Pushing-Induced Arrest Across Lattices and Dimensions

Abstract

Tracer-media interactions can give rise to transport phenomena beyond classical models; e.g., obstacle pushing can eliminate percolation. We demonstrate that the existing explanation of this effect fails in 3D. We show that confinement is governed by emergent trapping-rare door-closing events with constant probability per step-yielding exponential survival. This allows prediction of the time-dependent mean-squared displacement from short-time estimates of the diffusion constant and trapping probability, providing a minimal description of pushing-induced arrest across lattices and dimensions.

Figures (9)

• Figure 1: AIL vs Sokoban random walk. Examples of an (a) AIL and (b) Sokoban random walk on the Triangular lattice. Green nodes are occupied with obstacles, and white nodes are free. The orange tracers wander around obeying their respective laws of motion. While the AIL escapes, the Sokoban tracer happens to permanently confine itself. Two obstacles are marked with a circle and a cross, for traceability. They are moved by the Sokoban tracer during the walk, but are identical to all other obstacles.
• Figure 2: Existence of "Snowplow effect" for the Sokoban in different $2D$ lattices. The terminal MSD, $\mathrm{MSD}_{\infty}$, vs. the obstacle density $\rho$ for the Square, Triangular and Hexagonal lattices. Predictions coming from Eq. (\ref{['eq: snowplow prediction']}) (dashed lines) are in excellent agreement with Monte Carlo simulations (markers). The dotted vertical lines represent the percolation thresholds for the respective lattices.
• Figure 3: Trapping of the Sokoban random walk. (a) The Sokoban random walk ends its life in pocket (zoom-in) that can be orders of magnitude smaller than the size of the walk. Here, $\mathcal{A}$ and $\mathcal{P}$ that were defined below Eq. (\ref{['eq: snowplow prediction']}) are colored blue and pink, respectively. Obstacles confining the terminal pocket are colored green. (b) The survival probability, $S(n)$, of a Sokoban tracer for various lattices. Dashed lines are exponential fits. Additional densities are presented in appendix \ref{['appendix: trapping data']}. (c) An example of confinement via trapping. On the left, the tracer enters a pocket of obstacles, and then "closes the door", permanently confining itself to the six nodes inside the pocket. (d) Comparison of the directly measured trapping probability (markers) and the exponential decay rate extracted from the fits in (b).
• Figure 4: Successful prediction of $\mathrm{MSD}(n)$ across lattices and dimensions. The MSD of the Sokoban random walk (markers for various lattices) compared to the prediction provided by Eq. (\ref{['eq: MSD _t_ analytical']}) shown in dashed lines. Additional densities presented in appendix \ref{['appendix: trapping data']}.
• Figure A1: In $2D$, $\mathcal{A}$ and $\mathcal{P}$ are fractals.$\langle\mathcal{A}\rangle$ and $\langle\mathcal{P}\rangle$ from Eq. (\ref{['eq: snowplow definitions']}) for the Sokoban on (a) Square (b) Triangular and (c) Hexagonal lattices. In (a-c), the orange dataset corresponds to the average area, $\langle\mathcal{A}\rangle$, while the blue corresponds to the average perimeter, $\langle\mathcal{P}\rangle$. We also present the measured ratio $F_\mathcal{A}/F_\mathcal{P}$ for these lattices in panel (d), where different markers correspond to different lattices.