Mark Kac Seminar

March 6, 2009






Location: Utrecht, KNG80, room 230 (different room!)


speaker: Franz Merkl (München)

title: What are crystals? (1)

abstract (for the series of three lectures, to be continued on April 3 and May 8): 


As is well-known from everyday knowledge, interacting molecules at thermal equilibrium at low temperature form cristals. At higher temperature, they undergo a melting transition to a liquid phase. From the viewpoint of statistical mechanics, these well-known facts are by far not understood.


The talks are concerned with the notion of spontaneous symmetry breaking in interacting particle systems. On the one hand, I will present a (over-)simplified model for a two-dimensional continuum particle system that spontaneously breaks rotational symmetry. On the other hand, by the famous Mermin-Wagner phenomenon, two-dimensional particle systems frequently show preservation of continuous symmetry. For example, Richthammer has recently shown that translational symmetry is preserved in two-dimensional hard-core particle systems.


The Mermin-Wagner phenomenon plays also a role in the recent understanding of linearly edge-reinforced random walks. There, absence of spontaneous breaking of a certain scaling symmetry plays an essential role.


(Joint work with Silke Rolles.)



speaker: Anton Bovier (Bonn)

title: Kawasaki dynamics in large volumes


In this talk I report on recent work on metastability in large volumes at low temperatures for conservative
(Kawasaki) dynamics of a lattice gas.

Let \beta denote the inverse temperature and let \Lambda_\beta \subset Z^2 be a square box with periodic boundary conditions such that  
        \lim_{\beta\to\infty}  | \Lambda_\beta |  =  \infty.
We run the dynamics on \Lambda_\beta starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles) are small.

For large \beta, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \emph{critical droplet} somewhere in \Lambda_\beta.

Using potential-theoretic methods, we compute the average nucleation time. It turns out that this time grows as
       K e^{\Gamma\beta} / |\Lambda_\beta|
for Glauber dynamics and
       K \beta e^{\Gamma\beta} / |\Lambda_\beta|
for Kawasaki dynamics, where \Gamma is the local canonical, respectively, grand-canonical energy to create a critical droplet and K is a constant reflecting the geometry of the critical droplet provided these times tend to infinity (which puts a growth restriction on |\Lambda_\beta|).


Mark Kac Seminar 2008-2009


last updated: 18 dec 2008by Markus