Abstract
We investigate the structure of scaling solutions of Smoluchowski's coagulation equation, of the formc k (t)∼s(t)−τ′ ϕ(k/s(t)), wherec k (t) is the concentration of clusters of sizek at timet,s(t) is the average cluster size, andϕ(x) is a scaling function. For the rate constantK(i, j) in Smoluchowski's equation, we make the very general assumption thatK(i, j) is a homogeneous function of the cluster sizesi andj:K(i,j)=a −λ K(ai,aj) for alla>0, but we restrict ourselves to kernels satisfyingK(i, j)/j→0 asj→∞. We show that gelation occurs ifλ>1, and does not occur ifλ⩽1. For all gelling and nongelling models, we calculate the time dependence ofs(t), and we derive an equation forϕ(x). We present a detailed analysis of the behavior ofϕ(x) at large and small values ofx. For all models, we find exponential large-x behavior: ϕ(x)∼A x −λ e −δx asx→∞ and, for different kernelsK(i, j), algebraic or exponential small-x behavior: ϕ(x)∼Bx −τ or ϕ(x)=exp(−Cx −|μ| + ...) asx↓0.
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van Dongen, P.G.J., Ernst, M.H. Scaling solutions of Smoluchowski's coagulation equation. J Stat Phys 50, 295–329 (1988). https://doi.org/10.1007/BF01022996
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DOI: https://doi.org/10.1007/BF01022996