Active particle in a harmonic trap driven by a resetting noise: an approach via Kesten variables

Abstract

We consider the statics and dynamics of a single particle trapped in a one-dimensional harmonic potential, and subjected to a driving noise with memory, that is represented by a resetting stochastic process. The finite memory of this driving noise makes the dynamics of this particle ``active''. At some chosen times (deterministic or random), the noise is reset to an arbitrary position and restarts its motion. We focus on two resetting protocols: periodic resetting, where the period is deterministic, and Poissonian resetting, where times between resets are exponentially distributed with a rate $r$. Between the different resetting epochs, we can express recursively the position of the particle. The random relation obtained takes a simple Kesten form that can be used to derive an integral equation for the stationary distribution of the position. We provide a detailed analysis of the distribution when the noise is a resetting Brownian motion. In this particular instance, we also derive a renewal equation for the full time dependent distribution of the position that we extensively study. These methods are quite general and can be used to study any process harmonically trapped when the noise is reset at random times.

Dynamics of inertial particles under velocity resetting

“…while at late times it crosses over to normal diffusion ⟨x 2 ⟩ (0) t ∼ t. Hence, we expect an effective diffusivity that is initially constant before decaying as r −2 . This is indeed what is observed in figure 2, or equivalent in equation (32).…”

“…Furthermore, the run-and-tumble motion is in itself a velocity resetting process [29][30][31]. Recently, a similar problem where an overdamped particle in a potential is driven by a resetting noise was investigated using Kesten variables, where both propagators and moments were studied [32].…”

Noninteracting particles in a harmonic trap with a stochastically driven center

“…Thus u(t) represents the position of an RTP in a harmonic potential of stiffness µ, which has been widely studied first in chemical physics literature (see [33] for a review) and more recently in the context of active matter [27,28,30,34,35]. It is known that at late times, the position distribution h(u, t) approaches a stationary limit given by [27,28,34],…”

“…This external drive breaks the time reversal symmetry, and drives the system to a strongly correlated NESS. For N = 1 and z(t) = (v 0 /µ) σ(t), where σ(t) represents a telegraphic noise that switches between ±1 with rate γ, this model has been studied widely in the context of active systems [27][28][29][30] Our main results can be summarized as follows. For a general stochastic drive z(t), bounded in time, we show that the stationary JPDF has the CIID structure as in (5), where…”