Fermi-Dirac Wigner function for massive spin-1/2 particles in local equilibrium

TL;DR

This paper extends the Fermi-Dirac Wigner function for massive spin-1/2 particles to ensure proper normalization and thermodynamic consistency, linking microscopic quantum states to macroscopic hydrodynamic descriptions.

Contribution

It generalizes the local equilibrium Wigner function to Fermi-Dirac statistics, ensuring correct polarization and thermodynamic relations, and reveals its divergence-type structure.

Findings

Correct normalization of mean polarization vector

Reproduction of generalized thermodynamic relations with spin

Macroscopic currents derived from a generating function

Abstract

A recently proposed Boltzmann local equilibrium Wigner function for massive spin-1/2 particles is generalized to the case of Fermi-Dirac statistics. The resulting formula ensures the correct normalization of the mean polarization vector and reproduces the generalized thermodynamic relations with spin that were obtained in earlier studies. Moreover, we show that the macroscopic currents constructed from the Fermi-Dirac Wigner function can be obtained as derivatives of a suitably defined generating function with respect to the Lagrange multipliers (temperature, hydrodynamic flow, and chemical potentials). The identified generating function also indicates that the underlying framework can be classified as a divergence-type theory.