Running and Tumbling Localized Structures: A Non-Brownian Motion

Macroscopic systems present particle-type solutions. Spontaneous symmetry-breaking can cause these solutions to travel in different directions, and the inclusion of random fluctuations can induce them to run and tumble. We investigate the running and tumbling of localized structures observed on a prototype model of one-dimensional pattern formation with noise. Statistically, the dynamics of localized structures are examined, particularly the mean square displacement as a function of time. It initially shows a diffusive behavior, replaced by a ballistic one, and finally manifests itself as diffusive again. We derive a minimal model for the position and velocity of localized structures, which reveals the origin of the observed dynamics.