Rau, J; MÃ¼ller, B, *From reversible quantum microdynamics to irreversible quantum transport*,
Physics Reports, vol. 272 no. 1
(July, 1996),
pp. 1-59 [pdf], [doi] .
**Abstract:**

*The transition from reversible microdynamics to irreversible transport can be studied very efficiently and systematically with the help of the so-called projection method. After a concise introduction to that method we illustrate its power by using it to analyze the well-known rate and quantum Boltzmann equations, and we present, as a new application, the derivation of a source term which accounts for the spontaneous creation of electron-positron pairs in strong fields. Thereby we emphasize the pivotal role of time scales: Only if the various time scales exhibited by the dynamics are widely disparate can the evolution of the slower degrees of freedom be described by a conventional Markovian transport equation; otherwise one must account for finite memory effects. We show how the projection method can be employed to determine these time scales and how - if necessary - it allows one to include memory effects in a straightforward manner. There is also an appendix in which we discuss the concepts of entropy and macroscopic irreversibility.*