  In the previous paper [C. W. Kim and I. Franco, J. Chem. Phys. 160, 214111
(2024)], we developed a theory called MQME-D, which allows us to decompose the
overall energy dissipation process in open quantum system dynamics into
contributions by individual components of the bath when the subsystem dynamics
is governed by a Markovian quantum master equation (MQME). Here, we contrast
the predictions of MQME-D against the numerically exact results obtained by
combining hierarchical equations of motion (HEOM) with a recently reported
protocol for monitoring the statistics of the bath. Overall, MQME-D accurately
captures the contributions of specific bath components to the overall
dissipation while greatly reducing the computational cost as compared to exact
computations using HEOM. The computations show that MQME-D exhibits errors
originating from its inherent Markov approximation. We demonstrate that its
accuracy can be significantly increased by incorporating non-Markovianity by
exploiting time scale separations (TSS) in different components of the bath.
Our work demonstrates that MQME-D combined with TSS can be reliably used to
understanding how energy is dissipated in realistic open quantum system
dynamics.
