预测观察Mpemba效应的条件
Predicting the conditions for observing the Mpemba effect
AI总结：
通过一维过阻尼朗之万动力学，研究Mpemba效应的条件，发现边界（硬或软）的存在是主要驱动因素，而非势能面的内部结构，并利用Fokker-Planck算子的谱分解揭示其机制。
摘要：
Mpemba效应是一种反直觉现象，即较热系统比较冷系统弛豫更快，已在各种非平衡系统中被广泛观察到。尽管取得了这些进展，但出现该效应所需的能量景观的基本结构特征仍存在争议。在本研究中，我们在一维过阻尼朗之万动力学中探究Mpemba效应的条件。我们根据单阱或双阱的存在、对称性以及是否存在壁对势能面进行分类。我们确定，该效应的存在主要由边界（硬或软）的存在驱动，而非势能面的特定内部结构（如亚稳态或极小值数量）。通过采用Fokker-Planck算子的谱分解，我们分析了第一个非平凡本征模的行为，并证明其导数在低温区域表现为狄拉克δ峰。这有助于阐明Mpemba效应的机制：它表现为该行为与初始种群动力学之间由壁的存在以非平凡方式诱导的相互作用。我们的分析提供了跨单阱和双阱势的统一分类，突出了边界条件和不对称性的关键作用。此外，我们证明该框架允许设计能够产生多级Mpemba跃迁的势能面。
英文摘要：
The Mpemba effect, a counterintuitive phenomenon where a hotter system relaxes faster than a colder one, has been widely observed in various nonequilibrium systems. Despite this progress, the fundamental structural features of the energy landscape required for its emergence remain a subject of debate. In this study, we investigate the conditions for the Mpemba effect within one-dimensional overdamped Langevin dynamics. We classify the potential landscapes based on the presence of single or double wells, their symmetry properties, and the existence of walls. We establish that the existence of the effect is primarily driven by the presence of boundaries, either hard or soft, rather than the specific internal structure of the potential landscape, such as metastability or the number of minima. By employing a spectral decomposition of the Fokker-Planck operator, we analyze the behavior of the first nontrivial eigenmode and demonstrate that its derivative acts as a Dirac delta peak in the low-temperature regime. This helps us elucidate the mechanism underlying the Mpemba effect: it appears as the interplay between this behavior and the initial population dynamics in a non-trivial way induced by the presence of the wall. Our analysis provides a unified classification across single- and double-well potentials, highlighting the crucial role of boundary conditions and asymmetry. Furthermore, we demonstrate that this framework allows for the engineering of potential landscapes capable of producing multistage Mpemba transitions.
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