The Linear Boltzmann Equation as the Low Density Limit of a Random Schrodinger Equation

TL;DR

This paper demonstrates that the quantum evolution of a particle in a random potential converges to a linear Boltzmann equation in the low density limit, linking quantum scattering to classical kinetic theory.

Contribution

It establishes the weak convergence of the quantum phase space density to a classical Boltzmann equation with a collision kernel derived from quantum scattering cross sections.

Findings

Quantum particle evolution converges to a linear Boltzmann equation.

Collision kernel is explicitly given by the quantum scattering cross section.

Results connect quantum scattering theory with classical kinetic equations.

Abstract

We study the evolution of a quantum particle interacting with a random potential in the low density limit (Boltzmann-Grad). The phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation with collision kernel given by the full quantum scattering cross section.