Abstract
It has been conjectured by Lushnikov and Ziff that Smoluchowski's coagulation equation describes a gelation transition, i.e., the mean cluster size diverges within a finite timet c (gelpoint) if the coagulation rate constantsK(i,j) have the propertyK(ai,aj)=a λ K(i,j), with λ>1. The existing evidence was based on self-consistency arguments. Here we prove this conjecture for an appropriate class of physically acceptable rate constants by constructing a finite upper bound fort c and a nonvanishing lower bound. Apart from the exactly solved caseK(i,j)=ij this result provides the first solid proof of the occurrence of a gelation transition in a description based on Smoluchowski's coagulation equation.
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van Dongen, P.G.J., Ernst, M.H. On the occurrence of a gelation transition in Smoluchowski's coagulation equation. J Stat Phys 44, 785–792 (1986). https://doi.org/10.1007/BF01011907
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DOI: https://doi.org/10.1007/BF01011907