Thermodynamics of information


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.

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Figure 1: The Szilárd engine and recent experimental realizations.
Figure 2: Toy model of a memory.
Figure 3: Schematic of measurement, feedback and reset.


  1. 1

    Leff, H. S. & Rex, A. F. (eds) in Maxwell’s Demon: Entropy, Information, Computing (Princeton Univ. Press, 1990).

    Google Scholar 

  2. 2

    Maruyama, K., Nori, F. & Vedral, V. Colloquium: The physics of Maxwell’s demon and information. Rev. Mod. Phys. 81, 1–23 (2009).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 3

    Callen, H. B. Thermodynamics and an Introduction to Thermostatistics 2nd edn (John Wiley, 1985).

    Google Scholar 

  4. 4

    Szilárd, L. in Maxwell’s Demon: Entropy, Information, Computing (eds Leff, H. S. & Rex, A. F.) (Princeton Univ. Press, 1990).

    Google Scholar 

  5. 5

    Smoluchowski, M. v. Experimentell nachweisbare, der Üblichen Thermodynamik widersprechende Molekularphenomene. Phys. Zeitshur. 13, 1069–1089 (1912).

    MATH  Google Scholar 

  6. 6

    Feynman, R. P., Leighton, R. B. & Sands, M. The Feynman Lectures on Physics Vol. I, Ch. 46 (Addison-Wesley, 1966).

    Google Scholar 

  7. 7

    Penrose, O. Foundations of Statistical Mechanics: A Deductive Treatment (Pergmon Press, 1970).

    Google Scholar 

  8. 8

    Bennett, C. The thermodynamics of computation—a review. Int. J. Theor. Phys. 21, 905–940 (1982).

    Article  Google Scholar 

  9. 9

    Harris, R. J. & Schütz, G. M. Fluctuation theorems for stochastic dynamics. J. Stat. Mech. 2007, P07020 (2007).

    MathSciNet  Google Scholar 

  10. 10

    Jarzynski, C. Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Ann. Rev. Condens. Matter Phys. 2, 329–351 (2011).

    Article  ADS  Google Scholar 

  11. 11

    Sekimoto, K. Stochastic Energetics 799 (Lect. Notes Phys., Springer, 2010).

    Google Scholar 

  12. 12

    Seifert, U. Stochastic thermodynamics, fluctuation theorems, and molecular machines. Rep. Prog. Phys. 75, 126001 (2012).

    Article  ADS  Google Scholar 

  13. 13

    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, 988–992 (2010).

    Article  ADS  Google Scholar 

  14. 14

    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).

    Article  ADS  Google Scholar 

  15. 15

    Berut, A. et al. Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483, 187–189 (2011).

    Article  ADS  Google Scholar 

  16. 16

    Roldán, E., Martínez, I. A., Parrondo, J. M. R. & Petrov, D. Universal features in the energetics of symmetry breaking. Nature Phys. 10, 457–461 (2014).

    Article  ADS  Google Scholar 

  17. 17

    Plenio, M. B. & Vitelli, V. The physics of forgetting: Landauer’s erasure principle and information theory. Contemp. Phys. 42, 25–60 (2001).

    Article  ADS  Google Scholar 

  18. 18

    Sagawa, T. Thermodynamics of Information Processing in Small Systems (Springer Theses, Springer, 2012).

    Google Scholar 

  19. 19

    Sagawa, T. & Ueda, M. Information Thermodynamics: Maxwell’s Demon in Nonequilibrium Dynamics 181–211 (Wiley-VCH, 2013).

    Google Scholar 

  20. 20

    Parrondo, J. M. R. The Szilárd engine revisited: Entropy, macroscopic randomness, and symmetry breaking phase transitions. Chaos 11, 725–733 (2001).

    Article  ADS  MATH  Google Scholar 

  21. 21

    Cover, T. M. & Thomas, J. A. Elements of Information Theory 2nd edn (Wiley-Interscience, 2006).

    Google Scholar 

  22. 22

    Gaveau, B. & Schulman, L. A general framework for non-equilibrium phenomena: The master equation and its formal consequences. Phys. Lett. A 229, 347–353 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. 23

    Esposito, M. & Van den Broeck, C. Second law and Landauer principle far from equilibrium. Europhys. Lett. 95, 40004 (2011).

    Article  ADS  Google Scholar 

  24. 24

    Still, S., Sivak, D. A., Bell, A. J. & Crooks, G. E. Thermodynamics of prediction. Phys. Rev. Lett. 109, 120604 (2012).

    Article  ADS  Google Scholar 

  25. 25

    Horowitz, J. M., Sagawa, T. & Parrondo, J. M. R. Imitating chemical motors with optimal information motors. Phys. Rev. Lett. 111, 010602 (2013).

