Universal bounds on coarsening rates for some models of phase transitions

In this thesis, we prove one-sided, universal bounds on coarsening rates for three models of phase transitions by following a strategy developed by Kohn and Otto (Comm. Math. Phys. 229(2002),375-395). Our analysis for the phase-field model is performed in a regime in which the ratio between the transition layer thickness and the length scale of the pattern is small, and is also small compared to the square of the ratio between the pattern scale and the system size. The analysis extends the Kohn-Otto method to deal with both temperature and phase fields. For the mean-field models, we consider two kinds of them: one with a coarsening rate $l\sim t^{1/3}$ and the other with $l\sim t^{1/2}$. The $l\sim t^{1/2}$ rate is proved using a new dissipation relation which extends the Kohn-Otto method. In both cases, the dissipation relations are subtle and their proofs are based on a residual lemma (Lagrange identity) for the Cauchy-Schwarz inequality. The monopole approximation is a simplification of the Mullins-Sekerka model in the case when all particles are non-overlapping spheres and the centers of the particles do not move. We derive the monopole approximation and prove its well-posedness by considering a gradient flow restricted on collections of finitely many non-overlapping spheres. After that, we prove one-sided universal bounds on the coarsening rate for the monopole approximation

Phase separation in disordered exclusion models

The effect of quenched disorder in the one-dimensional asymmetric exclusion process is reviewed. Both particlewise and sitewise disorder generically induces phase separation in a range of densities. In the particlewise case the existence of stationary product measures in the homogeneous phase implies that the critical density can be computed exactly, while for sitewise disorder only bounds are available. The coarsening of phase-separated domains starting from a homogeneous initial condition is addressed using scaling arguments and extremal statistics considerations. Some of these results have been obtained previously in the context of directed polymers subject to columnar disorder.Comment: 15 pages, 4 figure

Cross-over in scaling laws: A simple example from micromagnetics

Scaling laws for characteristic length scales (in time or in the model parameters) are both experimentally robust and accessible for rigorous analysis. In multiscale situations cross--overs between different scaling laws are observed. We give a simple example from micromagnetics. In soft ferromagnetic films, the geometric character of a wall separating two magnetic domains depends on the film thickness. We identify this transition from a N\'eel wall to an Asymmetric Bloch wall by rigorously establishing a cross--over in the specific wall energy

Non-universal disordered Glauber dynamics

We consider the one-dimensional Glauber dynamics with coupling disorder in terms of bilinear fermion Hamiltonians. Dynamic exponents embodied in the spectrum gap of these latter are evaluated numerically by averaging over both binary and Gaussian disorder realizations. In the first case, these exponents are found to follow the non-universal values of those of plain dimerized chains. In the second situation their values are still non-universal and sub-diffusive below a critical variance above which, however, the relaxation time is suggested to grow as a stretched exponential of the equilibrium correlation length.Comment: 11 pages, 5 figures, brief addition

The Lifshitz-Slyozov-Wagner equation for reaction-controlled kinetics

We rigorously derive a weak form of the Lifshitz-Slyozov-Wagner equation as the homogenization limit of a Stefan-type problem describing reaction-controlled coarsening of a large number of small spherical particles. Moreover, we deduce that the effective mean-field description holds true in the particular limit of vanishing surface-area density of particles.Comment: 15 pages, LaTeX; minor revision, change of titl

Kinetics of step bunching during growth: A minimal model

We study a minimal stochastic model of step bunching during growth on a one-dimensional vicinal surface. The formation of bunches is controlled by the preferential attachment of atoms to descending steps (inverse Ehrlich-Schwoebel effect) and the ratio $d$ of the attachment rate to the terrace diffusion coefficient. For generic parameters ($d > 0$) the model exhibits a very slow crossover to a nontrivial asymptotic coarsening exponent $\beta \simeq 0.38$. In the limit of infinitely fast terrace diffusion ($d=0$) linear coarsening ($\beta$ = 1) is observed instead. The different coarsening behaviors are related to the fact that bunches attain a finite speed in the limit of large size when $d=0$, whereas the speed vanishes with increasing size when $d > 0$. For $d=0$ an analytic description of the speed and profile of stationary bunches is developed.Comment: 8 pages, 10 figure

Persistence and First-Passage Properties in Non-equilibrium Systems

In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.Comment: Review article submitted to Advances in Physics: 149 pages, 21 Figure

Recent advances in the evolution of interfaces: thermodynamics, upscaling, and universality

