A run-and-tumble particle moves in one direction until it randomly reverses. Unlike Brownian motion, it has an internal state — a direction of self-propulsion. Sezik, Britton, Touma, and Pruessner (arXiv: 2603.24277) ask: what does it paint?
They track not just whether a location is visited, but what the particle's internal state was when it visited. A position “painted” while the particle moved rightward carries different information from the same position painted during leftward motion. This is “polar deposition” — the particle's trajectory deposits directional information onto space.
Using Doi-Peliti field theory, they compute the state-dependent visit probability in one dimension: the likelihood that a particle reaches a given point before a deadline while in a specific propulsion state. The total painted volume — all positions visited in any state — has a clean analytical expression that extends known Brownian motion results.
The through-claim: the trail is richer than the trajectory. A Brownian walker and a run-and-tumble particle might visit the same total volume, but the run-and-tumble particle's trail carries directional information that the Brownian walker's doesn't. The internal state of the walker — invisible in position-only tracking — is written into the pattern of visits. Active matter doesn't just explore space; it annotates it.
For biological systems where cells secrete different signals depending on their motile state, the distinction matters. The chemical landscape created by a run-and-tumble bacterium carries information about how it was moving when it passed through.
Sezik, Britton, Touma & Pruessner, 2603.24277. Statistical mechanics / active matter / run-and-tumble / field theory.