Abstract
By its very nature, the second law of thermodynamics is probabilistic, in that its formulation requires a probabilistic description of the state of a system. This raises questions about the objectivity of the second law: does it depend, for example, on what we know about the system? For over a century, much effort has been devoted to incorporating information into thermodynamics and assessing the entropic and energetic costs of manipulating information. More recently, this historically theoretical pursuit has become relevant in practical situations where information is manipulated at small scales, such as in molecular and cell biology, artificial nano-devices or quantum computation. Here we give an introduction to a novel theoretical framework for the thermodynamics of information based on stochastic thermodynamics and fluctuation theorems, review some recent experimental results, and present an overview of the state of the art in the field.
This is a preview of subscription content, access via your institution
Access options
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Buy this article
- Purchase on SpringerLink
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Leff, H. S. & Rex, A. F. (eds) in Maxwellâs Demon: Entropy, Information, Computing (Princeton Univ. Press, 1990).
Maruyama, K., Nori, F. & Vedral, V. Colloquium: The physics of Maxwellâs demon and information. Rev. Mod. Phys. 81, 1â23 (2009).
Callen, H. B. Thermodynamics and an Introduction to Thermostatistics 2nd edn (John Wiley, 1985).
Szilárd, L. in Maxwellâs Demon: Entropy, Information, Computing (eds Leff, H. S. & Rex, A. F.) (Princeton Univ. Press, 1990).
Smoluchowski, M. v. Experimentell nachweisbare, der Ãblichen Thermodynamik widersprechende Molekularphenomene. Phys. Zeitshur. 13, 1069â1089 (1912).
Feynman, R. P., Leighton, R. B. & Sands, M. The Feynman Lectures on Physics Vol. I, Ch. 46 (Addison-Wesley, 1966).
Penrose, O. Foundations of Statistical Mechanics: A Deductive Treatment (Pergmon Press, 1970).
Bennett, C. The thermodynamics of computationâa review. Int. J. Theor. Phys. 21, 905â940 (1982).
Harris, R. J. & Schütz, G. M. Fluctuation theorems for stochastic dynamics. J. Stat. Mech. 2007, P07020 (2007).
Jarzynski, C. Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Ann. Rev. Condens. Matter Phys. 2, 329â351 (2011).
Sekimoto, K. Stochastic Energetics 799 (Lect. Notes Phys., Springer, 2010).
Seifert, U. Stochastic thermodynamics, fluctuation theorems, and molecular machines. Rep. Prog. Phys. 75, 126001 (2012).
Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E. & Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nature Phys. 6, 988â992 (2010).
Koski, J., Maisi, V., Sagawa, T. & Pekola, J. P. Experimental observation of the role of mutual information in the nonequilibrium dynamics of a Maxwell demon. Phys. Rev. Lett. 113, 030601 (2014).
Berut, A. et al. Experimental verification of Landauerâs principle linking information and thermodynamics. Nature 483, 187â189 (2011).
Roldán, E., MartÃnez, I. A., Parrondo, J. M. R. & Petrov, D. Universal features in the energetics of symmetry breaking. Nature Phys. 10, 457â461 (2014).
Plenio, M. B. & Vitelli, V. The physics of forgetting: Landauerâs erasure principle and information theory. Contemp. Phys. 42, 25â60 (2001).
Sagawa, T. Thermodynamics of Information Processing in Small Systems (Springer Theses, Springer, 2012).
Sagawa, T. & Ueda, M. Information Thermodynamics: Maxwellâs Demon in Nonequilibrium Dynamics 181â211 (Wiley-VCH, 2013).
Parrondo, J. M. R. The Szilárd engine revisited: Entropy, macroscopic randomness, and symmetry breaking phase transitions. Chaos 11, 725â733 (2001).
Cover, T. M. & Thomas, J. A. Elements of Information Theory 2nd edn (Wiley-Interscience, 2006).
Gaveau, B. & Schulman, L. A general framework for non-equilibrium phenomena: The master equation and its formal consequences. Phys. Lett. A 229, 347â353 (1997).
