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arXiv:2512.09948

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

Phil Attard

cond-mat.stat-mechcond-mat.quant-gasphysics.comp-phquant-phSubmitted Dec 8, 2025• Announced Dec 10, 2025• Revised Jan 26, 2026 (v2)
v1 · Dec 8, 2025v2 · Jan 26, 2026
arXivHTMLPDF

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.

Comments
5 pages, 2 figures. Version 2 has corrected heat capacity

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Quantum Monte Carlo in Classical Phase Space with the Wigner-Kirkwood Commutation Function. II. Diagonal Approximation in Position Space

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A third order expansion for Wigner-Kirkwood commutation function, a complex function in classical phase space that accounts for the Heisenberg uncertainty relation, is approximated and integrated over momentum to give a real function in position configuration space. Metropolis Monte Carlo computer simulation results are given for liquid Lennard-Jones $^4$He below 10\,K.

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Power Series for the Quantum Statistical Mechanics Probability with Results for the Second Virial Coefficient of Helium

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A power series for the Wigner-Kirkwood pair commutation function for quantum statistical mechanics in classical phase space is given with terms automatically generated by recursion. The calculated second virial coefficient agrees with the measured values of helium for temperatures greater than 65 K. Prospects for a general quantum Monte Carlo algorithm are discussed.

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

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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.

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Introduction

James E. Gubernatis, Naoki Kawashima, Philipp Werner

A quantum Monte Carlo method is simply a Monte Carlo method applied to a quantum problem. What distinguishes a quantum Monte Carlo method from a classical one is the initial effort necessary to represent the quantum problem in a form that is suitable for Monte Carlo simulation. It is in making this transformation that the quantum nature of the problem asserts itself not only through such obvious issues as the noncommutivity of the physical variables and the need to symmetrize or antisymmetrize the wave function, but also through less obvious issues such as the sign problem. Almost always, the transformation replaces the quantum degrees of freedom by classical ones, and it is to these classical degrees of freedom that the Monte Carlo method is actually applied. Succeeding chapters present and explain many of the quantum Monte Carlo methods being successfully used on a variety of quantum problems. In Chapters 1 and 2 we focus on discussing what the Monte Carlo method is and why it is useful. The Monte Carlo method The Monte Carlo method is not a specific technique but a general strategy for solving problems too complex to solve analytically or too intensive numerically to solve deterministically. Often a specific strategy incorporates several different Monte Carlo techniques. In what is likely the first journal article to use the phrase “Monte Carlo,” Metropolis and Ulam (1949) discuss this strategy. To paraphrase them, The Monte Carlo method is an iterative stochastic procedure, consistent with a defining relation for some function, which allows an estimate of the function without completely determining it. This is quite different from the colloquialism, “a method that uses random numbers.” Let us examine the definition piece by piece. A key point will emerge. Ulam and Metropolis were presenting the motivation and a general description of a statistical approach to the study of differential and integro-differential equations. These equations were their “defining relation for some function.” The “function” was the solution of these equations. This function is of course unknown a priori. Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) a few years later would propose a statistical approach to the study of equilibrium statistical mechanics. The defining relation there was a thermodynamic average of a physical quantity over the Boltzmann distribution. The function was the physical quantity, and the unknown its average.

Cambridge University Press eBooks•Jan 31, 2016•0 citations

Quantum Statistical Mechanics in Classical Phase Space

Phil Attard

Quantum and classical results are often presented as being dependent upon separate postulates as if the two are distinct and unrelated, and there is little attempt to show how the quantum implies the classical. The transformation to classical phase space gives researchers access to a range of algorithms derived from classical statistical mechanics that promise results on much more favourable numerical terms. Quantum Statistical Mechanics in Classical Phase Space offers not just a new computational approach to condensed matter systems, but also a unique conceptual framework for understanding the quantum world and collective molecular behaviour. A formally exact transformation, this revolutionary approach goes beyond the quantum perturbation of classical condensed matter to applications that lie deep in the quantum regime. It offers scalable computational algorithms and tractable approximations tailored to specific systems. Concrete examples serve to validate the general approach and demonstrate new insights. For example, the computer simulations and analysis of the λ-transition in liquid helium provide a new molecular-level explanation of Bose–Einstein condensation and a quantitative theory for superfluid flow. The intriguing classical phase space formulation in this book offers students and researchers a range of new computational algorithms and analytic approaches. It offers not just an efficient computational approach to quantum condensed matter systems, but also an exciting perspective on how the classical world that we observe emerges from the quantum mechanics that govern the behaviour of atoms and molecules. The applications, examples, and physical insights foreshadow new discoveries in quantum condensed matter systems. Key features • A computationally efficient approach to quantum condensed-matter many-particle systems based on a formally exact transformation of quantum statistical mechanics to classical phase space. • Shows how the observed classical world emerges from the underlying quantum mechanics of atoms and molecules. • Describes successive quantum corrections to classical statistical mechanics. • Derives a number of expansions that account for the non-commutativity of quantum position and momentum operators when transformed to classical phase space. • Tests various computer simulation algorithms against benchmark results for model condensed matter systems.

