  Variations of the bath energy are compared with the information flow in local
dephasing channels. Special correlated initial conditions are prepared from the
thermal equilibrium of the whole system, by performing a selective measurement
on the qubit. The spectral densities under study are ohmic-like at low
frequencies and include logarithmic perturbations of the power-law profiles.
The bath and the correlation energy alternately increase or decrease,
monotonically, over long times, according to the value of the ohmicity
parameter, following logarithmic and power laws. Consider initial conditions
such that the environment is in a thermal state, factorized from the state of
the qubit. In the super-ohmic regime the long-time features of the information
flow are transferred to the bath and correlation energy, by changing the
initial condition from the factorized to the specially correlated, even with
different temperatures. In fact, the low-frequency structures of the spectral
density that provide information backflow with the factorized initial
condition, induce increasing (decreasing) bath (correlation) energy with the
specially correlated initial configuration. By performing the same change of
initial conditions, the spectral properties providing information loss, produce
decrease (increase) of the bath (correlation) energy.
