Field-induced superdiffusion and dynamical heterogeneity

By analyzing two kinetically constrained models of supercooled liquids we show that the anomalous transport of a driven tracer observed in supercooled liquids is another facet of the phenomenon of dynamical heterogeneity. We focus on the Fredrickson-Andersen and the Bertin-Bouchaud-Lequeux models. By numerical simulations and analytical arguments we demonstrate that the violation of the Stokes-Einstein relation and the field-induced superdiffusion observed during a long preasymptotic regime have the same physical origin: while a fraction of probes do not move, others jump repeatedly because they are close to local mobile regions. The anomalous fluctuations observed out of equilibrium in the presence of a pulling force $\epsilon ,{\sigma }_x^2\left(t\right)=\langle x_{\epsilon }^2\left(t\right)\rangle -{\langle x_{\epsilon }\left(t\right)\rangle }^2\sim t^{3/2}$, which are accompanied by the asymptotic decay ${\alpha }_{\epsilon }\left(t\right)\sim t^{-1/2}$ of the non-Gaussian parameter from nontrivial values to zero, are due to the splitting of the probes population in the two (mobile and immobile) groups and to dynamical correlations, a mechanism expected to happen generically in supercooled liquids.

Figure 1

Data collapse for the non-Gaussian parameter ${\alpha }_{\epsilon }\left(t\right)$ obtained plotting $1+{\alpha }_{\epsilon }\left(t\right)$ vs $t/{\tau }_{\mathrm{eq}}$ for the different concentrations of mobility defects (different symbols) in (a) FA and (b) BBL. The continuous straight line emphasizes the behavior ${\alpha }_{\epsilon }\left(t\right)\sim t^{-1/2}$ in the preasymptotic regime. The different concentrations of mobility defects are $c={10}^{-2},6.7\times {10}^{-3},5.2\times {10}^{-3},3.8\times {10}^{-3},2.8\times {10}^{-3}$, and $1.9\times {10}^{-3}$ for the FA model; $c=2\times {10}^{-1},1.1\times {10}^{-1},5.1\times {10}^{-2},1.6\times {10}^{-2},2.7\times {10}^{-3}$, and $2.8\times {10}^{-4}$ for the BBL model. Also shown is the drift of the probe in (c) FA and (d) BBL; data are at the same concentrations of mobility defects of (a) and (b). Collapse is obtained plotting $\langle x_{\epsilon }\left(t\right)\rangle /\langle x_{\epsilon }\left({\tau }_{\mathrm{eq}}\right)\rangle$ vs $t/{\tau }_{\mathrm{eq}}$.

Figure 2

Mean square displacement around the drift ${\sigma }_x^2\left(t\right)$ for a driven probe $\left(\epsilon =1/2\right)$ in (a) FA and (b) BBL. Different symbols represent different concentrations $c$ of the mobility defects, which are, respectively for the two models, the same as in Fig. 1. Collapse of the curves is obtained by plotting ${\sigma }_x^2\left(t\right)/{\sigma }_x^2\left({\tau }_{\mathrm{eq}}\right)$ vs $t/{\tau }_{\mathrm{eq}}$. Full line is the $t^{3/2}$ scaling; the dashed line in (a) is the Fickian behavior ${\sigma }_x^2\left(t\right)\sim t$.