While a large number of studies have focused on the nonequilibrium dynamics of a system when it is quenched instantaneously from a disordered phase to an ordered phase, such dynamics have been relatively less explored when the quench occurs at a finite rate. Here, we study the slow quench dynamics in two paradigmatic models of classical statistical mechanics, a one-dimensional kinetic Ising model and a mean-field zero-range process, when the system is annealed slowly to the critical point. Starting from the time evolution equations for the spin–spin correlation function in the Ising model and the mass distribution in the zero-range process, we derive the Kibble–Zurek scaling laws. We then test a recent proposal that critical coarsening, which is ignored in the Kibble–Zurek argument, plays a role in the nonequilibrium dynamics close to the critical point. We find that the defect density in the Ising model and a scaled mass distribution in the zero-range process decay linearly to their respective values at the critical point with the time remaining until the end of the quench provided the final quench point is approached sufficiently fast, and sublinearly otherwise. As the linear scaling for the approach to the critical point also holds when a system following an instantaneous quench is allowed to coarsen for a finite time interval, we conclude that critical coarsening captures the scaling behavior in the vicinity of the critical point if the annealing is not too slow.

尽管大量研究着重于系统从无序相到有序相的瞬时淬火时系统的非平衡动力学，但是当以有限速率发生淬灭时，这种动力学的研究相对较少。在此，当系统缓慢退火至临界点时，我们研究了经典统计力学的两个范式模型（一维动力学伊辛模型和均值零范围过程）中的慢淬灭动力学。从伊辛模型中自旋-自旋相关函数的时间演化方程和零范围过程中的质量分布开始，我们得出了Kibble-Zurek缩放定律。然后，我们测试了最近的一项建议，即关键粗化（在Kibble–Zurek参数中被忽略），在接近临界点的非平衡动力学中发挥作用。我们发现，在Ising模型中的缺陷密度和零范围过程中的按比例分配质量分布在临界点处线性衰减至其各自的值，直到最终淬火点足够快地达到淬火结束时为止，还有剩余的时间，否则为次线性。当允许瞬时淬火的系统在有限的时间间隔内进行粗化时，由于逼近临界点的线性缩放也保持不变，因此我们得出结论：如果退火为零，则临界粗化会捕获临界点附近的缩放行为。不太慢。我们发现，在Ising模型中的缺陷密度和零范围过程中的按比例分配质量分布在临界点处线性衰减至其各自的值，直到最终淬火点足够快地达到淬火结束时为止，还有剩余的时间，否则为次线性。当允许瞬时淬火的系统在有限的时间间隔内进行粗化时，由于逼近临界点的线性缩放也保持不变，因此我们得出结论：如果退火为零，则临界粗化会捕获临界点附近的缩放行为。不太慢。我们发现，在Ising模型中的缺陷密度和零范围过程中的按比例分配质量分布在临界点处线性衰减至其各自的值，直到最终淬火点足够快地达到淬火结束时为止，还有剩余的时间，否则为次线性。当允许瞬时淬火的系统在有限的时间间隔内进行粗化时，由于逼近临界点的线性缩放也保持不变，因此我们得出结论：如果退火为零，则临界粗化会捕获临界点附近的缩放行为。不太慢。