Autocorrelation functions and ergodicity in diffusion with stochastic resetting
Abstract
Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) timeaveragedmeansquareddisplacement (TAMSD) which is of acute interest in physics. We find that both diffusion and driftdiffusion processes with resetting are ergodic at the mean level unlike their resetfree counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental setups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 March 2022
 DOI:
 10.1088/17518121/ac4ce9
 arXiv:
 arXiv:2107.11686
 Bibcode:
 2022JPhA...55j4003S
 Keywords:

 autocorrelations;
 ergodicity;
 diffusion;
 stochastic resetting;
 Condensed Matter  Statistical Mechanics
 EPrint:
 doi:10.1088/17518121/ac4ce9