Esposito, M. & Van den Broeck, C. Second law and Landauer principle far from equilibrium. Europhys. Lett. 95, 40004 (2011).
Still, S., Sivak, D. A., Bell, A. J. & Crooks, G. E. Thermodynamics of prediction. Phys. Rev. Lett. 109, 120604 (2012).
Horowitz, J. M., Sagawa, T. & Parrondo, J. M. R. Imitating chemical motors with optimal information motors. Phys. Rev. Lett. 111, 010602 (2013).
Deffner, S. & Lutz, E. Information free energy for nonequilibrium states. Preprint at http://arxiv.org/abs/1201.3888 (2012).
Hasegawa, H-H., Ishikawa, J., Takara, K. & Driebe, D. J. Generalization of the second law for a nonequilibrium initial state. Phys. Lett. A 374, 1001â1004 (2010).
Takara, K., Hasegawa, H-H. & Driebe, D. J. Generalization of the second law for a transition between nonequilibrium states. Phys. Lett. A 375, 88â92 (2010).
Lloyd, S. Use of mutual information to decrease entropy: Implications for the second law of thermodynamics. Phys. Rev. A 39, 5378â5386 (1989).
Sagawa, T. & Ueda, M. Second law of thermodynamics with discrete quantum feedback control. Phys. Rev. Lett. 100, 080403 (2008).
Ponmurugan, M. Generalized detailed fluctuation theorem under nonequilibrium feedback control. Phys. Rev. E 82, 031129 (2010).
Horowitz, J. M. & Vaikuntanathan, S. Nonequilibrium detailed fluctuation theorem for discrete feedback. Phys. Rev. E 82, 061120 (2010).
Sagawa, T. & Ueda, M. Generalized Jarzynski equality under nonequilibrium feedback control. Phys. Rev. Lett. 104, 090602 (2010).
Fujitani, Y. & Suzuki, H. Jarzynski equality modified in the linear feedback system. J. Phys. Soc. Jpn 79, 104003 (2010).
Horowitz, J. M. & Parrondo, J. M. R. Designing optimal discrete-feedback thermodynamic engines. New J. Phys. 13, 123019 (2011).
Horowitz, J. M. & Parrondo, J. M. R. Optimizing non-ergodic feedback engines. Acta Phys. Pol. B 44, 803â814 (2013).
Abreu, D. & Seifert, U. Extracting work from a single heat bath through feedback. Europhys. Lett. 94, 10001 (2011).
Bauer, M., Abreu, D. & Seifert, U. Efficiency of a Brownian information machine. J. Phys. A 45, 162001 (2012).
Landauer, R. Information is physical. Phys. Today 44 (5), 23â29 (1991).
Landauer, R. Maxwellâs Demon: Entropy, Information, Computing (Princeton Univ. Press, 1990).
Berut, A., Petrosyan, A. & Ciliberto, S. Detailed Jarzynski equality applied to a logically irreversible procedure. Europhys. Lett. 103, 60002 (2013).
Jun, Y., Gavrilov, M. & Bechhoefer, J. High-precision test of Landauerâs principle in a feedback trap. Phys. Rev. Lett. 113, 190601 (2014).
Sagawa, T. & Ueda, M. Minimal energy cost for thermodynamic information processing: Measurement and information erasure. Phys. Rev. Lett. 102, 250602 (2009).
Sagawa, T. Thermodynamic and logical reversibilities revisited. J. Stat. Mech. 2014, P03025 (2014).
Diana, G., Bagci, G. B. & Esposito, M. Finite-time erasing of information stored in fermionic bits. Phys. Rev. E 87, 012111 (2013).
Zulkowski, P. R. & DeWeese, M. R. Optimal finite-time erasure of a classical bit. Phys. Rev. E 89, 052140 (2014).
Dillenschneider, R. & Lutz, E. Memory erasure in small systems. Phys. Rev. Lett. 102, 210601 (2009).
Mandal, D. & Jarzynski, C. Work and information processing in a solvable model of Maxwellâs demon. Proc. Natl Acad. Sci. USA 109, 11641â11645 (2012).
