Semiclassical approximation of the Wigner function for the canonical ensemble

TL;DR

This paper introduces a semiclassical method to approximate the Wigner function of the canonical ensemble using classical trajectories, enabling efficient computation of quantum statistical properties across various temperatures.

Contribution

It develops a novel semiclassical approximation for the Wigner function in the canonical ensemble applicable at all temperatures, with a numerical scheme for broad system classes.

Findings

Thermodynamic averages are accurately reproduced for systems with one and two degrees of freedom.

The approximation performs well over a range of parameters.

The method bridges classical and quantum descriptions in phase space.

Abstract

The Weyl-Wigner representation of quantum mechanics allows one to map the density operator in a function in phase space - the Wigner function - which acts like a probability distribution. In the context of statistical mechanics, this mapping makes the transition from the classical to the quantum regimes very clear, because the thermal Wigner function tends to the Boltzmann distribution in the high temperature limit. We approximate this quantum phase space representation of the canonical density operator for general temperatures in terms of classical trajectories, which are obtained through a Wick rotation of the semiclassical approximation for the Weyl propagator. A numerical scheme which allows us to apply the approximation for a broad class of systems is also developed. The approximation is assessed by testing it against systems with one and two degrees of freedom, which shows that, for a considerable range of parameters, the thermodynamic averages are well reproduced.