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

## Abstracts for the talks

### On an SPDE describing amorphous surface growth

**Dirk Blömker** *(Universität Augsburg, Germany)*

Friday, January 30, 2009

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.

### Optimal paths for phase boundaries

**Nicolas Dirr** *(University of Bath, United Kingdom)*

Saturday, January 31, 2009

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.

### Hierarchical and multi-level coarse-graining methods

**Markos Katsoulakis** *(U. of Massachusetts at Amherst, USA)*

Thursday, January 29, 2009

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.

### Interface depinning in random media: a physicist's approach

**Thomas Nattermann** *(Universität zu Köln, Germany)*

Thursday, January 29, 2009

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.

### Motion by curvature in discrete media

**Matteo Novaga** *(University of Pisa, Italy)*

Saturday, January 31, 2009

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).

### Towards a (rough) pathwise theory of fully non-linear stochastic partial differential equations

**Harald Oberhauser** *(University of Cambridge, United Kingdom)*

Friday, January 30, 2009

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.

### Depinning transition in material failure, Barkhausen effect and contact line motion

**Laurent R. Ponson** *(California Institute of Technology, USA)*

Saturday, January 31, 2009

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.

### Randomness in macroscopic equations

**Errico Presutti** *(Universitá di Roma II, Italy)*

Thursday, January 29, 2009

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.

### Moving fronts in self averaging media

**Panagiotis Souganidis** *(University of Chicago, USA)*

Friday, January 30, 2009

I will review results about the homogenized/long time behavior of fronts
moving in self averaging (periodic/stationary) media.

### A Stochastic Phase-Field Model Derived from Molecular Dynamics

**Anders Szepessy** *(KTH Stockholm, Sweden)*

Thursday, January 29, 2009

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.

### Sharp interface limit for a stochastic Allen-Cahn Equation

**Hendrik Weber** *(Universität Bonn, Germany)*

Thursday, January 29, 2009

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.

### Interfacial Propagation in Inhomogeneous Medium, Some Results and Questions

**Nung Kwan Yip** *(Purdue University, USA)*

Friday, January 30, 2009

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.

## Date and Location

**January 29 - 31, 2009**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Patrick Dondl**

Max-Planck-Institute for Mathematics in the Sciences

**Stephan Luckhaus**

Leipzig University

**Max von Renesse**

Technische Universität Berlin

## Administrative Contact

**Katja Bieling**

Max Planck Institute for Mathematics in the Sciences