CEA, DEN, Service de Recherches de Métallurgie Physique, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France.
J Chem Phys. 2017 May 21;146(19):194101. doi: 10.1063/1.4983164.
Markov chain Monte Carlo methods are primarily used for sampling from a given probability distribution and estimating multi-dimensional integrals based on the information contained in the generated samples. Whenever it is possible, more accurate estimates are obtained by combining Monte Carlo integration and integration by numerical quadrature along particular coordinates. We show that this variance reduction technique, referred to as conditioning in probability theory, can be advantageously implemented in expanded ensemble simulations. These simulations aim at estimating thermodynamic expectations as a function of an external parameter that is sampled like an additional coordinate. Conditioning therein entails integrating along the external coordinate by numerical quadrature. We prove variance reduction with respect to alternative standard estimators and demonstrate the practical efficiency of the technique by estimating free energies and characterizing a structural phase transition between two solid phases.
马尔可夫链蒙特卡罗方法主要用于从给定的概率分布中进行抽样，并根据生成样本中包含的信息来估计多维积分。只要有可能，通过将蒙特卡罗积分和沿特定坐标的数值求积积分相结合，可以获得更精确的估计。我们表明，这种方差减少技术，在概率论中称为条件化，可以在扩展系综模拟中有利地实现。这些模拟旨在估计作为外部参数的函数的热力学期望，该外部参数像额外的坐标一样被采样。其中的条件化涉及通过数值求积沿外部坐标进行积分。我们证明了相对于替代标准估计量的方差减少，并通过估计自由能和描述两种固体相之间的结构相变来证明该技术的实际效率。
