Anomalous dimensions of the Smoluchowski coagulation equation

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) γ , but any homogeneous function can be treated using our methods. For sufficiently effective clumping 1 γ > 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. Apart from the solution branch which originates from the exactly solvable case γ = 1, we find a branch of solutions near γ = 1/2, which violates matching conditions for the limit of small cluster sizes, widely believed to hold on a universal basis.