# Kinetic theory of transport for inhomogeneous electron fluids

## TL;DR

This paper develops a kinetic theory approach to electron transport in inhomogeneous metals, accounting for interactions and disorder, and explains various resistivity behaviors in different regimes.

## Contribution

It introduces a unified solution to inhomogeneous Boltzmann equations that incorporates electron-electron interactions, revealing new transport mechanisms and explaining resistivity phenomena.

## Key findings

- Interactions can both increase and decrease momentum relaxation rates.
- Resistivity can be proportional to momentum-conserving collision rates.
- The theory explains T^2 and linear-in-T resistivity behaviors in metals.

## Abstract

The interplay between electronic interactions and disorder is neglected in the conventional Boltzmann theory of transport, yet can play an essential role in determining the resistivity of unconventional metals. When quasiparticles are long-lived, one can account for these intertwined effects by solving spatially inhomogeneous Boltzmann equations. Assuming smooth disorder and neglecting umklapp scattering, we solve these inhomogeneous kinetic equations and compute the electrical resistivity across the ballistic-to-hydrodynamic transition. An important consequence of electron-electron interactions is the modification of the momentum relaxation time; this effect is ignored in the conventional theory. We characterize precisely when interactions enhance the momentum scattering rate, and when they decrease it. Our approach unifies existing semiclassical theories of transport and reveals novel transport mechanisms. In particular, we explain how the resistivity can be proportional to the rate of momentum-conserving collisions. We compare this result with existing transport mysteries, including the disorder-independent $T^2$ resistivity of many Fermi liquids, and the linear-in-$T$ "Planckian-limited" resistivity of many strange metals.