We investigate the second time scale of the metastable behavior of the reversible inclusion process in an extension of the study by Bianchi et al. (Electron J Probab 22:1–34, 2017), which presented the first time scale of the same model and conjectured the scheme of multiple time scales. We show that \(N/d_{N}^{2}\) is indeed the correct second time scale for the most general class of reversible inclusion processes, and thus prove the first conjecture of the foresaid study. Here, N denotes the number of particles, and \(d_{N}\) denotes the small scale of randomness of the system. The main obstacles of this research arise in calculating the sharp asymptotics for the capacities, and in the fact that the methods employed in the former study are not directly applicable due to the complex geometry of particle configurations. To overcome these problems, we first thoroughly examine the landscape of the transition rates to obtain a proper test function of the equilibrium potential, which provides the upper bound for the capacities. Then, we modify the induced test flow and precisely estimate the equilibrium potential near the metastable valleys to obtain the correct lower bound for the capacities.

在Bianchi等人的研究扩展中，我们调查了可逆包涵过程的亚稳态行为的第二时间尺度。（Electron J Probab 22：1–34，2017年），提出了相同模型的第一个时间尺度，并推测了多个时间尺度的方案。我们证明\（N / d_ {N} ^ {2} \）确实是最通用的可逆包含过程类别的正确第二时间尺度，因此证明了上述研究的第一个猜想。在此，N表示粒子数，\（d_ {N} \）表示系统的小规模随机性。这项研究的主要障碍出现在计算能力的尖锐渐近性上，事实上，由于颗粒结构的几何形状复杂，因此先前研究中使用的方法不能直接应用。为了克服这些问题，我们首先彻底检查过渡速率的变化，以获得平衡势的适当测试函数，该函数为容量提供了上限。然后，我们修改诱导的测试流，并精确估计亚稳谷附近的平衡势，以获得正确的容量下限。