Project 3:
Phase
boundaries under the influence of
random perturbations
Participants
Stefan Adams, Jean-Dominique Deuschel, Nicolas Dirr, Roman
Kotecký, Stephan
Luckhaus, Barbara
Niethammer, Max
von Renesse, Michael
Scheutzow
Summary
We consider models for the formation and evolution of
interfaces in materials which are characterised by the interplay
of
- the interfacial energy
- quenched random and/or deterministic inhomogeneities on a very
small scale
- space-time dependent noise modelling thermal fluctuations.
The current project is focussed on two aspects of the dynamics of such systems:
The first aspect is the phenomenon of nucleation.
We are interested in exact mathematical results on large
time-space scales, which can be obtained by methods
from the theory of large deviations for continuum systems perturbed by
small additive noise.
The second aspect concerns the effective evolution of an (already
developed) interface on a large time-space scale.
We are interested in the influence of the inhomogeneities and the
noise
-- which originate on a smaller scale -- on these effective evolution
laws.
For both aspects we aim at rigorous mathematical results.
The models we consider are Mean Curvature Flow
with random perturbations and several simplified continuum models of
reaction-diffusion type
of gradient-flow structure with space-time noise and/or
with random or periodic coefficients.
Some related earlier preprints
Anna De Masi, Stephan Luckhaus and
Errico Presutti.
Two scale hydrodynamic limit for a model of malignant tumor cells.
Annales Henri Poincaré 43, 257-297 (2007).
MPI-MIS
Preprint 2/2005.
Anna De Masi, Nicolas Dirr and
Errico Presutti:
Interface Instability
under Forced Displacements
Annales Henri Poincaré 7:3, 471-511 (2006)
MPI-MIS
Preprint 5/2005.
Nicolas Dirr and Nung Kwan (Aaron) Yip.
Pinning
and de-pinning phenomena in front propagation in heterogeneous media.
Interfaces and Free Boundaries 8:1
79-109 (2006).
MPI-MIS
Preprint 50/2005.
Nicolas Dirr, Marcello Lucia and
Matteo Novaga.
Gamma-convergence of the Allen-Cahn energy with
an oscillating forcing term.
Interfaces and Free Boundaries 8:1, 57-78 (2006).
MPI-MIS
Preprint 38/2005.
Nicolas Dirr and Stephan Luckhaus.
Mesoscopic limit for non-isothermal phase transition.
Markov Process. Relat. Fields 7:3, 355-381 (2001).
MPI-MIS
Preprint 70/2001.
Achievements of the Research Group
M. v. Renesse and K.-T. Sturm:
Entropic measure and Wasserstein diffusion,
Ann. Probab., to appear
preprint
S. Andres and M. v. Renesse:
Particle approximation of the Wasserstein diffusion,
preprint
M. v. Renesse, M. Yor and L. Zambotti:
Quasi-invariance properties of a class of subordinators,
Stoch. Proc. Appl., to appear
preprint
M. v. Renesse:
An optimal transport view on Schrödinger's equation,
preprint
M. v. Renesse:
On local Poincare via transportation,
Math. Z. 259:1, 21-31 (2008).
preprint
G. Dimitroff and M. Scheutzow:
Dispersion of volume under the action of
isotropic Brownian flows.
Stoch. Proc. Appl, to appear.
preprint
M. Scheutzow:
Chaining techniques and their application to stochastic flows,
in: Trends in Stochastic Analysis , eds: Blath, J., Mörters, P., Scheutzow, M., Cambridge University Press, to appear.
preprint
Luca Mugnai and Matthias Röger:
The Allen-Cahn action functional in higher dimensions,
Interfaces and Free Boundaries 10:1, 45-78, (2008).
Nicolas Dirr, Georgia Karali and Aaron Nung Kwan Yip:
Pulsating Wave for Mean Curvature Flow in Inhomogeneous Medium,
European Journal of Applied Mathematics, to appear
preprint
Nicolas Dirr, Marcello Lucia and Matteo Novaga:
Gradient theory of phase transitions with a rapidly
oscillating forcing term,
Asymptotic Analysis, to appear
preprint
Nicolas Dirr and Enza Orlandi:
Sharp-interface limit of a mesoscopic free energy with
a random external field,
preprint
Giovanni Bellettini, Anna De Masi, Nicolas Dirr and
Errico Presutti:
Tunneling in two dimensions,
Commun. Math. Phys. 269, 715-763 (2007).
offprint,