From sol to gel exactly

The time evolution of a disperse system wherein the binary coagulation alone governs the temporal changes of particle mass spectra is studied under the assumption that the coagulation kernel is proportional to the product of masses of the coalescing particles. This model is known to reveal the sol-gel transition, i.e., the formation of a cluster with mass comparable to the total mass of the whole system. This Letter reports on the exact solution of this model. The single-particle mass spectrum is analyzed in the thermodynamic limit and it is demonstrated explicitly how the gel appears in the system.