关于亚椭圆三阶朗之万扩散的艾林 - 克莱默斯定律
An Eyring--Kramers Law for the Hypoelliptic Third-Order Langevin Diffusion
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中文总结 AI 辅助
研究亚椭圆三阶朗之万扩散在低温极限下亚稳态跃迁的艾林 - 克莱默斯定律，通过扩展策略并结合多种方法确定平均跃迁时间的相关尺度和因子，数值实验验证了阿累尼乌斯标度及因子比较。
中文摘要 AI 辅助
我们证明了在低温极限下亚椭圆三阶朗之万扩散亚稳态跃迁的艾林 - 克莱默斯定律。该模型受Mou等人（2021年，《机器学习研究杂志》，22(42)，1 - 41）的高阶朗之万扩散启发。对于具有唯一一阶跃迁鞍点的双阱势，我们确定了平均跃迁时间的阿累尼乌斯指数尺度和精确前置因子。我们的证明将Lee - Ramil - Seo（2026年，arXiv:2503.12610v2）的欠阻尼艾林 - 克莱默斯策略扩展到三阶情形，并结合了弱容量框架与鞍点适配边界层、显式高斯电流计算、转移概率定位和阱内平坦度。在匹配动力学归一化下，所得亚稳态前置因子严格小于其欠阻尼对应值。一维双阱的数值实验说明了阿累尼乌斯标度和预测的前置因子比较。
英文摘要
We prove an Eyring--Kramers law for metastable transitions of the hypoelliptic third-order Langevin diffusion in the low-temperature limit. This diffusion is a three-level Markovian lifting of Langevin dynamics: the Brownian noise acts only on the highest auxiliary variable and reaches the position variable through a third-order H"ormander chain. For a double-well potential with a unique index-one transition saddle, we determine both the Arrhenius exponential scale and the sharp prefactor of the mean transition time. The prefactor is governed by the unique positive unstable rate of the deterministic linearization at the saddle, equivalently the positive root of a cubic polynomial. Our proof combines a weak-capacity framework with a saddle-adapted boundary layer, an explicit Gaussian current calculation, committor localization, and intrawell flatness. Under matched kinetic normalizations, the resulting metastable prefactor is strictly smaller than its underdamped counterpart. A numerical experiment for a one-dimensional double well illustrates the Arrhenius scaling and the predicted prefactor comparison.
