We investigate numerically the three-dimensional three-state Potts models with nearest-neighbor (NN) ferromagnetic coupling and next-nearest-neighbor (NNN) antiferromagnetic coupling of relative strength \ensuremath{\gamma} on
L3 lattices with L=12, 16, 20, 32, and 40. For all the \ensuremath{\gamma} values that we studied, 0.0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}0.8, we find indications of a first-order phase transition between the ordered and disordered phases. In the neighborhood of \ensuremath{\gamma}=0.25, the latent heat becomes rather small, making it necessary to use still larger lattices to rule out a higher-order phase transition at \ensuremath{\gamma}=0.25. We also studied the boundary regimes of the three different ordered phases and find no criticality along them, thus suggesting a lack of criticality in these extended Potts models.
The three-state Potts model is numerically investigated on three-dimensional simple cubic lattices of up to
1283 volume, concentrating on the neighborhood of the first-order phase transition separating the ordered and disordered phases. In both phases clusters of like spins are observed with irregular boundaries. In the ordered phase the two different non-favored spins are found to attract each other with a long but finite range. As a result, the neighborhoods of the non-favored spins are interpreted as domains of the disordered phase. This explains why the first-order phase transitions associated with the global
Z3 symmetry, including the SU(3) pure-gauge one, are so weak.
In papers devoted to superfluidity, the generally accepted statement that the dynamics of the corresponding phase transition is described by the stochastic model F or E can be frequently found. Nevertheless, the dynamical critical index has not been found even in the leading order of the perturbation theory. It is also unknown which model, E or F, in fact corresponds to this system. We use two different approaches to study this problem. First, we study the dynamics of the critical behavior in a neighborhood of the λ point using the renormalization group method based on a quantum microscopic model in the formalism of the time-dependent Green’s functions at a finite temperature. Second, we study the stochastic model F to find whether it is stable under compressibility effects. Both approaches lead to the same very unexpected result: the dynamics of the phase transition to the superfluid state are described by the stochastic model A with a known dynamical critical index.