Electron random walk in ideal phonon gas. Spectra of density matrix evolution and electron mobility 1/f noise

TL;DR

This paper analyzes the quantum evolution of an electron in a phonon gas, revealing non-Gaussian behavior and 1/f noise in electron mobility, challenging traditional assumptions about electron walk statistics.

Contribution

It provides a detailed spectral analysis of the density matrix evolution equations, showing they imply non-Gaussian, long-range correlated electron walk behavior and 1/f noise in quantum systems.

Findings

Electron's path exhibits super-linear fourth cumulant growth.

Electron mobility shows 1/f-type low-frequency fluctuations.

Gaussian long-range asymptotics are invalid for this system.

Abstract

The previously derived exact evolution equations for density matrix of electron (quantum particle) in phonon field (boson thermostat) are qualitatively analysed. Their statistical interpretation is explained in detail, and their main symmetry and spectral properties are expounded. In application to the electron's random walk, it is shown that these properties certaimly forbid conventionally assumed Gaussian long-range asymptotic of the walk statistics. Instead, the exact equations imply super-linear dependence of fourth-order cumulant of total electron's path on observation time, which signifies existence of 1/f-type low-frequency fluctuations in electro's diffusivity and mobility. Physical meaning of this result is discussed, along with general origin of 1/f-noise in classical and quantum Hamiltonian many-particle systems.