We discuss the hydrodynamic limit for the Ginzburg-Landau interface model. Under the assumption that the microscopic interaction is strictly convex, the study of the asymptotic behavior for the stochastic dynamics, including the hydrodynamic limit, is highly developed. The aim of this talk is to discuss the behavior of the interface model without the assumption of strict convexity of the potential, and to derive the hydrodynamic limit. Our analysis is based on a recent paper with Codina Cotar where the unicity of the extremal gradient Gibbs measure and the strict convexity of the surface tension is shown in the high temperature regime.
This is a joint work with T. Nishikawa and Y. Vignaud.
There are only a few models of particle systems in Rd where a rigorous proof of phase transition is availalble. I will briefly describe the models, sketch the proofs and then focus on the approach proposed by Kac. Its implementation leads to an extension of the Pirogov Sinai theory to perturbations of mean field where techniques of probability theory are complemented with techniques of variational problems and control theory. I will then conclude with some open problems.
In the context of a tight-binding approximation of the Gross-Pitaevskii energy functional with a random background potential we want to discuss in dependence on the interaction coupling constant a criteria when the Gross-Pitaevskii ground state and the single particle ground state coincide.
Effective interface models - seen as gradient fields - enable one to study effective phase coexistence. In the probabilistic setting gradient fields involve the study of strongly correlated random variables. One major problem has been open for several decades. What can be proved for the free energy and the Gibbs states for non-convex interactions given a non-vanishing tilt at the boundary? We present in the talk the first break through for low temperature using Gaussian measures and renormalisation group techniques yielding an analysis in terms of dynamical systems. We outline also the connection to the Cauchy-Born rule which states that the deformation on the atomistic level is locally given by an affine deformation at the boundary. Work in cooperation with R. Kotecky and S Müller.
I will discuss some results on gradient models with non-convex interactions. In a specific example, which allows a reformulation by means of Gaussian gradient model with random coupling constants, I will show that in the absence of convexity, there could be more than one gradient Gibbs measures with zero tilt. Nonetheless, the large scale fluctuation structure in all such measures is still that of Gaussian Free Field. I will attempt to provide a sketch of some proofs. Based on joint works with R. Kotecky and H. Spohn.
We study existence and a priori estimates of invariant measures $\mu$ for a class of semilinear stochastic differential equation with additive noise on a separable Hilbert space. Furthermore, we discuss the corresponding parabolic Cauchy-problem in $L^1(mu)$. Particular emphasis will be put on stochastic reaction diffusion, and Cahn-Hilliard equations.
The electronic Schrödinger equation plays a fundamental role in molecular physics. It describes the stationary nonrelativistic behaviour of a quantum mechanical N-electron system in the electric field generated by the nuclei. The (Projected) Coupled Cluster Method has been developed for the numerical computation of the ground state energy and wave function. It provides a powerful tool for high accuracy electronic structure calculations. The talk aims to provide a rigorous analytical treatment and convergence analysis of this method. If the discrete Hartree-Fock solution is sufficiently good, the quasi-optimal convergence of the projected coupled cluster solution to the full CI (Configuration Interaction) solution is shown. Under reasonable assumptions also the convergence to the exact wave function can be shown in the Sobolev H^1-norm. The error of the ground state energy computation is estimated by an Aubin-Nitsche-type approach. Although the Projected Coupled Cluster method is nonvariational it shares advantages with the Galerkin or CI method. In addition it provides size consistency, which is considered as a fundamental property in many particle quantum mechanics.
There is broad agreement that global climate change may have substantial impacts on water resources. As multiple countries share a river, the likelihood of a water resource conflict stemming from climate change is higher in a transboundary setting. In this talk, using the framework of a stochastic Stackelberg differential game, we will explore the scope of cooperative bargain between an upstream and a downstream country over the level of transboundary water sharing by negotiating some non-water related issues of mutual interest to both the countries, given uncertainty in the river flow due to climate change. [Joint work with A.~Bhaduri (University of Bonn), J.~Liebe (University of Bonn), E.~Barbier (University of Wyoming)].
