Out-of-equilibrium spinodal-like scaling behaviors at the thermal first-order transitions of three-dimensional q-state Potts models

We study the out-of-equilibrium spinodal-like dynamics of three-dimensional q-state Potts systems driven across their thermal first-order transition in the thermodynamic limit, by a relaxational (heat-bath) dynamics. During the evolution, the inverse temperature β increases linearly with time, as δβ(t)≡β(t)-β_{fo}∼t/t_{s}, where β_{fo} is the inverse temperature at the transition point, t is the time, and t_{s} is a timescale. The dynamics starts at t_{i}<0 from an ensemble of disordered configurations equilibrated at an inverse temperature β(t_{i})<β_{fo} and ends at positive values of t, corresponding to β(t)>β_{fo} in the ordered phase (this is analogous to a standard Kibble-Zurek protocol). The time-dependent energy density shows an out-of-equilibrium scaling behavior in the large-t_{s} limit, in terms of the scaling variable σ≡t(lnt)^{κ}/t_{s}. The exponent κ turns out to be consistent with κ=3/2 (with good accuracy), which is the value obtained by assuming that the initial nucleation of ordered regions is the relevant mechanism providing the largest timescale. This scaling behavior implies a spinodal-like phenomenon close to the transition point: the passage from the disordered to the ordered phase, composed of large ordered regions of different color, occurs at δβ(t)=δβ_{*}>0, which decreases as δβ_{*}∼(lnt_{s})^{-κ} in the large-t_{s} limit.