Microscopic derivation of the Boltzmann equation for transport coefficients of resonating fermions at high temperature

TL;DR

This paper provides a microscopic derivation of the Boltzmann equation for transport coefficients of resonating fermions at high temperature, clarifying previous incomplete derivations and revealing nonmonotonic behavior of the Prandtl number.

Contribution

It completes the microscopic derivation of the Boltzmann equation beyond the relaxation-time approximation by resuming higher-order contributions, applicable to resonating fermions in high-temperature regimes.

Findings

Derived the Boltzmann equation from first principles in the high-temperature limit.

Calculated shear viscosity and thermal conductivity for arbitrary scattering lengths.

Discovered nonmonotonic behavior of the Prandtl number near the relaxation-time approximation value.

Abstract

Motivated by the recently observed failure of the kinetic theory for the bulk viscosity, we in turn revisit the shear viscosity and the thermal conductivity of two-component fermions with a zero-range interaction both in two and three dimensions. In particular, we show that their Kubo formula evaluated exactly in the high-temperature limit to the lowest order in fugacity is reduced to the linearized Boltzmann equation. Previously, such a microscopic derivation of the latter was achieved only incompletely corresponding to the relaxation-time approximation. Here, we complete it by resuming all contributions that are naively higher orders in fugacity but become comparable in the zero-frequency limit due to the pinch singularity, leading to a self-consistent equation for a vertex function identical to the linearized Boltzmann equation. We then compute the shear viscosity and the thermal conductivity in the high-temperature limit for an arbitrary scattering length and find that the Prandtl number exhibits a nonmonotonic behavior slightly below the constant value in the relaxation-time approximation.