Thermodynamics as a Consequence of Information Conservation

Manabendra Nath Bera1,2, Arnau Riera1,2, Maciej Lewenstein1,3, Zahra Baghali Khanian1,4, and Andreas Winter3,4

1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, ES-08860 Castelldefels, Spain
2Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany
3ICREA, Pg. Lluis Companys 23, ES-08010 Barcelona, Spain
4Departament de Física: Grup d'Informació Quàntica, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain

Thermodynamics and information have intricate interrelations. Often thermodynamics is considered to be the logical premise to justify that $\textit{information is physical}$ - through Landauer's principle -, thereby also linking information and thermodynamics. This approach towards information has been instrumental to understand thermodynamics of logical and physical processes, both in the classical and quantum domain. In the present work, we formulate thermodynamics as an exclusive consequence of information conservation. The framework can be applied to the most general situations, beyond the traditional assumptions in thermodynamics: we allow systems and thermal baths to be quantum, of arbitrary sizes and even possessing inter-system correlations.

Here, systems and baths are not treated differently, rather both are considered on an equal footing. This leads us to introduce a ''temperature''-independent formulation of thermodynamics. We rely on the fact that, for a fixed amount of information, measured by the von Neumann entropy, any system can be transformed to a state with the same entropy that possesses minimal energy. This state, known as a $\textit{completely passive}$ state, acquires Boltzmann-Gibbs canonical form with an $\textit{intrinsic temperature}$. We introduce the notions of bound and free energy and use them to quantify heat and work, respectively. Guided by the principle of information conservation, we develop universal notions of equilibrium, heat and work, Landauer's principle and universal fundamental laws of thermodynamics. We demonstrate that the maximum efficiency of a quantum engine with a finite bath is in general lower than that of an ideal Carnot engine. We introduce a resource theoretic framework for our $\textit{intrinsic temperature}$ based thermodynamics, within which we address the problem of work extraction and state transformations. Finally, the framework is extended to multiple conserved quantities.

Thermodynamics is a phenomenological theory developed during the 19th Century in the context of the industrial revolution. Steam engines were the engine of the industrial revolution and it was essential to understand its functioning as well as its efficiency. There is no doubt about the success of thermodynamics which has survived to the scientific revolutions of relativity and quantum mechanics. But, how is it possible that so many completely different systems irrespective of being classical or quantum fulfill the laws of thermodynamics? What is the common feature that makes all these different systems to respect thermodynamics?

In this work we show that the essential feature behind all the systems that fulfill thermodynamics is that they are described by underlying physical theories that are microscopically reversible, or more specifically, that conserve the amount of information quantified by the von Neumann entropy. Guided by this principle of information conservation, we develop a formalism of temperature independent thermodynamics and a notion of heat that does not require the environment to be thermal. This notion of temperature independent thermodynamics is expected to be very useful in the current revolution of nano-technology in which the miniaturisation of components and tools to the nano-scale baths will become of a small size and will be brought easily out of equilibrium. We show that the maximum efficiency of a quantum engine with small baths is in general lower than that of an ideal Carnot’s engine. We finally extend our formalism and results to systems with additional conserved quantities apart from the energy.

► BibTeX data

► References

[1] Jochen Gemmer, M. Michel, and Günther Mahler. Quantum Thermodynamics, volume 748 of Lecture Notes in Physics. Berlin, Heidelberg: Springer, 2009. ISBN 9783540705093. 10.1007/​978-3-540-70510-9. URL http:/​/​​book/​10.1007.

[2] Michael Flanders and Donald Swann. The First and Second Law of Thermodynamics. In At the Drop of Another Hat. Parlophone Ltd., 1964. URL https:/​/​​VnbiVw_1FN.

[3] Juan M. R. Parrondo, Jordan M. Horowitz, and Takahiro Sagawa. Thermodynamics of information. Nature Physics, 11: 131-139, 2015. doi:10.1038/​nphys3230. URL http:/​/​​10.1038/​nphys3230.

[4] James Clerk Maxwell. Theory of Heat. Longmans, Green, and Co.: London, New York, Bombay, 1908.

[5] Harvey S. Leff and Andrew F. Rex. Maxwell's Demon: Entropy, Information, Computing. Princeton University Press, 1990. ISBN 9780691605463. URL http:/​/​​titles/​4731.html.

