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

We derive and analyze an effective quantum Boltzmann equation in the kinetic regime for the interactions of four distinguishable types of fermionic spin-1/2 particles, starting from a general quantum field Hamiltonian. Each particle type is described by a time-dependent, 2 × 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 → ∞ thermal (FermiDirac) equilibrium state. We illustrate the approach to equilibrium by numerical simulations in the isotropic three-dimensional setting.