Abstract
We investigate the Brownian diffusion of particles in one spatial dimension and in the presence of finite regions within which particles can either evaporate or be reset to a given location. For open boundary conditions, we highlight the appearance of a Brownian yet non-Gaussian diffusion: at long times, the particle distribution is non-Gaussian but its variance grows linearly in time. Moreover, we show that the effective diffusion coefficient of the particles in such systems is bounded from below by $(1-2/\pi)$ times their bare diffusion coefficient. For periodic boundary conditions, i.e., for diffusion on a ring with resetting, we demonstrate a "gauge invariance" of the spatial particle distribution for different choices of the resetting probability currents, in both stationary and non-stationary regimes. Finally, we apply our findings to a stochastic biophysical model for the motion of RNA polymerases during transcriptional pauses, deriving analytically the distribution of the length of cleaved RNA transcripts and the efficiency of RNA cleavage in backtrack recovery.
Highlights
Stochastic processes whereby incremental changes are interspersed with sudden and large changes occurring at unpredictable times are common in nature [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
Our work provides analytical and numerical insights into the particle currents emerging in the presence of control mechanisms on an otherwise unbiased Brownian diffusion, for various boundary conditions
We have shown that resetting the particle position at stochastic times to a prescribed location leads to Brownian yet non-Gaussian diffusion
Summary
Stochastic processes whereby incremental changes are interspersed with sudden and large changes occurring at unpredictable times are common in nature [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. A variety of situations have been studied within this framework, e.g., Brownian particles resetting to a generic spatial distribution, under the action of an external potential [30], and for various choices of the resetting time probability distribution [31] Most of these studies consider resetting phenomena with open boundary conditions. We study minimal stochastic models of Brownian particles evaporating or resetting from a finite one-dimensional region with either open, periodic, and/or absorbing boundary conditions. Our key findings concern the transport properties of diffusion processes with resetting and periodic boundary conditions For these systems, a stationary state exists and we. We provide the exact analytical calculations and further numerical results in the Appendixes
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