Thermoreversible gelation is a Bose-Einstein condensation

On the basis of the classical tree statistics, being combined with the lattice theory of a polymer solution, we prove that within the mean-field approximation thermoreversible gelation of polyfunctional molecules is a third order phase transition analogous to the Bose-Einstein condensation of ideal Bose gases. The derivative of the osmotic compressibility with respect to the concentration of primary functional molecules is shown to reveal a discontinuity at the sol/gel transition point, whose width is directly related to the amplitude of the divergent term in the weight-average molecular weight of clusters. An inclusion of cycle formation, as well as concentration fluctuations, may, however, change the nature of the phase transition.