Mandal, D., Quan, H. T. & Jarzynski, C. Maxwellâs refrigerator: An exactly solvable model. Phys. Rev. Lett. 111, 030602 (2013).
Hoppenau, J. & Engel, A. On the energetics of information exchange. Europhys. Lett. 105, 50002 (2014).
Barato, A. C. & Seifert, U. An autonomous and reversible Maxwellâs demon. Europhys. Lett. 101, 60001 (2013).
Barato, A. C. & Seifert, U. Unifying three perspectives on information processing in stochastic thermodynamics. Phys. Rev. Lett. 112, 090601 (2014).
Deffner, S. & Jarzynski, C. Information processing and the second law of thermodynamics: An inclusive, Hamiltonian approach. Phys. Rev. X 3, 041003 (2013).
Deffner, S. Information-driven current in a quantum Maxwell demon. Phys. Rev. E 88, 062128 (2013).
Barato, A. & Seifert, U. Stochastic thermodynamics with information reservoirs. Phys. Rev. E 90, 042150 (2014).
Granger, L. & Kantz, H. Thermodynamics of measurements. Phys. Rev. E 84, 061110 (2011).
Sagawa, T. & Ueda, M. Role of mutual information in entropy production under information exchanges. New J. Phys. 15, 125012 (2013).
Horowitz, J. M. & Esposito, M. Thermodynamics with continuous information flow. Phys. Rev. X 4, 031015 (2014).
Allahverdyan, A. E., Janzing, D. & Mahler, G. Thermodynamic efficiency of information and heat flow. J. Stat. Mech. 2009, P09011 (2009).
Hartich, D., Barato, A. C. & Seifert, U. Stochastic thermodynamics of bipartite systems: Transfer entropy inequalities and a maxwellâs demon interpretation. J. Stat. Mech. 2014, P02016 (2014).
Shiraishi, N. & Sagawa, T. Fluctuation theorem for partially-masked nonequilibrium dynamics. Phys. Rev. E 91, 012130 (2015).
Barato, A., Hartich, D. & Seifert, U. Efficiency of cellular information processing. New J. Phys. 16, 103024 (2014).
Cao, F. J. & Feito, M. Thermodynamics of feedback controlled systems. Phys. Rev. E 79, 041118 (2009).
Sagawa, T. & Ueda, M. Nonequilibrium thermodynamics of feedback control. Phys. Rev. E 85, 021104 (2012).
Ito, S. & Sagawa, T. Information thermodynamics on causal networks. Phys. Rev. Lett. 111, 180603 (2013).
Barato, A. C., Hartich, D. & Seifert, U. Rate of mutual information between coarse-grained non-Markovian variables. J. Stat. Phys. 153, 460â478 (2013).
Sandberg, H., Delvenne, J-C., Newton, N. J. & Mitter, S. K. Maximum work extraction and implementation costs for nonequilibrium Maxwellâs demons. Phys. Rev. E 90, 042119 (2014).
Sagawa, T. & Ueda, M. Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics. Phys. Rev. Lett. 109, 180602 (2012).
Tasaki, H. Unified Jarzynski and SagawaâUeda relations for Maxwellâs demon. Preprint at http://arxiv.org/abs/1308.3776 (2013).
Horowitz, J. M. & Sandberg, H. Second-law-like inequalities with information and their interpretations. New J. Phys. 16, 125007 (2014).
Abreu, D. & Seifert, U. Thermodynamics of genuine nonequilibrium states under feedback control. Phys. Rev. Lett. 108, 030601 (2012).
Zurek, W. in Maxwellâs Demon: Entropy, Information, Computing (eds Leff, H. S. & Rex, A. F.) (Princeton Univ. Press, 1990).
Kim, S. W., Sagawa, T., De Liberato, S. & Ueda, M. Quantum Szilárd engine. Phys. Rev. Lett. 106, 070401 (2011).
Jacobs, K. The second law of thermodynamics and quantum feedback control: Maxwellâs demon with weak measurements. Phys. Rev. A 80, 012322 (2009).
Morkuni, Y. & Tasaki, H. Quantum JarzynskiâSagawaâUeda relations. J. Stat. Phys. 143, 1â10 (2011).