    Article  ADS  Google Scholar 

  26. 26

    Deffner, S. & Lutz, E. Information free energy for nonequilibrium states. Preprint at (2012).

  27. 27

    Hasegawa, H-H., Ishikawa, J., Takara, K. & Driebe, D. J. Generalization of the second law for a nonequilibrium initial state. Phys. Lett. A 374, 1001–1004 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. 28

    Takara, K., Hasegawa, H-H. & Driebe, D. J. Generalization of the second law for a transition between nonequilibrium states. Phys. Lett. A 375, 88–92 (2010).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. 29

    Lloyd, S. Use of mutual information to decrease entropy: Implications for the second law of thermodynamics. Phys. Rev. A 39, 5378–5386 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  30. 30

    Sagawa, T. & Ueda, M. Second law of thermodynamics with discrete quantum feedback control. Phys. Rev. Lett. 100, 080403 (2008).

    Article  ADS  Google Scholar 

  31. 31

    Ponmurugan, M. Generalized detailed fluctuation theorem under nonequilibrium feedback control. Phys. Rev. E 82, 031129 (2010).

    Article  ADS  Google Scholar 

  32. 32

    Horowitz, J. M. & Vaikuntanathan, S. Nonequilibrium detailed fluctuation theorem for discrete feedback. Phys. Rev. E 82, 061120 (2010).

    Article  ADS  Google Scholar 

  33. 33

    Sagawa, T. & Ueda, M. Generalized Jarzynski equality under nonequilibrium feedback control. Phys. Rev. Lett. 104, 090602 (2010).

    Article  ADS  Google Scholar 

  34. 34

    Fujitani, Y. & Suzuki, H. Jarzynski equality modified in the linear feedback system. J. Phys. Soc. Jpn 79, 104003 (2010).

    Article  ADS  Google Scholar 

  35. 35

    Horowitz, J. M. & Parrondo, J. M. R. Designing optimal discrete-feedback thermodynamic engines. New J. Phys. 13, 123019 (2011).

    Article  ADS  Google Scholar 

  36. 36

    Horowitz, J. M. & Parrondo, J. M. R. Optimizing non-ergodic feedback engines. Acta Phys. Pol. B 44, 803–814 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. 37

    Abreu, D. & Seifert, U. Extracting work from a single heat bath through feedback. Europhys. Lett. 94, 10001 (2011).

    Article  ADS  Google Scholar 

  38. 38

    Bauer, M., Abreu, D. & Seifert, U. Efficiency of a Brownian information machine. J. Phys. A 45, 162001 (2012).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. 39

    Landauer, R. Information is physical. Phys. Today 44 (5), 23–29 (1991).

    Article  ADS  Google Scholar 

  40. 40

    Landauer, R. Maxwell’s Demon: Entropy, Information, Computing (Princeton Univ. Press, 1990).

    Google Scholar 

  41. 41

    Berut, A., Petrosyan, A. & Ciliberto, S. Detailed Jarzynski equality applied to a logically irreversible procedure. Europhys. Lett. 103, 60002 (2013).

    Article  ADS  Google Scholar 

  42. 42

    Jun, Y., Gavrilov, M. & Bechhoefer, J. High-precision test of Landauer’s principle in a feedback trap. Phys. Rev. Lett. 113, 190601 (2014).

    Article  ADS  Google Scholar 

  43. 43

    Sagawa, T. & Ueda, M. Minimal energy cost for thermodynamic information processing: Measurement and information erasure. Phys. Rev. Lett. 102, 250602 (2009).

    Article  ADS  Google Scholar 

  44. 44

    Sagawa, T. Thermodynamic and logical reversibilities revisited. J. Stat. Mech. 2014, P03025 (2014).

    Article  Google Scholar 

  45. 45

    Diana, G., Bagci, G. B. & Esposito, M. Finite-time erasing of information stored in fermionic bits. Phys. Rev. E 87, 012111 (2013).

    Article  ADS  Google Scholar 

  46. 46

    Zulkowski, P. R. & DeWeese, M. R. Optimal finite-time erasure of a classical bit. Phys. Rev. E 89, 052140 (2014).

    Article  ADS  Google Scholar 

  47. 47

    Dillenschneider, R. & Lutz, E. Memory erasure in small systems. Phys. Rev. Lett. 102, 210601 (2009).

    Article  ADS  Google Scholar 

  48. 48

    Mandal, D. & Jarzynski, C. Work and information processing in a solvable model of Maxwell’s demon. Proc. Natl Acad. Sci. USA 109, 11641–11645 (2012).

