# Quantum Statistical Mechanics in Classical Phase Space. III. Mean Field Approximation Benchmarked for Interacting Lennard-Jones Particles

## TL;DR

This paper introduces a Monte Carlo simulation method for quantum systems in classical phase space using a mean field approach, validated against Lennard-Jones particle benchmarks, showing good accuracy at certain temperatures and densities.

## Contribution

It presents a novel Monte Carlo algorithm that accounts for quantum effects in classical phase space with a mean field approximation, including wave function symmetrization.

## Key findings

- Quantitative accuracy for Lennard-Jones benchmarks at moderate temperatures and densities.
- Lower energy for bosons compared to fermions at low temperatures.
- Algorithm expected to perform well in higher dimensions and scale sub-linearly with system size.

## Abstract

A Monte Carlo computer simulation algorithm in classical phase space is given for the treatment of quantum systems. The non-commutativity of position and momentum is accounted for by a mean field approach and instantaneous effective harmonic oscillators. Wave function symmetrization is included at the dimer and double dimer level. Quantitative tests are performed against benchmarks given by Hernando and Van\'i\v{c}ek (2013) for spinless neon--parahydrogen, modeled as interacting Lennard-Jones particles in a one dimensional harmonic trap. The mean field approach is shown to be quantitatively accurate for high to moderate temperatures $\beta \hbar \omega_\mathrm{LJ} < 7$, and moderate densities, $\rho \sigma \approx 1$. Results for helium show that at the lowest temperature studied, the average energy is about 4\% lower for bosons than for fermions. It is argued that the mean field algorithm will perform better in three dimensions than in one, and that it will scale sub-linearly with system size.