  Generalized quantum master equations (GQMEs) are an important tool in
modeling chemical and physical processes. For a large number of problems it has
been shown that exact and approximate quantum dynamics methods can be made
dramatically more efficient, and in the latter case more accurate, by
proceeding via the GQME formalism. However, there are many situations where
utilizing the GQME approach seems to offer no advantage over a direct
evaluation of the property of interest. Here we provide a more detailed
understanding of the conditions under which these methods will offer benefits.
In particular, we derive exact expressions for the memory kernel for systems
both in and out of equilibrium, and show the conditions under which these
expressions will be guaranteed to return a result identical to that obtained
from direct simulation. We also show the conditions which approximate methods
must satisfy if they are to offer different results when used in conjunction
with the GQME formalism. These exact analytical results thus provide new
insights as to when proceeding via the GQME approach can improve the accuracy
or efficiency of simulations.
