Mark Kac Seminar
March 6, 2009
Location:
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11:15-13:00 |
speaker: Franz Merkl (München) |
title: What
are crystals? (1) |
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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.) |
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14:15-16:00 |
speaker: Anton Bovier
( |
title: Kawasaki dynamics in large volumes |
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abstract: (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|). |
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Mark Kac Seminar 2008-2009 |
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last updated: 18 dec 2008by Markus |