= Constant-birth pair-annihilation process
A count-valued <Markov jump process> with constant single-particle births and pair annihilation must preserve $n\in\mathbb Z_{\geq0}$. Choose the rates
$$
n\longrightarrow n+1\text{ at rate }\lambda,\qquad
n\longrightarrow n-2\text{ at rate }\beta n(n-1),\qquad\lambda,\beta>0.
$$
The second rate vanishes at $n=0,1$, where there is no pair to remove; the factor $1/2$ for unordered pairs has been absorbed into $\beta$.
In the <Kramers-Moyal expansion>, the first two coefficients are the rate-weighted first and second jump moments:
$$
A(n)=\lambda-2\beta n(n-1),\qquad
B(n)=\lambda+4\beta n(n-1).
$$
Here $B$ is the noise <variance> per unit time in the local <diffusion approximation>. The positive drift root is
$$
n_* =\frac{1+\sqrt{1+2\lambda/\beta}}2,\qquad A'(n_*)=-2\beta(2n_*-1)<0.
$$
This is the stable fixed point of the drift equation, rather than an exact formula for the mean of the stochastic population.
For $n_*\gg1$, fluctuations far from the count boundary are approximately an <Ornstein-Uhlenbeck process>. Its stationary <variance> is
$$
\operatorname{Var}n\simeq\frac{B(n_*)}{-2A'(n_*)}=\frac{3n_*(n_*-1)}{2(2n_*-1)}\sim\frac{3n_*}{4}.
$$
At leading large-count order one may replace $n(n-1)$ by $n^2$ and $n_*$ by $\sqrt{\lambda/(2\beta)}$, recovering the simpler approximation. A rate proportional to $n^2$ used literally at $n=1$ would allow a negative population, so that replacement is an interior asymptotic approximation rather than the exact count model.
Back to article page
