Divergences and long-time tails in two- and three-dimensional quantum Lorentz gases

Abstract

We discuss divergences in the density expansion of the mobility of a quantum-mechanical particle moving in either two and three dimensions through a random array of hard-core scatterers. The most important divergences at low temperatures are of quantum-mechanical origin and involve the ratio of the de Broglie wavelength of the moving particle to its mean free path. When these divergent contributions are resummed they lead to both of the following: (1) logarithmic terms in the virial expansion of the mobility and (2) long-time tails in the integrand of the Green-Kubo formula for the mobility. In $d$ dimensions the long-time tails have the form $-\frac{{\alpha }^{\mathrm{}\left(d\right)\mathrm{}}}{t^{\frac{d}{2}}}$, where ${\alpha }^{\mathrm{}\left(d\right)\mathrm{}}$ is a positive constant. For $d=2\mathrm{}$, the long-time tail is not integrable and the implications of this are discussed. Similar results have been obtained by Abraham et al. For $d=3\mathrm{}$, the long-time tails lead to corrections to the zero-frequency mobility of $O\left({\omega }^{\frac{1}{2}}\right)$, where $\omega$ is the frequency. This result should be detectable in electron-mobility experiments in gases at low temperatures.