Short-time statistics of extinction and blowup in reaction kinetics
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917399451664384 |
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| author | Degany, Rotem Assaf, Michael Meerson, Baruch |
| author_facet | Degany, Rotem Assaf, Michael Meerson, Baruch |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04924 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Short-time statistics of extinction and blowup in reaction kinetics Degany, Rotem Assaf, Michael Meerson, Baruch Statistical Mechanics Probability 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. |
| title | Short-time statistics of extinction and blowup in reaction kinetics |
| topic | Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2601.04924 |