**Preface to "Fluctuation Relations and Nonequilibrium Thermodynamics in Classical and Quantum Systems"**

Out-of-equilibrium quantum thermodynamics is now establishing itself as a lively and productive area at the intersection of statistical mechanics and quantum information. This success has been spurred on, on one side, by the discovery of classical and quantum fluctuation theorems. On the other side, quantum information theoretic investigations on resource theories and information-powered engines have led to unexpected results. Moreover, advances in experimental quantum technologies have allowed for the demonstrations of thermodynamic devices with small quantum systems.

This Special Issue includes novel results on a diverse range of topics that provide an excellent showcase of the research activities in classical and quantum thermodynamics. In particular, a number of papers present schemes of engines or other thermal devices whose working substance is a small quantum system, e.g. a quantum harmonic oscillator [1], an electron in a quantum dot [2], a transmon qubit [3] and an atomic gas in an optical cavity [4]. Others explore the theory of classical and quantum fluctuation relations [5, 6], including the singular probability distribution of thermodynamic quantities [7]. One contribution investigates the concept of daemonic entropy, arising in schemes of work extraction through generalised quantum measurements [8]. Last but not least, one paper studies the thermalization of many-body systems by a collision model [9]. There are still many open problems, concerning the role of genuine quantum features in thermal devices, the emergence of the laws of thermodynamics from first principles, and the thermalization of closed systems. Moreover, quantum thermodynamics will also play a significant role in the realization of energy-efficient quantum technologies in the near future.

1. Deffner, S. Efficiency of Harmonic Quantum Otto Engines at Maximal Power. Entropy 2018, 20, 875.

2. Pena, F.J.; Negrete, O.; Alvarado Barrios, G.; Zambrano, D.; Gonz ˜ alez, A.; Nunez, A.S.; Orellana, ´ P.A.; Vargas, P. Magnetic Otto Engine for an Electron in a Quantum Dot: Classical and Quantum Approach. Entropy 2019, 21, 512.

3. Cherubim, C.; Brito, F.; Deffner, S. Non-Thermal Quantum Engine in Transmon Qubits. Entropy 2019, 21, 545.

4. Aljaloud, A.; Peyman, S.A.; Beige, A. A Quantum Heat Exchanger for Nanotechnology. Entropy 2020, 22, 379.

5. Holmes, Z.; Hinds Mingo, E.; Chen, C.-R.; Mintert, F. Quantifying Athermality and Quantum Induced Deviations from Classical Fluctuation Relations. Entropy 2020, 22, 111.

6. Santos, J.; Timpanaro, A.; Landi, G. Joint Fluctuation Theorems for Sequential Heat Exchange. Entropy 2020, 22, 763.

7. Corberi, F.; Sarracino, A. Probability Distributions with Singularities. Entropy 2019, 21, 312.

8. Bernards, F.; Kleinmann, M.; Guhne, O.; Paternostro, M. Daemonic Ergotropy: Generalised ¨ Measurements and Multipartite Settings. Entropy 2019, 21, 771.

9. Arısoy, O.; Campbell, S.; Mustecaplıo ¨ glu, ˘ O.E. Thermalization of Finite Many-Body Systems by a ¨ Collision Model. Entropy 2019, 21, 1182.

> **Gabriele De Chiara** *Editor*
