Diffusion, Long-Time Tails, and Localization in Classical and Quantum Lorentz Models: A Unifying Hydrodynamic Approach

TL;DR

This paper presents a unified, simple approach to understanding long-time tails in classical and quantum Lorentz models by modifying the diffusion equation to account for disorder, explaining decay exponents and prefactors.

Contribution

It introduces a unifying hydrodynamic framework that explains long-time tails in both classical and quantum Lorentz models through simple diffusion equation modifications.

Findings

Accounts for decay exponents of long-time tails

Predicts prefactors of velocity autocorrelation functions

Applicable to both classical and quantum disordered systems

Abstract

Long-time tails, or algebraic decay of time-correlation functions, have long been known to exist both in many-body systems and in models of non-interacting particles in the presence of quenched disorder that are often referred to as Lorentz models. In the latter, they have been studied extensively by a wide variety of methods, the best known example being what is known as weak-localization effects in disordered systems of non-interacting electrons. This paper provides a unifying, and very simple, approach to all of these effects. We show that simple modifications of the diffusion equation due to either a random diffusion coefficient, or a random scattering potential, accounts for both the decay exponents and the prefactors of the leading long-time tails in the velocity autocorrelation functions of both classical and quantum Lorentz models.