This book chapter is published open access.
In these notes, we review recent results for the limiting behavior of equilibrium fluctuations of interacting particle systems with one or several conserved quantities. Two main classes of models are considered. First, the weakly asymmetric simple exclusion process, a model with one conservation law, and whose fluctuations cross from the Edwards–Wilkinson (EW) universality class to the Kardar–Parisi–Zhang (KPZ) universality class. Second, we consider a class of Hamiltonian systems perturbed by a noise and conserving two quantities. In the case of an exponential potential, the transition occurs from diffusion to fractional behavior, while for a harmonic potential the fluctuations cross from diffusive to fractional behavior. We review two different methods which rigorously prove some of the aforementioned results.