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.
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.
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)−βfo∼t/ts, where βfo is the inverse temperature at the transition point, t is the time and ts is a time scale. The dynamics starts at ti<0 from an ensemble of disordered configurations equilibrated at inverse temperature β(ti)<βfo and ends at positive values of t, when the system is ordered (this is analogous to a standard Kibble-Zurek protocol). The time-dependent energy density shows an out-of-equilibrium scaling behavior in the large-ts limit, in terms of the scaling variable t(lnt)κ/ts. The corresponding exponent turns out to be consistent with κ=3/2 (with a good accuracy), which is the value obtained by assuming that the initial nucleation of ordered regions provides the relevant mechanism for the passage from one phase to the other. The 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, where δβ∗ decreases as 1/(lnts)3/2 in the large-ts limit.
We study the out-of-equilibrium scaling behavior of two-dimensional and three-dimensional Ising systems, when they are slowly driven across their magnetic first-order transitions at low temperature T<T_{c}, where T_{c} is the temperature of their continuous transition. We consider Kibble-Zurek (KZ) protocols in which a spatially homogenous magnetic field h varies as h(t)=t/t_{s} with a timescale t_{s}. The KZ dynamics starts from negatively magnetized configurations equilibrated at h_{i}<0 and stops at a positive value of h where the configurations acquire a positive average magnetization. We consider the Metropolis and the heat-bath dynamics, which are two specific examples of a purely relaxational dynamics. We focus on two different dynamic regimes. We consider the out-equilibrium finite-size scaling (OFSS) limit in which the system size L and the timescale t_{s} diverge simultaneously, keeping an appropriate combination fixed. Then, we analyze the KZ dynamics in the thermodynamic limit (TL), obtained by taking first the L→∞ limit at fixed t and t_{s}, and then considering the scaling behavior in the large-t_{s} limit. Our numerical results provide evidence of OFSS, as predicted by general scaling arguments. The results in the TL show the emergence of spinodal-like behaviors: The passage from the negatively magnetized phase to the positively magnetized one occurs at positive values h_{*}>0 of the magnetic field, which decrease as h_{*}∼1/(lnt_{s})^{κ}, with κ=2 and κ=1 in two and three dimensions, respectively, for t_{s}→∞. We identify σ≡t(lnt)^{κ}/t_{s} as the relevant scaling variable associated with the KZ dynamics in the TL.
We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field h(t)=t/ts (ts is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at h=0. The KZ protocol starts at time ti<0 from the negatively magnetized ground state for hi=ti/ts<0. Then, the system evolves unitarily up to a time tf>0, such that the magnetization of the state at time tf is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping t and ts fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at h(t)=h∗>0, where h∗ decreases with increasing ts, as h∗∼1/lnts. Moreover, in the large-ts limit, the time-dependence of the magnetization is described by a universal function of Ω=t/τs, with τs=ts/lnts.
We analyze the out-of-equilibrium relaxational dynamics of ferromagnetic Ising-like systems driven across their low-temperature magnetic first-order transition (FOT) line, by sudden and slow variations of an external homogenous magnetic field h. As a paradigmatic example, we consider a two-dimensional Ising system driven across its low-temperature FOT line by suddently quenching h from hi<0 to h>0, inducing a transition from the negatively to the positively magnetized phase. We show the emergence of a dynamic percolation transition at a finite critical time tc(h) along the post-quench evolution for finite (sufficiently small) values of h, which is marked by the percolation of the largest positive-spin cluster and the antipercolation of the largest negative-spin cluster. This out-of-equilibrium percolation transition displays a finite-size scaling behavior as in the standard random-percolation case. However, while the fractal dimension of the percolating clusters is consistent with the random-percolation value, the exponent controlling the approach to criticality differs and depends on h. We also show that the percolation transition marks the passage from the metastable negatively-magnetized phase to the stable positively-magnetized one. Therefore, in the small-h limit, the percolation critical behavior is related to the spinodal-like behavior of the magnetization, implying that the percolation time tc(h) exhibits a spinodal-like exponential dependence on h. The existence of percolation transitions is likely a generic phenomenon at magnetic Ising-like FOTs. For example, we observe an analogous behavior in dynamic (Kibble-Zurek-like) protocols entailing slow variations of the magnetic field driving the crossing of the FOT line.
We show that an out-of-equilibrium percolation transition occurs after quenching ferromagnetic Ising-like systems across their magnetic first-order transitions. As a paradigmatic example, we consider a two-dimensional Ising system driven across its low-temperature first-order transition line by a quench of the magnetic field h from hi<0 to h>0. In the thermodynamic limit and for finite values of h, the post-quench evolution under a purely relaxational dynamics is characterized by a dynamic transition at a finite critical time tc(h) from the metastable negatively magnetized phase to the positive one, marked by the percolation of the largest clusters of positive and negative spins. This out-of-equilibrium percolation transition displays a finite-size scaling behavior as in the standard random-percolation case. However, while the fractal dimension of the percolating clusters is consistent with the random-percolation value, the exponent controlling the approach to criticality differs and depends on h. We also show that the percolation critical behavior is related to the spinodal-like behavior of the magnetization in the small-h limit, which implies that the percolation time tc(h) exhibits a spinodal-like exponential dependence on h.
We study off-equilibrium behaviors at first-order transitions (FOTs) driven by a time dependence of the temperature across the transition point T(c), such as the linear behavior T(t)/T(c)=1±t/t(s) where t(s) is a time scale. In particular, we investigate the possibility of nontrivial off-equilibrium scaling behaviors in the regime of slow changes, corresponding to large t(s). We consider the two-dimensional Potts models, which provide an ideal theoretical laboratory to investigate issues related to FOTs driven by thermal fluctuations. We put forward general ansatzes for off-equilibrium scaling behaviors around the time t=0 corresponding to T(c). Then we present numerical results for the q=10 and 20 Potts models. We show that off-equilibrium scaling behaviors emerge at FOTs with relaxational dynamics, when appropriate boundary conditions are considered, such as mixed boundary conditions favoring different phases at the opposite sides of the system, which enforce an interface in the system.