Dimension-free Ergodicity of Path Integral Molecular Dynamics

TL;DR

This paper proves that both standard and Matsubara mode path integral molecular dynamics methods exhibit uniform ergodicity regardless of the number of modes or beads, enhancing theoretical understanding of their sampling efficiency.

Contribution

It establishes the first rigorous proof of dimension-free ergodicity for both standard and Matsubara mode PIMD using generalized Gamma calculus.

Findings

Uniform-in-N ergodicity for standard PIMD

Uniform-in-N ergodicity for Matsubara mode PIMD

Convergence rate independent of number of modes or beads

Abstract

The quantum thermal average plays a central role in describing the thermodynamic properties of a quantum system. Path integral molecular dynamics (PIMD) is a prevailing approach for computing quantum thermal averages by approximating the quantum partition function as a classical isomorphism on an augmented space, enabling efficient classical sampling, but the theoretical knowledge of the ergodicity of the sampling is lacking. Parallel to the standard PIMD with $N$ ring polymer beads, we also study the Matsubara mode PIMD, where the ring polymer is replaced by a continuous loop composed of $N$ Matsubara modes. Utilizing the generalized $\Gamma$ calculus, we prove that both the Matsubara mode PIMD and the standard PIMD have uniform-in-$N$ ergodicity, i.e., the convergence rate towards the invariant distribution does not depend on the number of modes or beads $N$.