Effective classical potential for quantum statistical averages

TL;DR

This work develops a local-starting-point quantum effective potential that enables quantum thermal averages of position-dependent observables to be estimated as classical ensemble averages. Grounded in path-integral theory, it builds on the variational framework and contrasts centroid-based potentials (Feynman-Hibbs, Feynman-Kleinert) with a new starting-point, mean-field approach that expands around the path start $x'$ to yield a closed-form LH potential. The key result is the LH-based density and potential, $\rho^{\text{LH}}_{\beta}(x')$ and $V_{\text{LH}}(x')= V(x')\Xi(x') - \tfrac{1}{2\beta}\ln\Xi(x')$ with $\Xi(x')=(\tanh \xi_a)/\xi_a$, $\xi_a=\beta\hbar\omega_a/2$, which, when renormalized and harmonically mapped, provides accurate quantum PDFs for harmonic-support anharmonic potentials and benchmarks on quartic, Morse, and double-well systems. The approach recovers exact classical and harmonic limits and offers a computationally efficient route to quantum statistics, with potential extensions to higher dimensions and machine-learning models of thermal density distributions. Limitations arise in strongly tunneling-dominated regimes (e.g., deep or shallow double wells at low temperature), suggesting future work in variational optimization of LH parameters and dynamical extensions (e.g., LSC-IVR/Wigner-based dynamics). Overall, the method provides a robust alternative to centroid-based potentials for estimating quantum statistical properties with direct observables calculable from the starting-point distribution.

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.

Figures (3)

• Figure 1: Comparison of the classical (green), quantum (red), Feynman-Hibbs (purple), Feynman-Kleinert (cyan) and local harmonic (dark blue) position distribution functions (from \ref{['PLH_final']}) at different temperatures for the quartic potential potential $V(x) = m\omega^2x^2/2 + gx^4/4$ in \ref{['Vquartic']} at different values of quartic perturbation strength $g$. The top row contains plots of the respective potential energy surfaces along with the reference harmonic oscillator of frequency $\omega=1.0$ (indicated in grey dashed line). Note that, for the purely harmonic oscillator with $g=0.0$, the Feynman-Kleinert and Feynman-Hibbs PDFs are hidden behind the classical PDF and the local harmonic PDF is hidden behind the quantum PDF.
• Figure 2: Comparison of the classical (green), quantum (red) and local harmonic (dark blue) position distribution functions (from \ref{['PLH_final']}) for the morse potential \ref{['Morsepot']} with parameters as in rossi2014remove at different temperatures. The dotted black lines correspond to the 'bare' PDF obtained from \ref{['Veff_prelim']} without the transformation in \ref{['magic']} (For T=1K, the unphysical divergence of the effective potential \ref{['Veff_prelim']} was too serious to result in a PDF).
• Figure 3: Same as in fig. \ref{['fig:morse_trpmd']} but for the double-well potential $-m\omega^2x^2/2 + gx^4/4 + {m\omega^4}/ {16g}$ as in \ref{['DWPES']} with (a) $g=0.1$ and (b) $g=0.5$ at different temperatures. The top row displays the potential energy surface indicating the well depths for both these parameter regimes.