## 5 Results
### Estimating Diffusivities
#### 5.2.1 Equally Spaced Obstacles Case  
We estimate the effective diffusivities from simulated results for the mean squared displacement in both the continuous (\(\kappa_{\text{cont}}\)) and surrogate (\(\kappa_{\text{surr}}\)) models via (18) and for each obstacle radius (Figure 13). Time averages for both effective diffusivities over the last 100,000 values are shown in Table 5, along with the commitment index \(M_{C}\) and the stay probability \(q\). The continuous and surrogate diffusivities exhibit a correlation coefficient of \(R=0.9784\) and percent differences between 0.02% and roughly 6%. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(\mathbf{R}_{\text{obs}_{\text{s}}}\)} & \multirow{2}{*}{\(\mathbf{M_{c}}\)} & \multirow{2}{*}{\(\mathbf{Sloep}\)} & \multirow{2}{*}{\(\mathbf{Sloep}\)} & \multirow{2}{*}{\(\mathbf{Sloep}\)} \\  & & & (Continuous) & & \\ \hline
0.5 & 6 & 2.9870 & 3.0059 & 0.632\% \\
1.0 & 6 & 2.7357 & 2.8531 & 4.291\% \\
1.5 & 6 & 2.7376 & 2.8939 & 5.709\% \\
2.0 & 6 & 2.6618 & 2.4751 & 7.014\% \\
2.5 & 5 & 2.4995 & 2.5904 & 3.637\% \\
3.0 & 5 & 2.2710 & 2.3651 & 4.144\% \\ \hline \end{tabular}
\end{table} 
Table 4: Slope values for \(\langle r^{2}\rangle\) vs. \(t\) in the multisize obstacles (continuous) model (\(R_{\text{part}}=1.0\), \(R_{\text{obs}_{\text{s}}}=1.0\)) and the lattice-based surrogate model. The best values of \(M_{C}\) are also shown. \begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Probability by Transition Type**} \\ \hline \(\mathbf{R}_{\text{obs}_{\text{s}}}\) & \(\mathbf{M_{c}}\) & \(\boldsymbol{p_{1}^{M}}\) & \(\boldsymbol{p_{2}^{M}}\) & \(\boldsymbol{p_{3}^{M}}\) & \(\boldsymbol{p_{4}^{M}}\) \\ \hline
0.5 & 6 & 0.99253 & \(1.8680\times 10^{-3}\) & \(2.0159\times 10^{-3}\) & \(1.7147\times 10^{-3}\) \\
1.0 & 6 & 0.99291 & \(1.7715\times 10^{-3}\) & \(1.7764\times 10^{-3}\) & \(1.7725\times 10^{-3}\) \\
1.5 & 6 & 0.99324 & \(1.6894\times 10^{-3}\) & \(1.5339\times 10^{-3}\) & \(1.8510\times 10^{-3}\) \\
2.0 & 6 & 0.99352 & \(1.6203\times 10^{-3}\) & \(1.2909\times 10^{-3}\) & \(1.9522\times 10^{-3}\) \\
2.5 & 5 & 0.99316 & \(1.7098\times 10^{-3}\) & \(1.1322\times 10^{-3}\) & \(2.2850\times 10^{-3}\) \\
3.0 & 5 & 0.99339 & \(1.6502\times 10^{-3}\) & \(8.1439\times 10^{-4}\) & \(2.4950\times 10^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Transition probabilities in the lattice-based surrogate model for the multisize obstacles (continuous) model for the best values of \(M_{c}\) with particle radius \(R_{\text{part}}=1.0\) and \(R_{\text{obs}_{\text{s}}}=1.0\). \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(\mathbf{R}_{\text{obs}_{\text{s}}}\)} & \multirow{2}{*}{\(\mathbf{M_{c}}\)} & \multirow{2}{*}{\(\mathbf{Sloep}\)} & \multirow{2}{*}{\(\mathbf{Sloep}\)} & \multirow{2}{*}{\(\mathbf{Sloep}\)} \\  & & & (Continuous) & & \\ \hline
0.5 & 6 & 2.9870 & 3.0059 & 0.632\% \\
1.0 & 6 & 2.7357 & 2.8531 & 4.291\% \\
1.5 & 6 & 2.7376 & 2.8939 & 5.709\% \\
2.0 & 6 & 2.6618 & 2.4751 & 7.014\% \\
2.5 & 5 & 2.4995 & 2.5904 & 3.637\% \\
3.0 & 5 & 2.2710 & 2.3651 & 4.144\% \\ \hline \end{tabular}
\end{table} 
Table 4: Slope values for \(\langle r^{2}\rangle\) vs. \(t\) in the multisize obstacles (continuous) model (\(R_{\text{part}}=1.0\), \(R_{\text{obs}_{\text{s}}}=1.0\)) and the lattice-based surrogate model. The best values of \(M_{C}\) are also shown. Estimated diffusivities for the surrogate models were used to compute values of \(\alpha^{2}\) and \(\Delta x(=\frac{L}{a})\), for each case. Comparisons of the effective diffusivities to the resulting theoretical diffusivities, determined via (33), are shown in Table 6. For comparison, the theoretical estimate of the diffusivity for the case \(L=10\), corresponding to \(\alpha^{2}=1\), is also shown in the table. The estimated values of both \(\alpha^{2}\) (mean 5.0466, st. dev. 0.0888) and \(\Delta x\) (mean 4.4519, st. dev. 0.0397) exhibited little variation or correlation (\(R=0.5532\)) across the range of obstacle sizes considered.