Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when one of the heat baths it operates between is finite. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.
One immediate application of our theoretical results is to the study of irreversible processes in the nanoscale regime. In particular, we show how the amount of ordered energy needed to drive a small system out of equilibrium is larger than the amount one could obtain in a reverse process. This affects reversibility of thermodynamic cycles and, in turn, deteriorates performance of nanoengines. Despite these negative finite-size effects, we find that in specially engineered conditions nanoscale engines can still achieve the ultimate limit of efficiency.
Our results expand the realm of applicability of thermodynamics beyond the constraint of macroscopic systems, and thus provide new tools to study the universe at the smallest scale.
 Robin Giles, Mathematical Foundations of Thermodynamics (Pergamon Press, 1964).
 I. Marvian and R. W. Spekkens, ``The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations,'' New J. Phys. 15, 033001 (2013).
 Iman Marvian, Symmetry, Asymmetry and Quantum Information, Ph.D. thesis, University of Waterloo (2012).
 D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and Th. Beth, ``Thermodynamic cost of reliability and low temperatures: tightening Landauer's principle and the second law,'' Int. J. Theor. Phys. 39, 2717-2753 (2000).
 Fernando G. S. L. Brandão, Michał Horodecki, Jonathan Oppenheim, Joseph M. Renes, and Robert W. Spekkens, ``Resource theory of quantum states out of thermal equilibrium,'' Phys. Rev. Lett. 111, 250404 (2013).
 Matteo Lostaglio, Kamil Korzekwa, David Jennings, and Terry Rudolph, ``Quantum coherence, time-translation symmetry, and thermodynamics,'' Phys. Rev. X 5, 021001 (2015a).
 Piotr Ć wikli ński, Michał Studziński, Michał Horodecki, and Jonathan Oppenheim, ``Limitations on the evolution of quantum coherences: towards fully quantum second laws of thermodynamics,'' Phys. Rev. Lett. 115, 210403 (2015).
 Gilad Gour, David Jennings, Francesco Buscemi, Runyao Duan, and Iman Marvian, ``Quantum majorization and a complete set of entropic conditions for quantum thermodynamics,'' arXiv:1708.04302 (2017).
 Charles H Bennett, Herbert J Bernstein, Sandu Popescu, and Benjamin Schumacher, ``Concentrating partial entanglement by local operations,'' Phys. Rev. A 53, 2046 (1996).
 Wataru Kumagai and Masahito Hayashi, ``Second-order asymptotics of conversions of distributions and entangled states based on Rayleigh-normal probability distributions,'' IEEE Trans. Inf. Theory 63, 1829-1857 (2017).
 Marco Tomamichel and Masahito Hayashi, ``A hierarchy of information quantities for finite block length analysis of quantum tasks,'' IEEE Trans. Inf. Theory 59, 7693-7710 (2013).
 Marco Tomamichel and Vincent Y. F. Tan, ``Second-order asymptotics for the classical capacity of image-additive quantum channels,'' Commun. Math. Phys. 338, 103-137 (2015).
 Nilanjana Datta, Marco Tomamichel, and Mark M. Wilde, ``On the second-order asymptotics for entanglement-assisted communication,'' Quantum Inf. Process. 15, 2569-2591 (2016).
 Nilanjana Datta and Felix Leditzky, ``Second-order asymptotics for source coding, dense coding, and pure-state entanglement conversions,'' IEEE Trans. Inf. Theory 61, 582-608 (2015).
 Mark M. Wilde, Joseph M. Renes, and Saikat Guha, ``Second-order coding rates for pure-loss bosonic channels,'' Quantum Inf. Process. 15, 1289-1308 (2016).
 Mark M. Wilde, Marco Tomamichel, Seth Lloyd, and Mario Berta, ``Gaussian hypothesis testing and quantum illumination,'' Phys. Rev. Lett. 119, 120501 (2017).
 John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk, ``The role of quantum information in thermodynamics - a topical review,'' J. Phys. A 49, 143001 (2016).
