1306.6500 (Oriane Blondel)

Tracer diffusion at low temperature in kinetically constrained models    [PDF]

Oriane Blondel

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 behaviour of $D$ when the density $1-q$ of the environment goes to 1 in two classes of KCSM. For non-cooperative 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 \cite{junggarrahanchandler}. 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 (\cite{junggarrahanchandler}), based on numerical simulations, that suggested $D\sim \gap^\xi$ with $\xi<1$.

View original: http://arxiv.org/abs/1306.6500