Moment Scaling at the Sol-Gel Transition

Two standard models of sol-gel transition are revisited here from the point of view of their fluctuations in various moments of both the mass-distribution and the gel-mass. Bond-percolation model is an at-equilibrium system and undergoes a static second-order phase transition, while Monte-Carlo Smoluchowski model is an off-equilibrium one and shows a dynamical critical phenomenon. We show that the macroscopic quantities can be splitted into the three classes with different scaling properties of their fluctuations, depending on whether they correspond to: (i) noncritical quantities, (ii) critical quantities or to (iii) an order parameter. All these three scaling properties correspond to a single form: 〈M〉δ P(M) = Φ((M − 〈M〉)/〈 M〉δ), with the values of δ respectively: = 1/2 (regime (i)), ≠ 1/2 and 1 (regime (ii)), and = 1 (regime (iii)). These new scalings are very robust and, in particular, they do not depend on the precise form of an Hamiltonian.