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(-tau) when s --> 0. tau 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 tau. For the specific kernel K-D(d)(x,y) = (x(1/D) + y(1/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 tau and develop a variational approximation which is used to carry out a systematic study of tau(d,D) for K-D(d). 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.