arXiv subjects

Uwe Grimm

Publications and source records attributed to Uwe Grimm.

At least 55 records · Page 3Linked to original sources

arXiv2009

Wave Packet Dynamics, Ergodicity, and Localization in Quasiperiodic Chains

In this paper, we report results for the wave packet dynamics in a class of quasiperiodic chains consisting of two types of weakly coupled clusters. The dynamics are studied by means of the return probability and the mean square displacement. The wave packets show anomalous diffusion in a stepwise process of fast expansion followed by time intervals of confined wave packet width. Applying perturbation theory, where the coupling parameter v is treated as perturbation, the properties of the eigenstates of the system are investigated and related to the structure of the chains. The results show the appearance of non-localized states only in sufficiently high orders of the perturbation expansions. Further, we compare these results to the exact solutions obtained by numerical diagonalization. This shows that eigenstates spread across the entire chain for v>0, while in the limit v->0 ergodicity is broken and eigenstates only spread across clusters of the same type, in contradistinction to trivial localization for v=0. Caused by this ergodicity breaking, the wave packet dynamics change significantly in the presence of an impurity offering the possibility to control its long-term dynamics.

cond-mat.mes-hall↗

arXiv2009

Surprises in aperiodic diffraction

Mathematical diffraction theory is concerned with the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Moreover, the phenomenon of homometry shows various unexpected new facets. Here, we report on some of the recent results in an exemplary and informal fashion.

math-ph↗

arXiv2009

Can Kinematic Diffraction Distinguish Order from Disorder?

Diffraction methods are at the heart of structure determination of solids. While Bragg-like scattering (pure point diffraction) is a characteristic feature of crystals and quasicrystals, it is not straightforward to interpret continuous diffraction intensities, which are generally linked to the presence of disorder. However, based on simple model systems, we demonstrate that it may be impossible to draw conclusions on the degree of order in the system from its diffraction image. In particular, we construct a family of one-dimensional binary systems which cover the entire entropy range but still share the same purely diffuse diffraction spectrum.

math-ph↗

arXiv2008

On the entropy and letter frequencies of powerfree words

We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution obtained by an enumeration of binary cubefree words up to length 80.

math.CO↗

arXiv2008

The singular continuous diffraction measure of the Thue-Morse chain

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product.

math.DS↗

arXiv2008

Homometric Point Sets and Inverse Problems

The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the idealised situation of perfect diffraction from an infinite structure. Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron's covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane. The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We expand on this example by constructing a family of mixed systems with fixed diffraction image but varying entropy. We also outline how this generalises to higher dimension.

math.MG↗

arXiv2007

Coincidence rotations of the root lattice $A_4$

The coincidence site lattices of the root lattice $A_4$ are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of $A_4$ into the icosian ring with its rich arithmetic structure, which recently (arXiv:math.MG/0702448) led to the classification of the similar sublattices of $A_4$.

math.MG↗

arXiv2006

A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns

Diffraction images with continuous rotation symmetry arise from amorphous systems, but also from regular crystals when investigated by powder diffraction. On the theoretical side, pinwheel patterns and their higher dimensional generalisations display such symmetries as well, in spite of being perfectly ordered. We present first steps and results towards a general frame to investigate such systems, with emphasis on statistical properties that are helpful to understand and compare the diffraction images. An alternative substitution rule for the pinwheel tiling, with two different prototiles, permits the derivation of several combinatorial and spectral properties of this still somewhat enigmatic example. These results are compared with properties of the square lattice and its powder diffraction.

math.SP↗

arXiv2006

Homometric model sets and window covariograms

Two Delone sets are called homometric when they share the same autocorrelation or Patterson measure. A model set LAMBDA within a given cut and project scheme is a Delone set that is defined through a window W in internal space. The autocorrelation measure of LAMBDA is a pure point measure whose coefficients can be calculated via the so-called covariogram of W. Two windows with the same covariogram thus result in homometric model sets. On the other hand, the inverse problem of determining LAMBDA from its diffraction image ultimately amounts to reconstructing W from its covariogram. This is also known as Matheron's covariogram problem. It is well studied in convex geometry, where certain uniqueness results have been obtained in recent years. However, for non-convex windows, uniqueness fails in a relevant way, so that interesting applications to the homometry problem emerge. We discuss this in a simple setting and show a planar example of distinct homometric model sets.

