Molecule & Life Nonlinear Sciences Laboratory, Research Institute for Electronic Science, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan.
Phys Chem Chem Phys. 2010 Dec 21;12(47):15382-91. doi: 10.1039/c0cp00543f. Epub 2010 Oct 20.
A framework recently developed for the extraction of a dynamic reaction coordinate to mediate reactions buried in a multidimensional Langevin equation is extended to the generalized Langevin equations without a priori assumption of the forms of the potential (in general, nonlinearly coupled systems) and the friction kernel. The equation of motion with memory effect can be transformed into an equation without memory at the cost of an increase in the dimensionality of the system, and hence the theoretical framework developed for the (nonlinear) Langevin formulation can be generalized to the non-Markovian process with colored noise. It is found that the increased dimension can be physically interpreted as effective modes of the fluctuating environment. As an illustrative example, we apply this theory to a multidimensional generalized Langevin equation for motion on the Müller-Brown potential surface with an exponential friction kernel. Numerical simulations find a boundary between the highly reactive region and the less reactive region in the space of initial conditions. The location of the boundary is found to depend significantly on both the memory kernel and the nonlinear couplings. The theory extracts a reaction coordinate whose sign determines the fate of the reaction taking into account thermally fluctuating environments, memory effect, and nonlinearities. It is found that the location of the boundary of reactivity is satisfactorily reproduced as the zero of the statistical average of the new reaction coordinate, which is an analytical functional of both the original position coordinates and velocities of the system, and of the properties of the environment.
最近开发的一种从多维 Langevin 方程中提取动态反应坐标来介导反应的框架被扩展到没有势能（通常是非线性耦合系统）和摩擦核函数形式先验假设的广义 Langevin 方程。具有记忆效应的运动方程可以通过增加系统的维度转化为无记忆的方程，因此为（非线性） Langevin 公式开发的理论框架可以推广到具有有色噪声的非马尔可夫过程。发现增加的维度可以物理地解释为波动环境的有效模式。作为一个说明性的例子，我们将该理论应用于具有指数摩擦核函数的 Müller-Brown 势能表面上运动的多维广义 Langevin 方程。数值模拟在初始条件空间中发现了高反应区域和低反应区域之间的边界。边界的位置发现与记忆核函数和非线性耦合都有显著的关系。该理论提取了一个反应坐标，其符号决定了在考虑热波动环境、记忆效应和非线性的情况下反应的命运。结果发现，反应性边界的位置可以通过新反应坐标的统计平均值的零点来很好地再现，该平均值是系统的原始位置坐标和速度以及环境性质的解析函数。
