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 formula4 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.
displaymath21
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 formula27 is equally distributed over all wave numbers formula29.

Unfortunately, PDE theory is far from a rigorous understanding of these phenomena. Over the past 20 years, bounds on the space-time average formula31 of (fractional) derivatives formula33 of u in terms of L have been established and improved. The best available result states that formula39 for all formula41.

In this talk, I shall present the new bound
displaymath43
for formula45. 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 formula47. >From this extension we learn that the quadratic term formula49, which is conservative, effectively behaves like a coercive term in the sense that we obtain a priori estimates as if formula51.


 

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 formula3; V is the deterministic part of the energy, W is a realization of the energy fluctations and formula9 is the typical distance between local minimal of E.

If the evolution of x for given formula9 and W is deterministic one obtains classical rate independent evolution in the limit where formula9 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

05.04.2017, 12:42