A quantum master equation of the Lindblad form is obtained in this paper by considering the reduction of wave packets. Different classical equations are derived after exactly mapping such a quantum master equation to a continuous time random walk (CTRW). Although this CTRW is a quasiclassical walk, the effects due to the quantum interference can still be important in such a walk. It is shown that macroscopically we shall consider the uncertainty of the potential and determine the effective kernel (transition probability) of this CTRW by a family of Schr\\"{o}dinger equations (or operators).
It is shown in this paper that the quantum master equation can be mapped to a modified continuous time random walk (CTRW) if the relaxation term is composed of transitions over a set of states. When the Hamiltonian is time-independent and transitions are between its eigenlevels, such a modified CTRW reduces to the Markovian walk equivalent to the Pauli master equation. On the other hand, the memory in such a modified CTRW is composed of a temporal factor and the probability determined by the Liouville flow when the relaxation term is reduced as a special dephasing term.
The continuous time random walk (CTRW) framework (Montroll and Weiss 1965, Montroll and Scher 1973) was described in our earlier Chapters 6 through 9. The CTRW scheme is conceptually appealing and at the same time mathematically flexible to facilitate incorporation of various generalizations. Thus, although CTRW is usually set in the context of a Poisson processdriven pulse sequence leading to stationary Markov systems, incorporation of non-Poissonian pulsing with concomitant waiting-time distributions was seen in Chapter 7 to yield anomalous diffusion.