Kinetically Constrained Lattice Gases

Abstract

ABSTRACT. Kinetically constrained lattice gases (KCLG) are interacting particle systems which show some of the key features of the liquid/glass transition and, more generally, of glassy dynamics. Their distintictive signature is the following: i) reversibility w.r.t. product i.i.d. Bernoulli measure at any particle density and ii) vanishing of the exchange rate across any edge unless the particle configuration around the edge satisfies a proper constraint besides hard core. Because of degeneracy of the exchange rates the models can show anomalous time decay in the relaxation process w.r.t. the usual high temperature lattice gas models particularily in the so called cooperative case, when the vacancies have to collectively cooperate in order for the particles to move through the systems. Here we focus on the Kob-Andersen (KA) model, a cooperative example widely analyzed in the physics literature, both in a finite box with particle reservoirs at the boundary and on the infinite lattice. In two dimensions (but our techniques extend to any dimension) we prove a diffusive scaling O(L 2 ) (apart from logarithmic corrections) of the relaxation time in a finite box of linear size L. We then use the above result to prove a diffusive decay 1/t (again apart from logarithmic corrections) of the spin-spin time autocorrelation function at any particle density, a result that has been sometimes questioned on the basis of numerical simulations. The techniques that we devise, based on a novel combination of renormalization and comparison with a long-range Glauber type constrained model, are robust enough to easily cover other choices of the kinetic constraints.

Kinetically constrained lattice gases: Tagged particle diffusion

“…The following result, proved in [20,4], shows that on a sufficiently large lengthscale frameable configurations are typical.…”

“…[4, Lemma 3.4] For any dimension d, any ρ < 1 and any ε > 0, there exists Ξ = Ξ(ρ, ε, d) < ∞ such that, for the KA process in Z d with facilitation parameter d, for L ≥ Ξ it holds…”

Hydrodynamic limit of the Kob-Andersen model

“…• The bounds on the diffusion coefficient may have consequences other than the hydrodynamic limit -in general, we expect the correlation µ(η(0)e tL η(x)) − ρ 2 to behave like ρ(1 − ρ)(4πt D) −d/2 e − x 2 4tD (see, e.g., [20]). It has been shown in [4] that, for x = 0, this correlation decays at least as fast as C (log t) 5 /t for some unidentified constant C, and any progress towards the predicted ρ(1 − ρ)(4πt D) −d/2 e − x 2 4tD would be an interesting result.…”

“…KCLGs could be either cooperative or non-cooperative (see [4,Definition 1.1]). We remind here that a non-cooperative model is model in which there exists a mobile cluster, defined as follows:…”

Dynamical Heterogeneities in a Two-Dimensional Driven Glassy Model: Current Fluctuations and Finite Size Effects

“…In the context of the study of the glass transition and the jamming transition, the slow, frustrated dynamics of real glasses has been modeled under strong simplifications by particles/spin models where the moves/spin flips have some dynamical restrictions. Such toy models (referred to as kinetically constrained models or KCM [3,4]) allow to explore elementary properties of glassy systems, such as long relaxation times, aging dynamics, anomalous diffusion etc. with a minimum number of ingredients.…”