Short-time statistics of extinction and blowup in reaction kinetics

TL;DR

This paper analyzes the short-time behavior of extinction and blowup times in stochastic reaction systems, using advanced WKB methods to derive accurate asymptotics for the distribution tails.

Contribution

It introduces a novel WKB-based approach to accurately compute the pre-exponential factors in the short-time tail asymptotics of reaction time distributions.

Findings

WKB approximation captures the essential singularity at T=0 in the distribution tail.

Method accurately computes the pre-exponential factor using Laplace-transformed equations.

Validated approach on three reaction examples with exact solutions.

Abstract

We study the statistics of extinction and blowup times in well-mixed systems of stochastically reacting particles. We focus on the short-time tail, $T \to 0$, of the extinction- or blowup-time distribution $\mathcal{P}_m(T)$, where $m$ is the number of particles at $t=0$. This tail often exhibits an essential singularity at $T=0$, and we show that the singularity is captured by a time-dependent WKB (Wentzel-Kramers-Brillouin) approximation applied directly to the master equation. This approximation, however, leaves undetermined a large pre-exponential factor. We show how to calculate this factor by applying a leading- and a subleading-order WKB approximation to the Laplace-transformed backward master equation. Accurate asymptotic results can be obtained when this WKB solution can be matched to another approximate solution (the ``inner" solution), valid for not too large $m$. We demonstrate and verify this method on three examples of reactions which are also solvable without approximations.