On the construction of different forms of exact and asymptotic master equations

The general method of construction of exact master equations (MEqs) of required form: non-markovian, time-convolutionless, with partial memory, etc. is presented. The method is based on the transformation of the Liouville-von Neumann equation (LEq) by properly constructed superoperators K and K′ such that K′K = 1. The same method can also be used for the derivation of different asymptotic in time forms of the LEq (asymptotic MEqs). The long-time limits considered here are defined by means of the Gell-Mann-Goldberger procedure. Among others, it is shown that it is possible to cast a general-either exact or asymptotic - MEq into a form in which the homogeneous term (or one of such terms) has the properties of the (irreversible) markovian MEq, the whole exact MEq still being fully equivalent to the (reversible) LEq. The long-time properties of such a MEq as a whole depend, however, also on the asymptotic behaviour of inhomogeneous terms, and these latter must be investigated for every specific physical system separately.