Normal and anomalous diffusion in the disordered wind-tree model

Ehrenfests' wind-tree model (EWTM) refers to a two-dimensional system where noninteracting point tracer particles move through a random arrangement of overlapping or nonoverlapping square-shaped scatterers. Here, extensive event-driven molecular dynamics simulations of the EWTM at different reduced scatterer densities $\rho$ are presented. For nonoverlapping scatterers, the asymptotic motion of the tracer particles is diffusive. We compare their diffusion coefficient $D$, as obtained from the simulation, with that predicted by kinetic theory where $D^{-1}$ is expanded up to the second order in the scatterer density. While at low density quantitative agreement between theory and simulation is found, we show that beyond the low-density regime deviations to the theory are associated with the emergence of a maximum in the non-Gaussian parameter at intermediate times. For the case of overlapping scatterers, in agreement with a theoretical prediction, the asymptotic motion of the tracer particles is subdiffusive, i.e., the mean-squared displacement at long times $t$ grows like $t^{1-2\rho /3}$. We propose a model of the van Hove correlation function that describes the density dependence of the tracer particles' asymptotic subdiffusive transport on a quantitative level.

Figure 1

The obstacle in the wind-tree model is a square with linear dimension $a$. As illustrated by the dashed lines and arrows, a tracer particle can approach the obstacle from four possible directions (north, east, south, west) and is scattered by the obstacle to one of these directions.

Figure 2

Trajectory of a tracer particle in the wind-tree model with nonoverlapping (nov) squares at the density $\rho =0.3$. In this case, small escape channels between the obstacles (see zoomed-in region) are important for the diffusive transport of the tracer.

Figure 3

(a) MSD $\delta r^2\left(t\right)$ for the wind-tree model with nonoverlapping (nov) squares for different densities. The inset shows the MSD scaled with the squared mean-free path ${\lambda }^2$ as a function of $t/t_0$ (with $t_0$ the mean-free time). (b) Exponent parameter $\gamma \left(t\right)$, as obtained from the MSDs in panel (a) [for the definition of $\gamma \left(t\right)$ see text]. The inset shows the same data as a function of $t/t_0$.

Figure 4

(a) Reduced diffusion coefficient $D^{*}$ as a function of $\rho$, as obtained from the simulation with obstacle configurations, generated by random sequential adsorption (RSA, blue circles) and annealing via Monte Carlo (green diamonds), in comparison to the theoretical prediction by Hauge and Cohen [33] (dashed line). For the details on the fit function (red line) see text. The inset shows ${\mathrm{\Delta }}_f$, as defined by Eq. (7), as a function of $\rho$. Here, the horizontal dashed line marks the value of $a_3$, as used in the fit (red line) in the main plot. (b) Static structure factor $S\left(q\right)$ of the obstacles at $\rho =0.55$ for the RSA and the annealed samples.

Figure 5

Non-Gaussian parameter ${\alpha }_2\left(t\right)$ at different densities for the wind-tree model with nonoverlapping (nov) squares. The inset shows the same data as a function of $t/t_0$.

Figure 6

Trajectories of the wind-tree model with overlapping (ov) squares at the densities $\rho =0.2$ (top), $\rho =0.4$ (bottom left), and $\rho =0.6$ (bottom right). The three trajectories are obtained over a time of ${10}^5/t_0$. For $\rho =0.6$, a small region of the trajectory is magnified.

Figure 7

The upper panel illustrates a basic retracing (or retroreflective) path in the wind-tree model with overlapping (ov) squares. Lower panel: Detailed view on a trajectory at $\rho =0.3$. The magnified regions indicate the occurrence of retroreflections.

Figure 8

(a) Scaled MSD $\delta r^2\left(t\right)/{\lambda }^2$, for the wind-tree model with overlapping (ov) squares at different densities, starting at $\rho =0.05$ (top curve in red) and then in density steps of 0.1 from $\rho =0.1$ to $\rho =0.9$ (lowest blue curve). The inset shows $D\left(t\right)/D_{\mathrm{B}}$, calculated from the derivative of the MSDs with respect to time. (b) Exponent parameter $\gamma \left(t\right)$, as obtained from the MSDs.

Figure 9

(a) Asymptotic exponent ${\gamma }_a={lim}_{t\to \infty }\gamma \left(t\right)$ as a function of $\rho$ versus the low density prediction of kinetic theory, ${\gamma }_a=1-\frac{2}{3}\rho$ (red dashed line). (b) Amplitude ratio $A/A_0$ as a function of density. The lines represent fits to Eq. (15) with different values of $n$.

Figure 10

Scaled van Hove correlation function $G^{★}\left(r^{★}\right)$ of the wind-tree model with overlapping squares at different densities, calculated from $G(r,t)$ for times in the asymptotic subdiffusive regime (see text).

Figure 11

Non-Gaussian parameter ${\alpha }_2\left(t\right)$ for densities $\rho =0.1$ (lowest curve) to $\rho =0.9$ (top curve) in density steps of 0.1. The dashed red lines mark the asymptotic values of ${\alpha }_2\left(t\right)$.

Figure 12

Averaged correlation function $G^{★}\left(r^{★}\right)$ for densities from $\rho =0.1$ to $\rho =0.8$ in steps of 0.1. The dashed green line is the Gaussian function $exp[-{\left(r^{★}\right)}^2/2]/\left(2\pi \right)$. The inset shows the exponent $\beta$ as a function of density. Here, the solid black line is the function $\beta \left(\rho \right)=2/(1+\frac{4}{3}\rho )$.