Abstract
Consider the system of particles onℤd where particles are of two types—A andB—and execute simple random walks in continuous time. Particles do not interact with their own type, but when anA-particle meets aB-particle, both disappear, i.e., are annihilated. This system serves as a model for the chemical reactionA+B→ inert. We analyze the limiting behavior of the densitiesρ A (t) andρ B (t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densitiesρ A (0)=ρ B (0) there is a change in behavior fromd⩽4, whereρ A (t)=ρ B (t)∼C/t d/4, tod⩾4, whereρ A (t)=ρ B (t)∼C/tast→∞. For unequal initial densitiesρ A (0)<ρ B (0),ρ A (t)∼e −c√l ind=1,ρ A (t)∼e −Ct/logt ind=2, andρ A (t)∼e −Ct ind⩾3. The termC depends on the initial densities and changes withd. 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.
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Bramson, M., Lebowitz, J.L. Asymptotic behavior of densities for two-particle annihilating random walks. J Stat Phys 62, 297–372 (1991). https://doi.org/10.1007/BF01020872
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DOI: https://doi.org/10.1007/BF01020872