Non-Abelian linear Boltzmann equation and quantum correction to Kramers and Smoluchowski equation

A quantum linear Boltzmann equation is proposed, constructed in terms of the operator-valued dynamic structure factor of the macroscopic system the test particle is interacting with. Due to this operator structure it is a non-Abelian linear Boltzmann equation and when expressed through the Wigner function it allows for a direct comparison with the classical one. Considering a Brownian particle the corresponding Fokker-Planck equation is obtained in a most direct way taking the limit of small energy and momentum transfer. A typically quantum correction to the Kramers equation thus appears, describing diffusion in position and further implying a correction to Einstein's diffusion coefficient in the high temperature and friction limit in which Smoluchowski equation emerges.