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Start date: 1984 × Subject: phase transitions ×

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Now showing 1 - 10 of 28
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    Uniformly positive correlations in the dimer model and phase transition in lattice permutations in $mathbbZ^d, d > 2$, via reflection positivity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Taggi, Lorenzo
    Our first main result is that correlations between monomers in the dimer model in ℤd do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices.
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    Entropic solutions to a thermodynamically consistent PDE system for phase transitions and damage
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Rocca, Elisabetta; Rossi, Riccarda
    In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as entropic, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its entropic formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.
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    In-situ transmission electron microscopy on high-temperature phase transitions of Ge-Sb-Te alloys
    (Berlin : Humboldt-Universität zu Berlin, 2018) Berlin, Katja
    Das Hochtemperaturverhalten beeinflusst viele verschiedene Prozesse von der Materialherstellung bis hin zur technologischen Anwendung. In-situ Transmissionselektronenmikroskopie (TEM) bietet die Möglichkeit, die atomaren Prozesse während struktureller Phasenübergänge direkt und in Realzeit zu beobachten. In dieser Arbeit wurde in-situ TEM angewendet, um die Reversibilität des Schmelz- und Kristallisationsprozesses, sowie das anisotropen Sublimationsverhaltens von Ge-Sb-Te (GST) Dünnschichten zu untersuchen. Die gezielte Probenpräparation für die erfolgreiche Beobachtung der Hochtemperatur-Phasenübergänge wird hervorgehoben. Die notwendige Einkapselung für die Beobachtung der Flüssigphase unter Vakuumbedingungen und die erforderliche sauberer Oberfläche für den Sublimationsprozess werden detailliert beschrieben. Außerdem wird die Elektronenenergieverlustspektroskopie eingesetzt um die lokale chemische Zusammensetzung vor und nach den Übergängen zu bestimmen. Die Untersuchung der Grenzflächenstruktur und Dynamik sowohl beim Phasenübergang fest-flüssig als auch flüssig-fest zeigt Unterschiede zwischen den beiden Vorgängen. Die trigonale Phase von GST weist beim Schmelzen eine teilweise geordnete Übergangszone an der fest-flüssig-Grenzfläche auf, während ein solcher Zwischenzustand bei der Erstarrung nicht entsteht. Außerdem läuft der Schmelzvorgang zeitlich linear ab, während die Kristallisation durch eine Wurzelabhängigkeit von der Zeit mit überlagerter Start-Stopp-Bewegung beschrieben werden kann. Der Einfluss der Substrat-Grenzfläche wird diskutiert und die Oberflächenenergie von GST bestimmt. Die anisotrope Dynamik führt beim Phasenübergang fest-gasförmig der kubischen Phase von GST zur Ausbildung stabiler {111} Facetten. Dies erfolgt über die Bildung von Kinken und Stufen auf stabilen Terrassen. Die Keimbildungsrate und die bevorzugten Keimbildungsorte der Kinken wurden identifiziert und stimmen mit den Voraussagen des Terrassen-Stufen-Kinken Modells überein.
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    Macroscopic loops in the $3d$ double-dimer model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Quitmann, Alexandra; Taggi, Lorenzo
    The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in ℤ d, d>2, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from [Betz, Taggi] that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when d >2, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.
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    Simulation der Strahlhärtung von Stahl mit WIAS-SHarP
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2002) Buchwalder, A.; Hömberg, D.; Jurke, Th.; Spies, H.-J.; Weiss, W.
    Die Software WIAS-SHarP zur Simulation der Oberflaechenhaertung von Stahl mit Laser- und Elektronenstrahl wurde im Rahmen eines zweijaehrigen interdisziplinaeren Forschungsprojektes entwickelt. Das zugrunde liegende mathematische Modell besteht aus einem System gewoehnlicher Differentialgleichungen zur Beschreibung der Gefuegeumwandlungen, gekoppelt mit einer nichtlinearen Waermeleitungsgleichung sowie Komponenten zur Beschreibung der Energieeinkopplung. Um eine moeglichst breite Anwendbarkeit der Software zu gewaehrleisten, wurden werkstoffspezifische Kennwerte zum Umwandlungsverhalten fuer eine grosse Anzahl praxisrelevanter Staehle bereitgestellt. Zur Modellverifikation wurden experimentelle Untersuchungen bei beteiligten Industriepartnern durchgefuehrt und mit den entsprechenden Simulationsrechnungen verglichen.
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    Intrinsische Eigenschaften ferromagnetischer Phasen : Magnetwerkstoffe ; Abschlussbericht
    (Dresden : Leibniz-Institut für Festkörper- und Werkstoffforschung, 2000) Handstein, Axel; Müller, Karl-Hartmut
    [no abstract available]
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    Hysteresis and phase transition in many-particle storage systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Dreyer, Wolfgang; Guhlke, Clemens; Herrmann, Michael
    We study the behavior of systems consisting of ensembles of interconnected storage particles. Our examples concern the storage of lithium in many-particle electrodes of rechargeable lithium-ion batteries and the storage of air in a system of interconnected rubber balloons. We are particularly interested in those storage systems whose constituents exhibit non-monotone material behavior leading to transitions between two coexisting phases and to hysteresis. In the current study we consider the case that the time to approach equilibrium of a single storage particle is much smaller than the time for full charging of the ensemble. In this regime the evolution of the probability to find a particle of the ensemble in a certain state, may be described by a nonlocal conservation law of Fokker-Planck type. Two constant parameter control whether the ensemble transits the 2-phase region along a Maxwell line or along a hysteresis path or if the ensemble shows the same non-monotone behavior as its constituents.
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    Hysteresis in the context of hydrogen storage and lithium-ion batteries
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Dreyer, Wolfgang; Guhlke, Clemens; Huth, Robert
    The processes of reversible storage of hydrogen in a metal by loading and unloading and of charging and discharging of lithium-ion batteries have many things in common. The both processes are accompanied by a phase transition and loading and unloading run along different paths, so that hysteretic behavior is observed. For hydrogen storage we consider a fine powder of magnesium (Mg) particles and lithium storage is studied for iron phosphate (FePO_4) particles forming the cathode of a lithium-ion battery. The mathematical models that are established in citeDGJ08 and citeDGH09a, describe phase transitions and hysteresis exclusively in a single particle and on that basis they can predict the observed hysteretic plots with almost horizontal plateaus. Interestingly the models predict that the coexistence of a 2-phase system in an individual particle disappears, if its size is below a critical value. However, measurements reveal that this is qualitatively not reflected by the mentioned hysteretic plots of loading and unloading. In other words: The behavior of a storage system consisting of many particles is qualitatively independent of the fact whether the individual particles itself develop a 2-phase system or if they remain in a single phase state. This apparent paradoxical observation will be resolved in this article. It will be shown that if each of the individual particles homogeneously distributes the supplied matter, nevertheless the many particle ensemble exhibits phase transition and hysteresis, because one of the two phases is realized in some part of the particles while the remaining part is in the other phase.
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    Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Cozzi, Matteo; Valdinoci, Enrico
    We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.
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    Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Cozzi, Matteo; Dipierro, Serena; Valdinoci, Enrico
    This paper contains three types of results: the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.