Inhomogeneous Two-Species Annihilation in the Steady State,
E. Ben-Naim and S. Redner
We investigate steady-state geometrical properties of the reaction
interface in the two-species annihilation process, A+B--> 0,
when a flux j of A and B particles is injected at opposite
extremities of a finite domain. By balancing the input flux with the
number of reactions, we determine that the width $w$ of the reaction
zone scales as $j^{-1/3}$ in the large flux limit, and that the
concentration in this zone is proportional to $j^{2/3}$. This same
behavior is deduced from the solution to the reaction-diffusion
equation. In the low flux limit, the concentration is almost
independent of position and is proportional to $j^{1/2}$. In the
latter case, the local reaction rate reaches maximum at the edges of the
system rather than at the midpoint. When the two species approach at a
finite velocity, there exists a critical velocity, above which the
reactants essentially pass through each other. Results similar to those
in one dimension are found in two- and three-dimensional radial
geometries. Finally, we apply the quasistatic approximation to our
steady-state solution to recover the known time dependence for the
reaction zone width for the case of initially separated components with
no external input.
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