Microscopic Description of Quantum Lorentz Gas by Complex Spectral Representation of the Liouville-von Neumann Equation: A Limitation of the Boltzmann Approximation

TL;DR

This paper analyzes the irreversible dynamics of a one-dimensional quantum Lorentz gas using complex spectral methods, revealing limitations of the Boltzmann approximation in describing certain spatial distribution shifts.

Contribution

It introduces a spectral analysis approach to quantum Lorentz gas dynamics, highlighting the failure of the Boltzmann equation for moderate scales and identifying distinct mechanisms behind distribution shifts.

Findings

Discovered a space-shifting motion in the Wigner distribution not captured by hydrodynamic models.

Identified two mechanisms: one from eigenvalue imaginary parts at small wavenumbers, another from real parts at larger wavenumbers.

Showed the Boltzmann approximation's limitations in describing quantum irreversible processes.

Abstract

Irreversible processes of one-dimensional quantum perfect Lorentz gas is studied on the basis of the fundamental laws of physics in terms of the complex spectral analysis associated with the resonance state of the Liouville-von Neumann operator. A limitation of the usual phenomenological Boltzmann equation is discussed from this dynamical point of view. For a Wigner distribution function that spreads over moderately small scale comparative to the mean-free-path, we found a shifting motion in space of the distribution that cannot be described by the hydrodynamic approximation of the kinetic equation. The mechanism of the shifting has two completely different origins: one is due to different value of the imaginary part of the eigenvalue of the Liouvillian and predominates in moderately small wavenumber associated to the spatial distribution, while the other is due to the existence of the real part of the eigenvalue associated to a wave propagation and predominates in moderately large wavenumber.