Continuum limit and preconditioned Langevin sampling of the path integral molecular dynamics

TL;DR

This paper explores the continuum limit of ring polymer representations in path integral molecular dynamics, proposing preconditioned Langevin methods that improve sampling stability and accuracy, especially for large bead numbers.

Contribution

It introduces two novel preconditioned Langevin sampling dynamics with enhanced stability, analyzes their modes, and demonstrates their exponential convergence in the quadratic potential case.

Findings

Preconditioned Langevin dynamics improve sampling stability.

Convergence to equilibrium is exponential for quadratic potentials.

Techniques are applicable to multi-level quantum systems in nonadiabatic regimes.

Abstract

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.