32 research outputs found
Ultracold atoms in optical lattices with random on-site interactions
We consider the physics of lattice bosons affected by disordered on-site
interparticle interactions. Characteristic qualitative changes in the zero
temperature phase diagram are observed when compared to the case of randomness
in the chemical potential. The Mott-insulating regions shrink and eventually
vanish for any finite disorder strength beyond a sufficiently large filling
factor. Furthermore, at low values of the chemical potential both the
superfluid and Mott insulator are stable towards formation of a Bose glass
leading to a possibly non-trivial tricritical point. We discuss feasible
experimental realizations of our scenario in the context of ultracold atoms on
optical lattices.Comment: 4 pages, 3 eps figure
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Adaptive time domain boundary element methods and engineering applications
Time domain Galerkin boundary elements provide an efficient tool for the numerical solution of boundary value problems for the homogeneous wave equation. We review recent advances in their a posteriori error analysis and the resulting adaptive mesh refinement procedures, as well as basic algorithmic aspects of these methods. Numerical results for adaptive mesh refinements are discussed in 2 and 3 dimensions, as are benchmark problems in a half space related to the transient emission of traffic noise.Parts of this work were funded by BMWi under the project SPERoN 2020, part II, Leiser StraB enverkehr, grant number 19 U 10016 F. H. G. acknowledges support by ERC Advanced Grant HARG 268105
Space-Time fractional diffusion in cell movement models with delay
The movement of organisms and cells can be governed by occasional long distance runs, according to an approximate Lévy walk. For T cells migrating through chronically-infected brain tissue, runs are further interrupted by long pauses and the aim here is to clarify the form of continuous model equations that describe such movements. Starting from a microscopic velocity-jump model based on experimental observations, we include power-law distributions of run and waiting times and investigate the relevant parabolic limit from a kinetic equation for resting and moving individuals. In biologically relevant regimes we derive nonlocal diffusion equations, including fractional Laplacians in space and fractional time derivatives. Its analysis and numerical experiments shed light on how the searching strategy, and the impact from chemokinesis responses to chemokines, shorten the average time taken to find rare targets in the absence of direct guidance information such as chemotaxis
Spin effects in Bose-Glass phases
We study the mechanism of formation of Bose glass (BG) phases in the spin-1
Bose Hubbard model when diagonal disorder is introduced. To this aim, we
analyze first the phase diagram in the zero-hopping limit, there disorder
induces superposition between Mott insulator (MI) phases with different filling
numbers. Then BG appears as a compressible but still insulating phase. The
phase diagram for finite hopping is also calculated with the Gutzwiller
approximation. The bosons' spin degree of freedom introduces another scattering
channel in the two-body interaction modifying the stability of MI regions with
respect to the action of disorder. This leads to some peculiar phenomena such
as the creation of BG of singlets, for very strong spin correlation, or the
disappearance of BG phase in some particular cases where fluctuations are not
able to mix different MI regions
Space-time enriched finite elements for transient wave problems
We investigate a generalised finite element method for the time-dependent wave equation based on enriching the approximation space with travelling plane waves. Our approach is based on a first-order discontinuous Galerkin formulation for the wave equation, discretised with continuous elements in space and discontinuous elements in time. Enrichment in space and time allows us to circumvent the limitations due to the small time steps required for time-independent enrichments. We obtain good approximation for coarse spatial meshes and large time steps, with a corresponding reduction in computational effort. Numerical results indicate the advantages of the proposed approach and investigate the attained accuracy of our scheme as the number of enrichment functions are increased
Nonclassical Spectral Asymptotics and Dixmier Traces: from Circles to Contact Manifolds
We consider the spectral behavior and noncommutative geometry of commutators [P, f], where P is an operator of order 0 with geometric origin and f a multiplication operator by a function. When f is Holder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes\u27 residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Holder continuous functions f, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori
Optimal operator preconditioning for pseudodifferential boundary problems.
We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain Ω, where Ω is either in Rn or in a Riemannian manifold. For linear systems of equations arising from low-order Galerkin discretizations, we obtain condition numbers that are independent of the mesh size and of the choice of bases for test and trial functions. The basic ingredient is a classical formula by Boggio for the fractional Laplacian, which is extended analytically. In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach gives a unified, independent proof for a series of recent results by Hiptmair, Jerez-Hanckes, Nédélec and Urzúa-Torres. We also study the increasing relevance of the regularity assumptions on the mesh with the order of the operator. Numerical examples validate our theoretical findings and illustrate the performance of the proposed preconditioner on quasi-uniform, graded and adaptively generated meshes