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Quantum Monte Carlo in Classical Phase Space with the Wigner-Kirkwood Commutation Function. Results for the Saturation Liquid Density of $^4$He

Phil Attard

TL;DR

This work develops a Metropolis Monte Carlo framework for quantum corrections in classical phase space using the Wigner-Kirkwood commutation function, enabling simulations with complex phase-space weights $e^{W}$ and symmetrization factor $\eta$. It applies a fluctuation-based expansion of $W$ up to third order to Lennard-Jones $^4$He on the saturation curve, demonstrating that including second- and third-order corrections brings the saturated-liquid density from the classical value in line with experimental measurements. The approach preserves the advantages of wavefunction symmetrization and offers a scalable route to quantum effects in large-scale simulations, though symmetrization is not yet implemented in the reported results. Overall, the method provides a practical path to incorporate quantum corrections in classical Monte Carlo without resorting to full path-integral methods, with potential for expansion to larger systems and more complete quantum treatments.

Abstract

A Metropolis Monte Carlo algorithm is given for the case of a complex phase space weight, which applies generally in quantum statistical mechanics. Computer simulations using Lennard-Jones $^4$He near the $λ$-transition, including an expansion to third order of the Wigner-Kirkwood commutation function, give a saturation liquid density in agreement with measured values.

Quantum Monte Carlo in Classical Phase Space with the Wigner-Kirkwood Commutation Function. Results for the Saturation Liquid Density of $^4$He

TL;DR

This work develops a Metropolis Monte Carlo framework for quantum corrections in classical phase space using the Wigner-Kirkwood commutation function, enabling simulations with complex phase-space weights and symmetrization factor . It applies a fluctuation-based expansion of up to third order to Lennard-Jones He on the saturation curve, demonstrating that including second- and third-order corrections brings the saturated-liquid density from the classical value in line with experimental measurements. The approach preserves the advantages of wavefunction symmetrization and offers a scalable route to quantum effects in large-scale simulations, though symmetrization is not yet implemented in the reported results. Overall, the method provides a practical path to incorporate quantum corrections in classical Monte Carlo without resorting to full path-integral methods, with potential for expansion to larger systems and more complete quantum treatments.

Abstract

A Metropolis Monte Carlo algorithm is given for the case of a complex phase space weight, which applies generally in quantum statistical mechanics. Computer simulations using Lennard-Jones He near the -transition, including an expansion to third order of the Wigner-Kirkwood commutation function, give a saturation liquid density in agreement with measured values.

Paper Structure

This paper contains 11 sections, 16 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formalism and Algorithm
  3. Phase Space Weight
  4. Commutation Function
  5. Symmetrization Function
  6. Fluctuation Expansion for the Commutation Function
  7. Temperature Derivatives
  8. Computational Results
  9. Model and Simulation Details
  10. Results
  11. Conclusion

Figures (2)

  • Figure 1: Radial distribution function for Lennard-Jones $^4$He at $k_{\rm B}T/\varepsilon =0.5$ at the respective liquid saturation densities. The dotted curve is $n_W^{\rm max} = 0$ (classical, $\rho\sigma^3=0.9331$), the dashed curve is $n_W^{\rm max} = 2$ ($\rho\sigma^3=0.483(4)$, $q_{\rm min} = 1.1\sigma$), and the dotted curve is $n_W^{\rm max} = 3$ ($\rho\sigma^3=0.2790(2)$, $q_{\rm min} = 1.3\sigma$).
  • Figure 2: Density profile along an axis through the system at $k_{\rm B}T/\varepsilon =0.5$ (dashed curve) and at $k_{\rm B}T/\varepsilon =0.45$ (solid curve) for $n_W^{\rm max}=3$. The statistical error (95% confidence) is about 4% for the former, and about 20% for the latter.