Workshop on
"
Metastability"
January 9-11, 2008

EURANDOM, Eindhoven, The Netherlands

ABSTRACTS

Gerard Barkema, Utrecht University

Statistical physics of polymer translocation

Transport of molecules across cell membranes is an essential mechanism for life processes. These molecules are often long and flexible, and the pores in the membranes are too narrow to allow them to pass through as a single unit. In such circumstances, the passage of a molecule through the pore --- translocation --- proceeds through an activated process in which polymer segments sequentially move through the pore.

For statistical physicists, interesting quantities of translocation are mainly scaling properties, for instance: how does the time it takes for a polymer to translocate scale with its length? How long does a translocating polymer block the pore during translocation, as a function of its length?

We will show that the translocation process can be viewed as a random-walk process with anomalous diffusion, characterized by a squared displacement scaling with time as $\langle r^2 \rangle \sim t^{(1+\nu)/(1+2\nu)}$, and that the dwell time $\tau_d$ scales with polymer length as $\tau_d \sim N^{2+\nu}$. Here, $\nu\approx 0.588$ is the Flory exponent of self-avoiding walks. We will also discuss further applications of the theoretical framework that yielded these results.

Alessandra Bianchi, WIAS Berlin

Sharp asymptotics for metastability in the random field Curie-Weiss model.

We will discuss the metastable behavior of the random field Curie-Weiss model under Glauber dynamics. As an application of the potential theoretic approach which is going to be presented in a talk by Dima Ioffe, we will prove sharp estimates on capacities and metastable exit times for any temperature and also in the case where the random field has continuous distribution.

Raphael Cerf, Université Paris-Sud
Joint work with Francesco Manzo

Nucleation and growth for the Ising model in $d$ dimensions at very low temperatures

We study the metastability of the Ising model in dimension $d$ with Metropolis dynamics under a small magnetic field when the temperature goes to~$0$.

Emilio Cirillo, La Sapienza Università di Roma

Metastable behavior of reversible Probabilistic Cellular Automata with self-interaction

The metastable behavior of reversible Probabilistic Cellular Automata can be studied with the support of recent general results. Peculiar difficulties that must be overcome will be discussed. Particular attention will be given to the effect of self-interaction.

Alexandre Gaudilliere, Università Roma 3

Homogeneous nucleation for Kawaski dynamics -

In the low temperature limit, we give exponential asymptotics for the escape time from metastability for Kawasaki dynamics and we describe
the typical nucleation pattern.

Dima Ioffe, Technion Haifa

Upper and lower bounds on escape times from metastable states

In this talk I shall try to present a potential theoretical approach to deriving matching upper and lower bounds on metastable escape times. This is an ongoing project with Alessandra Bianchi and Anton Bovier. The approach is specifically aimed at studying systems with large entropy of microscopic states which could not be canceled out. In particular, calculus of microscopic point to point capacities does not make sense, and one needs to trace the evolution of the system over a mesoscopic landscape of carefully chosen order parameters. However, the induced mesoscopic dynamics is typically non-Markovian and the crux of the matter is to understand how it should be controlled in terms of a tractable effective Markovian evolution on the mesoscopic level. Our approach yields sharp results for the simplest non-trivial model of this type: Curie-Weiss model in an iid continuously distributed random field.

Malwina Luczak, London School of Economics

Glauber dynamics for the meant-field ising model: cut-off, critical power law, and metastability

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For $\beta < 1$, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near $1$ to near $0$ in a window of order $n$ centered at $[2(1-\beta)]^{-1} n\log n$. For $\beta = 1$, we prove that the mixing time is of order $n^{3/2}$. For $\beta > 1$, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time $O(n \log n)$.

Francesco Manzo, Università Roma 1

Droplets shape and relaxation time in infinite volume: a simple model

I will discuss some features of the relaxation from a metastable state in the infinite volume, vanishing temperature regime with the help of a simple spin model.

Fabio Martinelli, Università di Roma 3

Dynamical relaxation of a 1D pinning model

I will present some result on the relaxation behavior of a 1D pinning model

Enzo Olivieri, Università Roma 2

Escape from metastability:the pathwise viewpoint for conservative dynamics.

We give an elementary introduction to metastability for a lattice gas evolving with conservation of particles.
We try to explain the difficulties arising from conservativeness.

Elisabetta Scoppola, Università Roma 3

How to work out the puzzle of gas-cluster interaction in the Kawasaki dynamics

The gas-cluster interaction is one of the main problem in the discussion of metastability for conservative dynamics.
We review the main ideas and tools used to control this problem.

Cristian Spitoni, Leiden University/EURANDOM

Homogeneous nucleation for Glauber dynamics

We study metastability in large volumes at low temperatures for Ising spins subject to Glauber spin-flip dynamics. We run the dynamics starting from a random initial configuration where all the droplets are small. In the low temperature regime, we investigate how the transition from the metastable state (with small droplets) to the stable state (with large droplets) takes place under the dynamics. This transition is triggered by the occurrence of a single critical droplet, occurring somewhere in the lattice. Using potential-theoretic methods, we derive sharp estimates on the average transition time. Since it is inversely proportional to the volume of the lattice, this type of behaviour is called homogeneous nucleation.

Eulalia Vares, CBPF-Centro Brasileiro de Pesquisas Fisicas

On a randomized PNG model with a columnar defect Maria Eulalia Vares (Centro Brasileiro de Pesquisas Fisicas)

The talk is based on an joint work with V. Beffara and V. Sidoravicius. We study a variant of poly-nuclear growth where the level boundaries perform continuous-time, discrete-space random walks, and study how its asymptotic behavior is affected by the presence of a columnar defect on the line. We show that there is a non-trivial phase transition in the strength of the perturbation, above which the law of large numbers for the height function is modified.