Arrival statistics and exploration properties of mortal walkers

TL;DR

This paper investigates how mortal random walkers, which can die during their movement, influence exploration and arrival statistics, with implications for chemical kinetics and defect diffusion models.

Contribution

It extends classical methods for immortal walkers to mortal walkers, analyzing exponential and power-law evanescence effects on exploration properties.

Findings

Evanescence significantly alters the number of visited sites.

Methods for immortal walkers are applicable to mortal walkers.

Results have implications for chemical kinetics and defect diffusion.

Abstract

We study some of the salient features of the arrival statistics and exploration properties of mortal random walkers, that is, walkers that may die as they move, or as they wait to move. Such evanescence or death events have profound consequences for quantities such as the number of distinct sites visited which are relevant for the computation of encounter-controlled rates in chemical kinetics. We exploit the observation that well-known methods developed decades ago for immortal walkers are widely applicable to mortal walkers. The particular cases of exponential and power-law evanescence are considered in detail. Finally, we discuss the relevance of our results to the target problem with mortal traps and a particular application thereof, namely, the defect diffusion model. Evanescence of defects is postulated as a possible complementary contribution or perhaps even an alternative to anomalous diffusion to explain observed stretched exponential relaxation behavior.