# Smoothed generalized free energies for thermodynamics

## TL;DR

This paper introduces a new family of smoothed generalized free energies for nanoscale quantum thermodynamics, enabling operational statements for approximate state transitions and converging to standard free energy in large systems.

## Contribution

The authors define explicit smoothing procedures for generalized free energies, allowing for operational interpretation of approximate thermodynamic transitions at the quantum scale.

## Key findings

- Smoothed free energies enable operational statements for approximate state transitions.
- The new smoothed quantities converge to standard free energy in the thermodynamic limit.
- Explicit smoothing procedures are constructed for quantum states within an epsilon-ball.

## Abstract

In the study of thermodynamics for nanoscale quantum systems, a family of quantities known as generalized free energies have been derived as necessary and sufficient conditions that govern state transitions. These free energies become important especially in the regime where the system of interest consists of only a few (quantum) particles. In this work, we introduce a new family of smoothed generalized free energies, by constructing explicit smoothing procedures that maximize/minimize the free energies over an $ \varepsilon$-ball of quantum states. In contrast to previously known smoothed free energies, these quantities now allow us to make an operational statement for approximate thermodynamic state transitions. We show that these newly defined smoothed quantities converge to the standard free energy in the thermodynamic limit.