Postcritical behavior of a gelling system

The formation of a gel in a disperse coagulating system, wherein binary coagulation governs the temporal changes of the particle mass spectrum is studied for the coagulation kernel proportional to K(g,l)=gl with g,l being the numbers of monomers in the coalescing pair of particles. This model is known to reveal the sol-gel transition, i.e., the formation of one giant cluster with the mass comparable to the total mass of the whole system. This paper reports on some exact results that are valid for arbitrary initial particle mass spectra. The particle spectrum is shown to be expressed through a contour integral of the exponentiated initial generating function of the particle mass spectrum. The exact formula for the gel mass is derived for the scenario of passive gelation. Two scenarios of gelation, where the gel is either active or passive, are considered.