Quantum Boltzmann equation for spin-dependent reactions in the kinetic regime

TL;DR

This paper derives a quantum Boltzmann equation for spin-dependent fermionic interactions, demonstrating conservation laws, the H-theorem, and equilibrium behavior through numerical simulations in three dimensions.

Contribution

It introduces a novel quantum Boltzmann framework for spin-dependent fermionic reactions, including conservation laws and equilibrium analysis from a general quantum Hamiltonian.

Findings

Conservation of density and energy in the derived equation

Validation of the H-theorem for the system

Numerical illustration of approach to equilibrium in 3D

Abstract

We derive and analyze an effective quantum Boltzmann equation in the kinetic regime for the interactions of four distinguishable types of fermionic spin-$\frac{1}{2}$ particles, starting from a general quantum field Hamiltonian. Each particle type is described by a time-dependent, $2 \times 2$ spin-density ("Wigner") matrix. We show that density and energy conservation laws as well as the H-theorem hold, and enumerate additional conservation laws depending on the interaction. The conserved quantities characterize the $t \to \infty$ thermal (Fermi-Dirac) equilibrium state. We illustrate the approach to equilibrium by numerical simulations in the isotropic three-dimensional setting.