Fully funded by DFG Research Group 718 "Analysis and Stochastics in Complex Physical Systems", see <link http: www.math.uni-leipzig.de www fg.html external>Homepage.
In physical systems near criticality stochastic effects may have macroscopic size. I shall discuss examples where a rigorous analysis indeed shows persistence of randomness after the macroscopic limit. The examples refer to interface fluctuations in one dimensional models, spinodal decomposition and dynamical hysteresis.
In this talk I introduce and 'solve' a minimal model for the depinning of a driven interface in random media. It is shown by qualitative methods and a functional renormalization group calculation that the interface close to the depinning threshold is self affine and characterized by a number of critical exponents which obey scaling relations.
The extension of these consideration to interfaces driven by an oscillating force is briefly discussed.
The phase-field method is widely used to study evolution of microstructural phase transformations on a continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landau equation modeling the dynamics of an order parameter determining the solid and liquid phases, including also stochastic fluctuations to obtain the qualitatively correct result of dendritic side branching. This work presents a method to derive stochastic phase-field models from atomistic formulations by coarse-graining molecular dynamics. It has four steps: (1) derivation of stochastic molecular dynamics from the time-indepedent SchrÃdinger equation; (2) a precise quantitative atomistic definition of the phase-field variable, based on the local potential energy; (3) derivation of its coarse-grained dynamics model, from microscopic Smoluchowski molecular dynamics; and (4) numerical computation of the coarse-grained model functions.
We will discuss a variety of coarse-graining methods for many-body microscopic systems.
We focus on mathematical, numerical and statistical methods allowing us to assess the parameter regimes where such approximations are valid. We also demonstrate, with direct comparisons between microscopic (DNS) and coarse-grained simulations, that the derived mesoscopic models can provide a substantial CPU reduction in the computational effort.
Furthermore, we discuss the feasibility of spatiotemporal adaptivity methods for the coarse-graining of microscopic simulations, having the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. Here we will show that in some cases the adaptivity criterion can be based on a posteriori estimates on the loss of information in the transition from a microscopic to a coarse-grained system.
Finally, motivated by related problems in the simulation of macromolecular systems, we discuss mathematical strategies for reversing the coarse-graining procedure. The principal purpose of such a task is recovering local microscopic information in a large system by first employing inexpensive coarse-grained solvers.
A description of the short time behavior of solutions of the Allen-Cahn equation with a smoothened additive noise is presented. The key result is that in the sharp interface limit solutions move according to motion by mean curvature with an additional stochastic forcing. This extends a similar result of Funaki in spatial dimension 2 to arbitrary dimensions.
We discuss a stochastic partial differential equation arising as a phenomenological model in amorphous surface growth. The dynamics shows the formation of parabola shaped hills of a characteristic length scale, which then slowly coarsen.
Although numerical approximations seem to converge very fast, the equation exhibits similar problems than 3D-Navier Stokes, and it is an open question, whether the model has solutions that blow up and therefore non-uniqueness.
We will present some results for the interfacial propagation in inhomogeneous medium. The prototype equation is motion by mean curvature. The key feature is the interaction between the mean curvature of the interface and the underlying spatial inhomogeneity. We will describe the transition between the pinning and de-pinning of the interface and the existence of pulsating waves. Some recent investigations on the pinning threshold and the front propagations between patterns will also be discussed.
We return to seminal work of P.L.Lions and P.Souganidis on nonlinear stochastic partial differential equations in viscosity sense and present some evidence that rough path analysis a la T.Lyons may allow to continue, and perhaps complete, the program they started in a series of papers from 1998-2003.
The motion of an elastic manifold in a random medium is involved in many physical situations. If the driving force is increased, the interplay between elasticity and disorder results in a transition from a stable manifold pinned by the defects to a propagating one, also referred to as a depinning transition.
Here, we show how the theoretical concepts associated with such a transition can be applied to describe the behavior of three different systems: the propagation of cracks in heterogeneous materials, the motion of ferromagnetic domain walls leading to the Barkhausen effect and the motion of contact lines of liquids on solid substrates.
We introduce a multi-scale model for a two-phase material. The model is on the finest scale a stochastic process. The effective behaviour on larger scales is governed by deterministic nonlinear evolution equations. Due to the stochasticity on the finest scale, deviations from these limit evolution laws can happen with small probability. We describe the most likely among those deviations when we enforce a fast motion on a manifold of stationary solutions. The most likely path is the minimiser of an appropriate action functional.
Joint work with Giovanni Bellettini, Anna DeMasi, Dimitrios Tsagkarogiannis and Errico Presutti.
I describe the motion of interfaces in a discrete environment, obtained by coupling the minimizing movements approach of Almgren, Taylor and Wang and a discrete-to-continuous analysis. I show that, below a critical ratio of the time and space scalings there is no motion of interfaces (pinning), while above that ratio the discrete motion is approximately described by the crystalline motion by curvature on the continuum. The critical regime is quite richer, exhibiting a pinning threshold, partial pinning, quantization of the interface velocity, and non-uniqueness effects.
This is a joint work with A. Braides (Rome) and M.S. Gelli (Pisa).
Participants
Sebastian Andres
Lorenzo Bertini
Dirk Blömker
Rainer Buckdahn
Jerome Coville
Giuseppe Da Prato
Jean-Dominique Deuschel
Nicolas Dirr
Abdelhadi Es-Sarhir
Markos Katsoulakis
Wolfgang König
Thomas Nattermann
Matteo Novaga
Harald Oberhauser
Laurent R. Ponson
Errico Presutti
Manfred Salmhofer
Michael Scheutzow
Panagiotis Souganidis
Anders Szepessy
Dimitrios Tsagkarogiannis
Yvon Vignaud
Max von Renesse
Hendrik Weber
Lihu Xu
Nung Kwan Yip
Scientific Organizers
Patrick Dondl
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Stephan Luckhaus
Universität Leipzig
Max von Renesse
Technische Universität Berlin
Administrative Contact
Katja Bieling
Max Planck Institute for Mathematics in the Sciences
Contact by email
