Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field

TL;DR

This paper demonstrates that the long-term behavior of a quantum electron weakly coupled to phonons can be described by the linear Boltzmann equation, linking quantum dynamics with classical kinetic theory.

Contribution

It establishes the convergence of the electron's Wigner distribution to the linear Boltzmann equation in the weak coupling limit over infinite time.

Findings

Wigner distribution converges to the Boltzmann solution

Collision kernel includes emission and absorption terms

Results hold globally in time

Abstract

We consider the long time evolution of a quantum particle weakly interacting with a phonon field. We show that in the weak coupling limit the Wigner distribution of the electron density matrix converges to the solution of the linear Boltzmann equation globally in time. The collision kernel is identified as the sum of an emission and an absorption term that depend on the equilibrium distribution of the free phonon modes.