## 5 Results
### Estimating Diffusivities
#### 5.2.2 Multisize Obstacles Case  
We also estimated effective diffusivities from simulated results for the continuous and surrogate models via (18) and for each obstacle radius combination (Figure 14). Time averages for each of these effective diffusivities over the last 100,000 values are shown in Table 7 along with the commitment index \(M_{C}\) and the stay probability \(q\). The continuous and surrogate diffusivities exhibit a correlation coefficient of \(R=0.8503\) and differences ranging between roughly 1.1% and 6.8%. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(R_{\text{obs}}\) & \(q\) & \(\mathbf{a^{2}}\) & \(\Delta\mathbf{x}\) & \(\mathbf{\kappa_{\text{cont}}}\) & \(\mathbf{\kappa_{\text{surr}}}\) & \(D\) & \(D|_{\Delta\mathbf{x=10}}\) \\ \hline
0.5 & 0.99276 & 5.1048 & 4.4260 & 0.7550 & 0.7100 & 0.7100 & 3.6243 \\
1.0 & 0.99291 & 5.0027 & 4.4709 & 0.7314 & 0.7084 & 0.7084 & 3.5440 \\
1.5 & 0.99297 & 5.1224 & 4.4184 & 0.6759 & 0.6862 & 0.6862 & 3.5149 \\
2.0 & 0.99371 & 5.0907 & 4.4321 & 0.6146 & 0.6165 & 0.6165 & 3.1384 \\
2.5 & 0.99384 & 5.0728 & 4.4399 & 0.5885 & 0.6016 & 0.6016 & 3.0517 \\
3.0 & 0.99493 & 4.8863 & 4.5239 & 0.5188 & 0.5187 & 0.5187 & 2.5343 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Theoretical diffusivity (\(D\)) and grid size (\(\Delta x\)) in the equally spaced obstacles case. The stay probability (\(q\)), best commitment index (\(M_{C}\)) and theoretical diffusivity with \(L=10\) (\(D|_{\Delta x=10}\)) are also shown for each obstacle radius (\(R_{\text{obs}}\)). Recalling Equation (31) (Section 4.2.2), for the theoretical diffusivities in the multisize obstacles case, we write  
\[D_{H}=\frac{q_{x}(\Delta x_{M})^{2}}{2(\Delta t)},\ \ \ D_{V}=\frac{q_{y}(\Delta x_{M})^{2}}{2( \Delta t)}. \tag{38}\]  
Here, \(\Delta x_{M}\) denotes the effective grid size for the multisize obstacles case, and its value remains to be calibrated. Note, however, that the ratio of these two diffusivities is independent of this parameter. Consequently, we first examined this ratio using (36) by evaluating the relevant mean squared displacements (Figure 15) and their ratio (Figure 16). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(R_{\text{obs}}\) & \(\mathbf{M_{c}}\) & \(\mathbf{q_{1}}\) & \(\mathbf{\kappa_{\text{cont}}}\) & \(\mathbf{\kappa_{\text{surr}}}\) &
\begin{tabular}{c} **Percent** \\ **Difference** \\ \end{tabular} \\ \hline
0.5 & 6 & 0.9925 & 0.7509 & 0.7428 & 1.084\% \\
1.0 & 6 & 0.9929 & 0.6842 & 0.7261 & 6.133\% \\
1.5 & 6 & 0.9932 & 0.6943 & 0.7418 & 6.845\% \\
2.0 & 6 & 0.9935 & 0.6746 & 0.6290 & 6.755\% \\
2.5 & 5 & 0.9932 & 0.6205 & 0.6440 & 3.785\% \\
3.0 & 5 & 0.9939 & 0.5667 & 0.5942 & 4.859\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Effective diffusivities in the multisize obstacles case for the continuous model (\(\kappa_{\text{cont}}\)) and the lattice-based surrogate model (\(\kappa_{\text{surr}}\)). The resulting stay probability (\(q_{1}\)) in the square lattice surrogate model and the best commitment index (\(M_{\text{C}}\)) are also shown for each obstacle radius (\(R_{\text{obs}}\)). The multisize obstacles model has the property that the RVE is more open in either the horizontal (\(R_{\text{obs}_{2}}<1\)) or vertical (\(R_{\text{obs}_{2}}>1\)) direction, depending on the value of \(R_{\text{obs}_{2}}\). This effect is subtle for \(R_{\text{obs}_{2}}<1\) due to the large amount of available space for particle diffusion (Figures 15a and 16a). Significant effects on the ratio of diffusivities are observed for larger values in the regime \(R_{\text{obs}_{2}}>1\), where available space in the horizontal direction becomes more limited (Figures 15e,f and 16e,f). An increased time to achieve a stationary response is also observed for these cases. The effective grid size \(\Delta x_{M}\) can be computed in one of two ways, by approximating \(D_{H}\) and \(D_{V}\) by \(\kappa_{H}\) and \(\kappa_{V}\), respectively, and solving the corresponding relation in (38) for \(\Delta x_{M}\) (Table 8).