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

Abstract

We introduce and study a Levy walk 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 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 Levy walk 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 Levy walks 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 Levy walks is characterised by the first-passage time statistic.

Stochastic dynamics with multiplicative dichotomic noise: Heterogeneous telegrapher’s equation, anomalous crossovers and resetting

“…Moreover, in view of the importance of resetting phenomena in the context of search problems, we can speculate that the heterogeneous telegrapher's equation with resetting represents a toy model of random search in a turbulent environment. We also note that run-and-tumble particle motion under stochastic resetting [82,[86][87][88] and more generalized models of Lévy walks under resetting [89,90] are of current interest. By Laplace transform of equation ( 48), it follows that Pr (x, s) = s + r s P (x, s + r).…”

Thermodynamic work of partial resetting

“…Here a resetting trap is turned on, presumably at a random instance of time, in order to bring a particle to a prescribed position. This resetting scheme has been studied both in theory [52][53][54][55][56][57][58] and in recent experimental implementations of resetting using optical tweezers [27,59,60]. Other types of costs that do not necessarily correspond to thermodynamic quantities has also been considered recently [61,62].…”

Time-dependent probability density function for partial resetting dynamics

“…Various aspects beyond Brownian SR have been discussed. Inter alia, non-instantaneous returns [49] and soft resetting by switching harmonic potentials [50] were studied. SR of anomalous diffusion processes include heterogeneous diffusion processes with distance-dependent diffusion coefficient [51][52][53], scaled Brownian motion with time-dependent diffusivity in renewal and non-renewal settings [54,55], and continuous time random walk processes with complete and incomplete [56] as well as with power-law [57] resetting.…”