Effective classical potential for quantum statistical averages

TL;DR

This paper introduces an effective potential enabling quantum thermal averages to be estimated via classical ensemble averages, using a mean-field approach that balances accuracy and computational robustness.

Contribution

It develops a new closed-form effective potential based on mean-field quantum fluctuations, improving robustness over full variational methods.

Findings

Exact in classical and harmonic limits

Good agreement with exact distributions for benchmark potentials

Robust numerical performance across different potential types

Abstract

We present an effective potential that allows quantum thermal expectation values of a position-dependent observable to be estimated as a classical ensemble average of the corresponding function. We follow the approach of Feynman and Hibbs, but perform the mean-field treatment of quantum fluctuations about the path starting point rather than the path centroid. Furthermore, rather than performing a full variational optimization of the potential, we explore approximate functional forms that yield a numerical robustness. The resulting closed-form potential is exact in the classical and harmonic limits; benchmarks against exact position distributions for one-dimensional quartic, Morse, and double-well potentials, show good agreement for potentials with harmonic support.