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- ItemDistribution of Cracks in a Chain of Atoms at Low Temperature(Cham (ZG) : Springer International Publishing AG, 2021) Jansen, Sabine; König, Wolfgang; Schmidt, Bernd; Theil, FlorianWe consider a one-dimensional classical many-body system with interaction potential of Lennard–Jones type in the thermodynamic limit at low temperature 1/β∈(0,∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of Nexp(−βesurf/2) with esurf>0 a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations.
- ItemLocal Well-Posedness of Strong Solutions to the Three-Dimensional Compressible Primitive Equations(Berlin ; Heidelberg : Springer, 2021) Liu, Xin; Titi, Edriss S.This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. © 2021, The Author(s).
- ItemFrom Large Deviations to Semidistances of Transport and Mixing: Coherence Analysis for Finite Lagrangian Data(New York, NY : Springer, 2018) Koltai, Péter; Renger, D.R. MichielOne way to analyze complicated non-autonomous flows is through trying to understand their transport behavior. In a quantitative, set-oriented approach to transport and mixing, finite time coherent sets play an important role. These are time-parametrized families of sets with unlikely transport to and from their surroundings under small or vanishing random perturbations of the dynamics. Here we propose, as a measure of transport and mixing for purely advective (i.e., deterministic) flows, (semi)distances that arise under vanishing perturbations in the sense of large deviations. Analogously, for given finite Lagrangian trajectory data we derive a discrete-time-and-space semidistance that comes from the “best” approximation of the randomly perturbed process conditioned on this limited information of the deterministic flow. It can be computed as shortest path in a graph with time-dependent weights. Furthermore, we argue that coherent sets are regions of maximal farness in terms of transport and mixing, and hence they occur as extremal regions on a spanning structure of the state space under this semidistance—in fact, under any distance measure arising from the physical notion of transport. Based on this notion, we develop a tool to analyze the state space (or the finite trajectory data at hand) and identify coherent regions. We validate our approach on idealized prototypical examples and well-studied standard cases.
- ItemSymmetries in transmission electron microscopy imaging of crystals with strain(London : [Verlag nicht ermittelbar], 2022) Koprucki, Thomas; Maltsi, Anieza; Mielke, AlexanderTransmission electron microscopy (TEM) images of strained crystals often exhibit symmetries, the source of which is not always clear. To understand these symmetries, we distinguish between symmetries that occur from the imaging process itself and symmetries of the inclusion that might affect the image. For the imaging process, we prove mathematically that the intensities are invariant under specific transformations. A combination of these invariances with specific properties of the strain profile can then explain symmetries observed in TEM images. We demonstrate our approach to the study of symmetries in TEM images using selected examples in the field of semiconductor nanostructures such as quantum wells and quantum dots.
- ItemSpectral Theory of Infinite Quantum Graphs(Cham (ZG) : Springer International Publishing AG, 2018) Exner, Pavel; Kostenko, Aleksey; Malamud, Mark; Neidhardt, HagenWe investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.
- ItemSurface Energy and Boundary Layers for a Chain of Atoms at Low Temperature(Berlin ; Heidelberg : Springer, 2021) Jansen, Sabine; König, Wolfgang; Schmidt, Bernd; Theil, FlorianWe analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature β- 1 goes to zero. Our main results are: (1) As β→ ∞ at fixed positive pressure p> 0 , the Gibbs measures μβ and νβ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals E¯ bulk and E¯ surf. The minimizer of the surface functional corresponds to zero temperature boundary layers; (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of E¯ surf; (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts; (4) Bounds on the decay of correlations are provided, some of them uniform in β. © 2020, The Author(s).
- ItemTopology- and Geometry-Controlled Functionalization of Nanostructured Metamaterials(Basel : MDPI, 2023) Fomin, Vladimir M.; Marquardt, Oliver[no abstract available]
- ItemThermoviscoelasticity in Kelvin–Voigt Rheology at Large Strains(Berlin ; Heidelberg : Springer, 2020) Mielke, Alexander; Roubíček, TomášThe frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration by using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back to the reference configuration. The existence of weak solutions in the quasistatic setting, that is inertial forces are ignored, is shown by time discretization. © 2020, The Author(s).
- ItemCouplings via Comparison Principle and Exponential Ergodicity of SPDEs in the Hypoelliptic Setting(Berlin ; Heidelberg : Springer, 2020) Butkovsky, Oleg; Scheutzow, MichaelWe develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise of the stochastic reaction–diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer and Mattingly (Electron J Probab 16:658–738, 2011). © 2020, The Author(s).
- ItemEDP-convergence for nonlinear fast–slow reaction systems with detailed balance*(Bristol : IOP Publ., 2021) Mielke, Alexander; Peletier, Mark A.; Stephan, ArturWe consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP-convergence, i.e. convergence in the sense of the energy-dissipation principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.