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

Jianfeng Lu, Yulong Lu, Zhennan Zhou

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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.

Original languageEnglish (US)
Article number109788
JournalJournal of Computational Physics
StatePublished - Dec 15 2020
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2020 Elsevier Inc.


  • Continuum limit
  • Dimension-independent convergence rate
  • Path integral molecular dynamics
  • Preconditioned Langevin dynamics


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