Fractal time random walk and subrecoil laser cooling considered as renewal processes with infinite mean waiting times

TL;DR

This paper presents a unified theoretical framework for understanding fractal time random walks and subrecoil laser cooling, both involving infinite mean waiting times, using renewal theory and the generalized central limit theorem.

Contribution

It introduces a unified approach combining renewal theory and the generalized central limit theorem to analyze processes with infinite mean waiting times, applicable to diverse physical phenomena.

Findings

Derived key physical properties without technical difficulties

Highlighted the role of sums with infinite means in process dynamics

Provided a pedagogic, unified treatment of two seemingly dissimilar processes

Abstract

There exist important stochastic physical processes involving infinite mean waiting times. The mean divergence has dramatic consequences on the process dynamics. Fractal time random walks, a diffusion process, and subrecoil laser cooling, a concentration process, are two such processes that look qualitatively dissimilar. Yet, a unifying treatment of these two processes, which is the topic of this pedagogic paper, can be developed by combining renewal theory with the generalized central limit theorem. This approach enables to derive without technical difficulties the key physical properties and it emphasizes the role of the behaviour of sums with infinite means.