[Paper Review] Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Viktor Stojkoski, Trifce Sandev+2 more|arXiv (Cornell University)|Jul 24, 2021

Diffusion and Search Dynamics97 references30 citations

TL;DR

This paper introduces a novel trajectory-based approach to compute autocorrelation functions and study ergodicity in diffusion with stochastic resetting. By applying the law of total expectation to single trajectories, it analytically derives moments and autocorrelations, revealing that resetting induces ergodicity in mean-based observables like the sample mean, while breaking ergodicity in the time-averaged mean squared displacement (TAMSD), with a critical resetting rate r* that separates regimes of increased or decreased particle spread.

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 in the level of 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) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift-diffusion processes are ergodic at the mean level unlike their reset-free 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. The current study provides an important baseline that unifies two different approaches, used ubiquitously in economics and physics, for studying the ergodic properties in diffusion with resetting. We believe that our results can be verified in single particle experimental set-ups and thus have strong implications in the field of resetting.

Motivation & Objective

• To develop a tractable method for computing two-time correlation functions in diffusion with stochastic resetting, where exact time-dependent probability densities are generally unknown.
• To unify two distinct approaches—sample mean in economics and TAMSD in physics—for analyzing ergodic properties in resetting systems.
• To investigate how stochastic resetting affects ergodicity in canonical diffusion and drift-diffusion processes.
• To identify critical resetting rates that govern the transition between regimes of enhanced or constrained particle spread in single-particle dynamics.

Proposed method

• A stochastic solution is derived directly at the single-trajectory level for the resetting Langevin equation, avoiding reliance on ensemble-level renewal formalism.
• The law of total expectation is applied to the position variable along a single trajectory to compute moments and autocorrelation functions analytically.
• The approach leverages the renewal property of resetting events to express time-dependent statistics in terms of conditional expectations over reset intervals.
• Large deviation analysis is used to study relaxation to the non-equilibrium steady state, with the β-mixing coefficient quantifying convergence to stationarity.
• The method is applied to both simple diffusion and drift-diffusion processes with Poissonian resetting to evaluate ergodicity of the sample mean and TAMSD.
• Numerical simulations are used to corroborate analytical results, particularly for TAMSD and ergodicity breaking parameters.

Experimental results

Research questions

• RQ1 How can autocorrelation functions be computed in diffusion with stochastic resetting when the full time-dependent probability density is unknown?
• RQ2 Does stochastic resetting restore ergodicity in the sample mean for processes that are non-ergodic in the absence of resetting?
• RQ3 How does resetting affect the ergodicity of the time-averaged mean squared displacement (TAMSD) in diffusion and drift-diffusion processes?
• RQ4 Is there a critical resetting rate r* that separates regimes where resetting increases or decreases the effective spread of a particle in single-particle tracking?
• RQ5 How do the ergodic properties of the sample mean and TAMSD interrelate under general resetting dynamics?

Key findings

• Stochastic resetting renders the sample mean ergodic in both simple and drift-diffusion processes, in contrast to their reset-free counterparts which are non-ergodic.
• The TAMSD is ergodic in the absence of resetting for both diffusion and drift-diffusion, but resetting induces ergodicity breaking in this observable.
• A critical resetting rate r* exists such that for r < r*, resetting increases the TAMSD, while for r > r*, resetting suppresses the TAMSD, indicating a transition in particle spread dynamics.
• The system exhibits β-mixing behavior, with β(t) ∼ e^{-(r + μ²/(2σ²))t}, confirming convergence to the non-equilibrium steady state and long-time relaxation.
• The proposed trajectory-based method enables analytical computation of autocorrelation functions without requiring knowledge of the full time-dependent PDF, offering a general framework for Markovian resetting processes.
• The results are consistent across analytical derivations and extensive numerical simulations, supporting potential verification in single-particle experiments using optical tweezers.