Stochastic harmonic trapping of a Levy walk: transport and first-passage dynamics under soft resetting strategies

We introduce and study a Levy walk (LW) model of particle spreading with a finite propagation speed combined with soft resets, stochastically occurring periods in which an harmonic external potential is switched on and forces the particle towards a specific position. Soft resets avoid instantaneous relocation of particles that in certain physical settings may be considered unphysical. Moreover, soft resets do not have a specific resetting point but lead the particle towards a resetting point by a restoring Hookean force. Depending on the exact choice for the LW waiting time density and the probability density of the periods when the harmonic potential is switched on, we demonstrate a rich emerging response behaviour including ballistic motion and superdiffusion. When the confinement periods of the soft-reset events are dominant, we observe a particle localisation with an associated non-equilibrium steady state. In this case the stationary particle probability density function turns out to acquire multimodal states. Our derivations are based on Markov chain ideas and LWs with multiple internal states, an approach that may be useful and flexible for the investigation of other generalised random walks with soft and hard resets. The spreading efficiency of soft-rest LWs is characterised by the first-passage time statistic.

Automated summary

We introduce and study a Levy walk model with a finite propagation speed combined with soft resets. The model exhibits a rich emerging response behavior, including ballistic motion, superdiffusion, and particle localization. Our research findings suggest that the soft-reset Levy walk model has potential applications in investigating other generalized random walks with soft and hard resets.