    Article  ADS  Google Scholar 

  49. 49

    Mandal, D., Quan, H. T. & Jarzynski, C. Maxwell’s refrigerator: An exactly solvable model. Phys. Rev. Lett. 111, 030602 (2013).

    Article  ADS  Google Scholar 

  50. 50

    Hoppenau, J. & Engel, A. On the energetics of information exchange. Europhys. Lett. 105, 50002 (2014).

    Article  ADS  Google Scholar 

  51. 51

    Barato, A. C. & Seifert, U. An autonomous and reversible Maxwell’s demon. Europhys. Lett. 101, 60001 (2013).

    Article  ADS  Google Scholar 

  52. 52

    Barato, A. C. & Seifert, U. Unifying three perspectives on information processing in stochastic thermodynamics. Phys. Rev. Lett. 112, 090601 (2014).

    Article  ADS  Google Scholar 

  53. 53

    Deffner, S. & Jarzynski, C. Information processing and the second law of thermodynamics: An inclusive, Hamiltonian approach. Phys. Rev. X 3, 041003 (2013).

    Google Scholar 

  54. 54

    Deffner, S. Information-driven current in a quantum Maxwell demon. Phys. Rev. E 88, 062128 (2013).

    Article  ADS  Google Scholar 

  55. 55

    Barato, A. & Seifert, U. Stochastic thermodynamics with information reservoirs. Phys. Rev. E 90, 042150 (2014).

    Article  ADS  Google Scholar 

  56. 56

    Granger, L. & Kantz, H. Thermodynamics of measurements. Phys. Rev. E 84, 061110 (2011).

    Article  ADS  Google Scholar 

  57. 57

    Sagawa, T. & Ueda, M. Role of mutual information in entropy production under information exchanges. New J. Phys. 15, 125012 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  58. 58

    Horowitz, J. M. & Esposito, M. Thermodynamics with continuous information flow. Phys. Rev. X 4, 031015 (2014).

    Google Scholar 

  59. 59

    Allahverdyan, A. E., Janzing, D. & Mahler, G. Thermodynamic efficiency of information and heat flow. J. Stat. Mech. 2009, P09011 (2009).

    Article  Google Scholar 

  60. 60

    Hartich, D., Barato, A. C. & Seifert, U. Stochastic thermodynamics of bipartite systems: Transfer entropy inequalities and a maxwell’s demon interpretation. J. Stat. Mech. 2014, P02016 (2014).

    Article  MathSciNet  Google Scholar 

  61. 61

    Shiraishi, N. & Sagawa, T. Fluctuation theorem for partially-masked nonequilibrium dynamics. Phys. Rev. E 91, 012130 (2015).

    Article  ADS  Google Scholar 

  62. 62

    Barato, A., Hartich, D. & Seifert, U. Efficiency of cellular information processing. New J. Phys. 16, 103024 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  63. 63

    Cao, F. J. & Feito, M. Thermodynamics of feedback controlled systems. Phys. Rev. E 79, 041118 (2009).

    Article  ADS  Google Scholar 

  64. 64

    Sagawa, T. & Ueda, M. Nonequilibrium thermodynamics of feedback control. Phys. Rev. E 85, 021104 (2012).

    Article  ADS  Google Scholar 

  65. 65

    Ito, S. & Sagawa, T. Information thermodynamics on causal networks. Phys. Rev. Lett. 111, 180603 (2013).

    Article  ADS  Google Scholar 

  66. 66

    Barato, A. C., Hartich, D. & Seifert, U. Rate of mutual information between coarse-grained non-Markovian variables. J. Stat. Phys. 153, 460–478 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  67. 67

    Sandberg, H., Delvenne, J-C., Newton, N. J. & Mitter, S. K. Maximum work extraction and implementation costs for nonequilibrium Maxwell’s demons. Phys. Rev. E 90, 042119 (2014).

    Article  ADS  Google Scholar 

  68. 68

    Sagawa, T. & Ueda, M. Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics. Phys. Rev. Lett. 109, 180602 (2012).

    Article  ADS  Google Scholar 

  69. 69

    Tasaki, H. Unified Jarzynski and Sagawa–Ueda relations for Maxwell’s demon. Preprint at (2013).

  70. 70

    Horowitz, J. M. & Sandberg, H. Second-law-like inequalities with information and their interpretations. New J. Phys. 16, 125007 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  71. 71

    Abreu, D. & Seifert, U. Thermodynamics of genuine nonequilibrium states under feedback control. Phys. Rev. Lett. 108, 030601 (2012).

