Johel Beltran (Lima), Alexandre Gaudillière (Marseille), Frank den Hollander (Leiden), Sabine Jansen (Leiden), Michail Loulakis (Athens), Fabio Martinelli (Roma), Francesca Nardi (Eindhoven), Augusto Teixeira (Rio de Janeiro)

Johel Beltran (Lima): Nucleation for the zero range process on a finite set of sites. pdf

We consider a zero-range process with decreasing rates on a finite set of sites. Let N be a scaling parameter, which will be sent to infinity, and let us put initially N particles at each site. Because of the attraction between particles, after a time of order N^2, the particles finally concentrate on a single site and remains there for a time of order larger than N^2. The trajectory followed by the process until this absorbing state is called nucleation. In our work, we prove a scaling limit for the evolution of this zero range process on this stage. The Markov process limit provides a precise description of nucleation.

Alexandre Gaudillière (Marseille): $(\kappa, \lambda)$-capacities and soft measures.

We prove a Poincaré inequality based on variational principles inherited from a slight generalization of standard capacities of standard potential theory, that, in the case of metastable systems give the precise asymptotic of the spectral gap. We also explain the connection between these $(\kappa, \lambda)$-capacities and soft measures that allow to describe metastable states.

Sabine Jansen (Leiden): Metastability at low temperature for continuum interacting particle systems

In this talk we consider a system of point particles living in a finite box in Euclidean space, interacting with each other through a finite-range stable pair potential, and moving according to three possible random dynamics that allow for particles to be created and annihilated: (1) Glauber dynamics (without particle hopping); (2) Kawasaki dynamics (with particle hopping); (3) gradient dynamics (with particle diffusion). The parameters of the dynamics are such that the system is metastable: starting from the vacuum configuration where the box is empty, the system wants to nucleate (i.e., fill up the box with particles), but has to overcome an energetic threshold to do so. We are interested in the nucleation time in the limit as the temperature tends to zero.

Using the potential-theoretic approach to metastability, we compute the average nucleation time and show that the nucleation time divided by its average is exponentially distributed. The three dynamics exhibit the same scaling for the nucleation time, but with different prefactors. We compute these prefactors and identify how they depend on the pair potential.

Our results extend earlier work for lattice systems. The difficulty of working in the continuum is that it is hard to identify the shape of the critical droplets triggering the nucleation and to control the energy landscape in the vicinity of these critical droplets. We rely on properties derived in the literature for minimal energy configurations at fixed particle numbers. We identify which properties are crucial for the metastable behavior. Some of these are expected to be true for a large class of pair potentials, but as yet remain unproven.

Michail Loulakis (Athens): Scaling limit of the condensate dynamics in a reversible zero-range process.

Zero range processes with decreasing jump rates can equilibrate in a condensed phase when the particle density exceeds a critical value. In this phase a non-trivial fraction of the mass in the system concentrates on a single site, the condensate. At suitably long time scales, the location of this site changes. Beltrán and Landim have studied the motion of the condensate for zero range processes on finite sets and have shown that - observed at the right time scale - it converges to a random walk on this set. In this work we consider a supercritical nearest neighbor symmetric zero range process on the discrete torus (1/L)Z/Z. We show that the scaling limit of the condensate dynamics is a Lévy process on the unit torus with jump rates inversely proportional to the jump length. Joint work with Inés Armendáriz and Stefan Grosskinsky

Fabio Martinelli (Roma): Metastable dynamics in two examples of a surface interacting with a wall.

In this talk we shall consider two models of a surface interacting with a wall and evolving with a Glauber dynamics. The first example consists of a (2+1)-dimensional Solid-On-Solid model above a hard wall, in an box of side L and with zero boundary conditions, at large inverse temperature. It has been recently proved that the floor constraint induces an entropic repulsion effect which lifts the surface to an average height H=c log(L) with a sharp constant c.

If the Glauber dynamics starts from a flat initial condition at zero height, then, because of the entropic repulsion from the wall, the surface will rise to its final height H through a sequence of metastable transitions between consecutive levels. If n is a positive fraction of the final height H, then the time for a transition from height n to height n+1 is roughly doubly-exponentially large in the height n because of the presence of an effective energy barrier. In particular, the mixing time of the dynamics is exponentially large in L. We emphasize that without a floor constraint at height zero the mixing time is no longer exponentially large in L (and probably poly(L) ). Ref: arXiv:1207.3580 and arXiv:1205.6884

The second example consists of a (1+1)-dimensional polymer with endpoints at height zero and distance L, interacting via a repulsive potential with the line at zero height. At equilibrium the system exhibits a double well structure: the polymer lies (essentially) either above or below the repulsive line. As a consequence one expects a metastable behavior with rare jumps between the two phases combined with a fast thermalization inside each phase. However the energy barrier between these two phases is only logarithmic in the system size L and therefore the two relevant time scales are only polynomial in L with no clear-cut separation between them. The whole evolution is governed by a subtle competition between the diffusive behavior inside one phase and the jumps across the energy barriers. In particular the usual scenario in which the tunneling time coincides with the exponential of the energy barrier breaks down. Ref: Probability Theory and Related Fields August 2012, Volume 153, Issue 3-4 , pp 587-641 also arXiv:1007.4470

Frank den Hollander (Leiden): Metastability for Kawasaki dynamics at low temperature with two types of particles

We study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between neighboring particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite gas reservoir. We start the dynamics from the empty box and study metastability, i.e., compute the transition time to the full box and nucleation. This transition is indeed triggered by a critical droplet appearing somewhere in the box. Joint work with Francesca Nardi and Alessio Troiani.

Augusto Teixeira (Rio de Janeiro): Metastability for a polymer close to an interface

In this talk we will consider a polymer of length N in 1+1 dimensions. This polymer receives an energetic reward each time it visits height zero. But on the other hand, its left and rightmost points are pinned at height bN for some positive b. This competition between the attractive wall at level zero and the high boundary constraints gives rise to an interesting behavior: for certain values of the temperature and the constant b, the polymer is metastable in either the pinned or the free phases. In this talk we will further discuss this phenomenon, elaborating on the difficulties and new techniques involved.