[6] Harvey S. Leff and Andrew F. Rex. Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing. Taylor and Francis, London, 2002. ISBN 9780750307598. URL https:/​/​​books/​9780750307598.

[7] Koji Maruyama, Franco Nori, and Vlatko Vedral. Colloquium: The physics of Maxwell's demon and information. Reviews in Modern Physics, 81: 1-23, Jan 2009. 10.1103/​RevModPhys.81.1. URL http:/​/​​doi/​10.1103/​RevModPhys.81.1.

[8] Leonard Szilard. Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Zeitschrift für Physik, 53 (11): 840-856, 1929. 10.1007/​BF01341281. URL http:/​/​​10.1007/​BF01341281.

[9] Rolf Landauer. Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development, 5 (3): 183-191, July 1961. ISSN 0018-8646. 10.1147/​rd.53.0183.

[10] Charles H. Bennett. The thermodynamics of computation-a review. International Journal of Theoretical Physics, 21 (12): 905-940, 1982. ISSN 1572-9575. 10.1007/​BF02084158. URL http:/​/​​10.1007/​BF02084158.

[11] Martin B. Plenio and V. Vitelli. The physics of forgetting: Landauer's erasure principle and information theory. Contemporary Physics, 42 (1): 25-60, 2001. 10.1080/​00107510010018916. URL http:/​/​​10.1080/​00107510010018916.

[12] Lidia del Rio, Johan Åberg, Renato Renner, Oscar C. O. Dahlsten, and Vlatko Vedral. The thermodynamic meaning of negative entropy. Nature, 474: 61-63, 2011. 10.1038/​nature10123. URL http:/​/​​10.1038/​nature10123.

[13] David Reeb and Michael M. Wolf. An improved Landauer principle with finite-size corrections. New Journal of Physics, 16 (10): 103011, 2014. 10.1088/​1367-2630/​16/​10/​103011. URL http:/​/​​article/​10.1088/​1367-2630/​16/​10/​103011.

[14] Claude E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27 (3): 379-423, July 1948. ISSN 0005-8580. 10.1002/​j.1538-7305.1948.tb01338.x.

[15] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge: Cambridge University Press, 2000. ISBN 978-0521635035.

[16] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. John Wiley and Sons, Inc., 2 edition, 2005. ISBN 9780471748823. 10.1002/​047174882X. URL http:/​/​​10.1002/​047174882X.

[17] Robert Alicki and Mark Fannes. Entanglement boost for extractable work from ensembles of quantum batteries. Physical Review E, 87: 042123, Apr 2013. 10.1103/​PhysRevE.87.042123. URL http:/​/​​doi/​10.1103/​PhysRevE.87.042123.

[18] Martí Perarnau-Llobet, Karen V. Hovhannisyan, Marcus Huber, Paul Skrzypczyk, Nicolas Brunner, and Antonio Acín. Extractable Work from Correlations. Physical Review X, 5: 041011, Oct 2015. 10.1103/​PhysRevX.5.041011. URL http:/​/​​doi/​10.1103/​PhysRevX.5.041011.

[19] Manabendra Nath Bera, Arnau Riera, Maciej Lewenstein, and Andreas Winter. Generalized laws of thermodynamics in the presence of correlations. Nature Communications, 8: 2180, 2017a. doi:10.1038/​s41467-017-02370-x. URL https:/​/​​articles/​s41467-017-02370-x.

[20] Anthony J. Short. Equilibration of quantum systems and subsystems. New Journal of Physics, 13 (5): 053009, 2011. 10.1088/​1367-2630/​13/​5/​053009. URL http:/​/​​article/​10.1088/​1367-2630/​13/​5/​053009.

[21] John Goold, Marcus Huber, Arnau Riera, Lidia del Rio, and Paul Skrzypczyk. The role of quantum information in thermodynamics: a topical review. Journal of Physics A: Mathematical and Theoretical, 49 (14): 143001, 2016. 10.1088/​1751-8113/​49/​14/​143001. URL http:/​/​​article/​10.1088/​1751-8113/​49/​14/​143001.

[22] Lídia del Rio, Adrian Hutter, Renato Renner, and Stephanie Wehner. Relative thermalization. Physical Review E, 94: 022104, Aug 2016. 10.1103/​PhysRevE.94.022104. URL http:/​/​​doi/​10.1103/​PhysRevE.94.022104.

