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 ${\mathcal{P}}_m\left(T\right)$, 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.

Close

PrevNext

View figure in article

• Fluctuations & noise
• Stochastic processes
• Population dynamics
• First passage problems
• Large deviation & rare event statistics
• Master equation
• Monte Carlo methods
• Semiclassical methods