Anomalous dimensions of the Smoluchowski coagulation equation

TL;DR

This paper investigates the anomalous scaling behavior in the Smoluchowski coagulation equation with specific kernels, revealing new solution branches and challenging previously assumed universal scaling relations.

Contribution

It introduces analytical and numerical methods to compute anomalous scaling dimensions, identifying a new solution branch near b3 = 1/2 that defies established scaling laws.

Findings

Discovery of a new solution branch near b3 = 1/2

Challenging of universal scaling relations in coagulation dynamics

Analytical and numerical calculation of anomalous dimensions

Abstract

The coagulation (or aggregation) equation was introduced by Smoluchowski in 1916 to describe the clumping together of colloidal particles through diffusion, but has been used in many different contexts as diverse as physical chemistry, chemical engineering, atmospheric physics, planetary science, and economics. The effectiveness of clumping is described by a kernel $K(x,y)$, which depends on the sizes of the colliding particles $x,y$. We consider kernels $K = (xy)^{\gamma}$, but any homogeneous function can be treated using our methods. For sufficiently effective clumping $1 \ge \gamma > 1/2$, the coagulation equation produces an infinitely large cluster in finite time (a process known as the gel transition). Using a combination of analytical methods and numerics, we calculate the anomalous scaling dimensions of the main cluster growth, calling into question results much used in the literature. Apart from the solution branch which originates from the exactly solvable case $\gamma = 1$, we find a new branch of solutions near $\gamma = 1/2$, which violates scaling relations widely believed to hold universal.