Hydrodynamics of degenerate Fermi gases on spherical Fermi surfaces
Benjamin Anwasia, Diogo Arsénio
TL;DR
This work develops a rigorous low-temperature hydrodynamic limit for a degenerate Fermi gas on a spherical Fermi surface, derived from the Boltzmann--Fermi--Dirac equation with Pauli blocking. It proves that fluctuations concentrate on the Fermi sphere and, in the macroscopic limit, yield a coupled plasma-wave system for density and velocity, while the dilated energy fluctuations converge to an infinitesimal Fermi--Dirac distribution with a time-invariant energy density. Crucially, the analysis lifts the classical upper bound on the Knudsen number scaling, allowing arbitrarily fast convergence and revealing an anisotropic energy distribution on $|v|=R$ contrary to naive isotropy expectations. The results bridge kinetic theory and compressible fluid-like behavior in a quantum degenerate regime, offering a pathway toward a full Euler-type limit for quantum kinetic equations and enhancing understanding of Langmuir-like plasma waves in metals.
Abstract
We consider the description of a Fermi gas of free electrons given by the Boltzmann--Fermi--Dirac equation, and aim at providing a precise mathematical understanding of the Fermi ground state and its first-order approximation of excited states on the Fermi sphere. In order to achieve that, using the framework of hydrodynamic limits in collisional kinetic theory, we identify the low-temperature regimes in which charge-density fluctuations concentrate on the Fermi sphere. In three spatial dimensions or higher, we also characterize the thermodynamic equilibra and energy spectra of fluctuations. This allows us to derive the macroscopic hydrodynamic equations describing how charge densities flow and propagate in metals, thereby providing a precise description of plasma oscillations in conductors. The two-dimensional case is fundamentally different and is handled in a companion article. Remarkably, our results establish that excited electrons and their energy can be distributed on the Fermi sphere anisotropically, which deviates from the common intuitive assumption that electrons and their energy should be distributed uniformly in all directions. The hydrodynamic regimes featured in this work are akin to the acoustic limit of the classical Boltzmann equation. However, we emphasize that our derivation holds for arbitrarily fast rates of convergence of the Knudsen number, which significantly extends the applicability of the known proofs of the classical acoustic limit. This suggest that low-temperature limits of Fermi gases provide a promising avenue of research toward a complete understanding of the compressible Euler limit.