Stochastic thermodynamics in the strong coupling regime: An unambiguous approach based on coarse graining

We consider a classical and possibly driven composite system X ⊗ Y weakly coupled to a Markovian thermal
<br /><br /><br /><br /><br /><br />reservoir R so that an unambiguous stochastic thermodynamics ensues for X ⊗ Y . This setup can be equivalently
<br /><br /><br /><br /><br /><br />seen as a system X strongly coupled to a non-Markovian reservoir Y ⊗ R. We demonstrate that only in the limit
<br /><br /><br /><br /><br /><br />where the dynamics of Y is much faster than X, our unambiguous expressions for thermodynamic quantities,
<br /><br /><br /><br /><br /><br />such as heat, entropy, or internal energy, are equivalent to the strong coupling expressions recently obtained in
<br /><br /><br /><br /><br /><br />the literature using the Hamiltonian of mean force. By doing so, we also significantly extend these results by
<br /><br /><br /><br /><br /><br />formulating them at the level of instantaneous rates and by allowing for time-dependent couplings between X and
<br /><br /><br /><br /><br /><br />its environment. Away from the limit where Y evolves much faster than X, previous approaches fail to reproduce
<br /><br /><br /><br /><br /><br />the correct results from the original unambiguous formulation, as we illustrate numerically for an underdamped
<br /><br /><br /><br /><br /><br />Brownian particle coupled strongly to a non-Markovian reservoir.