Asymptotic behavior of densities for two-particle annihilating random walks

Maury Bramson, Joel L. Lebowitz

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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.

Original languageEnglish (US)
Pages (from-to)297-372
Number of pages76
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Jan 1 1991


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


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