Finite-time blowup of a Brownian particle in a repulsive potential

TL;DR

This paper investigates the finite-time blowup behavior of a Brownian particle in rapidly growing repulsive potentials, revealing universal exponential decay in blowup time distribution tails and detailed analysis for the quartic case.

Contribution

It provides a universal characterization of blowup time distributions for particles in super-quadratic potentials, including detailed analysis of the quartic potential case.

Findings

Blowup time distribution tails decay exponentially.

Short-time tail exhibits an essential singularity.

Results are universal across all rapidly growing power-law potentials.

Abstract

We consider a Brownian particle performing an overdamped motion in a power-law repulsive potential. If the potential grows with the distance faster than quadratically, the particle escapes to infinity in a finite time. We determine the average blowup time and study the probability distribution of the blowup time. In particular, we show that the long-time tail of this probability distribution decays purely exponentially, while the short-time tail exhibits an essential singularity. These qualitative features turn out to be quite universal, as they occur for all rapidly growing power-law potentials in arbitrary spatial dimensions. The quartic potential is especially tractable, and we analyze it in more detail.