We consider the evolution of interfaces in binary mixtures permeating strongly heterogeneous systems such as porous media. To this end, we first review available thermodynamic formulations for binary mixtures based on \emph{general reversible-irreversible couplings} and the associated mathematical attempts to formulate a \emph{non-equilibrium variational principle} in which these non-equilibrium couplings can be identified as minimizers. Based on this, we investigate two microscopic binary mixture formulations fully resolving heterogeneous/perforated domains: (a) a flux-driven immiscible fluid formulation without fluid flow; (b) a momentum-driven formulation for quasi-static and incompressible velocity fields. In both cases we state two novel, reliably upscaled equations for binary mixtures/multiphase fluids in strongly heterogeneous systems by systematically taking thermodynamic features such as free energies into account as well as the system's heterogeneity defined on the microscale such as geometry and materials (e.g. wetting properties). In the context of (a), we unravel a \emph{universality} with respect to the coarsening rate due to its independence of the system's heterogeneity, i.e. the well-known ${\cal O}(t^{1/3})$-behaviour for homogeneous systems holds also for perforated domains. Finally, the versatility of phase field equations and their \emph{thermodynamic foundation} relying on free energies, make the collected recent developments here highly promising for scientific, engineering and industrial applications for which we provide an example for lithium batteries

Upper bounds on the coarsening rate of discrete, ill-posed nonlinear diffusion equations

We prove a weak upper bound on the coarsening rate of the discrete-in-space version of an ill-posed, nonlinear diffusion equation. The continuum version of the equation violates parabolicity and lacks a complete well-posedness theory. In particular, numerical simulations indicate very sensitive dependence on initial data. Nevertheless, models based on its discrete-in-space version, which we study, are widely used in a number of applications, including population dynamics (chemotactic movement of bacteria), granular flow (formation of shear bands), and computer vision (image denoising and segmentation). Our bounds have implications for all three applications. © 2008 Wiley Periodicals, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/61227/1/20259_ftp.pd

Scaling laws in two models for thermodynamically driven fluid flows

In this thesis, we consider two models from physics, which are characterized by the interplay of thermodynamical and fluid mechanical phenomena: demixing (spinodal decomposition) and Rayleigh--Bénard convection. In both models, we investigate the dependencies of certain intrinsic quantities on the system parameters. The first model describes a thermodynamically driven demixing process of a binary viscous fluid. During the evolution, the two components of the mixture separate into two domains of the different equilibrium volume fractions. One observes a clear tendency: Larger domains grow at the expense of smaller ones, and thus, the average domain sizes increases --- a phenomenon called coarsening. It turns out that two mechanisms are relevant for the coarsening process. At an early stage of the evolution, material transport is essentially mediated by diffusion; at a later stage, when the typical domain size exceeds a certain value, due to the viscosity of the mixture, a fluid flow sets in and becomes the relevant transport mechanism. In both regimes, the growth rates of the typical domain size obey certain power laws. In this thesis, we rigorously establish one-sided bounds on these growth rates via a priori estimates. The second model, Rayleigh--Bénard convection, describes the behavior of a fluid between two rigid horizontal plates that is heated from below and cooled from above. There are two competing heat transfer mechanisms in the system: On the one hand, thermodynamics favors a state in which temperature variations are locally minimized. Thus, in our model, the thermodynamical equilibrium state is realized by a temperature with a linearly decreasing profile, corresponding to pure conduction. On the other hand, due to differences in the densities of hot and cold fluid parcels, buoyancy forces act on the fluid. This results in an upward motion of hot parcels and a downward motion of cold parcels. We study the dependence of the average upward heat flux, measured in the so-called Nusselt number, on the temperature forcing encoded by the container height. It turns out that the efficiency of the heat transport is independent of the height of the container, and thus, the Nusselt number is a constant function of height. Using a priori estimates, we prove an upper bound on the Nusselt number that displays this dependency --- up to logarithmic errors. Further investigations on the flow pattern in Rayleigh--Bénard convection show a clear separation of length scales: Along the horizontal top and bottom plates one observes thin boundary layers in which heat is essentially conducted, whereas the large bulk is characterized by a convective heat flow. We give first rigorous results in favor of linear temperature profiles in the boundary layers, which indicate that heat is indeed essentially conducted close to the boundaries.:1 Introduction 2 Coarsening rates in binary viscous fluids 2.1 Background from physics 2.2 Background from mathematics 2.3 The model 2.4 The gradient flow structure 2.5 Heuristics 2.6 Numerical simulations 2.7 Main results 2.8 Preliminaries 2.9 Proof of upper bounds on coarsening rates 2.10 Appendix: Well-posedness and regularity of solutions 3 Scaling of the Nusselt number 3.1 Background from physics 3.2 The model and the Nusselt number 3.3 Heuristics 3.4 Main results 3.5 Scaling law in the linear regime 3.6 Preliminaries and review 3.7 Upper bound using the background field method 3.8 Upper bound using the maximum principle 3.9 Appendix: Some elementary estimates 4 The laminar boundary layer 4.1 Background, model, and motivation 4.2 Main results 4.3 Preparation: Bounds on the velocity field 4.4 On the energy distribution 4.5 Bounds on the second order derivatives of the temperature field 4.6 Bounds on the third order derivatives of the temperature fiel