Power laws and stretched exponentials in a noisy finite-time-singularity model

TL;DR

This paper investigates how white noise affects a finite-time-singularity model, transforming the singularity into a distribution with power law or stretched exponential tails, relevant to various physical sciences.

Contribution

It demonstrates that noise replaces the finite-time singularity with a probabilistic distribution characterized by power laws and stretched exponentials.

Findings

Noise resolves the finite-time singularity into a first-passage-time distribution.

The distribution has a peak at the original singularity time.

Long-time tails follow power law or stretched exponential behavior.

Abstract

We discuss the influence of white noise on a generic dynamical finite-time-singularity model for a single degree of freedom. We find that the noise effectively resolves the finite-time-singularity and replaces it by a first-passage-time or absorbing state distribution with a peak at the singularity and a long time tail exhibiting power law or stretched exponential behavior. The study might be of relevance in the context of hydrodynamics on a nanometer scale, in material physics, and in biophysics.