Kinetically constrained lattice gases: Tagged particle diffusion

Abstract

Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice Z d with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for all dimensions d ≥ 2 and for any equilibrium particle density, under diffusive rescaling the motion of the tracer converges to a d-dimensional Brownian motion with non-degenerate diffusion matrix. Therefore we disprove the occurrence of a diffusive/non diffusive transition which had been conjectured in physics literature. Our technique is flexible enough and can be extended to analyse the tracer behavior for other choices of constraints.MSC 2010 subject classifications:60K35 60J27

Self-diffusion coefficient in the Kob-Andersen model

“…At large times, the path of a marked particle converges to a Brownian motion with a coefficient called the self-diffusion, which is the subject of this paper. In [1] it has been proven that this coefficient is strictly positive for all q ∈ (0, 1), in contrast to the conjecture in the physics literature that below some non-zero critical q the path of tagged particles is no longer diffusive. In this work we find the dependence of this diffusion coefficient in q.…”

“…The proof of the lower bound will closely follow the proof of [1], Sections 4 and 5. However, we use more refined combinatorial properties of the KA model in order to obtain the correct scaling.…”

“…In [1], it was first proven that D(q) > 0 for all q > 0. We will give in this paper the appropriate scale of D(q) when q → 0.…”

“…. Similarly to [1], the purpose of this coarse grained lattice is to define an auxiliary dynamics that has diffusive behavior on a larger scale.…”

“…The main tool we use is a variational formula of [16] for the diffusion coefficient. In order to bound it from below, as in [1], we compare the Kob-Andersen dynamics with a random walk on an infinite percolation cluster. The upper bound is obtained by identifying an appropriate test function related to the bootstrap percolation, a process which is closely related to the Kob-Andersen model.…”