Moment scaling at the sol - gel transition

TL;DR

This paper investigates the fluctuation scaling laws at the sol-gel transition using two models, revealing a universal form of fluctuation behavior across different quantities and regimes.

Contribution

It introduces a unified scaling framework for fluctuations in sol-gel models, highlighting robustness and independence from Hamiltonian specifics.

Findings

Identification of three fluctuation regimes with distinct scaling exponents.

Universal fluctuation scaling form applicable to different quantities.

Robustness of scaling laws across equilibrium and off-equilibrium models.

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.