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The presence of fluctuating local relaxation times, $\tau_r(t)$ has been used for some time as a conceptual tool to describe dynamical heterogeneities in glass-forming systems. However, until now no general method is known to extract the full space and time dependent $\tau_r(t)$ from experimental or numerical data. Here we report on a new method for determining the local phase field, $\phi_r(t) = \int^{t} dt'/\tau_r(t')$ from snapshots $\{r(t_i)\}_{i=1...M}$ of the positions of the particles in a system, and we apply it to extract $\phi_r(t)$ and $\tau_r(t)$ from numerical simulations. 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.

Fluctuating phases and fluctuating relaxation times in glass forming liquids