Subdiffusive motion in kinetically constrained models

We discuss a kinetically constrained model in which real-valued local densities fluctuate in time, as introduced recently by Bertin, Bouchaud, and Lequeux. We show how the phenomenology of this model can be reproduced by an effective theory of mobility excitations propagating in a disordered environment. Both excitations and probe particles have subdiffusive motion, characterized by different exponents and operating on different time scales. We derive these exponents, showing that they depend continuously on one of the parameters of the model.

Figure 1

Typical realizations of the propensity, with times such that the spatially averaged persistence function satisfies $1-⟨p\left(t\right)⟩\simeq 0.4$. Large values of the propensity indicate sites that are very likely to have relaxed, on this time scale. The models with subdiffusive dynamics have large jumps in the propensity, which arise from sites that relax very infrequently. (a) BBL model at $\mu =2$ and $\eta =0.01$. (b) Bond-disordered FA model at $\mu =2$ and $c=0.01$. (c) Pure FA model at $c=0.01$. In the pure case, there are no barriers for excitation propagation, and the propensity is smooth [except near sites with $n_i(t=0)=1$, where it is maximal].

Figure 2

Subdiffusion in the FA and BBL models is apparent in the four-point susceptibility. (a) We show ${\chi }_{4\mathrm{n}}\left(t\right)$ in the BBL model at various $\mu$ and $\eta$. For the case $\mu =0$, we use a binary distribution of on-site densities $P_{\mathrm{s}}\left(\rho \right)\propto \alpha \delta \left(\rho \right)+\delta (2-\rho -0^{+})$, so that $\eta =\alpha (2+\alpha )∕{(1+\alpha )}^2$; the small offset $0^{+}$ ensures that the facilitation constraint is well defined. The dashed lines show the power law predictions of Eq. (24), and simple diffusive scaling (23) for $\mu =0$. At large times, ${\chi }_{4\mathrm{n}}\left(t\right)$ saturates at a value of order ${\eta }^{-1}$: this is apparent in the data for $\mu =0$. The label $\left(G\right)$ denotes data for the grand canonical variant of the BBL model. (b) Bond-disordered FA model data with the same values of $\mu$, showing that this effective model captures the four point correlations of the BBL model. The labels $\left(Q\right)$ and $\left(S\right)$ denote data from the model with quenched bond disorder and fluctuating site disorder respectively; the different variants have very similar behavior.

Figure 3

Dynamics of a random walker in a disordered environment of random energy barriers, where the distribution of barrier-crossing rates is $P_r\left(r\right)$, with $\mu =3$. We compare quenched and fluctuating disorder: in the fluctuating case, the rate for hopping between sites $i$ to $i+1$ is randomized whenever the random walker hops between these sites. We compare these two dynamical models with the effective dynamics discussed in the main text. (a) Mean square displacement of the random walker, $x^2\left(t\right)$. In the long-time regime, we find dynamical scaling $x^2\left(t\right)\simeq {\left(\Omega t\right)}^{2∕(1+\mu )}={\left(\Omega t\right)}^{1∕2}$. The power law $x^2\sim t^{1∕2}$ is shown as a light dashed line. The agreement between the effective dynamics and the model with quenched disorder was discussed in [26]. Here we emphasize that the only effect of introducing fluctuating disorder is a small reduction in the prefactor $\Omega$. (b) Scaling form of the diffusion front, normalized to equal height at the origin. That is, we write the distribution of the particle displacement as $P(x\mid t)=P(0\mid t)f\left[xP(0\mid t)\right]$ and we plot the function $f\left(u\right)$. Within the scaling regime, $f\left(u\right)$ does not depend on the time $t$: the data shown were obtained at $t=2^{22}$, for which this condition is satisfied. The results for models with quenched and fluctuating disorder are very similar. As discussed in [26], deviations between the models and the effective dynamics are expected for finite $\mu$, but vanish in the limit of large $\mu$.

Figure 4

(a) Persistence function, plotted against the scaling variable $\lambda t$ in the site-disordered FA model. We show data for $c=0.5$, 0.27, 0.12, 0.047, 0.018 (decreasing from left to right; these are inverse temperatures $\beta =0$, 1, 2, 3, 4). As discussed in the text, the persistence time scales as ${\tau }_p\sim c^{-2-\mu }\sim c^{-4}$, but we use the variable variable $\lambda t$ to verify the long time prediction of Eq. (35): the dashed line shows a power law with exponent $1∕2$. (b) Persistence at infinite temperature and varying $\mu$. Again, dashed lines show power laws with exponents predicted by Eq. (35).

Figure 5

Comparison of root mean square probe displacement $X_{\mathrm{rms}}\left(t\right)={⟨X{\left(t\right)}^2⟩}^{1∕2}$ with the inverse of the persistence function $1∕p\left(t\right)$, for the site-disordered FA model. We show data at $\mu =3$ (a) and 4 (b). We have offset the data at $c=0.25$ for clarity: the ordinate is $10t$ in that case. In the long-time limit, the persistence scales as $t^{-1∕\mu }$ (this power law is illustrated with dashed lines). At long times the persistence sets a bound on the scaling of the mean square displacement, which appears to saturate for long times.

Figure 6

Mean square probe displacements in the BBL and FA models, for times shorter than ${\tau }_p$. (a) BBL model. For $\mu >2$, the dashed lines show the predictions of Eq. (40); for $\mu =1$, we show a diffusive law $⟨X^2\left(t\right)⟩\sim t$. (b) Site-disordered FA model, with the same power law predictions. In both cases, we expect a crossover at long times to the asymptotic scaling of Eq. (38).