Do hydrodynamic models misestimate exchange effects? Comparison with kinetic theory for electrostatic waves

TL;DR

This paper compares kinetic theory and hydrodynamic models for exchange effects in electrostatic waves, revealing hydrodynamics approximates well at high phase velocities but underestimates at low velocities, especially for ion-acoustic waves.

Contribution

It extends kinetic results to arbitrary frequencies and wavenumbers, providing a comprehensive comparison with hydrodynamic exchange models in low-temperature plasmas.

Findings

Hydrodynamic models are accurate when phase velocity exceeds 2.5 times the Fermi velocity.

At low phase velocities, exchange corrections are significantly enhanced due to wave-particle interactions.

Hydrodynamic expressions underestimate exchange effects by an order of magnitude for ion-acoustic waves.

Abstract

We have extended previous kinetic results to compute the exchange correction to the electrostatic electron susceptibility for arbitrary frequencies and wavenumbers in the low temperature limit. This has allowed us to make a general comparison with a much used hydrodynamic expression for exchange effects. It is found that the hydrodynamic expression gives a useful approximation when the phase velocity is roughly a factor $2.5$ larger than the Fermi velocity. For low phase velocities, as for ion-acoustic waves, wave-particle interaction leads to a strong enhancement of the exchange correction and the hydrodynamic results is smaller by an order of magnitude.