Lukas Jahnke

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Steven J. Miller

Williams College

Sandra (A.C.C.) van Wijk

American University of Sharjah

Jeffrey Townsend

Yale University

Samit Chakrabarty

University of Leeds

Islam AbdulMaksoud

Alexandria University

Viacheslav Kuleshov

Stockholm University

Adebayo D Agunbiade

Olabisi Onabanjo University, Ago-Iwoye, Nigeria

María Cecilia Tomasini

Universidad de Buenos Aires

Adnan Awad

The University Of Jordan

Tesfaye Abera

Haramaya university

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Papers by Lukas Jahnke

Equidistant band formation of precipitation in a reaction–diffusion process

New Journal of Physics, 2010

... Equidistant band formation of precipitation in a reactiondiffusion process. Lukas Jahnke and... more

Consequences of fluctuations in Liesegang pattern formation

EPL (Europhysics Letters), 2008

ABSTRACT Using lattice-gas simulations we show that Liesegang pattern formation in reaction-diffu... more ABSTRACT Using lattice-gas simulations we show that Liesegang pattern formation in reaction-diffusion systems on small length scales can be described by a mean-field model, supplemented with three first-order corrections. We find that fluctuations due to thermal motion of the particles and due to impurities in the sample increase the number of Liesegang bands or even initiate patterns not forming without fluctuations. I.e., noise enhances or even induces ordering. Therefore, we suggest that it is possible to tune the number or distance of Liesegang bands by changing the impurities in the sample. Moreover, it might be possible to measure the nucleation threshold when changing the level of impurities in a micro- or nano-scale experimental setting.

The effect of a small magnetic flux on the metal-insulator transition

We numerically show that very small magnetic flux can significantly shift the metal-insulator tra... more We numerically show that very small magnetic flux can significantly shift the metal-insulator transition point in a disordered electronic system. The shift we observe for the 3d Anderson model obeys a power law as predicted by Larkin and Khmel'nitskii (1981). We compute the exponent and find good agreement with the prediction. However, the power law holds only for much smaller magnetic fields than has been previously assumed, and is accompanied by a large prefactor, leading to a surprising strong dependence of the transition point on the applied magnetic field. Furthermore, we show that the critical level-spacing distribution is identical in the presence and absence of a magnetic field if hard-wall boundary conditions are applied. This result is surprising since both cases belong to different universality classes and different distributions have been reported for periodic boundary conditions.

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Dissertation zur Erlangung des akademischen Grades

doctor rerum naturalium (Dr. rer. nat.) Influencing pattern formation and localization through di... more

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for Liesegang patterns

Comparison of models and lattice-gas simulations for Liesegang patterns

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Influencing pattern formation and localization through disorder

