Diffusion/Subdiffusion in the Pushy Random Walk

Abstract

We introduce the pushy random walk, where a walker can push multiple obstacles, thereby penetrating large distances in environments with finite obstacle density. This process gives a more realistic depiction of experimentally observed interactions of active particles in dense media. In one dimension, the walker carves out an obstacle-free cavity whose length grows subdiffusively over time. In two dimensions, increasing obstacle density drives a transition from free diffusion to localized behavior, where the walker is trapped within a cavity whose radius again grows subdiffusively with time.

Figures (7)

• Figure 1: (a) Illustration of a pushy random walk (circle) in one dimension. An isolated unit-mass obstacle (square) is pushed one lattice spacing at rate 1 by the walk. When a composite obstacle of mass 3 is created, it moves with rate $1/3^\alpha$ when hit by a pushy random walk. (b) A cavity of length $L$ and the crust of thickness magenta $z=4$ on one side.
• Figure 2: Cavity length versus time for a pushy random walk in $1D$ for different obstacle densities $\rho$. Solid lines give the theoretical prediction from Eq. \ref{['L-1d sol']} and symbols come from simulations averaged over $10^3$ walk trajectories. The dashed line shows the asymptotic behavior of $t^{1/3}$.
• Figure 3: The result after a random walk in 2D pushes against a composite obstacle. Only the horizontal row of 7 obstacles moves one lattice spacing.
• Figure 4: A cavity in two dimensions with a surrounding crust. The random walk is about to push an obstacle of 4 contiguous particles horizontally to the left by 1 lattice spacing.
• Figure 5: Cavity radius $R(t)$ vs. time, for the pushy random walk on the 2D square lattice with different obstacle densities. The dashed and solid lines indicate $t^{1/4}$ and $t^{1/2}$ asymptotics, respectively. Symbols represent simulation data based on $2560$ walk trajectories.