As recently demonstrated [1], the entropy production in open systems is a result of generation of both the correlations between the system and the bath and the intra-bath correlations; the latter contribution predominates when the system is driven far from equilibrium.
Here we investigate the behavior of these contributions to the entropy production during thermalization of a single resonant level coupled to a fermionic bath using the correlation matrix method [2]. This approach provides a numerically exact description of dynamics and thermodynamics of both the system and the bath.
Applying the tools of the quantum information theory, we demonstrate that after the initial built-up of the correlations between the system and the bath they are later reconverted into the correlations between the energy levels within the bath, as indicated, e.g., by change of its von Neumann entropy. Interestingly, the rearrangement of correlations takes place at the time scales longer than the relaxation time of the system; therefore, we refer to this phenomenon as the post-thermalization.
Surprisingly, this phenomenon is observed also for a weak coupling to the bath, when the reduced dynamics of the system is effectively Markovian; this is contrary to the expectation that in the weak coupling regime the bath decorrelates from the state of the system much faster than the relaxation time. We also show, that the post-thermalization process can be suppressed by the presence of the system-environment bound state or the system-bath Coulomb interaction. Our results demonstrate the non-trivial nature of the dynamics of the bath during the relaxation process.

[1] K. Ptaszynski and M. Esposito, Phys. Rev. Lett. 123, 200603 (2019)
[2] I. Peschel, J. Phys. A: Math. Gen. 36, L205 (2003); A. Sharma and E. Rabani, Phys. Rev. B 91, 085121 (2015)