## 5 Results
### Building the Lattice-Based Surrogate Models  
#### 5.1.1 Equally Spaced Obstacles Case  
Obstacle radii: Continuous model simulations were performed on the RVE for the equally spaced obstacles model (see Section 3.2.1 and Figure 3a). The particle radius was first fixed at \(R_{\text{part}}=1.0\), and the obstacle radius (\(R_{\text{obs}}\)) was varied from 0.5 to 3.0 by increments of 0.5. Stationarity and linearity of \(\langle r^{2}\rangle\) versus time (Section 2.1) for an increasing number of realizations is illustrated, for a representative case (\(R_{\text{part}}=1.0\), \(R_{\text{obs}}=1.0\)), in Figure 7.  
Determining transition probabilities: Based on the approach illustrated in Figure 4, continuous model results were used to determine transition probabilities for the equally spaced obstacles model via (14). These probabilities are illustrated for transitions from subdomain 6 over the time course of one realization in Figure 8b-d; rescaling the range more effectively illustrates the stay and leave probabilities (Figure 8c,d, respectively). This figure also demonstrates the stationary nature of these probabilities at later times. In determining the final stay (\(p^{E}(1)\)) and leave (\(p^{E}(2)\)) transition probabilities for the lattice-based surrogate model, values of each type were pooled together (see (14)) and then averaged across all realizations. Note that our results are consistent with the improbability of diagonal transitions (Figure 8d, blue curve). Figure 7: Mean squared displacement versus time for the equally spaced obstacles model. Effects of an increasing number of realizations are shown for the case \(R_{\text{part}}=1.0\) and \(R_{\text{obs}}=1.0\): (**a**) curves; (**b**) linear fits to the curves. Commitment index calibration: Lattice-based surrogate models were built using the transition probabilities obtained from the continuous model (see (14), Section 3.2.2) by first calibrating the commitment index \(M_{\rm C}\) (Section 4.1). \(M_{\rm C}\) was varied from 3 to 8 and calibrated by comparing the continuous and surrogate plots of \(\langle r^{2}\rangle\) versus time. This procedure was repeated for varying obstacle radii, with the particle radius fixed at \(R_{\rm part}=1.0\), illustrated in Figure 9 for \(R_{\rm part}=1.0\) and \(R_{\rm obs}=2.0\). Here, \(M_{\rm C}=5\) yields the best result; 2000 realizations were sufficient to observe stationary values sufficient to calibrate \(M_{\rm C}\) (Figure 9--right). Probabilities and commitment index values are shown in Table 1. We observe that the sum of the first probability and four times the second probability is very close to 1 (based on symmetries in the RVE); the largest discrepancy was observed to be \(3.6\times 10^{-6}\). We also observe that the probability of staying in the current subdomain increases with the obstacle radius. Comparisons of slopes for \(\langle r^{2}\rangle\) versus time for the continuous and surrogate models are compared in Table 2 and Figure 10. We observe differences in slopes ranging from 0.004% to roughly 4%. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{3}{c}{**Probability by Transition Type**} \\ \hline \(R_{\rm obs}\) & \(M_{\rm c}\) & \(p_{1}^{\rm E}\) & \(p_{2}^{\rm E}\) \\ \hline
0.5 & 7 & 0.99276 & \(1.8099\times 10^{-3}\) \\
1.0 & 6 & 0.99291 & \(1.7724\times 10^{-3}\) \\
1.5 & 5 & 0.99297 & \(1.7573\times 10^{-3}\) \\
2.0 & 5 & 0.99371 & \(1.5734\times 10^{-3}\) \\
2.5 & 4 & 0.99384 & \(1.5401\times 10^{-3}\) \\
3.0 & 4 & 0.99493 & \(1.2676\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Transition probabilities in the lattice-based surrogate model for the equally spaced obstacles (continuous) model for the best values of \(M_{\rm c}\) with particle radius \(R_{\rm part}=1.0\). Figure 8: Transition probabilities for building the lattice-based surrogate model from the equally spaced obstacles (continuous) model.