Is there a fractional breakdown of the Stokes-Einstein relation in Kinetically Constrained Models at low temperature?

TL;DR

This paper rigorously analyzes the Stokes-Einstein relation in kinetically constrained models near the glass transition, revealing a non-fractional logarithmic decay of diffusion with relaxation time at low temperatures.

Contribution

It provides rigorous proofs showing the absence of fractional Stokes-Einstein violation in East models at low temperature, and confirms fractional behavior in FA1f models in one dimension.

Findings

Logarithmic relation between diffusion and relaxation time in East models

Fractional Stokes-Einstein relation in FA1f model in 1D

Extension of results to a broader class of non-cooperative models

Abstract

We study the motion of a tracer particle injected in facilitated models which are used to model supercooled liquids in the vicinity of the glass transition. We consider the East model, FA1f model and a more general class of non-cooperative models. For East previous works had identified a fractional violation of the Stokes-Einstein relation with a decoupling between diffusion and viscosity of the form $D\sim\tau^{-\xi}$ with $\xi\sim 0.73$. We present rigorous results proving that instead $\log(D)=-\log(\tau)+O(\log(1/q))$, which implies at leading order $\log(D)/\log(\tau)\sim -1$ for very large time-scales. Our results do not exclude the possibility of SE breakdown, albeit non fractional. Indeed extended numerical simulations by other authors show the occurrence of this violation and our result suggests $D\tau\sim 1/q^\alpha$, where $q$ is the density of excitations. For FA1f we prove fractional Stokes Einstein in dimension $1$, and $D\sim\tau^{-1}$ in dimension $2$ and higher, confirming previous works. Our results extend to a larger class of non-cooperative models.