Modified stochastic fragmentation of an interval as an ageing process

TL;DR

This paper introduces a modified stochastic fragmentation model of an interval that exhibits aging, record dynamics, and universal power-law fragment size distributions, with exact analytical results and numerical validation.

Contribution

It presents a novel fragmentation process with a reset mechanism, providing exact distributions, correlation functions, and insights into aging phenomena in such systems.

Findings

Fragment size distribution follows a universal inverse power law with logarithmic corrections.

Two-time correlation functions exhibit aging and scaling behavior.

Reset mechanism impedes aging beyond a certain time scale.

Abstract

We study a stochastic model based on a modified fragmentation of a finite interval. The mechanism consists in cutting the interval at a random location and substituting a unique fragment on the right of the cut to regenerate and preserve the interval length. This leads to a set of segments of random sizes, with the accumulation of small fragments near the origin. This model is an example of record dynamics, with the presence of "quakes" and slow dynamics. The fragment size distribution is a universal inverse power law with logarithmic corrections. The exact distribution for the fragment number as function of time is simply related to the unsigned Stirling numbers of the first kind. Two-time correlation functions are defined and computed exactly. They satisfy scaling relations and exhibit aging phenomena. In particular the probability that the same number of fragments is found at two different times $t>s$ is asymptotically equal to $[4\pi\log(s)]^{-1/2}$ when $s\gg 1$ and the ratio $t/s$ fixed, in agreement with the numerical simulations. The same process with a reset impedes the aging phenomena beyond a typical time scale defined by the reset parameter.