The quantum thermal average plays a central role in describing the thermodynamic properties of a quantum system. From the computational perspective, the quantum thermal average can be computed by the path integral molecular dynamics (PIMD), but the knowledge on the quantitative convergence of such approximations is lacking. We propose an alternative computational framework named the continuous loop path integral molecular dynamics (CL-PIMD), which replaces the ring polymer beads by a continuous loop in the spirit of the Feynman--Kac formula. By truncating the number of normal modes to a finite integer $N\in\mathbb N$, we quantify the discrepancy of the statistical average of the truncated CL-PIMD from the true quantum thermal average, and prove that the truncated CL-PIMD has uniform-in-$N$ geometric ergodicity. These results show that the CL-PIMD provides an accurate approximation to the quantum thermal average, and serves as a mathematical justification of the PIMD methodology.

翻译：量子热平均值在描述量子系统热力学性质中扮演着核心角色。从计算角度来看，量子热平均值可通过路径积分分子动力学（PIMD）计算，但此类近似方法的定量收敛性认识尚不充分。我们提出了一种名为连续圈路径积分分子动力学（CL-PIMD）的新型计算框架，该框架基于费曼-卡克公式的思想，用连续圈替换环聚合物珠。通过将正则模数量截断为有限整数$N\in\mathbb N$，我们量化了截断CL-PIMD统计平均值与真实量子热平均值之间的偏差，并证明截断CL-PIMD具有关于$N$的一致几何遍历性。这些结果表明，CL-PIMD能精确逼近量子热平均值，并为PIMD方法论提供了数学依据。