We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. We propose two preconditioned Langevin sampling dynamics, which are shown to have improved stability and sampling accuracy. We present a careful mode analysis of the preconditioned dynamics and show their connections to the normal mode, the staging coordinate and the Matsubara mode representation for ring polymers. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite dimensional counterpart has a dimension-independent convergence rate. In addition, the preconditioning techniques can be naturally applied to the multi-level quantum systems in the nonadiabatic regime, which are compatible with various numerical approaches.

我们研究了连续数极限，即在平均数的环状聚合物表示中，珠子的数量达到无穷大。对轨迹采样方程的连续极限的研究揭示了可能的预处理技术，用于对具有大量珠子的环状聚合物构型进行采样。我们提出了两个预处理的Langevin采样动力学，它们被证明具有更高的稳定性和采样精度。我们对预处理的动力学进行了仔细的模式分析，并显示了它们与正常模式，阶段坐标和环状聚合物的Matsubara模式表示的联系。在电势为二次方的情况下，我们表明，经预处理的质量修正兰格文动力学的连续极限以指数速度快速收敛至其平衡，这表明有限维对应物具有与维无关的收敛速度。另外，预处理技术可以自然地应用于非绝热状态下的多级量子系统，该系统与各种数值方法兼容。