Extension and dynamical phases in random walkers depositing and following chemical trails

TL;DR

This paper introduces a model of a random walker depositing and following a chemical trail, revealing structural transitions and re-entrant behavior in its trajectory, with implications for understanding active biological systems.

Contribution

It presents a simple yet novel model of a chemically influenced random walk, analyzing structural and dynamical phase transitions through simulations and mean field theory.

Findings

Non-monotonic coil-globule transition with deposition rate

Re-entrant behavior in trajectory extension

Deviation from diffusive scaling at intermediate times

Abstract

Active walker models have proved to be extremely effective in understanding the evolution of a large class of systems in biology like ant trail formation and pedestrian trails. We propose a simple model of a random walker which modifies its local environment that in turn influences the motion of the walker at a {\em later} time. We perform direct numerical simulations of the walker in a discrete lattice with the walker actively depositing a chemical which attracts the walker trajectory and also evaporates in time. We propose a method to look at the structural transitions of the trajectory using radius of gyration for finite time walks. The extension over a definite time-window shows a non-monotonic change with the deposition rate characteristic of a coil-globule transition. At certain regions of the parameter space of the chemical deposition and evaporation rates, the extensions of the walker shows a re-entrant behavior. The dynamics, characterised by the mean-squared displacement, shows deviation from diffusive scaling at intermediate time scales, returning to diffusive behavior asymptotically. A mean field theory captures the variation of the asymptotic diffusivity.