# Free energy for non-equilibrium quasi-stationary states

## TL;DR

This paper explores the concept of free energy in non-equilibrium quasi-stationary states of Markov systems interacting with two thermal baths, identifying conditions for its existence and properties.

## Contribution

It introduces conditions under which a free energy can be defined in non-equilibrium systems, extending thermodynamic concepts beyond equilibrium.

## Key findings

- Free energy exists under specific conditions like time-scale separation and partial controllability.
- Non-equilibrium free energy can increase with temperature, unlike in equilibrium.
- External fields can more efficiently reduce entropy at higher temperatures.

## Abstract

We study a class of non-equilibrium quasi-stationary states for a Markov system interacting with two different thermal baths. We show that the work done under a slow, external change of parameters admits a potential, i.e., the free energy. Three conditions are needed for the existence of free energy in this non-equilibrium system: time-scale separation between variables of the system, partial controllability (external fields couple only with the slow variable), and an effective detailed balance. These conditions are facilitated in the continuous limit for the slow variable. In contrast to its equilibrium counterpart, the non-equilibrium free energy can increase with temperature. One example of this is that entropy reduction by means of external fields (cooling) can be easier (in the sense of the work cost) if it starts from a higher temperature.