[23] Christian Gogolin and Jens Eisert. Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems. Reports on Progress in Physics, 79 (5): 056001, 2016. 10.1088/​0034-4885/​79/​5/​056001. URL http:/​/​​article/​10.1088/​0034-4885/​79/​5/​056001.

[24] Sandu Popescu, Anthony J. Short, and Andreas Winter. Entanglement and the foundations of statistical mechanics. Nature Physics, 2: 745-758, 2006. 10.1038/​nphys444. URL http:/​/​​10.1038/​nphys444.

[25] 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. Physical Review Letters, 111: 250404, Dec 2013. 10.1103/​PhysRevLett.111.250404. URL http:/​/​​doi/​10.1103/​PhysRevLett.111.250404.

[26] Oscar C. O. Dahlsten, Renato Renner, Elisabeth Rieper, and Vlatko Vedral. Inadequacy of von Neumann entropy for characterizing extractable work. New Journal of Physics, 13 (5): 053015, 2011. 10.1088/​1367-2630/​13/​5/​053015. URL http:/​/​​article/​10.1088/​1367-2630/​13/​5/​053015.

[27] Johan Åberg. Truly work-like work extraction via a single-shot analysis. Nature Communications, 4: 1925, 2013. 10.1038/​ncomms2712. URL http:/​/​​10.1038/​ncomms2712.

[28] Michał Horodecki and Jonathan Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nature Communications, 4: 2059, 2013. 10.1038/​ncomms3059. URL http:/​/​​10.1038/​ncomms3059.

[29] Paul Skrzypczyk, Anthony J. Short, and Sandu Popescu. Work extraction and thermodynamics for individual quantum systems. Nature Communications, 5: 4185, 2014. 10.1038/​ncomms5185. URL http:/​/​​10.1038/​ncomms5185.

[30] Fernando G. S. L. Brandao, Michał Horodecki, Nelly Ng, Jonathan Oppenheim, and Stephanie Wehner. The second laws of quantum thermodynamics. Proceedings of the National Academy of Sciences, 112: 3275-3279, 2015. doi:10.1073/​pnas.1411728112. URL http:/​/​​content/​112/​11/​3275.

[31] Piotr Ć wikliński, Michał Studziński, Michał Horodecki, and Jonathan Oppenheim. Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics. Physical Review Letters, 115: 210403, Nov 2015. 10.1103/​PhysRevLett.115.210403. URL http:/​/​​doi/​10.1103/​PhysRevLett.115.210403.

[32] Matteo Lostaglio, Kamil Korzekwa, David Jennings, and Terry Rudolph. Quantum Coherence, Time-Translation Symmetry, and Thermodynamics. Physical Review X, 5: 021001, Apr 2015a. 10.1103/​PhysRevX.5.021001. URL http:/​/​​doi/​10.1103/​PhysRevX.5.021001.

[33] Dario Egloff, Oscar C. O. Dahlsten, Renato Renner, and Vlatko Vedral. A measure of majorization emerging from single-shot statistical mechanics. New Journal of Physics, 17 (7): 073001, 2015. 10.1088/​1367-2630/​17/​7/​073001. URL http:/​/​​article/​10.1088/​1367-2630/​17/​7/​073001.

[34] Matteo Lostaglio, David Jennings, and Terry Rudolph. Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nature Communications, 6: 6383, 2015b. 10.1038/​ncomms7383. URL http:/​/​​10.1038/​ncomms7383.

[35] Manabendra Nath Bera, Antonio Acín, Marek Kuś, Morgan Mitchell, and Maciej Lewenstein. Randomness in quantum mechanics: Philosophy, physics and technology. Reports on Progress in Physics, 80 (12): 124001, 2017b. 10.1088/​1361-6633/​aa8731. URL http:/​/​​article/​10.1088/​1361-6633/​aa8731.

[36] Florian Hulpke, Uffe V. Poulsen, Anna Sanpera, Aditi Sen(De), Ujjwal Sen, and Maciej Lewenstein. Unitarity as preservation of entropy and entanglement in quantum systems. Foundations of Physics, 36 (4): 477-499, 2006. ISSN 1572-9516. 10.1007/​s10701-005-9035-7. URL http:/​/​​10.1007/​s10701-005-9035-7.

[37] Christopher Jarzynski. Hamiltonian derivation of a detailed fluctuation theorem. Journal of Statistical Physics, 98 (1): 77-102, Jan 2000. ISSN 1572-9613. 10.1023/​A:1018670721277. URL https:/​/​​10.1023/​A:1018670721277.

