# Third Law of Thermodynamics as a Single Inequality

## TL;DR

This paper derives a unified criterion for approximate cooling in thermodynamics, linking non-equilibrium resources to the third law, and introduces new mathematical insights into Renyi-divergences.

## Contribution

It establishes a single necessary and sufficient condition for cooling using non-equilibrium resources, generalizing the third law of thermodynamics.

## Key findings

- Derived a single criterion for approximate cooling.
- Connected the third law to a function similar to free energy.
- Provided new results on Renyi-divergences' convexity/concavity.

## Abstract

The third law of thermodynamics in the form of the unattainability principle states that exact ground-state cooling requires infinite resources. Here we investigate the amount of non-equilibrium resources needed for approximate cooling. We consider as resource any system out of equilibrium, allowing for resources beyond the i.i.d. assumption and including the input of work as a particular case. We establish in full generality a sufficient and a necessary condition for cooling and show that for a vast class of non-equilibrium resources these two conditions coincide, providing a single necessary and sufficient criterion. Such conditions are expressed in terms of a single function playing a similar role for the third law to the one of the free energy for the second law. From a technical point of view we provide new results about concavity/convexity of certain Renyi-divergences, which might be of independent interest.