Kinetically constrained spin models, glasses and coalescence processes

Tuesday 25 January 2011, ESPCI

Moderators: Fabio Martinelli (Roma) and Cristina Toninelli (Paris)

J. Bertoin (Paris), G. Biroli (Saclay), T. Bodineau (Paris), B. Derrida (Paris), A. Faggionato (Roma), P. Sollich (London).

Jean Bertoin (Paris): Burning cars in a parking.

Knuth's parking scheme is a model in computer science for hashing with linear probing. One may imagine a circular parking with $n$ sites; cars arrive at each site with unit rate. When a car arrives at a vacant site, it parks there; otherwise it turns clockwise and parks at the first vacant site which is found. It is known from a work by Chassaing and Louchard that the formation of large occupied blocks is governed by the so-called additive coalescent.
We incorporate fires to this model by throwing Molotov cocktails on each site at a smaller rate $n^{-\alpha}$. When a car is hit by a Molotov cocktail, it burns and the fire propagates to the entire occupied interval which turns vacant. We show that with high probability when the size of the parking is large, the parking becomes saturated at a time close to 1 (i.e. as in the absence of fire) for $\alpha>2/3$, whereas for $\alpha<2/3$, the mean occupation approaches $1$ at time $1$ but then quickly drops to $0$ before the parking is ever saturated.

Giulio Biroli (Saclay): Glass Transition and Kinetically Constrained Models.

The first part of this talk will be devoted to an overview of the glass transition and glassy dynamics. I will then discuss the main ideas that motivated the introduction and the analysis of Kinetically Constrained Models and present some of the main outcomes of these studies, in particular concerning growing dynamic and static lengthscales. Finally, I will make a "where do we stand" critical discussion about the application of Kinetically Constrained Models to glassy dynamics. In particular I shall discuss some decisive tests that should be able to unveil whether they indeed provide a correct description of the dynamics close to the glass and jamming transition.

Thierry Bodineau (Paris): Phase transition in kinetically constrained models.

An important issue is to understand the structure of the dynamical heterogeneity in kinetically constrained models, i.e. the regions which are mobile (active) vs the regions which are blocked (inactive). The activity of the system measures the microscopic number of moves per unit time and it has been proposed as a relevant parameter to characterize glassiness. We will focus on the large deviations of the activity and show that it leads to a first order phase transition (Joint work with C. Toninelli).

Bernard Derrida (Paris): Universal distributions in one dimensional coarsening models.

When spin systems with ferromagnetic interactions are quenched to zero temperature, one observes the growth of domains. In the long time limit, the distribution of the size of these domains becomes universal, irrespective of their initial distribution. This talk will review a few exactly soluble models where the limiting distribution of these sizes as well as the persistence exponents can be computed.

Alessandra Faggionato (Roma): Scaling limits in one dimensional hierarchical coalescence processes.

Motivated by several models introduced in the physical literature to study the non-equilibrium coarsening dynamics of one-dimensional systems, we consider a large class of "hierarchical coalescence processes'' (HCP). An HCP consists of an infinite sequence of coalescence processes, each process occurs in a different "epoch'' and evolves for an infinite time, while the evolution in subsequent epochs are linked in such a way that the initial distribution of an epoch coincides with the asymptotic distribution of the preceding epoch. Inside each epoch the process evolves by coalescence of pairs or triples of consecutive domains. Our main result concerns the asymptotic scaling limit of the above HCP's and the analysis of universality classes. (Joint work with F. Martinelli, C. Roberto, C. Toninelli).

Peter Sollich (London): Space-time phase transitions and biased steady states in kinetically constrained models.

I will review results from the last few years which support the view that kinetically constrained models operate at or near a space-time phase transition between a phase that is inactive and one that is active. At the transition, domains of these phases can coexist, thus providing a natural interpretation of dynamical heterogeneities occurring in these models.
The bulk space-time phases can be explored by studying ensembles of trajectories which are biased towards high or low activity (number of spin flips or particle moves per site and unit time). I will explain how the steady states of such trajectory ensembles correspond to Markov chains which are modified by an additional effective interaction, and will describe some initial results from an exploration of the properties of this interaction in one-dimensional kinetically constrained models.

Cristina Toninelli (Paris): East model: rigorous results for the low temperature non-equilibrium dynamics.

We consider a special example of one dimensional kinetically constrained model, the East model, and focus on the low temperature non-equilibrium dynamics which follows a quench from an initial distribution which is different from the reversible one. This setting has been extensively studied in physics literature and on the basis of heuristic arguments and numerical simulations it was observed that dynamics can be approximated by an irreversible coarsening process for the domains (intervals separating consecutive vacancies) with a peculiar hierarchical structure. We will explain how, provided the initial distribution of the domains is a renewal process, this approximation can be made rigorous and how, by analyzing the asymptotic behavior of the coalescence process, one can prove a staircase behavior for the persistence function, an aging behavior for the correlation function and give a sharp description on the statistics of the intervals separating consecutive vacancies . These results confirm the results of the physicists in the case of an initial distribution which is Bernoulli and show that a different behavior appears if instead the initial domain distribution is in the attraction basin of an $\alpha$-stable law. (Joint work with A. Faggionato, F. Martinelli, C. Roberto).