[38] Massimiliano Esposito, Upendra Harbola, and Shaul Mukamel. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys., 81: 1665-1702, Dec 2009. 10.1103/​RevModPhys.81.1665. URL https:/​/​​doi/​10.1103/​RevModPhys.81.1665.

[39] Massimiliano Esposito, Katja Lindenberg, and Christian Van den Broeck. Entropy production as correlation between system and reservoir. New Journal of Physics, 12 (1): 013013, 2010. 10.1088/​1367-2630/​12/​1/​013013. URL http:/​/​​article/​10.1088/​1367-2630/​12/​1/​013013.

[40] Philipp Strasberg, Gernot Schaller, Tobias Brandes, and Massimiliano Esposito. Quantum and Information Thermodynamics: A Unifying Framework Based on Repeated Interactions. Physical Review X, 7: 021003, Apr 2017. 10.1103/​PhysRevX.7.021003. URL https:/​/​​doi/​10.1103/​PhysRevX.7.021003.

[41] Takahiro Sagawa. Second law-like inequalities with quantum relative entropy: an introduction. Lectures on Quantum Computing, Thermodynamics and Statistical Physics, volume 8. World Scientific, Singapore, 2012. 10.1142/​9789814425193_0003. URL https:/​/​​10.1142/​9789814425193_0003.

[42] A. Puglisi, A. Sarracino, and A. Vulpiani. Temperature in and out of equilibrium: A review of concepts, tools and attempts. Physics Reports, 709-710: 1-60, 2017. ISSN 0370-1573. https:/​/​​10.1016/​j.physrep.2017.09.001. URL https:/​/​​journal/​physics-reports/​vol/​709/​suppl/​C.

[43] Carlo Sparaciari, Jonathan Oppenheim, and Tobias Fritz. Resource theory for work and heat. Physical Review A, 96: 052112, Nov 2017. 10.1103/​PhysRevA.96.052112. URL https:/​/​​doi/​10.1103/​PhysRevA.96.052112.

[44] Henrik Wilming, Rodrigo Gallego, and Jens Eisert. Axiomatic characterization of the quantum relative entropy and free energy. Entropy, 19 (6), 2017. ISSN 1099-4300. 10.3390/​e19060241. URL http:/​/​​1099-4300/​19/​6/​241.

[45] Markus P. Müller. Correlating thermal machines and the second law at the nanoscale. 10.1103/​PhysRevX.8.041051. URL https:/​/​​prx/​abstract/​10.1103/​PhysRevX.8.041051.

[46] Edwin T. Jaynes. Information Theory and Statistical Mechanics. Physical Review, 106: 620-630, May 1957a. 10.1103/​PhysRev.106.620. URL https:/​/​​doi/​10.1103/​PhysRev.106.620.

[47] Edwin T. Jaynes. Information Theory and Statistical Mechanics. II. Physical Review, 108: 171-190, Oct 1957b. 10.1103/​PhysRev.108.171. URL https:/​/​​doi/​10.1103/​PhysRev.108.171.

[48] Wiesław Pusz and Stanisław L. Woronowicz. Passive states and KMS states for general quantum systems. Communications in Mathematical Physics, 58 (3): 273-290, 1978. ISSN 1432-0916. 10.1007/​BF01614224. URL http:/​/​​10.1007/​BF01614224.

[49] Andrew Lenard. Thermodynamical proof of the Gibbs formula for elementary quantum systems. Journal of Statistical Physics, 19 (6): 575-586, 1978. ISSN 1572-9613. 10.1007/​BF01011769. URL http:/​/​​10.1007/​BF01011769.

[50] Lluís Masanes and Jonathan Oppenheim. A general derivation and quantification of the third law of thermodynamics. Nature Communications, 8: 14538, 2017. 10.1038/​ncomms14538. URL http:/​/​​10.1038/​ncomms14538.

[51] Nicole Yunger Halpern and Joseph M. Renes. Beyond heat baths: Generalized resource theories for small-scale thermodynamics. Physical Review E, 93: 022126, Feb 2016. 10.1103/​PhysRevE.93.022126. URL https:/​/​​doi/​10.1103/​PhysRevE.93.022126.

[52] Nicole Yunger Halpern. Beyond heat baths II: Framework for generalized thermodynamic resource theories. Journal of Physics A: Mathematical and Theoretical, 51: 094001, 2018. 10.1088/​1751-8121/​aaa62f. URL http:/​/​​article/​10.1088/​1751-8121/​aaa62f/​meta.

