# On work and heat in time-dependent strong coupling

## TL;DR

This paper analyzes the effects of time-dependent strong coupling between a system and a thermal bath, revealing additional forces and correction terms in work and heat functionals relevant to stochastic thermodynamics and quantum entropy changes.

## Contribution

It introduces the impact of time-dependent coupling on work, heat, and forces in system-bath interactions, extending stochastic thermodynamics to strong, dynamic couplings.

## Key findings

- Additional harmonic force due to time-dependent coupling.
- Large correction terms in work and heat during rapid coupling changes.
- Finite contributions to work in Jarzynski's equality despite large corrections.

## Abstract

This paper revisits the classical problem of representing a thermal bath interacting with a system as a large collection of harmonic oscillators initially in thermal equilibrium. As is well known the system then obeys an equation, which in the bulk and in the suitable limit tends to the Kramers-Langevin equation of physical kinetics. I consider time-dependent system-bath coupling and show that this leads to an additional harmonic force acting on the system. When the coupling is switched on and switched off rapidly the force has delta-function support at the initial and final time, an effect that has been observed previously. I further show that the work and heat functionals as recently defined in stochastic thermodynamics at strong coupling contain additional terms depending on the time derivative of the system-bath coupling. I discuss these terms and show that while they can be very large if the system-bath coupling changes quickly they only give a finite contribution to the work that enter in Jarzynski's equality. I also discuss that these corrections to standard work and heat functionals provide an explanation for non-standard terms in the change of the von Neumann entropy of a quantum bath interacting with a quantum system found in an earlier contribution (Aurell \& Eichhorn, 2015).