Generalized quantum master equations (GQMEs) are an important tool inmodeling chemical and physical processes. For a large number of problems it hasbeen shown that exact and approximate quantum dynamics methods can be madedramatically more efficient, and in the latter case more accurate, byproceeding via the GQME formalism. However, there are many situations whereutilizing the GQME approach seems to offer no advantage over a directevaluation of the property of interest. Here we provide a more detailedunderstanding of the conditions under which these methods will offer benefits.In particular, we derive exact expressions for the memory kernel for systemsboth in and out of equilibrium, and show the conditions under which theseexpressions will be guaranteed to return a result identical to that obtainedfrom direct simulation. We also show the conditions which approximate methodsmust satisfy if they are to offer different results when used in conjunctionwith the GQME formalism. These exact analytical results thus provide newinsights as to when proceeding via the GQME approach can improve the accuracyor efficiency of simulations.
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