Nontrivial polydispersity exponents in aggregation models

We consider the scaling solutions of Smoluchowski's equation of irreversible aggregation, for a nongelling collision kernel. The scaling mass distribution f(s) diverges as s^-τ when s{→}0. {τ} is nontrivial and could, until now, only be computed by numerical simulations. We develop here general methods to obtain exact bounds and good approximations of {τ}. For the specific kernel KDᵈ(x,y)=(x1/D+y1/D{)}ᵈ$, describing a mean-field model of particles moving in d dimensions and aggregating with conservation of ``mass'' s=${R}D$ (R is the particle radius), perturbative and nonperturbative expansions are derived. For a general kernel, we find exact inequalities for {τ} and develop a variational approximation which is used to carry out a systematic study of {τ}(d,D) for KDᵈ. The agreement is excellent both with the expansions we derived and with existing numerical values. Finally, we discuss a possible application to 2d decaying turbulence.