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

## TL;DR

This paper develops an unambiguous stochastic thermodynamics framework for strongly coupled systems by coarse-graining, valid when the auxiliary system evolves faster than the primary system, extending previous results to time-dependent couplings.

## Contribution

It introduces a coarse-graining approach that unambiguously defines thermodynamic quantities in strongly coupled systems, including time-dependent interactions and instantaneous rates.

## Key findings

- The framework matches strong coupling results in the fast auxiliary limit.
- Numerical examples show previous approaches fail outside the fast auxiliary limit.
- The method extends thermodynamic analysis to non-Markovian reservoirs with time-dependent couplings.

## Abstract

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