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.
We investigate the dynamic properties of systems at thermal first-order transitions (FOT), when boundary conditions favor one of the two phases. In particular, we study the dynamic scaling properties arising from a slow heating of systems with disordered boundary conditions, starting in the ordered phase and moving across the FOT. As a theoretical laboratory we consider the two-dimensional Potts model. We show that a power-law dynamic scaling emerges, associated with a mixed regime where the two phases are spatially separated. We argue that these features generally apply to systems at FOTs, when boundary conditions favor one of the two phases. In particular, they should be relevant for the experimental search of FOTs of the quark-gluon plasma in heavy-ion collisions.
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.
Abstract We investigate how the scaling behavior of finite systems at magnetic first-order transitions (FOTs) with relaxational dynamics changes in correspondence of various boundary conditions. As a theoretical laboratory, we consider the two-dimensional Ising model in the low-temperature phase. When the boundary conditions do not favor any specific phase of the system, we show that a dynamic finite-size scaling (DFSS) theory can be developed to describe the dynamic behavior in the coexistence region, where different phases coexist. When the boundary conditions at two opposite sides of the system generate a planar interface separating the phases, we show that the autocorrelation times are characterized by a power-law behavior, related to the dynamics enforced by the interface. Numerical results for a purely relaxational dynamics confirm the general picture.
We investigate the dynamic behavior of finite-size systems close to a first-order transition (FOT). We develop a dynamic finite-size scaling (DFSS) theory for the dynamic behavior in the coexistence region where different phases coexist. This is characterized by an exponentially large time scale related to the tunneling between the two phases. We show that, when considering time scales of the order of the tunneling time, the dynamic behavior can be described by a two-state coarse-grained dynamics. This allows us to obtain exact predictions for the dynamical scaling functions. To test the general DFSS theory at FOTs, we consider the two-dimensional Ising model in the low-temperature phase, where the external magnetic field drives a FOT, and the 20-state Potts model, which undergoes a thermal FOT. Numerical results for a purely relaxational dynamics fully confirm the general theory.
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 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.