Diffusion/Subdiffusion in the Pushy Random Walk

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.

💡 Research Summary

The authors introduce the “pushy random walk,” a stochastic tracer model in which a moving particle can push not only single obstacles but also clusters of obstacles, thereby creating a macroscopic cavity in a medium of finite obstacle density ρ. Each obstacle carries a unit mass; when a cluster of total mass M is encountered, the tracer and the cluster move together by one lattice spacing a with a rate proportional to M^{‑α}. The paper focuses primarily on the case α = 1, but the analysis is readily generalized to arbitrary α ≥ 0.

In one dimension the tracer clears a linear cavity of length L(t). On either side of the cavity a “crust” of obstacles of thickness z forms. Mass conservation gives z = ρL/