We are now able to verify that the coagulation process and the annihilation process are in the same universality class [76, 155, 156]. Consider the coagulation process,A+A→A, with rate ˆλ and annihilation A+A → ∅ with equal rate ˆλ. The microscopic action is given by
Eq. (3.53) with λ = ˆλ, λ0 = 0 for coagulation, and λ= 0, λ0 = ˆλ for annihilation. Thus it follows that for the microscopic scale Λ we have the identity Γ(1Λ,,coag2) = 1
2Γ (1,2)
Λ,annih. As remarked
in the previous section, c.f. Eq. (4.23), there is a simple relation between the (1,2)- and (2,2)- vertex functions for the coagulation process. In complete analogy, for the annihilation process one has instead that
Γ(1κ,annih (,2) p
1,ω1;p2,ω2;p3,ω3;p4,ω4)= (2π)
d+1Γ(2,2)
κ,annih (p1,ω1;p2,ω2;p3,ω3;p4,ω4). (4.24)
Therefore, when the microscopic rates for coagulation and annihilation are equal, we have that the identity Γ(1κ,,coag2) = 12Γ(1κ,,annih2) is conserved along the flow.
More generally we assert that
Γ(κ,m,ncoag) = 2m−nΓ(κ,m,nannih) , (4.25) exactly at all scales κ. This can be proven by induction: At the microscopic scale Λ the equation is certainly true. The flow of the (m, n)-point vertex functions, determined by the corresponding one-loop diagrams, is a sum of products of vertex-functions with n1, . . . , nj (n1+. . .+nj =n) incoming and m1, . . . , mj (m1+. . .+mj =m) outgoing legs, such that (n1−m1) +. . .+ (nj−mj) =n−m. Therefore, assuming that Eq. (4.25) holds, the flow of the vertex functions for annihilation is accelerated by the factor 2(n−m) as compared to the
flow for coagulation. Thus, Eq. (4.25) holds for all vertex functions and at all scalesκ. It follows that, due to the extremal princple, c.f. Eq. (3.93), the kinetics for annihilation and coagulation are simply related. If Γ[ ¯ψ = 0, ψ = ρ] is a solution of the extremal principle for the coagulation process with particle input J, then Γ[ ¯ψ = 0, ψ = ρ/2] is a solution for the annihilation process with particle inputJ/2. Of course one can proceed in an analogous fashion for a mixed annihilation and coagulation process, the “α-process”,A+A→ ∅with
rate (α −1)ˆλ and A+A → A with rate (2−α)ˆλ (α ∈ [1,2]), to find that the decay is accelerated by a factorα instead,
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1
Figure 4.2:Comparison of the decay for a pure coagulation process, A+A → A with rate ˆλ, and a pure annihilation process, A+A → ∅ with the same rate ˆλ. For Poissonian initial
conditions the result agrees with Eq. (4.26) (black line,α= 1 for coagulation, α= 2 for annihilation). When the particles are distributed in pairs on the lattice (on each site the probability to find 2nparticles is cosh(ρ)(2ρ2nn)!, with no correlations between the sites), this symmetry between coagulation (blue line) and annihilation (red line) is broken. It has been shown that strongly correlated initial condition affect the long time behavior for the annihilation process in one dimension. The long time decay for the coagulation process, however, is universal [156].
if the equation holds initially. Thus, not only are the processes in the same universality class, but there is a simple, exact relation between their dynamics which is always true and not just in the long time and low density limit (also see [66, 156–158]).
However, we must add an important caveat to this neat result. There are initial conditions which cannot be created by particle input. This is the case for instance, when particles are initially distributed in pairs (two particles on the lattice site), with the probability cosh(ρ)(2ρ2nn)! to have npairs, giving an initial density ρ. This state can be created by linear combination of the coherent states|ρiand|−ρi. The former state,|ρi, would just imply a Poissonian dis- tribution, which can be created by particle input J(x, t) =ρ δ(t). Our above analysis applies to both of these coherent states |ρi, |−ρi separately. For the sum, however, the computer simulations clearly show that there are strong deviations from Eq. (4.26), c.f. Figure 4.2. In fact, one can show that Eq. (4.26) only holds if initially the n-point correlation functions fulfill [156]
hψ(x1)·. . .·ψ(xn)icoag=αnhψ(x1)·. . .·ψ(xn)iα. (4.27) (Notice that this is true, in particular, for Poissonian initial conditions, so long as ρcoag =
2ρannih). These can be generated in our model by particle input of the formJ(t) =ρcoagδ(t)
for the coagulation process andJ(t) =ραδ(t) for the α-process, withρcoag=αρα.
time behavior depends on the initial conditions, can perhaps be best understood for the one-dimensional annihilation model, after a mapping to a zero temperature, dynamic Ising model [159]. It was found that on condition that the initial magnetization m0 vanishes, the
long time behavior coincides with the long time behavior for Poissonian initial conditions. However, when m0 is non-zero, corresponding to short range correlations in the annihilation
model, one observes different behavior, which can be expressed as a lowering of the ampli- tude A for the long time decay At−1/2 of the annihilation model. The annihilation model
corresponds to the α-model with α = 2. In contrast, as soon as there is a non-zero rate for coagulation, 1≤α <2, the decay amplitudeAis independent of the initial state [156].