math.MG↗

arXiv2006

Pinwheel patterns and powder diffraction

Pinwheel patterns and their higher dimensional generalisations display continuous circular or spherical symmetries in spite of being perfectly ordered. The same symmetries show up in the corresponding diffraction images. Interestingly, they also arise from amorphous systems, and also from regular crystals when investigated by powder diffraction. We present first steps and results towards a general frame to investigate such systems, with emphasis on statistical properties that are helpful to understand and compare the diffraction images. We concentrate on properties that are accessible via an alternative substitution rule for the pinwheel tiling, based on two different prototiles. Due to striking similarities, we compare our results with the toy model for the powder diffraction of the square lattice.

math-ph↗

arXiv2006

Freely forming groups: Trying to be rare

A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components.

q-bio.PE↗

arXiv2006

Surprises in approximating Levenshtein distances

The Levenshtein distance is an important tool for the comparison of symbolic sequences, with many appearances in genome research, linguistics and other areas. For efficient applications, an approximation by a distance of smaller computational complexity is highly desirable. However, our comparison of the Levenshtein with a generic dictionary-based distance indicates their statistical independence. This suggests that a simplification along this line might not be possible without restricting the class of sequences. Several other probabilistic properties are briefly discussed, emphasizing various questions that deserve further investigation.

q-bio.QM↗

arXiv2006

Bravais colourings of planar modules with N-fold symmetry

The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N-fold symmetry that emerge as the rings of integers in cyclotomic fields with class number one. The counting functions are multiplicative and can be encapsulated in Dirichlet series generating functions, which turn out to be the Dedekind zeta functions of the corresponding cyclotomic fields.

math.CO↗

arXiv2006

Multiple planar coincidences with N-fold symmetry

Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.

math.MG↗

arXiv2005

Averaged coordination numbers of planar aperiodic tilings

We consider averaged shelling and coordination numbers of aperiodic tilings. Shelling numbers count the vertices on radial shells around a vertex. Coordination numbers, in turn, count the vertices on coordination shells of a vertex, defined via the graph distance given by the tiling. For the Ammann-Beenker tiling, we find that coordination shells consist of complete shelling orbits, which enables us to calculate averaged coordination numbers for rather large distances explicitly. The relation to topological invariants of tilings is briefly discussed.

math-ph↗

arXiv2004

Level-spacing distributions of the Gaussian unitary random matrix ensemble

Level-spacing distributions of the Gaussian Unitary Ensemble (GUE) of random matrix theory are expressed in terms of solutions of coupled differential equations. Series solutions up to order 50 in the level spacing are obtained, thus providing a very good description of the small-spacing part of the level-spacing distribution, which can be used to make comparisons with experimental or numerical data. The level-spacing distributions can be obtained by solving the system of differential equations numerically.

cond-mat.dis-nn↗

arXiv2004

Maximum principle and mutation thresholds for four-letter sequence evolution

A four-state mutation-selection model for the evolution of populations of DNA-sequences is investigated with particular interest in the phenomenon of error thresholds. The mutation model considered is the Kimura 3ST mutation scheme, fitness functions, which determine the selection process, come from the permutation-invariant class. Error thresholds can be found for various fitness functions, the phase diagrams are more interesting than for equivalent two-state models. Results for (small) finite sequence lengths are compared with those for infinite sequence length, obtained via a maximum principle that is equivalent to the principle of minimal free energy in physics.

q-bio.PE↗

arXiv2003

The modified XXZ Heisenberg chain, conformal invariance, surface exponents of c<1 systems, and hidden symmetries of the finite chains

The spin-1/2 XXZ Heisenberg chain with two types of boundary terms is considered. For the first type, the Hamiltonian is hermitian but not for the second type which includes the U_{q}[SU(2)] symmetric case. It is shown that for a certain `tuning' between the anisotropy angle and the boundary terms the spectra present unexpected degeneracies. These degeneracies are related to the structure of the irreducible representations of the Virasoro algebras for c<1.

hep-th↗