Moment scaling at the sol - gel transition

Abstract

Two standard models of sol-gel transition are revisited here from the point of view of their fluctutations 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 wheather they correspond to : (i) non-critical quantities, (ii) critical quantities or to (iii) an order parameter. All these three scaling properties correspond to a single form : $<M>^{\delta} P(M) = \Phi ((M-<M>)/<M>^{\delta})$, with the values of $\delta$ respectively : =1/2 (regime (i)), \neq 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.