# Short-time statistics of extinction and blowup in reaction kinetics

Source: [Short-time statistics of extinction and blowup in reaction kinetics](<https://www.sota2.com/research/paper/69608b7553fdd5c361744352>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-04-15T05:00:04.591Z

Authors: Rotem Degany, Michael Assaf, Baruch Meerson

Published: 2026-01-08T13:22:07.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2601.04924>)

[PDF](<https://arxiv.org/pdf/2601.04924.pdf>)