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.

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.

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 $H^{1/2}$ 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.

We study deterministic and stochastic gradient descents in random energies $E_\varepsilon(x) = V(x) + \varepsilon W(x/\varepsilon)$; $V$ is the deterministic part of the energy, $W$ is a realization of the energy fluctations and $\varepsilon$ is the typical distance between local minimal of $E$. If the evolution of $x$ for given $\varepsilon$ and $W$ is deterministic one obtains classical rate independent evolution in the limit where $\varepsilon$ 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).

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).

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.

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).

The Kuramoto-Sivashinsky equation, i. e. $$ \partial_t u+\partial_x({\textstyle\frac{1}{2}}u^2) +\partial_x^2u+\partial_x^4u\;=\;0 $$ 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 $\int u^2\,dx$ is equally distributed over all wave numbers $|k|\ll 1$. Unfortunately, PDE theory is far from a rigorous understanding of these phenomena. Over the past 20 years, bounds on the space-time average $\langle\langle(|\partial_x|^\alpha u)^2\rangle\rangle^{1/2}$ of (fractional) derivatives $|\partial_x|^\alpha u$ of $u$ in terms of $L$ have been established and improved. The best available result states that $\langle\langle(|\partial_x|^\alpha u)^2\rangle\rangle^{1/2}=o(L)$ for all $0\le\alpha\le 2$.In this talk, I shall present the new bound $$ \langle\langle(|\partial_x|^\alpha u)^2\rangle\rangle^{1/2}\;=\;O(\ln^{5/3} L) $$ for $1/3< \alpha\le 2$. 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 $\partial_tu+\partial_x(\frac{1}{2}u^2)\;=\;f$. From this extension we learn that the quadratic term $\partial_x(\frac{1}{2}u^2)$, which is conservative, effectively behaves like a coercive term in the sense that we obtain a priori estimates as if $\int \partial_x(\frac{1}{2}u^2)\,u\,dx\;\sim\; \int||\partial_x|^{1/3}u|^3\,dx$.

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.

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.

Participants

Stefan Adams

University of Warwick

Nicolas Alikakos

University of Athens

Volker Betz

University of Warwick

Francis Comets

Université Paris 7

Sergio Conti

Universität Bonn

Jerome Coville

INRA

Nicholas Dirr

University of Bath

Gero Friesecke

Technische Universität München

Roman Kotecky

Charles University Praha

Noemi Kurt

TU Berlin

Juan J. L. Velazquez

Universidad Complutense

Stephan Luckhaus

Universität Leipzig

Luca Mugnai

Max Planck Institute

Felix Otto

Universität Bonn

Errico Presutti

Universitá di Roma

Senya Shlosman

Université de Provence

Takis Souganidis

University of Chicago

Florian Theil

University of Warwick

Johannes Zimmer

University of Bath

Scientific Organizers

Stephan Luckhaus

Universität Leipzig

Errico Presutti

Universitá di Roma

Luca Mugnai

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Administrative Contact

Katja Bieling

Max Planck Institute for Mathematics in the Sciences
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