# Workshop on Phase Transitions

## Abstracts for the talks

**Nicolas Alikakos**

Monday, August 24, 2009

**On Entire Solutions of the System Laplacian of u - grad W(u)=0, u from Rn to Rn**

We establish existence for potentials with a finite number of global minima and symmetric under a finite reflection group, plus other conditions. We also make some basic observations that do not require any symmetry: A Sress Tensor T that allows the system to be written as DivT=0, a weak monotonocity formula, and a Liouville type Theorem.

**Volker Betz**

Tuesday, August 25, 2009

**Bose-Einstein condensation and infinite cycles in random permutation**

I present a model of random permutations on a set with spatial structure. The probability of obtaining a given permutation is determined by a Gibbs factor, and the energy is higher when the permutation contains more jumps between distant points of the underlying set. So, the jump length of a typical random permutation will be small. For this model I show the existence of a phase transition: Depending on the density of the points forming the spatial structure, there either exist exclusively finite cycles (for low density), or a coexistence of finite and macroscopic cycles (for high density). The physical relevance of the model comes from its connections to Bose-Einstein condensation; I will briefly explain these connections and highlight open question. This is joint work with Daniel Ueltschi.

**Francis Comets**

Thursday, August 27, 2009

**Diffusivity of Stochastic Billiards in a random tube**

Consider a random tube which stretches to infinity in the direction of the first coordinate, and which is stationary and ergodic, and also well-behaved in some sense. When strictly inside the tube, the particle ("ball") moves straight with constant speed. Upon hitting the boundary, it is reflected randomly: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. We also consider the discrete-time random walk formed by the particle's positions when hitting the boundary. Under the condition of existence of the second moment of the projected jump length with respect to the stationary measure for the environment seen from the particle, we prove the quenched invariance principles for the projected trajectories of the random walk and the stochastic billiard. Joint work with S.Popov, G.Schütz and M. Vachkovskaia.

**Sergio Conti**

Monday, August 24, 2009

**Multiscale decomposition of dislocation microstructures**

Dislocations are topological defects in crystals which generate long-range elastic stresses. We consider a model in which the elastic interactions are represented via a singular kernel behaving as the norm of the slip. We obtain a sharp-interface limit within the framework of Gamma convergence. One key ingredient is a proof of the fact that the presence of infinitely many equivalent length scales gives strong restrictions on the geometry of the microstructure. In particular we show that the micrustructure must be one-dimensional on most length scales, and that only few are available for the relaxation. This talk is based on joint work with Adriana Garroni and Stefan Müller.

**Gero Friesecke**

Tuesday, August 25, 2009**From interatomic potentials to Wulff shapes, via Gamma convergence**

We investigate ground state configurations of atomic pair
potential systems in two dimensions as the number of particles tends to infinity.
Assuming crystallization (which has been proved for some cases such
as the Radin potential, and is believed to hold more generally),
we show that after suitable rescaling, the ground states converge to
a unique macroscopic Wulff shape. Moreover, sharp estimates on the
microscopic fluctuations about the Wulff shape are obtained.
Joint work with Yuen Au-Yeung and Bernd Schmidt (TU Munich).

**Felix Otto**

Thursday, August 27, 2009**Optimal bounds on the Kuramoto-Sivashinsky equation**

The Kuramoto-Sivashinsky equation, i. e.

is a ``normal form''
for many processes which lead to complex dynamics in space and time
(one example is the roughening of the crystal surface in epitaxial growth).
Numerical simulations show that after an initial layer, the
statistical properties of the solution are independent of the
initial data and the system size *L* (defined by the period *u*(*t*,*x*+*L*)=*u*(*t*,*x*)).
More precisely, the energy is equally distributed over
all wave numbers .

Unfortunately, PDE theory is far from a rigorous understanding
of these phenomena. Over the
past 20 years, bounds on the space-time average
of (fractional) derivatives of *u*
in terms of *L* have been established and improved.
The best available result states that
for all .

