![]() Outside of the macroscopic classical regime, quantum systems will generically possess coherence or entanglement, and the ordering of states typically displays a much richer structure 20.Ī unique additive entropic function implies that such assumptions must hold 19. In addition a ‘Comparison Hypothesis’ 18, 19 is required to hold (or derived from other axioms), which in itself makes a highly non-trivial assumption on the structure of the thermodynamic partial order. For example, a scaling hypothesis is required, which is no longer valid for small systems. The existence of an essentially unique entropic form of the second law is found to be equivalent to assumptions that fail to hold in small-scale systems or high correlation quantum environments. Of central importance is the partial order of thermodynamic states, from which an entropy function can then be derived in a rigorous manner. However, more rigorous derivations of the entropic form of the second law exist, such as by Carathéodory 17, Giles 18 and more recently by Lieb and Yngvason 19. Alternative approaches follow a statistical mechanical treatment of the system based on underlying microstates, and provide an explanation of the thermodynamics in terms of microscopic degrees of freedom. This thermodynamic entropy function is then assumed (but often not proved) to completely describe the irreversible constraints on the system at hand. The textbook treatments of classical, macroscopic equilibrium thermodynamics are typically based on notions such as Carnot cycles, with the entropy function generically defined via an integral in terms of heat flow 16. It is increasingly apparent that the traditional entropic formulation that emerges as an essentially unique description of the irreversibility of classical, macroscopic systems, will only place necessary, but not sufficient, constraints on the physics of small-scale systems manifesting coherence or quantum correlations. As such, a crucial question is: to what degree do traditional thermodynamic formulations and techniques encapsulate this regime? This is a broad, foundational question about thermodynamics. The physics of these remarkable small-scale systems, displaying coherence or entanglement, constitute extreme quantum regimes. In a similar way, the phenomenon of thermality due to entanglement and the thermodynamics of area laws reveal deep connections between thermodynamics and the theory of entanglement 14, 15. Within quantum information science, the question of thermodynamically robust quantum memories, and thermodynamic constraints on quantum computation are still only partially understood and provide deep questions in the overlap between thermodynamics and quantum theory 12, 13. Conversely, dissipative quantum thermodynamics offers the possibility of on-demand generation of quantum information resources essential for future quantum technologies (communication, encryption, metrology and computing) 11. Such coherence has been shown to play important roles in thermal to electrical power conversion, heat dissipation in atomic-scale junctions and the engineering toolkit of quantum dots 10. Electrical conductance of molecular-scale components no longer obey Kirchhoff's laws and phase coherence can provide both destructive as well as constructive interference effects on electrical transport 9. ![]() Towards the lower-end of the nanoscale, quantum mechanical effects such as quantum coherence and entanglement increasingly make their presence felt. There is also increasing evidence for the role of quantum effects within biological systems 6, 7, 8. With operating scales between 1 and 10 2 nm, there has been remarkable progress in the development of molecular information ratchets, molecular motors, optical thermal ratchets and artificial bipedal nanowalkers inspired by naturally occurring biomolecular walkers 1, 2, 3, 4, 5. This has led to an explosion of work in the field of nanotechnology, with a myriad of applications to areas in industry, information technology, medicine and energy technologies. ![]() We are increasingly able to probe and manipulate the physics of micro- and nanoscale systems.
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