Damped finite-time-singularity driven by noise

TL;DR

This paper investigates how noise and damping influence finite-time singularities in dynamical systems, showing noise can prevent singularities and alter the timing and distribution of system states, relevant across physics and biophysics.

Contribution

It introduces a model combining damping and noise effects on finite-time singularities, revealing how noise resolves singularities and affects system behavior over time.

Findings

Noise replaces the singularity with a first-passage-time distribution.

Power law behavior in early time regimes depends on system parameters.

Damping influences the late-time dynamics and distribution tail.

Abstract

We consider the combined influence of linear damping and noise on a 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. The damping introduces a characteristic cross-over time. In the early time regime the probability distribution and first-passage-time distribution show a power law behavior with scaling exponent depending on the ratio of the non linear coupling strength to the noise strength. In the late time regime the behavior is controlled by the damping. The study might be of relevance in the context of hydrodynamics on a nanometer scale, in material physics, and in biophysics.