Short-time statistics of extinction and blowup in reaction kinetics

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. Here 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.

Figures (5)

• Figure 1: The universal ($m\to \infty$) annihilation time probability distribution $\mathcal{P}(T)$ (black dotted-dashed line), the long-time tail (\ref{['longtime']}) (red dashed line) and the short-time tail (\ref{['shorttime']}) (blue solid line).
• Figure 2: (a) Solutions of Eq. (\ref{['annihilation_laplace']}) for the annihilation: exact (solid line), inner (red dashed line), and the leading and subleading WKB (blue dotted line), for $s=10^2$. (b) Probability distribution $\mathcal{P}(T)$ of extinction times. The short-time distribution tail (\ref{['shorttime']}) (black dashed line) is compared with numerical simulations (blue dots), with $10^7$ realizations. The inset shows the simulation results for $\mathcal{P}(T)$ (in linear scale) over a longer time scale.
• Figure 3: The MTE $\bar{T}$ vs $\mu$ for the initial number of particles $m=1$ (dashed red line), $m=10$ (dashed blue line) and $m=100$ (dashed green line). Also shown is the MTE in the universal limit $m\to\infty$ (solid black line).
• Figure 4: (a) Solutions of Eq. (\ref{['comp_death']}): Exact (solid line), inner (red dashed line) and WKB (blue dotted line), with $s=10^2$. (b) The probability distribution of extinction times $\mathcal{P}(T)$ for $\mu=1$ (orange squares) and $\mu=1/2$ (blue circles): the short-time tail (\ref{['short_tail_comp_death']}) (line) versus numerical simulations (circles) with $2\times 10^7$ realizations. The inset shows simulation results for $\mathcal{P}(T)$ (in linear scale) over a longer time scale.
• Figure 5: (a) Solutions of Eq. (\ref{['Pieq']}): exact (solid line), WKB (blue dotted line) and inner (red dashed line) for $s=10^2$. (b) The short-time tail (\ref{['PsmallT']}) of the blowup time distribution ${\cal P}(T)$ for the binary branching $2A \to 3A$ (dashed line) versus simulations (symbols) over $10^7$ realizations. The inset shows simulation results for ${\cal P}(T)$ (in linear scale) over a longer time scale.