A quantum leap in solid state physics was the understanding that macroscopic objects can be descr... more A quantum leap in solid state physics was the understanding that macroscopic objects can be described using the symmetries of the Hamiltonian. For instance, if translational symmetry is present, the atomic structure of the macroscopic objects is described by the solution of a unit cell. Disorder, for example a random potential landscape, breaks translational symmetry. Other symmetries are untouched by disorder, such as orthogonal or unitary transformations of the Hamiltonian. Physicists in the seventies and eighties of the last century realized that phase transitions even in disordered solids can be described in a scaling ansatz. This scaling is due to the underlying symmetry of the Hamiltonian and is therefore universal. One type of phase transition in disordered solids is the disorder-induced metal-insulator transition. A metallic solid is transformed to an insulator by an increase of disorder. Therefore the initially extended electronic wave function localizes. Disorder also has an impact on classical systems, for example, reaction-diffusion systems. Especially interesting is the effect of disorder and fluctuations in systems where ordering mechanisms occur, as in the case of phase separation of particles. In the last years, trend words as self-assembly, self-organization, and pattern formation were used to describe different kinds of ordering reactions. These are investigated driven by the motivation to design devices using reaction-diffusion dynamics. The Liesegang pattern formation is a prominent example, due to the simple descriptions of pattern distances and widths. Commonly, Liesegang patterns are described by mean-field approximations. In such scenarios, the concentrations of particles are approximated as smooth functions and the reaction is described by differential equations. In nature, fluctuations of the particle concentrations are present due to disorders such as defects, impurities etc. An important progress achieved in the last decade is the understanding that fluctuations do not necessarily destroy pattern formation. In contrary, they are able to enhance or induce pattern formation and order. The publications presented in Part III are the core of this thesis. In Part I the general context of these publications is introduced, while in Part II they are summarized and discussed in relation with each other. Part I is divided into three chapters. In Chapter 1 different examples of compositionally and structural-topologically disordered systems are described, starting with percolation. Realizations of the tight-binding approximation for different types of disorders are introduced, for instance, the Anderson model, a binary alloy system, quantum percolation, and the random phase model. Gels and glasses are mesoscopic examples of structurally disordered systems. Their properties, especially their local structures, are described. At last, the general properties of complex networks are introduced with emphasis on the description of clustering. In Chapter 2 the transport dynamics of particles in disordered systems are discussed. At first the tight-binding model for electron transport in disordered media is derived. Results for applications are presented, for example, for quantum percolation, and for the Anderson model with and without magnetic field. Furthermore it is shown that the tight-binding model is not only valid for quantum wave transport in disordered media, but also for classical waves. Finally reactiondiffusion dynamics are derived and discussed with focus on the influence of disorder. Chapter 3 is devoted to the understanding of Liesegang pattern formation and the localization of electronic wave functions in disordered media. In the context of Liesegang pattern formation vii four important laws are introduced and mean-field predictions are discussed. The section on localization deals with a scaling ansatz to describe the metal-insulator transition. Additionally, a common statistical method is introduced called level spacing statistic. Eventually, numerical and experimental studies dealing with the critical parameters of the metal-insulator transition are discussed. In Part II the publication presented in Part III are summarized and discussed. Chapters 4 to 6 deal with investigations of Liesegang pattern formation in disordered materials, whereas the aim of Chapters 7 to 9 is to understand the affect of a small magnetic field and of topologicalstructural disorder for metal-insulator phase transitions. Chapter 4 is a detailed review of different models on Liesegang pattern formation. In addition, a lattice-gas model to simulate Liesegang patterns is introduced. Different methods to analyze simulation and experimental results are presented. In Chapter 5 the lattice-gas model is used to investigate how disorder and fluctuations effect the pattern formation process. Additionally, effective mean-field descriptions incorporating disorder and fluctuations are derived. In Chapter 6 these results are applied for designing equidistant and more complex patterns. Chapter 7 is the first publication dealing with calculations on localization effects. A prediction for the shift of the metal-insulator transition induced by a small magnetic field is tested in extensive numerical calculation and confirmed. Finally, the topic of Chapters 8 and 9 is the influence of topological disorder on metal-insulator and optical localization transitions. Topological disorder is introduced, using different types of networks and variations of local topological structures in complex networks. At last, the bibliography and the acknowledgments complete this thesis.

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Comparison of models and lattice-gas simulations for Liesegang patterns

The European Physical Journal Special Topics, 2008

Page 1. Eur. Phys. J. Special Topics 161, 121–141 (2008) c EDP Sciences, Springer-Verlag 2008 DOI... more

Complete Relativistic Description of the N*(1520)

Wave Localization on Complex Networks

Towards an Information Theory of Complex Networks, 2011

In this chapter we consider the role played by Anderson localization in transport through complex... more In this chapter we consider the role played by Anderson localization in transport through complex networks, for example, an optical network. The network is described by a tight binding Hamiltonian, which may be used to determine the properties of the Anderson transition according to the statistical properties of its eigenvalues. The Anderson transition properties of different complex networks will be studied, emphasizing the role played by clustering on the localization of waves. We shall show that new complex topologies lead to novel physics, specifically clustering may lead to localization.

Wave Localization Transitions in Complex Systems

by Lukas Jahnke and Jan Kantelhardt

Volume 3, 2010

Wave Localization in Complex Networks with High Clustering

Physical Review Letters, 2008

We show that strong clustering of links in complex networks, i.e., a high probability of triadic ... more We show that strong clustering of links in complex networks, i.e., a high probability of triadic closure, can induce a localization-delocalization quantum phase transition (Anderson-like transition) of coherent excitations. For example, the propagation of light wave packets between two distant nodes of an optical network (composed of fibers and beam splitters) will be absent if the fraction of closed triangles exceeds a certain threshold. We suggest that such an experiment is feasible with current optics technology. We determine the corresponding phase diagram as a function of clustering coefficient and disorder for scalefree networks of different degree distributions PðkÞ $ k À . Without disorder, we observe no phase transition for < 4, a quantum transition for > 4, and an additional distinct classical transition for > 4:5. Disorder reduces the critical clustering coefficient such that phase transitions occur for smaller .

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Relativistic description of the

Nuclear Physics A, 2006

A relativistic description of spin 3/2 resonances and their decay channels is presented by calcul... more A relativistic description of spin 3/2 resonances and their decay channels is presented by calculating their selfenergies and spectral functions. The full vector-spinor structure is taken into account. Special emphasis is put on the N * (1520) and its decay channels πN , ρN and π∆. All interactions are formulated such that only the correct number of degrees of freedom of a spin 3/2 state is propagated. The obtained results are compared with several approximations frequently used to avoid the complicated vector-spinor structure. Since this structure is taken fully into account here, the quality of the approximations can be judged.

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Consequences of fluctuations in Liesegang pattern formation

by Lukas Jahnke and Jan Kantelhardt

EPL (Europhysics Letters), 2008