The crossover from classical to quantum transport in a weakly-interacting Fermi gas

TL;DR

This work tackles the crossover from quantum degenerate Fermi-liquid to classical Boltzmann gas by solving the linearized quantum Boltzmann equation for a weakly interacting Fermi gas. The authors develop an exact, non-variational approach based on angular-momentum–specific orthogonal polynomials (Q_n^l) to decompose the phase-space distribution, enabling fast, systematically improvable calculations of transport coefficients. They obtain leading-order expressions for shear viscosity $\eta$, thermal diffusivity $\kappa$, and spin diffusivity $D$ in terms of the scattering length $a$ and collision dynamics, and show substantial failures of the relaxation-time approximation at low temperature. The method provides a numerically efficient benchmark for transport in strongly correlated regimes and a path to simulate kinetics beyond hydrodynamics, including time-dependent and nonlinear effects.

Abstract

We present an exact solution of the quantum kinetic equation of a weakly interacting Fermi gas in the crossover from the degenerate Fermi-liquid regime to the classical Boltzmann gas. We construct families of orthogonal polynomials tailored to each angular momentum channel, enabling a fast and systematically improvable decomposition of the phase-space distribution. This approach yields accurate, non-variational predictions for the shear viscosity, thermal diffusivity, and spin diffusivity to leading order in the scattering length. We demonstrate that the commonly used relaxation-time approximation fails dramatically at low temperature--by up to 25%. Our method provides a numerically efficient framework for benchmarking transport in strongly correlated regimes and for simulating the kinetics of quantum gases beyond hydrodynamics.

Figures (4)

• Figure 1: The reduced mean collision time $\bar{a}^2 \tau_{\rm m}/t_{\rm F}$ (with $\bar{a}=k_{\rm F} a$) as function of $T/T_{\rm F}$ at fixed density. It decreases monotonically first as $\tau_{\rm m}\sim (3/4\pi) (T_{\rm F}/T)^2 t_{\rm F}/\bar{a}^2$ in the low temperature (Fermi liquid) regime (blue dashed curve), then slower, as the collision time $\tau_{\rm m}\sim 1/\sqrt{2} v_{\rm m}\rho_\uparrow \sigma$ of a hard sphere Boltzmann gas of crosssection $\sigma=4\pi a^2$, mean velocity $v_{\rm m}=\sqrt{8T/\pi m}$ and only $\uparrow\downarrow$ collisions (corresponding to $\bar{a}^2 \tau_{\rm m}/t_{\rm F} \sim (3\pi^2/2)\sqrt{\pi T_{\rm F}/2T}$ in the Fermi units, red dashed curve). Data for Figs. \ref{['figtaum']}--\ref{['figeta']}--\ref{['figkappa']}--\ref{['figD']} available in vtemp.
• Figure 2: (Top pannel) The reduced viscosity $\bar{\eta}\equiv\bar{a}^2 \eta/\rho_{\rm eq}$ (with $\rho_{\rm eq}=k_{\rm F}^3/3\pi^2$ the total density and $\bar{a}=k_{\rm F} a$) as a function of $\theta=T/T_{\rm F}$ at fixed density. The limiting behaviors are $\bar{\eta}\underset{\theta\to0}{\sim} (2\pi/5)(\tau_\eta/\tau)/\theta^2\approx 0.1288/\theta^2$ (blue dashed curve) and $\bar{\eta}\underset{\theta\to+\infty}{\sim} (3\pi^{3/2}/8)(\tau_\eta/\tau_{\rm HT}) \sqrt{\theta}\approx 1.875\sqrt{\theta}$ (red dashed curve). (Bottom) Ratio of the exact value to the first approximation $\eta/\eta_0=\tau_\eta/\tau_{\eta}^0$ (see Eq. \ref{['premiereapprox']}), showing the sharp deviation from unity at low temperature.
• Figure 3: (Top pannel) The reduced thermal conductivity $\bar{\kappa} \equiv m\bar{a}^2 \kappa/\rho_{\rm eq}$ as a function of $\theta=T/T_{\rm F}$ at fixed density. The limiting behaviors are $\bar{\kappa}\underset{\theta\to0}{\sim} (\pi^3/3)(\tau_\kappa/\tau)/\theta\approx 0.616/\theta$ (blue dashed curve) and $\bar{\kappa}\underset{\theta\to+\infty}{\sim} (15\pi^{3/2}/16)(\tau_\kappa/\tau_{\rm HT}) \sqrt{\theta}\approx 7.09\sqrt{\theta}$. (Bottom) Ratio of the exact value to the first approximation $\kappa/\kappa_0=\tau_\kappa/\tau_{\kappa}^0$ (see Eq. \ref{['premiereapprox']}).
• Figure 4: (Top pannel) The reduced spin diffusivity $\bar{D} \equiv m\bar{a}^2 D/2$ as a function of $\theta=T/T_{\rm F}$ at fixed density. The limiting behaviors are $\bar{D} \underset{\theta\to0}{\sim} (\pi/3)(\tau_D/\tau)/\theta^2\approx 0.067/\theta^2$ (blue dashed curve) and $\bar{D} \underset{\theta\to+\infty}{\sim} (3\pi^{3/2}/16)(\tau_D/\tau_{\rm HT}) \sqrt{\theta}\approx 0.57\sqrt{\theta}$ (red dashed curve). (Bottom) Ratio of the exact value to the first approximation $D/D_0=\tau_D/\tau_{D}^0$ (see Eq. \ref{['premiereapprox']}).