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Dimension-free Ergodicity of Path Integral Molecular Dynamics

Xuda Ye, Zhennan Zhou

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 $Γ$ 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 $N$.

Dimension-free Ergodicity of Path Integral Molecular Dynamics

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 ring polymer beads, we also study the Matsubara mode PIMD, where the ring polymer is replaced by a continuous loop composed of Matsubara modes. Utilizing the generalized calculus, we prove that both the Matsubara mode PIMD and the standard PIMD have uniform-in- ergodicity, i.e., the convergence rate towards the invariant distribution does not depend on the number of modes or beads .
Paper Structure (22 sections, 8 theorems, 148 equations, 3 figures, 1 table)

This paper contains 22 sections, 8 theorems, 148 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Derivation of the Matsubara mode PIMD
  3. Quantum equilibrium as the Boltzmann distribution of a continuous loop
  4. Represent the energy function with Matsubara modes
  5. Formulation of the Matsubara mode PIMD
  6. Implementation of the Matsubara mode PIMD in finite dimensions
  7. Relation to the standard PIMD
  8. Uniform-in-$N$ ergodicity of the Matsubara mode PIMD
  9. Assumptions and ergodicity results
  10. Uniform ergodicity of overdamped Matsubara mode PIMD
  11. Uniform ergodicity of underdamped Matsubara mode PIMD
  12. Extension of the the standard PIMD
  13. Numerical tests
  14. Setup of parameters and the time average estimator
  15. 1D model potential
  16. ...and 7 more sections

Key Result

Theorem 3.1

Under Assumption (i), let $(P_t)_{t\geqslant0}$ be the Markov semigroup of the overdamped Matsubara mode PIMD over, then for any positive smooth function $f(\xi)$ in $\mathbb R^{dN}$, where the convergence rate $\lambda_1 = \exp(-4\beta M_1)$.

Figures (3)

  • Figure 1: Time average error in computing the quantum thermal average for the 1D model potential \ref{['model: 1']}. Left: Matsubara mode PIMD. Right: standard PIMD. Top to bottom: the inverse temperature $\beta = 1,2,4,8$.
  • Figure 2: Correlation functions in the numerical simulation of the 1D potential \ref{['model: 1']}. Left: Matsubara mode PIMD. Right: standard PIMD. Top to bottom: $\beta = 1,2,4,8$. The centroid mode $\xi_0$ is colored in blue, $\xi_1,\xi_2$ are colored in red, and $\xi_3,\xi_4$ are colored in yellow.
  • Figure 3: Probability density of $|q|$ in the simulation of the PIMD. Left: Matsubara mode PIMD. Right: standard PIMD. The top and bottom graphs use different scales.

Theorems & Definitions (25)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 15 more