Generalisation of the Eyring-Kramers Transition Rate Formula to Irreversible Diffusion Processes
Generalisation of the Eyring-Kramers Transition Rate Formula to Irreversible Diffusion Processes
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DOI：
10.1007/s00023-016-0507-4
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发表时间：
2016-12-01
影响因子：
1.5
通讯作者：
Reygner, Julien
中科院分区：
文献类型：
--
作者：
Bouchet, Freddy;Reygner, Julien
在小噪声条件下，可逆扩散过程亚稳态间的平均跃迁时间用阿累尼乌斯定律在对数尺度上描述。Eyring-Kramers公式经典地为这种大偏差估计提供了一个次指数预因子。对于不可逆扩散过程，Freidlin-Wentzell理论给出了等效的Arrhenius定律。本文计算了相关的前因子，从而将Eyring-Kramers公式推广到不可逆扩散过程。在我们的公式中，势的作用由Freidlin-Wentzell的准势起作用，并且根据系统沿最小作用路径的非gibbsianness进行了更正。我们的研究假设了向量场的一些性质：(1)吸引子是孤立的点；(2)限制于吸引边界盆地的动力学被吸引到单点（即向量场的鞍点）上。此外，我们假设连接吸引子和相邻鞍点（瞬子）的最小作用路径具有在结论中总结的一般性质。在技术层面上，我们的推导结合了围绕实例的一阶WKB展开的精确计算和接近鞍点的一阶匹配渐近展开的精确计算。虽然一旦假设了形式展开式，结果是准确的，但这些渐近展开式的有效性仍有待证明。
In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius' law. The Eyring-Kramers formula classically provides a subexponential prefactor to this large deviation estimate. For irreversible diffusion processes, the equivalent of Arrhenius' law is given by the Freidlin-Wentzell theory. In this paper, we compute the associated prefactor and thereby generalise the Eyring-Kramers formula to irreversible diffusion processes. In our formula, the role of the potential is played by Freidlin-Wentzell's quasipotential, and a correction depending on the non-Gibbsianness of the system along the minimum action paths is highlighted. Our study assumes some properties for the vector field: (1) attractors are isolated points, (2) the dynamics restricted to basin of attraction boundaries are attracted to single points (which are saddle-points of the vector field). We moreover assume that the minimum action paths that connect attractors to adjacent saddle-points (the instantons) have generic properties that are summarised in the conclusion. At a technical level, our derivation combines an exact computation for the first-order WKB expansion around the instanton and an exact computation of the first-order match asymptotics expansion close to the saddle-point. While the results are exact once a formal expansion is assumed, the validity of these asymptotic expansions remains to be proven.