Active cage model of glassy dynamics

We build up a phenomenological picture in terms of the effective dynamics of a tracer confined in a cage experiencing random hops to capture some characteristics of glassy systems. This minimal description exhibits scale invariance properties for the small-displacement distribution that echo experimental observations. We predict the existence of exponential tails as a crossover between two Gaussian regimes. Moreover, we demonstrate that the onset of glassy behavior is controlled only by two dimensionless numbers: the number of hops occurring during the relaxation of the particle within a local cage and the ratio of the hopping length to the cage size.

Figure 1

Time-evolution of (a) the mean-square displacement (MSD), and (b) the non-Gaussian parameter (NGP). The MSD saturates for a steady cage (solid black line), corresponding to a vanishing NGP. (c) Comparison of the theoretical prediction of the distribution of displacement with that of colloidal particles at density $\phi =0.429$ taken from Ref. [10]. (d) Theoretical distribution of displacement scaled by the standard deviation of the central Gaussian part corresponding to a binary Lennard-Jones glass-forming mixture for different temperatures at the $\alpha$ relaxation time [35]. See the details in Appendix pp2.

Figure 2

Distribution of small displacement scaled by $x^{*}\left(t\right)=\sigma \sqrt{2f_t/(1-{\tau }_{\text{R}}/{\tau }_0)}$ in Eq. (18). The central part is Gaussian (solid black line) with power-law tails (dashed lines). (Inset) Crossover value $X$ between the central Gaussian part and the tails of the distribution as a function of ${\tau }_0/{\tau }_{\text{R}}$.

Figure 3

Flow curves of the Fourier-displacement distribution at $q=1/\sigma$. (a) The transition from a single to a two-step relaxation as $D_{\text{A}}/D$ decreases is the hallmark of glassy systems. There is no terminal relaxation for a steady cage (solid black line). The curves are Gaussian at all times when $\varepsilon \ll \sigma$ (dotted lines). (Inset) For the two-step dynamics, we scale the time by the terminal relaxation time to reveal an exponential master curve (dot-dashed line). (b) The non-Gaussian statistics affects the relaxation when $\varepsilon \gtrsim \sigma$ (dashed lines). (c) The curves such as $\varepsilon \gtrsim \sigma$ fall into the left wall $e^{-t/{\tau }_0+f_t}$ (dashed line), reflecting the dynamical slowing down as the cage hops become less frequent.