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：量子热平均在描述量子系统的热力学性质中起着核心作用。从计算角度来看，量子热平均可以通过路径积分分子动力学（PIMD）进行计算，但关于这种近似方法的定量收敛性知识尚缺乏。我们提出了一种替代计算框架，称为连续回路路径积分分子动力学（CL-PIMD），该方法根据费曼-卡克公式的精神，用连续回路取代环形聚合物珠点。通过将正则模截断为有限整数$N\in\mathbb N$，我们量化了截断CL-PIMD的统计平均与真实量子热平均之间的偏差，并证明了截断CL-PIMD具有一致于$N$的几何遍历性。这些结果表明，CL-PIMD提供了对量子热平均的精确逼近，并为PIMD方法提供了数学依据。