# Finite-bath corrections to the second law of thermodynamics

## TL;DR

This paper derives finite-bath corrections to the second law of thermodynamics, showing how reversibility is lost with finite heat capacity and connecting these corrections to information theory and single-shot thermodynamics.

## Contribution

It provides a general framework for finite-bath corrections to the second law without specific bath models, linking thermodynamics with information theory.

## Key findings

- Finite baths cause deviations from ideal reversible processes.
- Connections established between thermodynamics and Shannon information theory.
- Finite-bath corrections to min and max free energies in single-shot thermodynamics.

## Abstract

The second law of thermodynamics states that a system in contact with a heat bath can undergo a transformation if and only if its free energy decreases. However, the "if" part of this statement is only true when the effective heat bath is infinite. In this article we remove this idealization and derive corrections to the second law in the case where the bath has a finite size, or equivalently finite heat capacity. This can also be translated to processes lasting a finite time, and we show that thermodynamical reversibility is lost in this regime. We do so in full generality, that is without assuming any particular model for the bath, the only parameters defining the bath are its temperature and heat capacity. We find connections with second order Shannon information theory, in particular in the case of Landauer erasure. We also consider the case of non-fluctuating work, and derive finite-bath corrections to the min and max free energies employed in single-shot thermodynamics.