Institute of Physics eBooks•Nov 1, 2021•0 citations
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.
A third order expansion for Wigner-Kirkwood commutation function, a complex function in classical phase space that accounts for the Heisenberg uncertainty relation, is approximated and integrated over momentum to give a real function in position configuration space. Metropolis Monte Carlo computer simulation results are given for liquid Lennard-Jones 4^44He below 10\,K.
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.
A quantum Monte Carlo method is simply a Monte Carlo method applied to a quantum problem. What distinguishes a quantum Monte Carlo method from a classical one is the initial effort necessary to represent the quantum problem in a form that is suitable for Monte Carlo simulation. It is in making this transformation that the quantum nature of the problem asserts itself not only through such obvious issues as the noncommutivity of the physical variables and the need to symmetrize or antisymmetrize the wave function, but also through less obvious issues such as the sign problem. Almost always, the transformation replaces the quantum degrees of freedom by classical ones, and it is to these classical degrees of freedom that the Monte Carlo method is actually applied. Succeeding chapters present and explain many of the quantum Monte Carlo methods being successfully used on a variety of quantum problems. In Chapters 1 and 2 we focus on discussing what the Monte Carlo method is and why it is useful. The Monte Carlo method The Monte Carlo method is not a specific technique but a general strategy for solving problems too complex to solve analytically or too intensive numerically to solve deterministically. Often a specific strategy incorporates several different Monte Carlo techniques. In what is likely the first journal article to use the phrase “Monte Carlo,” Metropolis and Ulam (1949) discuss this strategy. To paraphrase them, The Monte Carlo method is an iterative stochastic procedure, consistent with a defining relation for some function, which allows an estimate of the function without completely determining it. This is quite different from the colloquialism, “a method that uses random numbers.” Let us examine the definition piece by piece. A key point will emerge. Ulam and Metropolis were presenting the motivation and a general description of a statistical approach to the study of differential and integro-differential equations. These equations were their “defining relation for some function.” The “function” was the solution of these equations. This function is of course unknown a priori. Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) a few years later would propose a statistical approach to the study of equilibrium statistical mechanics. The defining relation there was a thermodynamic average of a physical quantity over the Boltzmann distribution. The function was the physical quantity, and the unknown its average.
Quantum and classical results are often presented as being dependent upon separate postulates as if the two are distinct and unrelated, and there is little attempt to show how the quantum implies the classical. The transformation to classical phase space gives researchers access to a range of algorithms derived from classical statistical mechanics that promise results on much more favourable numerical terms. Quantum Statistical Mechanics in Classical Phase Space offers not just a new computational approach to condensed matter systems, but also a unique conceptual framework for understanding the quantum world and collective molecular behaviour. A formally exact transformation, this revolutionary approach goes beyond the quantum perturbation of classical condensed matter to applications that lie deep in the quantum regime. It offers scalable computational algorithms and tractable approximations tailored to specific systems. Concrete examples serve to validate the general approach and demonstrate new insights. For example, the computer simulations and analysis of the λ-transition in liquid helium provide a new molecular-level explanation of Bose–Einstein condensation and a quantitative theory for superfluid flow. The intriguing classical phase space formulation in this book offers students and researchers a range of new computational algorithms and analytic approaches. It offers not just an efficient computational approach to quantum condensed matter systems, but also an exciting perspective on how the classical world that we observe emerges from the quantum mechanics that govern the behaviour of atoms and molecules. The applications, examples, and physical insights foreshadow new discoveries in quantum condensed matter systems. Key features • A computationally efficient approach to quantum condensed-matter many-particle systems based on a formally exact transformation of quantum statistical mechanics to classical phase space. • Shows how the observed classical world emerges from the underlying quantum mechanics of atoms and molecules. • Describes successive quantum corrections to classical statistical mechanics. • Derives a number of expansions that account for the non-commutativity of quantum position and momentum operators when transformed to classical phase space. • Tests various computer simulation algorithms against benchmark results for model condensed matter systems.