovsky, New J. Phys. 15, 053015 (2013)], and a relation between microscopic chaos and macroscopic diffusion is established. (C) 2014 AIP Publishing LLC. Y1 - 2014 U6 - https://doi.org/10.1063/1.4868259 SN - 1054-1500 SN - 1089-7682 VL - 24 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - THES A1 - Mulansky, Mario T1 - Chaotic diffusion in nonlinear Hamiltonian systems T1 - Chaotische Diffusion in nichtlinearen Hamiltonschen Systemen N2 - 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. N2 - Diese Arbeit beschäftigt sich mit dem Phänomen der Diffusion in nichtlinearen Systemen. Unter Diffusion versteht man normalerweise die zufallsmä\ss ige Bewegung von Partikeln durch den stochastischen Einfluss einer thermodynamisch beschreibbaren Umgebung. Dieser Prozess ist mathematisch beschrieben durch die Diffusionsgleichung. In dieser Arbeit werden jedoch abgeschlossene Systeme ohne Einfluss der Umgebung betrachtet. Dennoch wird eine Art von Diffusion, üblicherweise bezeichnet als Subdiffusion, beobachtet. Die Ursache dafür liegt im chaotischen Verhalten des Systems. Vereinfacht gesagt, erzeugt das Chaos eine intrinsische Pseudo-Zufälligkeit, die zu einem gewissen Grad mit dem Einfluss einer thermodynamischen Umgebung vergleichbar ist und somit auch diffusives Verhalten provoziert. Zur quantitativen Beschreibung dieses subdiffusiven Prozesses wird eine Verallgemeinerung der Diffusionsgleichung herangezogen, die Nichtlineare Diffusionsgleichung. Desweiteren wird die mikroskopische Dynamik des Systems mit analytischen Methoden untersucht, und Schlussfolgerungen für den makroskopischen Diffusionsprozess abgeleitet. Die Technik der Verbindung von mikroskopischer Dynamik und makroskopischen Beobachtungen, die in dieser Arbeit entwickelt wird und detailliert beschrieben ist, führt zu einem tieferen Verständnis von hochdimensionalen chaotischen Systemen. Die mit mathematischen Mitteln abgeleiteten Ergebnisse sind darüber hinaus durch ausführliche Simulationen verifiziert, welche teilweise auf einem der leistungsfähigsten Supercomputer Europas durchgeführt wurden, dem sp6 in Bologna, Italien. Desweiteren können die in dieser Arbeit vorgestellten Erkenntnisse und Techniken mit Sicherheit auch in anderen Fällen bei der Untersuchung chaotischer Systeme Anwendung finden. KW - Chaos KW - Diffusion KW - Thermalisierung KW - Energieausbreitung KW - chaos KW - diffusion KW - thermalization KW - energy spreading Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63180 ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Dynamical thermalization of disordered nonlinear lattices N2 - We study numerically how the energy spreads over a finite disordered nonlinear one-dimensional lattice, where all linear modes are exponentially localized by disorder. We establish emergence of dynamical thermalization characterized as an ergodic chaotic dynamical state with a Gibbs distribution over the modes. Our results show that the fraction of thermalizing modes is finite and grows with the nonlinearity strength. Y1 - 2009 UR - http://pre.aps.org/ U6 - https://doi.org/10.1103/Physreve.80.056212 SN - 1539-3755 ER - TY - JOUR A1 - Mulansky, Mario A1 - Pikovskij, Arkadij T1 - Scaling properties of energy spreading in nonlinear Hamiltonian two-dimensional lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonlinear coupling, for which the scaling exponent is estimated analytically. This shows that the process of chaotic diffusion in such lattices does not require disorder. Y1 - 2012 U6 - https://doi.org/10.1103/PhysRevE.86.056214 SN - 1539-3755 VL - 86 IS - 5 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mulansky, Mario A1 - Pikovskij, Arkadij T1 - Energy spreading in strongly nonlinear disordered lattices JF - New journal of physics : the open-access journal for physics N2 - We study the scaling properties of energy spreading in disordered strongly nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled local linear or nonlinear oscillators, and demonstrate a rather slow, subdiffusive spreading of initially localized wave packets. We use a fractional nonlinear diffusion equation as a heuristic model of this process, and confirm that the scaling predictions resulting from a self-similar solution of this equation are indeed applicable to all studied cases. We show that the spreading in nonlinearly coupled linear oscillators slows down compared to a pure power law, while for nonlinear local oscillators a power law is valid in the whole studied range of parameters. Y1 - 2013 U6 - https://doi.org/10.1088/1367-2630/15/5/053015 SN - 1367-2630 VL - 15 IS - 5 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Strong and weak chaos in weakly nonintegrable many-body hamiltonian systems JF - Journal of statistical physics N2 - We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology. KW - Lyapunov exponent KW - Arnold diffusion KW - Chaos spreading Y1 - 2011 U6 - https://doi.org/10.1007/s10955-011-0335-3 SN - 0022-4715 VL - 145 IS - 5 SP - 1256 EP - 1274 PB - Springer CY - New York ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij T1 - Scaling of energy spreading in strongly nonlinear disordered lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation, we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to the pure power law. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.026205 SN - 1539-3755 VL - 83 IS - 2 PB - American Physical Society CY - College Park ER -