Asymptotic Behavior of Densities in Diffusion-Dominated Annihilation Reactions

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21 November 1988

journal article
research article
Published by American Physical Society (APS) in Physical Review Letters

Vol. 61 (21) , 2397-2400
https://doi.org/10.1103/physrevlett.61.2397

Abstract

We obtain rigorous bounds on the long-time behavior of the densities

ρ_{A} (t)

and

ρ_{B} (t)

of species

A

and

B

, which diffuse and annihilate upon meeting, i.e.,

A + B \to inert

. For equal initial densities

ρ_{A} (0) = ρ_{B} (0)

, the density goes to zero asymptotically as

t^{\frac{- d}{4}}

for dimensions

d \leq 4

and as

t^{- 1}

for

d \geq 4

. When

ρ_{A} (0) < ρ_{B} (0)

,

ρ_{A} (t) \sim \exp ({- λ}_{1} \sqrt{t})

,

ρ_{A} (t) \sim \exp (\frac{{- λ}_{2} t}{In t})

for

d = 1, 2, r e s p e c t i v e l y, a n d ρ_{A} (t) \sim \exp ({- λ}_{d} t)

for

d \geq 3

.

Keywords

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