    Article  ADS  Google Scholar 

  72. 72

    Zurek, W. in Maxwell’s Demon: Entropy, Information, Computing (eds Leff, H. S. & Rex, A. F.) (Princeton Univ. Press, 1990).

    Google Scholar 

  73. 73

    Kim, S. W., Sagawa, T., De Liberato, S. & Ueda, M. Quantum Szilárd engine. Phys. Rev. Lett. 106, 070401 (2011).

    Article  ADS  Google Scholar 

  74. 74

    Jacobs, K. The second law of thermodynamics and quantum feedback control: Maxwell’s demon with weak measurements. Phys. Rev. A 80, 012322 (2009).

    Article  ADS  Google Scholar 

  75. 75

    Morkuni, Y. & Tasaki, H. Quantum Jarzynski–Sagawa–Ueda relations. J. Stat. Phys. 143, 1–10 (2011).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  76. 76

    Funo, K., Watanabe, Y. & Ueda, M. Integral quantum fluctuation theorems under measurement and feedback control. Phys. Rev. E 88, 052121 (2013).

    Article  ADS  Google Scholar 

  77. 77

    Albash, T., Lidar, D. A., Marvian, M. & Zanardi, P. Fluctuation theorems for quantum processes. Phys. Rev. E 88, 032146 (2013).

    Article  ADS  Google Scholar 

  78. 78

    Zurek, W. Qauntum discord and Maxwell’s demons. Phys. Rev. A 67, 012320 (2003).

    Article  ADS  Google Scholar 

  79. 79

    Funo, K., Watanabe, Y. & Ueda, M. Thermodynamic work gain from entanglement. Phys. Rev. A 88, 052319 (2013).

    Article  ADS  Google Scholar 

  80. 80

    Park, J. J., Kim, K. H., Sagawa, T. & Kim, S. W. Heat engine driven by purely quantum information. Phys. Rev. Lett. 111, 230402 (2013).

    Article  ADS  Google Scholar 

  81. 81

    Reimann, P. Brownian motors: Noisy transport far from equilibrium. Phys. Rep. 361, 57–265 (2002).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  82. 82

    Serreli, V., Lee, C-F., Kay, E. R. & Leigh, D. A. A molecular information ratchet. Nature 445, 523–527 (2007).

    Article  ADS  Google Scholar 

  83. 83

    Esposito, M. & Schaller, G. Stochastic thermodynamics for “Maxwell demon” feedbacks. Europhys. Lett. 99, 30003 (2012).

    Article  ADS  Google Scholar 

  84. 84

    Andrieux, D. & Gaspard, P. Nonequilibrium generation of information in copolymerization processes. Proc. Natl Acad. Sci. USA 105, 9516–9521 (2008).

    Article  ADS  Google Scholar 

  85. 85

    Jarzynski, C. The thermodynamics of writing a random polymer. Proc. Natl Acad. Sci. USA 105, 9451–9452 (2008).

    Article  ADS  Google Scholar 

  86. 86

    Bennett, C. H. Dissipation-error tradeoff in proofreading. Biosystems 11, 85–91 (1979).

    Article  Google Scholar 

  87. 87

    Hopfield, J. J. Kinetic proofreading: A new mechanism for reducing errors in biosynthetic processes requiring high specificity. Proc. Natl Acad. Sci. USA 71, 4135–4139 (1974).

    Article  ADS  Google Scholar 

  88. 88

    Murugan, A., Huse, D. A. & Leibler, S. Discriminatory proofreading regimes in nonequilibrium systems. Phys. Rev. X 4, 021016 (2014).

    Google Scholar 

  89. 89

    Sartori, P. & Pigolotti, S. Kinetic versus Energetic Discrimination in Biological Copying. Phys. Rev. Lett. 110, 188101 (2013).

    Article  ADS  Google Scholar 

  90. 90

    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, 521–530 (2013).

    Article  Google Scholar 

  91. 91

    Sartori, P., Granger, L., Lee, C. F. & Horowitz, J. M. Thermodynamic costs of information processing in sensory adaptation. PLOS Comp. Biol. 10, e1003974 (2014).

    Article  ADS  Google Scholar 

  92. 92

    Ito, S. & Sagawa, T. Maxwell’s demon in biochemical signal transduction. Preprint at (2014).

  93. 93

    Mlodinow, L. & Brun, T. A. Relation between the psychological and thermodynamic arrows of time. Phys. Rev. E 89, 052102 (2014).

    Article  ADS  Google Scholar 

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This work was partially supported by the COST Action MP1209 ‘Thermodynamics in the quantum regime’. 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.

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Parrondo, J., Horowitz, J. & Sagawa, T. Thermodynamics of information. Nature Phys 11, 131–139 (2015).

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