Consider the system of particles onℤd where particles are of two types-A and B-and execute simple random walks in continuous time. Particles do not interact with their own type, but when an A-particle meets a B-particle, both disappear, i.e., are annihilated. This system serves as a model for the chemical reaction A+B→ inert. We analyze the limiting behavior of the densities ρA(t) and ρB(t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densities ρA(0)=ρB(0) there is a change in behavior from d≤4, where ρA(t)=ρB(t)∼C/td/4, to d≥4, where ρA(t)=ρB(t)∼C/tas t→∞. For unequal initial densities ρA(0)<ρB(0), ρA(t)∼e-c√l in d=1, ρA(t)∼e-Ct/log t in d=2, and ρA(t)∼e-Ct in d≥3. The term C depends on the initial densities and changes with d. Techniques are from interacting particle systems. The behavior for this two-particle annihilation process has similarities to those for coalescing random walks (A+A→A) and annihilating random walks (A+A→inert). The analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Diffusion-dominated reaction
- annihilating random walks
- asymptotic densities
- exact results