  The foraging problem asks how a collective of particles with limited
computational, communication and movement capabilities can autonomously
compress around a food source and disperse when the food is depleted or
shifted, which may occur at arbitrary times. We would like the particles to
iteratively self-organize, using only local interactions, to correctly gather
whenever a food particle remains in a position long enough and search if no
food particle has existed recently. Unlike previous approaches, these search
and gather phases should be self-induced so as to be indefinitely repeatable as
the food evolves, with microscopic changes to the food triggering macroscopic,
system-wide phase transitions. We present a stochastic foraging algorithm based
on a phase change in the fixed magnetization Ising model from statistical
physics: Our algorithm is the first to leverage self-induced phase changes as
an algorithmic tool. A key component of our algorithm is a careful token
passing mechanism ensuring a dispersion broadcast wave will always outpace a
compression wave. We also present a highly structured alternative algorithm
that gathers by incrementally building a spiral tightly wrapped around the food
particle.
