Tracer diffusion at low temperature in kinetically constrained models

TL;DR

This paper investigates tracer particle diffusion in kinetically constrained spin models at low temperature, establishing conditions for diffusion, analyzing the asymptotic behavior of the diffusion coefficient, and comparing theoretical predictions with numerical results.

Contribution

It provides rigorous analysis of the diffusion coefficient in KCSM, confirming some predictions and contradicting others from physics literature.

Findings

Diffusion occurs when the environment's spectral gap is positive.

In noncooperative models, D scales as a power of q with an explicit exponent.

In the East model, D is comparable to the spectral gap, contradicting previous predictions.

Abstract

We describe the motion of a tracer in an environment given by a kinetically constrained spin model (KCSM) at equilibrium. We check convergence of its trajectory properly rescaled to a Brownian motion and positivity of the diffusion coefficient $D$ as soon as the spectral gap of the environment is positive (which coincides with the ergodicity region under general conditions). Then we study the asymptotic behavior of $D$ when the density $1-q$ of the environment goes to $1$ in two classes of KCSM. For noncooperative models, the diffusion coefficient $D$ scales like a power of $q$, with an exponent that we compute explicitly. In the case of the Fredrickson-Andersen one-spin facilitated model, this proves a prediction made in Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205]. For the East model, instead we prove that the diffusion coefficient is comparable to the spectral gap, which goes to zero faster than any power of $q$. This result contradicts the prediction of physicists (Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205; J. Chem. Phys. 123 (2005) 084509]), based on numerical simulations, that suggested $D\sim \operatorname {gap}^{\xi}$ with $\xi<1$.