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arXiv:2608.21451

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

Phil Attard

cond-mat.stat-mechcond-mat.quant-gasphysics.comp-phquant-phSubmitted Aug 20, 2026• Announced Aug 21, 2026
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Abstract

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.

Comments
8 pages, 2 figures, 1 idea

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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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Quantum Monte Carlo in Classical Phase Space with the Wigner-Kirkwood Commutation Function. Results for the Saturation Liquid Density of 4^44He
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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 4^44He below 10\,K.
An expansion for quantum statistical mechanics is derived that gives classical statistical mechanics as the leading term. Each quantum correction comes from successively larger permutation loops, which arise from the factorization of the symmetrization of the wave function with respect to localized particle interchange. Explicit application of the theory yields the full fugacity expansion for the quantum ideal gas, and the second fugacity coefficient for interacting quantum particles, which agree with known results. Compared to the Lee-Yang virial cluster expansion, the present expansion is expected to be more rapidly converging and the individual terms appear to be simpler to evaluate. The results obtained in this paper are intended for practical computer simulation algorithms for terrestrial condensed matter quantum systems.
In a previous article \cite{kn:anirban1} a method has been introduced to derive the all order Bose-Einstein distribution of the non interacting Bosons as the solution of the Wigner equation. The process was a perturbative one where the Bose-Einstein distribution was taken as the unperturbed solution. In this article it is shown that the same formalism is also applicable in the case of interacting Bosons. The formalism has been applied to calculate the quantum second virial coefficient of the Bosons interacting pairwise via Lenard-Jones potential and compared with the previous result.