Fluctuating Relaxation Times in Glass-forming Liquids

The presence of fluctuating local relaxation times, $\tau(\vec{r},t)$ has been used for some time as a conceptual tool to describe dynamical heterogeneities~\cite{Ediger-arpc-2000}. Here we report on a new method for determining the local phase field, $\phi(\vec{r},t)\equiv\int^{t}\frac{dt'}{\tau(\vec{r},t')}$ from snapshots $\{\vec{r}(t_i)\}_{i=1...M}$ of the positions of the particles in a system, and we apply it to extract $\phi(\vec{r},t)$ from simulations of glass forming models. By studying how the phase field depends on the number of snapshots, we find that it is a well defined quantity. By studying fluctuations of the phase field, we find that they describe heterogeneities well at long distance scales. We also determine how the stretching exponent $\beta$ depends on the coarse graining volume, in order to test the hypothesis that relaxation in small regions is exponential and it only becomes non-exponential when considering large regions of the system. \\[4pt] [1] M.~D. Ediger, 2000 Annu. Rev. Phys. Chem. \textbf{51} 99