 Matteo Lostaglio, David Jennings, and Terry Rudolph, ``Description of quantum coherence in thermodynamic processes requires constraints beyond free energy,'' Nat. Commun. 6, 6383 (2015b).
 Raam Uzdin, Amikam Levy, and Ronnie Kosloff, ``Equivalence of quantum heat machines, and quantum-thermodynamic signatures,'' Phys. Rev. X 5, 031044 (2015).
 Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, and David Jennings, ``The extraction of work from quantum coherence,'' New J. Phys. 18, 023045 (2016).
 Martí Perarnau-Llobet, Karen V Hovhannisyan, Marcus Huber, Paul Skrzypczyk, Nicolas Brunner, and Antonio Acín, ``Extractable work from correlations,'' Phys. Rev. X 5, 041011 (2015).
 Jonatan Bohr Brask, Géraldine Haack, Nicolas Brunner, and Marcus Huber, ``Autonomous quantum thermal machine for generating steady-state entanglement,'' New J. Phys. 17, 113029 (2015).
 Kosuke Ito, Wataru Kumagai, and Masahito Hayashi, ``Asymptotic compatibility between local-operations-and-classical-communication conversion and recovery,'' Phys. Rev. A 92, 052308 (2015).
 Nelly Huei Ying Ng, Mischa Prebin Woods, and Stephanie Wehner, ``Surpassing the Carnot efficiency by extracting imperfect work,'' New J. Phys. 19, 113005 (2017).
 F. G. S. L. Brandão, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, ``The second laws of quantum thermodynamics,'' Proc. Natl. Acad. Sci. U.S.A. 112, 3275 (2015).
 Kamil Korzekwa, Coherence, thermodynamics and uncertainty relations, Ph.D. thesis, Imperial College London (2016).
 Rajendra Bhatia, Matrix analysis (Springer Science & Business Media, 2013).
 Remco Van Der Meer, Nelly Huei Ying Ng, and Stephanie Wehner, ``Smoothed generalized free energies for thermodynamics,'' Phys. Rev. A 96, 062135 (2017).
 Michał Horodecki, Karol Horodecki, Paweł Horodecki, Ryszard Horodecki, Jonathan Oppenheim, Aditi Sen(De), and Ujjwal Sen, ``Local information as a resource in distributed quantum systems,'' Phys. Rev. Lett. 90, 100402 (2003).
 Guifré Vidal, Daniel Jonathan, and MA Nielsen, ``Approximate transformations and robust manipulation of bipartite pure-state entanglement,'' Phys. Rev. A 62, 012304 (2000).
 Marco Tomamichel, Quantum information processing with finite resources: mathematical foundations, Vol. 5 (Springer, 2015).
 Dario Egloff, Oscar CO Dahlsten, Renato Renner, and Vlatko Vedral, ``A measure of majorization emerging from single-shot statistical mechanics,'' New J. Phys. 17, 073001 (2015).
 Chris Perry, Piotr Ć wikliński, Janet Anders, Michał Horodecki, and Jonathan Oppenheim, ``A sufficient set of experimentally implementable thermal operations,'' arXiv:1511.06553 (2015).
 Jonathan G Richens, Álvaro M Alhambra, and Lluis Masanes, ``Finite-bath corrections to the second law of thermodynamics,'' Phys. Rev. E 97, 062132 (2018).
 Kosuke Ito and Masahito Hayashi, ``Optimal performance of generalized heat engines with finite-size baths of arbitrary multiple conserved quantities based on non-iid scaling,'' arXiv:1612.04047 (2016).
 P. Faist, J. Oppenheim, and R. Renner, ``Gibbs-preserving maps outperform thermal operations in the quantum regime,'' New J. of Phys. 17, 043003 (2015).
 Masahito Hayashi, Quantum Information - An Introduction (Springer, 2006).
 C.T. Chubb, V.Y.F. Tan, and M. Tomamichel, ``Moderate Deviation Analysis for Classical Communication over Quantum Channels,'' Commun. Math. Phys. 355 (2017).
Crossref's cited-by service has no data on citing works. Unfortunately not all publishers provide suitable citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.