Abstract
How do different reset protocols affect ergodicity of a diffusion process in single-particle-tracking experiments? We here address the problem of resetting of an arbitrary stochastic anomalous-diffusion process (ADP) from the general mathematical points of view and assess ergodicity of such reset ADPs for an arbitrary resetting protocol. The process of stochastic resetting describes the events of the instantaneous restart of a particle's motion via randomly distributed returns to a preset initial position (or a set of those). The waiting times of such resetting events obey the Poissonian, Gamma, or more generic distributions with specified conditions regarding the existence of moments. Within these general approaches, we derive general analytical results and support them by computer simulations for the behavior of the reset mean-squared displacement (MSD), the new reset increment-MSD (iMSD), and the mean reset time-averaged MSD (TAMSD). For parental nonreset ADPs with the MSD($t$)$\ensuremath{\propto}{t}^{\ensuremath{\mu}}$ we find a generic behavior and a switch of the short-time growth of the reset iMSD and mean reset TAMSDs from $\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{\mu}}$ for subdiffusive to $\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{1}$ for superdiffusive reset ADPs. The critical condition for a reset ADP that recovers its ergodicity is found to be more general than that for the nonequilibrium stationary state, where obviously the iMSD and the mean TAMSD are equal. The consideration of the new statistical quantifier, the iMSD---as compared to the standard MSD---restores the ergodicity of an arbitrary reset ADP in all situations when the $\ensuremath{\mu}\mathrm{th}$ moment of the waiting-time distribution of resetting events is finite. Potential applications of these new resetting results are, inter alia, in the area of biophysical and soft-matter systems.
Highlights
How do different reset protocols affect ergodicity of a diffusion process in single-particle-tracking experiments? We here address the problem of resetting of an arbitrary stochastic anomalous-diffusion process (ADP) from the general mathematical points of view and assess ergodicity of such reset Anomalous-diffusion processes (ADPs) for an arbitrary resetting protocol
We propose a timescale-decomposition approach to obtain the approximate mean-time-averaged MSD (TAMSD) expression in the limit of short lag times. This enables us to reveal the generic form of the mean TAMSD, with a switching between the two exponents for arbitrary stationary-increment ADPs under the conditions of stochastic resetting with arbitrary reset waiting-time density (WTD)
We note that the iMSD for a reset ADP—at short lag times and at long enough times t1—has two distinct exponents which are switching from ∝ μ to ∝ 1 for subdiffusive and superdiffusive nonreset ADPs
Summary
Stochastic processes under restart have attracted scientific interest in the statistical-physics community in recent years [1–4], gaining considerable momentum recently due to a series of theoretical works [5–23] which uncover a number of nonintuitive features of various resetting strategies. Resetting are, inter alia, applicable to (for brevity, without explicit references here) (i) optimization of generic problems with “‘returns,” (ii) mathematical description of foraging strategies employed by animals and their movement-ecology data, (iii) stochastic-switching mechanisms between different phenotypes in a simple organism, (iv) certain classes of computational search-and-recognition algorithms, (v) description of principles of eye movements upon perception of art paintings and images, (vi) interrupted/manipulated diffusion of micron-sized beads in optical traps, and (vii) modeling of prices of reset- and barrier-type options in financial mathematics
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