Springer Lecture Notes in Physics 716

336 U. C. Täuber
difference of particle numbers (even locally), i.e., there is a conservation law
for c(t) =a(t) − b(t) =c(0) [61]. The rate equations for the concentrations
ȧ(t) =−λa(t) b(t) =ḃ(t) (7.181)
are for equal initial densities a(0) = b(0) solved by the single-species pair
annihilation mean-field power law
a(t) =b(t) ∼ (λt) −1 , (7.182)
whereas for unequal initial densities c(0) = a(0) − b(0) > 0, say, the majority
species a(t) → a ∞ = c(0) > 0ast →∞, and the minority population disappears,
b(t) → 0. From Eq. (7.181) we obtain for d>d c = 2 the exponential
approach
a(t) − a ∞ ∼ b(t) ∼ e −c(0) λt . (7.183)
Mapping the associated master equation onto a continuum field theory
(7.167), the reaction term now reads (with the fields ψ and ϕ representing the
A and B particles, respectively) [62]
H reac ( ˆψ,
(
ψ, ˆϕ, ϕ) =−λ 1 − ˆψ
)
ˆϕ ψϕ . (7.184)
As in the single-species case, there is no propagator renormalisation, and
moreover the Feynman diagrams for the renormalised reaction vertex are of
precisely the same form as for A + A →∅, see Fig. 7.6. Thus, for unequal
initial densities, c(0) > 0, the mean-field power law ∼ λt in the exponent of
Eq. (7.183) becomes again replaced with (Dt) d/2 in dimensions d ≤ d c =2,
leading to stretched exponential time dependence,
d