Wigner distribution functions for complex dynamical systems: the emergence of the Wigner-Boltzmann equation

TL;DR

This paper derives an equation of motion for the reduced Wigner function in complex quantum systems coupled to harmonic oscillators, connecting influence functional theory with the Wigner-Boltzmann equation under specific approximations.

Contribution

It introduces a derivation of the Wigner-Boltzmann equation from influence functional theory for systems coupled to harmonic oscillators, linking self-energy and influence functional.

Findings

Derived explicit expressions for the Wigner function's equation of motion.

Connected the influence functional with the self-energy in the Wigner framework.

Showed conditions under which the equation simplifies to the Wigner-Boltzmann form.

Abstract

The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.