Resetting with stochastic return through linear confining potential

Abstract

We consider motion of an overdamped Brownian particle subject to stochastic resetting in one dimension. In contrast to the usual setting where the particle is instantaneously reset to a preferred location (say, the origin), here we consider a finite time resetting process facilitated by an external linear potential V (x) = λ|x| (λ > 0). When resetting occurs, the trap is switched on and the particle experiences a force −∂ x V (x) which helps the particle to return to the resetting location. The trap is switched off as soon as the particle makes a first passage to the origin. Subsequently, the particle resumes its free diffusion motion and the process keeps repeating. In this set-up, the system attains a non-equilibrium steady state. We study the relaxation to this steady state by analytically computing the position distribution of the particle at all time and then analysing this distribution using the spectral properties of the corresponding Fokker-Planck operator. As seen for the instantaneous resetting problem, we observe a 'cone spreading' relaxation with travelling fronts such that there is an inner core region around the resetting point that reaches the steady state, while the region outside the core still grows ballistically with time. Notably, our method, based on spectral properties, complements the existing renewal formalism and reveals the intricate mathematical structure responsible for such relaxation phenomena. We verify our analytical results against extensive numerical simulations.

Brownian motion under intermittent harmonic potentials

“…In the other experiment [36] the particle diffuses freely and after exponentially distributed time intervals the particles are driven back to the starting position mimicking resetting events. Similar theoretical models with different return protocols have been proposed recently [37][38][39][40][41][42].…”