In this talk, I shall present the new bound

for . This seems the first
result in favor of an extensive behavior -- albeit only
up to a logarithm and for a restricted range of fractional derivatives.

The proof estentially relies on an extension of Oleinik's principle to the inhomogeneous inviscid Burgers' equation . >From this extension we learn that the quadratic term , which is conservative, effectively behaves like a coercive term in the sense that we obtain a priori estimates as if .

**Errico Presutti**

Thursday, August 27, 2009**Coexistence of metastable phases in states with a current**

Such states are obtained as stationary solutions of
an integro-differential equation derived by Giacomin and Lebowitz from
Ising systems with Kac potential and Kawasaki dynamics. In the
thermodynamic limit they
are described by magnetization profiles whose values lie almost everywhere in
the metastable region, positive and negative magnetization regions
being separated by
a sharp interface. When instead no currents are imposed the state
relaxes to a Wulff shape.

**Senya Shlosman**

Monday, August 24, 2009**The Ising rain stays mainly on the plane**

We discuss a simple model describing coexistence of solid and vapour
phases. The two phases are separated by an interface. We show that when the
concentration of supersaturated vapour reaches the dew-point, the droplet of
solid is created spontaneously on the interface, adding to it a monolayer of
a
visible size, while the vapour concentration drops down below the dew-point.
Further increase of concentration results thus in the sequence of new
monolayer creations. We discuss the resulting geometry.
This is a joint work with Dima Ioffe.

**Florian Theil**

Monday, August 24, 2009**Relaxation dynamics in thermal and athermal systems**

We study deterministic and stochastic gradient descents in random energies
; *V* is the
deterministic part of the energy,
*W* is a realization of the energy fluctations and is the typical
distance between local minimal of *E*.

If the evolution of *x* for given and
*W* is deterministic one obtains classical rate
independent evolution in the limit where tends to 0. We extend
the analysis to the stochastic case and find a generalization of
rate-independent evolution which exhibits a nontrivial relaxation dynamics.
Our results can potentially explain well-known creep phenomena such as
Andrade creep in plasticity.

This is joint work with Michael Ortiz (Caltech), Marisol Koslowski
(Purdue) and Tim Sullivan (Caltech).

**Johannes Zimmer**

Tuesday, August 25, 2009**Dynamic lattice models for elastic phase transitions and dislocations**

Martensitic phase transitions can dissipate energy, but are
often described by conservative (Hamiltonian) equations on the lattice
scale. How can conservative lattice models generate dissipation on the
continuum scale? To understand this, we consider a model problem,
namely travelling waves in a one-dimensional chain of atoms with
nearest neighbour interaction. The elastic potential is piecewise
quadratic and the model is thus capable of describing phase
transitions. A solution which explores both wells of the energy will
then have a phase boundary, moving with the speed of the wave. We show
that for suitable fixed subsonic velocities, there is a family of
"heteroclinic" travelling waves (heteroclinic means here that these
solutions connect both wells of the energy). Though the microscopic
picture is Hamiltonian, we derive non-trivial so-called kinetic
relations on the continuum scale. Kinetic relations in turn can be
related to the dissipation generated by a moving phase boundary. We
then investigate the question of when the kinetic relation does not
vanish (dissipation is generated). It turns out that a microscopic
asymmetry determines here the macroscopic dissipation.
The technique we developed for lattice problem above can also be
applied to the model of disclation dynamics proposed by Frenkel and
Kontorova in 1939 (for piecewise quadratic on-site potential), and
this results and some implications will be discussed in the second
part of the talk
The first part is joint work with Hartmut Schwetlick (Bath), the
second one with Carl-Friedrich Kreiner (Munich).

## Date and Location

**August 23 - 28, 2009**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Stephan Luckhaus**

Leipzig University

Contact by Email

**Errico Presutti**

Universitá di Roma

Contact by Email

**Luca Mugnai**

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Contact by Email

## Administrative Contact

**Katja Bieling**

Max Planck Institute for Mathematics in the Sciences

Contact by Email