2003/06/24 by Govind Menon, Menon, Govind, Robert L. Pego +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #FOS: Physical sciences #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #nlin.AO
paper · pdf · doi:10.48550/arxiv.nlin/0306047
Latex2e, 42 pages with 1 figure
arxiv created 2003/06/24 · openalex publication_date 2003/06/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels K(x,y)=2, x+y and xy. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K=2 the size distribution is Mittag-Leffler, and for K=x+y and K=xy it is a power-law rescaling of a maximally skewed α-stable Levy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.