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Events
Analysis of Stochastic Surface Evolution - From Microscopic Models to Large Scale Behaviour

MPI für Mathematik in den Naturwissenschaften Leipzig

G3 10 (Lecture hall)

conference

29.01.09 31.01.09

Analysis of Stochastic Surface Evolution - From Microscopic Models to Large Scale Behaviour

Fully funded by DFG Research Group 718 "Analysis and Stochastics in Complex Physical Systems".

Speakers

Lorenzo Bertini

Università di Roma La Sapienza

Dirk Blömker

Universität Augsburg

Rainer Buckdahn

Université de Bretagne Occidentale

Giuseppe Da Prato

SNS Pisa

Nicolas Dirr

University of Bath

Markos Katsoulakis

U. of Massachusetts at Amherst

Thomas Nattermann

Universität zu Köln

Matteo Novaga

University of Pisa

Harald Oberhauser

University of Cambridge

Laurent R. Ponson

California Institute of Technology

Errico Presutti

Universitá di Roma II

Panagiotis Souganidis

University of Chicago

Anders Szepessy

KTH Stockholm

Hendrik Weber

Universität Bonn

Nung Kwan Yip

Purdue University

Program

10:00 - 10:30	Coffee break
10:30 - 11:30	Errico Presutti (Universitá di Roma II, Italy) Randomness in macroscopic equations 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.
11:30 - 12:30	Thomas Nattermann (Universität zu Köln, Germany) Interface depinning in random media: a physicist's approach 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.
12:30 - 14:30	Lunch
14:30 - 15:30	Anders Szepessy (KTH Stockholm, Sweden) A Stochastic Phase-Field Model Derived from Molecular Dynamics 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.
15:30 - 16:30	Markos Katsoulakis (U. of Massachusetts at Amherst, USA) Hierarchical and multi-level coarse-graining methods 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.
16:30 - 17:00	Coffee break
17:00 - 18:00	Hendrik Weber (Universität Bonn, Germany) Sharp interface limit for a stochastic Allen-Cahn Equation 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.

09:00 - 10:00	Giuseppe Da Prato (SNS Pisa, Italy) Some results on stochastic porous media equations
10:00 - 10:30	Coffee break
10:30 - 11:30	Dirk Blömker (Universität Augsburg, Germany) On an SPDE describing amorphous surface growth 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.
11:30 - 12:30	Nung Kwan Yip (Purdue University, USA) Interfacial Propagation in Inhomogeneous Medium, Some Results and Questions 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.
12:30 - 14:30	Lunch
14:30 - 15:30	Panagiotis Souganidis (University of Chicago, USA) Moving fronts in self averaging media I will review results about the homogenized/long time behavior of fronts moving in self averaging (periodic/stationary) media.
15:30 - 16:30	Rainer Buckdahn (Université de Bretagne Occidentale, France) Pathwise Stochastic Taylor Expansions and Stochastic Viscosity Solution
16:30 - 17:00	Coffee break
17:00 - 18:00	Harald Oberhauser (University of Cambridge, United Kingdom) Towards a (rough) pathwise theory of fully non-linear stochastic partial differential equations 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.

09:00 - 10:00	Laurent R. Ponson (California Institute of Technology, USA) Depinning transition in material failure, Barkhausen effect and contact line motion 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.
10:00 - 10:30	Coffee break
10:30 - 11:30	Nicolas Dirr (University of Bath, United Kingdom) Optimal paths for phase boundaries 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.
11:30 - 12:30	Matteo Novaga (University of Pisa, Italy) Motion by curvature in discrete media 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).

Analysis of Stochastic Surface Evolution - From Microscopic Models to Large Scale Behaviour

Speakers

Lorenzo Bertini

Dirk Blömker

Rainer Buckdahn

Giuseppe Da Prato

Nicolas Dirr

Markos Katsoulakis

Thomas Nattermann

Matteo Novaga

Harald Oberhauser

Laurent R. Ponson

Errico Presutti

Panagiotis Souganidis

Anders Szepessy

Hendrik Weber

Nung Kwan Yip

Program

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

Organizers

Patrick Dondl

Stephan Luckhaus

Max von Renesse

Administrative Contact

Katja Bieling