Scaling solutions of Smoluchowski's coagulation equation

We investigate the structure of scaling solutions of Smoluchowski's coagulation equation, of the formc_k(t)s(t)^– (k/s(t)), wherec_k(t) is the concentration of clusters of sizek at timet,s(t) is the average cluster size, and(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^–K(ai,aj) for alla>0, but we restrict ourselves to kernels satisfyingK(i, j)/j0 asj. We show that gelation occurs if>1, and does not occur if1. For all gelling and nongelling models, we calculate the time dependence ofs(t), and we derive an equation for(x). We present a detailed analysis of the behavior of(x) at large and small values ofx. For all models, we find exponential large-x behavior: (x)Ax^–e^–x asx and, for different kernelsK(i, j), algebraic or exponential small-x behavior: (x)Bx^– or (x)=exp(–Cx^–|| + ...) asx0.