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Volume 38, Issue 5
Dimension-Free Ergodicity of Path Integral Molecular Dynamics

Xuda Ye & Zhennan Zhou

DOI: 10.4208/cicp.OA-2024-0004

Commun. Comput. Phys., 38 (2025), pp. 1355-1388.

Published online: 2025-09

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  • 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 $N$.

  • Keywords

Quantum thermal average, path integral molecular dynamics, Matsubara modes, ergodicity, generalized $\Gamma$ calculus.

  • AMS Subject Headings

37A30, 82B31, 81S40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-38-1355, author = {Ye , Xuda and Zhou , Zhennan}, title = {Dimension-Free Ergodicity of Path Integral Molecular Dynamics}, journal = {Communications in Computational Physics}, year = {2025}, volume = {38}, number = {5}, pages = {1355--1388}, 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 $N$.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2024-0004}, url = {http://global-sci.org/intro/article_detail/cicp/24460.html} }
TY - JOUR T1 - Dimension-Free Ergodicity of Path Integral Molecular Dynamics AU - Ye , Xuda AU - Zhou , Zhennan JO - Communications in Computational Physics VL - 5 SP - 1355 EP - 1388 PY - 2025 DA - 2025/09 SN - 38 DO - http://doi.org/10.4208/cicp.OA-2024-0004 UR - https://global-sci.org/intro/article_detail/cicp/24460.html KW - Quantum thermal average, path integral molecular dynamics, Matsubara modes, ergodicity, generalized $\Gamma$ calculus. AB -

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 $N$.

Ye , Xuda and Zhou , Zhennan. (2025). Dimension-Free Ergodicity of Path Integral Molecular Dynamics. Communications in Computational Physics. 38 (5). 1355-1388. doi:10.4208/cicp.OA-2024-0004
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