Extinction and survival in two-species annihilation

We study diffusion-controlled two-species annihilation with a finite number of particles. In this stochastic process, particles move diffusively, and when two particles of opposite type come into contact, the two annihilate. We focus on the behavior in three spatial dimensions and for initial conditions where particles are confined to a compact domain. Generally, one species outnumbers the other, and we find that the difference between the number of majority and minority species, which is a conserved quantity, controls the behavior. When the number difference exceeds a critical value, the minority becomes extinct and a finite number of majority particles survive, while below this critical difference, a finite number of particles of both species survive. The critical difference ${\mathrm{\Delta }}_c$ grows algebraically with the total initial number of particles $N$, and when $N\gg 1$, the critical difference scales as ${\mathrm{\Delta }}_c\sim N^{1/3}$. Furthermore, when the initial concentrations of the two species are equal, the average number of surviving majority and minority particles, $M_{+}$ and $M_{-}$, exhibit two distinct scaling behaviors, $M_{+}\sim N^{1/2}$ and $M_{-}\sim N^{1/6}$. In contrast, when the initial populations are equal, these two quantities are comparable $M_{+}\sim M_{-}\sim N^{1/3}$.

Figure 1

The average number of surviving particles $M$ versus system size $N$ for equal populations. The inset shows the quantity $\alpha \equiv dlnM/dlnN$ versus $N$. Fitting the data to the power law $M\sim N^{\alpha }$ in the range for $N>{10}^4$ yields the exponent $\alpha =0.34$.

Figure 2

The scaling function $F\left(z\right)$ defined in (13). Shown is the scaled number of particles $F\equiv n/N^{1/3}$ versus the scaled time $z\equiv t/N^{2/3}$ for the case of equal number of particles for three different system sizes.

Figure 3

The scaling function $G\left(x\right)$, defined in Eq. (18). Shown is the scaled number of surviving minority particles $G\equiv M_{-}/N^{1/3}$ versus the scaled difference $x\equiv \mathrm{\Delta }/N^{1/3}$ for the case of fixed population difference $\mathrm{\Delta }$. Inset shows the function $1+x/G$, which according to Eq. (19) should increase exponentially with $x$.

Figure 4

The surviving populations $M_{+}$ and $M_{-}$ versus system size $N$ for the special case $\mathrm{\Delta }=N^{1/3}$. A fit of $M_{+}$ and $M_{-}$ to a power law, $M_{\pm }\sim N^{{\beta }_{\pm }}$ in the range $N>{10}^4$ yields the exponents ${\beta }_{+}=0.34$ and ${\beta }_{-}=0.35$, respectively. The inset shows the quantity ${\beta }_{\pm }\equiv dln{M}_{\pm }/dlnN$ versus $N$.

Figure 5

The normalized population $M_{-}/N^{1/2}$ versus $N^{1/6}$ for the special case $\mathrm{\Delta }=N^{1/2}$. The inset shows the surviving population $M_{-}$ versus system size $N$.

Figure 6

The average number of surviving minority particles $M_{-}$ versus $N$ for the equal concentration case. The inset shows the quantity ${\beta }_{-}=dln{M}_{-}/dlnN$ versus $N$.

Figure 7

The average number of surviving minority particles $M_{-}$ versus $N$ when the initial difference $\mathrm{\Delta }$ is distributed according to (22). A fit of $M_{-}$ to the power-law, $M_{-}\sim N^{{\beta }_{-}}$ in the range $N>{10}^4$ yields the exponents ${\beta }_{-}=0.21$. The inset shows the quantity ${\beta }_{-}=dln{M}_{-}/dlnN$ versus $N$.

Figure 8

The reaction rate $-dn/dt$ versus time $t$ for the case $N=7,153$.