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 detail. 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.

Automated summary

This study explores the propagation of fast particles in a low-density gas at thermal equilibrium within the framework of quantum mechanics. A quantum master equation in the Redfield form, governing the reduced density matrix of the particle, is derived from first principles. The linearity of the Boltzmann equation is obtained under certain approximations. The article examines the positivity of time evolution using a Lindblad form and discusses the Born and Markov assumptions as well as other approximations related to the bath correlation function. Moreover, it demonstrates the equivalence of these master equations when the density matrix of the particle is diagonal in the momentum basis or if the collision rate is independent of the particle momentum.