Over the last ten years it has become understood how many diffusion equations can be equipped with a gradient-flow structure, based on the entropy as the driving force and the Wasserstein distance as the opposing 'brake'. In this talk I shall describe recent efforts to also bring reactions into this structure, by describing the reaction process as a Brownian motion in a chemical-energy landscape.
We give a brief survey on the construction and a few properties of Feller processes. We will then explain a (also numerically useful) approximation scheme for Feller processes. (Joint work with B. Böttcher, TU Dresden).
In this talk we discuss a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density $\frac{1}{2}$. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \in (0,1)$. The copolymer in the emulsion has an energy that is minus $\alpha$ times the number of $AA$-matches minus $\beta$ times the number of $BB$-matches. We will consider both the supercritical regime (oil droplets form an infinite cluster) and the subcritical regime (no infinite cluster).
We study the role of defect in the quasistatic evolution of martensitic phase boundaries. This transformation involves a change in shape of the underlying crystal. Therefore, the propagation of the phase boundary is accompanied by an evolving mechanical stress and strain field. We derive, in the sense of Gamma-convergence, an approximate model for a shallow slope phase boundary. We show that, in this quasilinear approximation, the evolution reduces to a one- dimensional problem that exhibits stick-slip behavior, and thus gives rise to hysteresis. We also present numerical simulations of the de- pinning transition showing a power-law behavior in the average velocity.
We study the relation between long cycles and Bose-Condensation in the Infinite range Bose-Hubbard model with a hard core. We calculate the density of particles on long cycles in the thermodynamic limit and find that the existence of a non-zero long cycle density coincides with the occurence of Bose-Einstein condensation but this density is not equal to that of the Bose condensate. We give some partial results for the Infinite range Bose-Hubbard model without a hard core.
Bose-Einstein condensation (BEC) of the free boson gas was predicted in 1925 and rehabilitated only in 1938 by F. London, who was motivated by superfluidity of the liquid Helium-4. BEC is a subtle collective quantum phenomenon which can be mathematically expressed as formation of a coherent state (vector) in the boson Fock space. Recent mathematical studies showed that the structure of the BEC may be more complicated (generalized BEC a la van den Berg-Lewis-Pule or a dynamical condensation) than the one predicted in 1925. This concerns the free Bose gas, as well as the systems with particle interactions or embedded in external potentials, like in the recent experiments with bosons in traps (Nobel Prize 2001) and an important progress in the mathematical description of these systems by Lieb-Seiringer-Solovej-Yngvason. In spite of that the BEC is still a challenging problem of Mathematical Physics. In my lecture I am going to discuss mathematical problems related to only one particular question: how can BEC be modified by external potentials? I give a review of some cases in which one can prove that it survives (and even amplifies), including the cases of traps, random potentials and magnetic/electric fields.
After an elementary general introduction to the subject, we describe the difficulties arising in the context of the microscopic Kawasaki dynamics. Then we try to give some ideas that should lead to the solution of the various problems.
Many material systems display fascinating structures that are related to the existence of orientational degrees of freedom. In this talk we address the analytical and numerical challenges in the simulation of these systems. Examples include nematic or smectic elastomers and membranes in the gel phase.
We construct a canonical reversible process $(\mu_t)_{t\ge0}$ on the $L^2$-Wasserstein space of probability measures P(R), regarded as an infinite dimensional Riemannian manifold. This process has an invariant measure $P^\beta$ which may be characterized as the 'uniform distribution' on P(R) with weight function $exp(-\beta Ent(.|m))/Z$ where m denotes a given finite measure on R.One of the key results is the quasi-invariance of this measure $P^\beta$ under push forwards $\mu\mapsto h_*\mu$ by means of smooth diffeomorphisms h of R.
