## Abstract

Consider the system of particles onℤ^{d} 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 ρ_{A}(t) and ρ_{B}(t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densities ρ_{A}(0)=ρ_{B}(0) there is a change in behavior from d≤4, where ρ_{A}(t)=ρ_{B}(t)∼C/t^{d}/4, to d≥4, where ρ_{A}(t)=ρ_{B}(t)∼C/tas t→∞. For unequal initial densities ρ_{A}(0)<ρ_{B}(0), ρ_{A}(t)∼e^{-c√l} in d=1, ρ_{A}(t)∼e^{-Ct/log t} in d=2, and ρ_{A}(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.

Original language | English (US) |
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Pages (from-to) | 297-372 |

Number of pages | 76 |

Journal | Journal of Statistical Physics |

Volume | 62 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 1991 |

## Keywords

- Diffusion-dominated reaction
- annihilating random walks
- asymptotic densities
- exact results