  We study the bath dynamics in the dephasing model of a two-state quantum
system (qubit) coupled to an environment of harmonic oscillators. This model
was shown [Morozov et al., Phys. Rev. A, 2012, 85, 022101] to admit the
analytic solution for the qubit and environment dynamics. Using this solution,
we derive the exact expression for the bath reduced density matrix in the
presence of initial qubit-environment correlations. We obtain the
non-equilibrium phonon distribution function and discuss in detail the time
behavior of the bath energy. It is shown that only the inclusion of dynamic
correlations between the qubit and the bath ensures the proper time behavior of
the quantity which may be interpreted as the "environment energy".