[53] Matteo Lostaglio, David Jennings, and Terry Rudolph. Thermodynamic resource theories, non-commutativity and maximum entropy principles. New Journal of Physics, 19 (4): 043008, 2017. 10.1088/​1367-2630/​aa617f. URL http:/​/​​article/​10.1088/​1367-2630/​aa617f.

[54] Yelena Guryanova, Sandu Popescu, Anthony J. Short, Ralph Silva, and Paul Skrzypczyk. Thermodynamics of quantum systems with multiple conserved quantities. Nature Communications, 7: 12049, 2016. 10.1038/​ncomms12049. URL http:/​/​​10.1038/​ncomms12049.

[55] Nicole Yunger Halpern, Philippe Faist, Jonathan Oppenheim, and Andreas Winter. Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges. Nature Communications, 7: 12051, 2016. 10.1038/​ncomms12051. URL http:/​/​​10.1038/​ncomms12051.

[56] Zahra B. Khanian, Manabendra Nath Bera, Arnau Riera, Maciej Lewenstein, and Andreas Winter. Resource theory of work and heat and everything else: basing thermodynamics of multiple non-commuting conserved quantities on an asymptotic equivalence principle. In preparation, 2018.

Cited by

[1] Małgorzata Bartkiewicz, Andrzej Grudka, Ryszard Horodecki, Justyna Łodyga, and Jacek Wychowaniec, "Closed timelike curves and the second law of thermodynamics", Physical Review A 99 2, 022304 (2019).

[2] Manabendra Nath Bera, Andreas Winter, and Maciej Lewenstein, "Thermodynamics from Information", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 799 (2018).

[3] Toshio Croucher, Jackson Wright, André R. R. Carvalho, Stephen M. Barnett, and Joan A. Vaccaro, "Information Erasure", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 713 (2018).

[4] Manabendra Nath Bera, Andreas Winter, and Maciej Lewenstein, "Thermodynamics from information", arXiv:1805.10282 (2018).

[5] Gonzalo Manzano, Jordan M. Horowitz, and Juan M. R. Parrondo, "Quantum Fluctuation Theorems for Arbitrary Environments: Adiabatic and Nonadiabatic Entropy Production", Physical Review X 8 3, 031037 (2018).

[6] Carlo Sparaciari, Lidia del Rio, Carlo Maria Scandolo, Philippe Faist, and Jonathan Oppenheim, "The first law of general quantum resource theories", arXiv:1806.04937 (2018).

[7] Nicole Yunger Halpern, "Beyond heat baths II: framework for generalized thermodynamic resource theories", Journal of Physics A Mathematical General 51 9, 094001 (2018).

[8] Mohammad Mehboudi, Anna Sanpera, and Luis A. Correa, "Thermometry in the quantum regime: Recent theoretical progress", arXiv:1811.03988 (2018).

[9] Raam Uzdin, "The Second Law and Beyond in Microscopic Quantum Setups", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 681 (2018).

[10] Sergi Julia-Farre, Tymoteusz Salamon, Arnau Riera, Manabendra N. Bera, and Maciej Lewenstein, "Bounds on Capacity and Power of Quantum Batteries", arXiv:1811.04005 (2018).

[11] Małgorzata Bartkiewicz, Andrzej Grudka, Ryszard Horodecki, Justyna Łodyga, and Jacek Wychowaniec, "Closed timelike curves and the second law of thermodynamics", arXiv:1711.08334 (2017).

[12] Paul Boes, Henrik Wilming, Jens Eisert, and Rodrigo Gallego, "Statistical ensembles without typicality", Nature Communications 9, 1022 (2018).

[13] Dario Ferraro, Michele Campisi, Gian Marcello Andolina, Vittorio Pellegrini, and Marco Polini, "High-Power Collective Charging of a Solid-State Quantum Battery", Physical Review Letters 120 11, 117702 (2018).

[14] Mir Alimuddin, Tamal Guha, and Preeti Parashar, "Bound on ergotropic gap for bipartite separable states", Physical Review A 99 5, 052320 (2019).

The above citations are from Crossref's cited-by service (last updated 2019-05-20 21:03:03) and SAO/NASA ADS (last updated 2019-05-20 21:03:04). The list may be incomplete as not all publishers provide suitable and complete citation data.