Gray Thomas H, Yong Ee Hou
Department of Chemical Engineering and Biotechnology, West Cambridge Site, University of Cambridge, Philippa Fawcett Drive, CB3 0AS Cambridge, United Kingdom.
Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371.
J Chem Phys. 2021 Feb 28;154(8):084103. doi: 10.1063/5.0040071.
Thermally activated escape processes in multi-dimensional potentials are of interest to a variety of fields, so being able to calculate the rate of escape-or the mean first-passage time (MFPT)-is important. Unlike in one dimension, there is no general, exact formula for the MFPT. However, Langer's formula, a multi-dimensional generalization of Kramers's one-dimensional formula, provides an approximate result when the barrier to escape is large. Kramers's and Langer's formulas are related to one another by the potential of mean force (PMF): when calculated along a particular direction (the unstable mode at the saddle point) and substituted into Kramers's formula, the result is Langer's formula. We build on this result by using the PMF in the exact, one-dimensional expression for the MFPT. Our model offers better agreement with Brownian dynamics simulations than Langer's formula, although discrepancies arise when the potential becomes less confining along the direction of escape. When the energy barrier is small our model offers significant improvements upon Langer's theory. Finally, the optimal direction along which to evaluate the PMF no longer corresponds to the unstable mode at the saddle point.
多维势场中的热激活逃逸过程在多个领域都备受关注，因此能够计算逃逸速率或平均首次通过时间（MFPT）非常重要。与一维情况不同，对于MFPT没有通用的精确公式。然而，兰格公式是克莱默斯一维公式的多维推广，当逃逸势垒较大时，它能提供一个近似结果。克莱默斯公式和兰格公式通过平均力势（PMF）相互关联：当沿特定方向（鞍点处的不稳定模式）计算并代入克莱默斯公式时，结果就是兰格公式。我们基于这一结果，在MFPT的精确一维表达式中使用PMF。我们的模型与布朗动力学模拟的吻合度比兰格公式更好，不过当势场在逃逸方向上的限制变小时会出现差异。当能量势垒较小时，我们的模型相对于兰格理论有显著改进。最后，评估PMF的最佳方向不再对应于鞍点处的不稳定模式。
