We investigated the equilibrium states of the self-assembly of amphiphilic molecules in water. The amphiphiles are represented by chains of the type H1T4, where H is the hydrophilic part of the molecule and T is its hydrophobic portion formed by four monomers. We have performed Monte Carlo simulations on a two-dimensional lattice, in which each water molecule occupies a single site, and the amphiphiles occupy five sites of the lattice. We have determined the aggregate distribution curves for the system at low concentration and fixed temperature. We have shown that the criterion to determine the equilibrium states of the system, based on the stabilization of energy curves as a function of the simulation time, is not reliable. The best way to ensure that the equilibrium state was reached was to follow the route to equilibrium of all aggregate sizes of the system.