Transient subdiffusion from an Ising environment

We introduce a model in which a particle performs a continuous-time random walk (CTRW) coupled to an environment with Ising dynamics. The particle shows locally varying diffusivity determined by the geometrical properties of the underlying Ising environment, that is, the diffusivity depends on the size of the connected area of spins pointing in the same direction. The model shows anomalous diffusion when the Ising environment is at critical temperature. We show that any finite scale introduced by a temperature different from the critical one, or a finite size of the environment, cause subdiffusion only during a transient time. The characteristic time, at which the system returns to normal diffusion after the subdiffusive plateau depends on the limiting scale and on how close the temperature is to criticality. The system also displays apparent ergodicity breaking at intermediate time, while ergodicity breaking at longer time occurs only under the idealized infinite environment at the critical temperature.

Figure 1

Schematic of the CTRW performed by the particle in an Ising environment. (a) We present a squared Ising lattice close to the critical temperature, $k_B{T}_c\approx 2.2691\text{J}$, with $k_B$ the Boltzmann constant, and side length of $L=500$. Here dark (light) blue is pointing up (down) spins. The white area is the domain in which the particle sits at time $t$. We show a zoom of a region in which two particles (red and blue curves) perform a CTRW with waiting-time PDF given by Eq. (1), assuming that the Ising chain does not evolve. We highlight the domains entered by the particles after leaving the initial domain with the same color code as the trajectories. (b) A scheme of a particle's motion through two different clusters. Here, $\kappa$ is determined from the size of the domain in which they sit, according to Eq. (3).

Figure 2

Ensemble-averaged mean-squared displacement (eMSD): comparison between the theoretical predictions and Monte Carlo simulations. (a) eMSD for different theoretical values of $\eta$ and $\mu =0$ for infinite-size Ising environments at critical temperature. The movement is subdiffusive, that is ${\mathrm{M}}_2\left(t\right)\propto t^{\alpha }$, with $0\le \alpha \le 1$. Dashed black line represents the Brownian motion limit. Dotted lines show the theoretical prediction of Eqs. (7) and (9) [same for (b)]. Inset: the corresponding probability distribution function $P_{\mathrm{new}}\left(\kappa \right)$ used for the simulations. (b) eMSD obtained for different values of $\mu$ and $\eta =0.05$. The expected subdiffusive behavior is observed. (c) Relation between the eMSD exponent $\alpha \left(\mu \right)$ and the time-scale parameter $\mu$. In this panel, symbols represent the numerical calculations while lines are theoretical predictions. We see that for $\mu >\xi -1=0.05$ normally diffusing Brownian motion occurs, i.e., $\alpha \left(\mu \right)=1$.

Figure 3

Ensemble-averaged mean-squared displacement out of criticality due to deviations from the critical temperature and finite-size effects. (a) eMSD for $\alpha =0.05$ and different deviations from the critical temperature, described by the typical size $S^{*}$. A transient subdiffusive behavior occurs for intermediate times. As $S^{*}$ is increased, the onset of diffusive dynamics occurs at a longer time. (b) eMSD for $\alpha =0.2$ and different values of the size $N$ of the Ising environment at critical temperature. As shown the movement is diffusive in the long term behavior. In intermediate times, the finite-size cases show the same subdiffusive behavior than the infinite size case (bottom curve). The position at which it departs from subdiffusion is larger as the size is increased. Dashed lines show the Brownian motion limit.

Figure 4

Time-averaged mean-squared displacement $\overline{{\delta }^2}(t,t_{\mathrm{lag}})$ in Ising environment of different sizes. (a) Time-averaged mean-squared displacement for $\alpha =0.4$ in the infinite-size Ising environment. As shown, the tMSD remains a random variable in time, as seen by the large scattering observed at all time lags. (b–d) Same in the presence of a maximum size $N=L^2={10}^{16},{10}^{12}$, and ${10}^6$, respectively.

Figure 5

Behavior of the ergodicity-breaking parameter $\mathcal{E}\left(t\right)$ for different processes and sizes of the environment. (a) Ergodicity-breaking parameter for different values of $\alpha$ in the infinite-size environment case. This parameter tends asymptotically to the value predicted by Eq. (15) [we indicate the asymptotic value of the ergodicity breaking parameter, $\overline{\mathcal{E}}$, for each $\alpha$ with horizontal lines]. (b) Ergodicity-breaking parameter for $\alpha =0.4$ both for the infinite- and finite-size environment cases. For all finite cases the curves tend to zero asymptotically for the maximum measurement time taken here, $t={10}^6$. This means that the behavior is ergodic at long times for finite size if the measurement time is large enough. The time at which it reaches the zero value is larger as the size of the environment is increased.

Figure 6

Distribution of the domain sizes and of the average number of steps $\overline{n}$ in a domain of $S$ obtained from Monte Carlo simulations. (a) Distribution of domain sizes for 200 patterns calculated at the critical temperature, $k_B{T}_c\approx 2.2691\text{J}$, for $L=500$. (b) Numerically calculated probability distribution of the numbers of steps taken before exiting a domain of size $S$ when the initial position is in the border. To do the calculations we select all the patterns of a given size $S$, perform $X$ simulations of a particle starting at random positions in the border of the domain, and retain the time at which the particle leaves the domain. Inset: average time $\overline{n}$ for leaving a domain of size $S$ in log-log scale, showing that $\overline{n}$ grows approximately as $S^{\mu }$, with $\mu \sim 0.3$.