Diffusion, long-time tails, and localization in classical and quantum Lorentz models: A unifying hydrodynamic approach

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 noninteracting particles in the presence of quenched disorder that are often referred to as Lorentz models. In classical Lorentz models, one manifestation is the long-time tail of the velocity autocorrelation function. In quantum systems, the best-known example is what is known as weak-localization effects in disordered systems of noninteracting electrons. A wide variety of techniques has been used to study these phenomena, including kinetic theory and mode-coupling theories for classical systems, and many-body theory and field-theoretic techniques for quantum mechanical ones. This paper provides a unifying, and very simple, approach to these effects. We show that simple modifications of the diffusion equation due to either a random diffusion coefficient or a random scattering potential account 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.