What has not yet been determined from this analysis, however, is whether the proposed relaxation method in Eq. A27a, Eq. A27b, and Eq. A28 corresponds to the effective time averaged Liouvillian in Eq. A24 or not. As will be shown in the next section, the proposed relaxation stage does not correspond to this Liouvillian, and the γ R decay rate must be modified in order to satisfy Eq. A26.