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arXiv:2607.16301

Gaussian Reformulation of the Feynman Path Integral for Quantum Statistical Mechanics with Results for the Second Virial Coefficient of $^4$He

Phil Attard

cond-mat.stat-mechcond-mat.quant-gasquant-phSubmitted Jul 14, 2026• Announced Jul 15, 2026
arXivHTMLPDF

Abstract

The Feynman path integral for quantum statistical mechanics is reformulated as Gaussian sampling of the neighborhood of each position configuration. The variance and mean are obtained from ring polymer statistics on a lattice, and from the high temperature expansion of the Wigner-Kirkwood commutation function, respectively. The algorithm avoids multiple temperature nodes for each configuration and the need for numerical cancelation in the statistical averages, which are problematic for conventional path integral quantum Monte Carlo. Analytic and simulation results for the second virial coefficient of helium are compared to laboratory measurements.

Comments
12 pages, 2 figures

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Gaussian Reformulation of the Feynman Path Integral for Quantum Statistical Mechanics with Results for the Second Virial Coefficient of 4^44He
The Feynman path integral for quantum statistical mechanics is reformulated as Gaussian sampling of the neighborhood of each position configuration. The variance and mean are obtained from ring polymer statistics on a lattice, and from the high temperature expansion of the Wigner-Kirkwood commutation function, respectively. The algorithm avoids multiple temperature nodes for each configuration and the need for numerical cancelation in the statistical averages, which are problematic for conventional path integral quantum Monte Carlo. Analytic and simulation results for the second virial coefficient of helium are compared to laboratory measurements.
Abstract The focus of Chapter 12 is the Feynman path integral formulations of quantum mechanics and quantum statistical mechanics. The connection between real-time quantum propagator and the density matrix is established via the Wick rotation, and a Trotter expansion is used to derive discrete path integral expressions for both. The continuous limit of the discrete path integral is derived to yield the functional integral formulation of the path integral. The construction of estimators and their averages for computing quantum observables is discussed. Next, the path integral is extended to incorporate Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, and a useful recursive formula for the partition function is derived that accelerates convergence by building up N-body partition functions from those of smaller systems. Quantum free energies are discussed, and algorithms for the evaluation of path integrals using molecular dynamics or Monte Carlo are presented. The chapter concludes with a discussion of higher-order factorizations and ring-polymer contraction methods to reduce the computational overhead of path-integral calculations
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An introduction is given to the methodology of the path-integral formulation of quantum Monte Carlo simulation. This allows us to reformulate a d −dimensional partition function of a quantum mechanical system into the form of a d +1−dimensional path-integral written in classical mechanical variables. In addition, the quantum mechanical partition function is shown to be related by a process of analytical continuation to the time-development operator of the quantum mechanical system. The evaluation of the path-integral representation is discussed for the Maxwell-Boltzmann ideal gas, a chain of atoms, and liquid 4 He . The properties of liquid Helium I and Helium II are discussed in terms of the two-fluid model, the Ginzburg theory, and recent quantum Monte Carlo Simulation studies. The properties of liquid 3 He are discussed in the context of recent analytical p-wave pairing theories which are compared to the theory of s-wave pairing in the theory of superconducting electron.
Quantum Monte Carlo in Classical Phase Space with the Wigner-Kirkwood Commutation Function. Results for the Saturation Liquid Density of 4^44He
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 4^44He near the λλλ-transition, including an expansion to third order of the Wigner-Kirkwood commutation function, give a saturation liquid density in agreement with measured values.