  We investigate an energy flow in an extended Holstein model describing
electron systems coupled to hot-phonons and heat-bath phonons. To analyze the
relaxation process after the photo-excitation of electrons, we employ the
nonequilibrium dynamical mean-field theory (DMFT). We find the backward energy
flow during the relaxation, where the direction of energy transfer between
electrons and hot-phonons is reversed. To clarify the microscopic mechanism of
the backward energy flow, we introduce the approximated energy flows, which are
calculated with the gradient and quasiparticle approximations and are related
to the nonequilibrium distribution functions. We compare these approximated
energy flows with the full energy flows calculated from the nonequilibrium
DMFT. We find that, in the weak electron-hot-phonon coupling regime, the full
and approximated energy flows are almost the same, meaning that the relaxation
dynamics can be correctly understood in terms of the nonequilibrium
distribution functions. As the strength of the electron-hot-phonon coupling
increases, the approximated energy flow fails to qualitatively reproduce the
full energy flow. This indicates that the microscopic origin of the energy flow
cannot be solely explained by the nonequilibrium distribution functions. By
comparing the energy flows with different levels of approximation, we reveal
the role of the gradient and quasiparticle approximations.