In the 1980ies, Diaconis asked whether linearly edge-reinforced random walk on Z^2 is recurrent. This problem is still open. In the talk, I will give an overview on related recent results on linearly edge-reinforced-random walks. In particular, the main ideas for proving recurrence of linearly edge-reinforced-random walks on some two-dimensional graphs will be presented.
The problem of convergence of cluster (Mayer) expansions has a long history, and several different methods were used by various authors to get reasonable estimates. The conclusion of some recent developments seems to be that possibly the most powerful, and at the same time the simplest, method is the purely combinatorial one. I will show the connection of this method with exactly soluble cases (determinants, Ising model) and will also suggest the possibility to apply the method to establish the nonabsolute convergence of some interesting cluster expansion series, appearing in the perturbation theory of masslesss Gaussians.
I will discuss models of directed d-dimensional polymers interacting with a 1-dimensional defect (e.g. (1+1)-dimensional wetting models) in presence of quenched randomness. These may be seen as renewal processes perturbed by disorder. These models undergo a localization/delocalization phase transition. I will discuss heuristic predictions and rigorous results concerning the relation between the (quenched) critical point and critical exponents to the critical point and critical exponents of the corresponding (easy) annealed model. In particular, I will present a simple method, based on the estimation of non-integer moments of the partition function, whici allows to prove that quenched and annealed critical points differ in some situations, and to find the asymptotics of the critical point for large disorder. In particular, in the large disorder limit this makes rigorous some heuristic renormalization-group predictions made previously in the physics literature. If time allows, I will also present related recent results (obtained in collaboration with Giambattista Giacomin and Hubert Lacoin) about a hierarchical version of these models.
We consider the real-valued Gaussian field on the $d$-dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. Such a field can be interpreted as a model for a d-dimensional interface in d+1-dimensional space. For the model we consider, d=4 is critical in the sense that in higher dimensions, the infinite volume Gibbs measure exists, but not in d=4 and below. Understanding the model requires good estimates on the Green's function of a discrete biharmonic boundary value problem. In this talk, I will present the analytical and probabilistic methods we use to address this problem, and hint at how these results are used to investigate the effect of a 'hard wall' on the interface, requiring the field to be positive inside a certain region.
We study probability distributions on cycles representating permutations of finitely many elements. These distributions are defined through Feynman-Kac formulae for traces of certain trace class operators studied in quantum statistical mechanics. Large deviations results are obtained for continuous and lattice systems. We discuss various definitions of Bose-Einstein condensation and their probabilistic interpretation.
Many rate-independent evolution systems can be described by an energy strorage functional and a dissipation functional. Thus, there is a similar geometric structure like in gradient flows; but now the dissipation is positively homogeneous of degree 1 and not 2 like for gradient flows. These functionals may depend on small parameters, for instance due to a periodic inhomogeneities, singular perturbations, regularizing terms or due to numerical discretization. We present abstract conditions that guarantee the convergence of solutions of these problems for the parameter going to 0 to the solutions associated with the limit functionals obtained as suitable Gamma limits. Application to a two-scale homogenization for elastoplasticity is discussed. (The talk is based on joint work with Michael Ortiz, Ulisse Stefanelli, Tomas Roubicek and Aida Timofte.)
We investigate spectral characteristics of Markov chains that exhibit ageing. We consider two rather systems with rather different properties, Bouchaud's trap model and Sinai's random walk, and show how in both cases it is possible to obtain enough information on eigennvalues and eigenfunctions to deduce in an easy way all relevant dynamical properties.
Bounds are obtained for the heat content of an open set in a complete Riemannian manifold, provided the Dirichlet Laplace Beltrami operator satisfies a strong Hardy inequality. Brownian motion tools are used to obtain asymptotic results in various examples.
We give a review on recent results concerning disordered models ranging form Anderson-like disorder to aperiodic order. A central theme will be the integrated density of states.