Prof. Jan Wehr (Department of Mathematics, The University of Arizona, USA).
Exploring a terrain with random direction changes.

An explorer moves in an inhomogeneous environment, changing the direction according to a random process, adapting its speed to the terrain changes with a delay. This is modeled by a system of stochastic delay equations, which is approximated by a system of stochastic differential equations (SDE) and studied in the limit in which the direction changes occur very fast. In the experimental realization of this system, the explorer is a light-sensitive mini-robot, programmed to react to the changes of the local light intensity with a desired delay. Within our approximation, the delay can be negative (peaking into the future). A qualitative change of behavior occurs at a certain negative value of the delay. Consequences for the behavior of a swarm of such explorers, interacting with each other follow from a mean field argument and are confirmed by experiment. In particular, at a critical value of the delay parameter, an aggregation-deaggregation transition occurs. The mathematical analysis of the problem is a particular case of a general theorem about singular limits of SDE systems.

‍Warsaw colloquium for theoretical physics at CTP PAS – 2024-06-12