Funo, K., Watanabe, Y. & Ueda, M. Integral quantum fluctuation theorems under measurement and feedback control. Phys. Rev. E 88, 052121 (2013).
Albash, T., Lidar, D. A., Marvian, M. & Zanardi, P. Fluctuation theorems for quantum processes. Phys. Rev. E 88, 032146 (2013).
Zurek, W. Qauntum discord and Maxwellâs demons. Phys. Rev. A 67, 012320 (2003).
Funo, K., Watanabe, Y. & Ueda, M. Thermodynamic work gain from entanglement. Phys. Rev. A 88, 052319 (2013).
Park, J. J., Kim, K. H., Sagawa, T. & Kim, S. W. Heat engine driven by purely quantum information. Phys. Rev. Lett. 111, 230402 (2013).
Reimann, P. Brownian motors: Noisy transport far from equilibrium. Phys. Rep. 361, 57â265 (2002).
Serreli, V., Lee, C-F., Kay, E. R. & Leigh, D. A. A molecular information ratchet. Nature 445, 523â527 (2007).
Esposito, M. & Schaller, G. Stochastic thermodynamics for âMaxwell demonâ feedbacks. Europhys. Lett. 99, 30003 (2012).
Andrieux, D. & Gaspard, P. Nonequilibrium generation of information in copolymerization processes. Proc. Natl Acad. Sci. USA 105, 9516â9521 (2008).
Jarzynski, C. The thermodynamics of writing a random polymer. Proc. Natl Acad. Sci. USA 105, 9451â9452 (2008).
Bennett, C. H. Dissipation-error tradeoff in proofreading. Biosystems 11, 85â91 (1979).
Hopfield, J. J. Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity. Proc. Natl Acad. Sci. USA 71, 4135â4139 (1974).
Murugan, A., Huse, D. A. & Leibler, S. Discriminatory proofreading regimes in nonequilibrium systems. Phys. Rev. X 4, 021016 (2014).
Sartori, P. & Pigolotti, S. Kinetic versus Energetic Discrimination in Biological Copying. Phys. Rev. Lett. 110, 188101 (2013).
Depken, M., Parrondo, J. M. R. & Grill, S. W. Intermittent transcription dynamics for the rapid production of long transcripts of high fidelity. Cell Rep. 5, 521â530 (2013).
Sartori, P., Granger, L., Lee, C. F. & Horowitz, J. M. Thermodynamic costs of information processing in sensory adaptation. PLOS Comp. Biol. 10, e1003974 (2014).
Ito, S. & Sagawa, T. Maxwellâs demon in biochemical signal transduction. Preprint at http://arxiv.org/abs/1406.5810 (2014).
Mlodinow, L. & Brun, T. A. Relation between the psychological and thermodynamic arrows of time. Phys. Rev. E 89, 052102 (2014).
Acknowledgements
This work was partially supported by the COST Action MP1209 âThermodynamics in the quantum regimeâ. J.M.R.P. is supported financially by the Spanish MINECO Grant ENFASIS (FIS2011-22644). J.M.H. is supported financially by the ARO MURI grant W911NF-11-1-0268. T.S. is supported financially by JSPS KAKENHI Grants No. 22340114 and No. 25800217.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
About this article
Cite this article
Parrondo, J., Horowitz, J. & Sagawa, T. Thermodynamics of information. Nature Phys 11, 131â139 (2015). https://doi.org/10.1038/nphys3230
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys3230
This article is cited by
-
Entropy, irreversibility and inference at the foundations of statistical physics
Nature Reviews Physics (2024)
-
The persistence of memory in ionic conduction probed by nonlinear optics
Nature (2024)
-
Geometric thermodynamics for the FokkerâPlanck equation: stochastic thermodynamic links between information geometry and optimal transport
Information Geometry (2024)
-
A thermodynamic model showing that information recording can drive active ion transport
Signal Transduction and Targeted Therapy (2023)
-
Thermodynamic fluctuation theorems govern human sensorimotor learning
Scientific Reports (2023)