Quantum master equations for a fast particle in a gas

The propagation of a fast particle in a low-density gas at thermal equilibrium is studied in the context of quantum mechanics. A quantum master equation in the Redfield form governing the reduced density matrix of the particle is derived explicitly from first principles. Under some approximations, this equation reduces to the linear Boltzmann equation. The issue of the positivity of the time evolution is also discussed by means of a Lindblad form. The Born and Markov assumptions underlying these equations, as well as other approximations regarding the bath correlation function, are discussed in details. Furthermore, all these master equations are shown to be equivalent with each other if the density matrix of the particle is diagonal in the momentum basis, or if the collision rate is independent of the particle momentum.