Extended Lagrangian molecular dynamics (XLMD) is a general method for performing molecular dynamics simulations using quantum and classical many-body potentials. Recently several new XLMD schemes have been proposed and tested on several classes of many-body polarization models such as induced dipoles or Drude charges, by creating an auxiliary set of these same degrees of freedom that are reversibly integrated through time. This gives rise to a singularly perturbed Hamiltonian system that provides a good approximation to the time evolution of the real mutual polarization field. To further improve upon the accuracy of the XLMD dynamics in the context of classical polarizable force field simulation, and to potentially extend it to other many-body potentials, we introduce a stochastic modification which leads to a set of singularly perturbed Langevin equations with degenerate noise. We prove that the resulting Stochastic-XLMD converges to the accurate dynamics, and the convergence rate is both sharp and is independent of the accuracy of the initial polarization field. We carefully study the scaling of the damping factor and numerical noise for efficient numerical simulation for Stochastic-XLMD, and we demonstrate the effectiveness of the method for water molecules described by a polarizable force field.

扩展拉格朗日分子动力学（XLMD）是使用量子和经典多体势进行分子动力学模拟的通用方法。最近，已经提出了几种新的XLMD方案，并在多类多体极化模型（例如感应偶极子或Drude电荷）上进行了测试，并通过创建可逆时间积分的相同自由度的辅助集进行了测试。这就产生了一个奇摄动的哈密顿系统，它为真实的互极化场的时间演化提供了一个很好的近似值。为了在经典可极化力场模拟的背景下进一步提高XLMD动力学的准确性，并有可能将其扩展到其他多体势，我们介绍了一种随机修正，它导致了一组带有退化噪声的奇摄动的Langevin方程。我们证明了所产生的随机XLMD收敛于精确的动力学，并且收敛速率既清晰又与初始极化场的精度无关。我们仔细研究了阻尼因子和数值噪声的换算，以进行随机XLMD的高效数值模拟，并证明了该方法对极化力场描述的水分子的有效性。