Skip to main content
SpringerLink
Log in

Introduction to Information Thermodynamics on Causal Networks

  • Chapter
  • First Online:
Information Thermodynamics on Causal Networks and its Application to Biochemical Signal Transduction

Part of the book series: Springer Theses ((Springer Theses))

  • 733 Accesses

Abstract

This presents the main purpose of this thesis. In this thesis, we mainly discuss the relationship between nonequilibrium thermodynamics and information theory from the view point of Maxwell’s demon which is the thought experiment in the 19th century. As a generalization of the study of Maxwell’s demon, we propose the graphical thermodynamic theory of information processing which is applicable to quite a broad class of nonequilibrium dynamics. Characterizing the complex dynamics by the Bayesian networks, we obtain a nobel generalization of the second law of thermodynamics with complex information flow. We also discuss the biophysical meaning of the information flow inside the cell as an application of its theory. At the end of this chapetr, we summarize the organization of this thesis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  2. T.M. Cover, J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991)

    Book  MATH  Google Scholar 

  3. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010)

    Book  MATH  Google Scholar 

  4. R. Phillips, J. Kondev, J. Theriot, H. Garcia, Physical Biology of the Cell (Garland Science, New York, 2009)

    Google Scholar 

  5. B.D. Gomperts, I.M. Kramer, P.E. Tatham, Signal Transduction (Academic Press, Orlando, 2009)

    Google Scholar 

  6. F. Tostevin, P.R. ten Wolde, Mutual information between input and output trajectories of biochemical networks. Phys. Rev. Lett. 102, 218101 (2009)

    Article  ADS  Google Scholar 

  7. W.R. Atchley, K.R. Wollenberg, W.M. Fitch, W. Terhalle, A.W. Dress, Correlations among amino acid sites in bHLH protein domains: an information theoretic analysis. Mol. Biol. Evol. 17, 164–178 (2000)

    Article  Google Scholar 

  8. J.M. Skerker, B.S. Perchuk, A. Siryaporn, E.A. Lubin, O. Ashenberg, M. Goulian, M.T. Laub, Rewiring the specificity of two-component signal transduction systems. Cell 133, 1043–1054 (2008)

    Article  Google Scholar 

  9. P. Mehta, S. Goyal, T. Long, B.L. Bassler, N.S. Wingreen, Information processing and signal integration in bacterial quorum sensing. Mol. Syst. Biol. 5 (2009)

    Google Scholar 

  10. R. Cheong, A. Rhee, C.J. Wang, I. Nemenman, A. Levchenko, Information transduction capacity of noisy biochemical signaling networks. Science 334, 354–358 (2011)

    Article  ADS  Google Scholar 

  11. S. Uda, T.H. Saito, T. Kudo, T. Kokaji, T. Tsuchiya, H. Kubota, Y. Komori, Y. Ozaki, S. Kuroda, Robustness and compensation of information transmission of signaling pathways. Science 341, 558–5614 (2013)

    Article  ADS  Google Scholar 

  12. S. Carnot, Réflexions sur la pussance motrice du feu et sur les machines propresà développer atte puissance (Bachelier, 1824)

    Google Scholar 

  13. J.C. Maxwell, Theory of Heat (Appleton, London, 1871)

    Google Scholar 

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

    MATH  Google Scholar 

  15. E.H. Lieb, J. Yngvason, The physics and mathematics of the second law of thermodynamics. Phys. Rep. 310, 1 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. H. Touchette, S. Lloyd, Information-theoretic limits of control. Phys. Rev. Lett. 84, 1156 (2000)

    Article  ADS  Google Scholar 

  17. H. Touchette, S. Lloyd, Information-theoretic approach to the study of control systems. Phys. A 331, 140 (2004)

    Article  MathSciNet  Google Scholar 

  18. F.J. Cao, L. Dinis, J.M. Parrondo, Feedback control in a collective flashing ratchet. Phys. Rev. Lett. 93, 040603 (2004)

    Article  ADS  Google Scholar 

  19. L. Dinis, J.M. Parrondo, F.J. Cao, Closed-loop control strategy with improved current for a flashing ratchet. Europhys. Lett. 71, 536 (2005)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  22. F.J. Cao, M. Feito, H. Touchette, Information and flux in a feedback controlled Brownian ratchet. Phys. A 388, 113 (2009)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  26. Y. Fujitani, H. Suzuki, Jarzynski equality modified in the linear feedback system. J. Phys. Soc. Jpn. 79, 044708 (2010)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  29. Y. Morikuni, H. Tasaki, Quantum Jarzynski-Sagawa-Ueda Relations. J. Stat. Phys. 143, 1 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  31. S. Ito, M. Sano, Effects of error on fluctuations under feedback control. Phys. Rev. E 84, 021123 (2011)

    Article  ADS  Google Scholar 

  32. J.M. Horowitz, J.M. Parrondo, Thermodynamic reversibility in feedback processes. Europhys. Lett. 95, 10005 (2011)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  34. S. Vaikuntanathan, C. Jarzynski, Modeling Maxwell’s demon with a microcanonical Szilard engine. Phys. Rev. E 83, 061120 (2011)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  36. L. Granger, H. Kantz, Thermodynamic cost of measurements. Phys. Rev. E 84, 061110 (2011)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  39. T. Munakata, M.L. Rosinberg, Entropy production and fluctuation theorems under feedback control: the molecular refrigerator model revisited. J. Stat. Mech. P05010 (2012)

    Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

  42. S. Lahiri, S. Rana, A.M. Jayannavar, Fluctuation theorems in the presence of information gain and feedback. J. Phys. A 45, 065002 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  44. D. Mandal, C. Jarzynski, Work and information processing in a solvable model of Maxwell’s demon. Proc. Nat. Acad. Sci. 109, 11641 (2012)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  46. A. Kundu, Nonequilibrium fluctuation theorem for systems under discrete and continuous feedback control. Phys. Rev. E 86, 021107 (2012)

    Article  ADS  Google Scholar 

  47. P. Strasberg, G. Schaller, T. Brandes, M. Esposito, Thermodynamics of a physical model implementing a maxwell demon. Phys. Rev. Lett. 110, 040601 (2013)

    Article  ADS  Google Scholar 

  48. A.C. Barato, D. Hartich, U. Seifert, Information-theoretic versus thermodynamic entropy production in autonomous sensory networks. Phys. Rev. E 87, 042104 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  54. H. Tajima, Second law of information thermodynamics with entanglement transfer. Phys. Rev. E 88, 042143 (2013)

    Article  ADS  Google Scholar 

  55. H. Tasaki, Unified Jarzynski and Sagawa-Ueda relations for Maxwell’s demon (2013), arXiv:1308.3776

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

    Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

  59. S. Ito, T. Sagawa, Maxwell’s demon in biochemical signal transduction with feedback loop. Nat. Commun. 8, 7498 (2015)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  62. P. Sartori, L. Granger, C.F. Lee, J.M. Horowitz, Thermodynamic costs of information processing in sensory adaption. PLoS Comput. Biol. 10, e1003974 (2014)

    Article  ADS  Google Scholar 

  63. S. Bo, M. Del Giudice, A. Celani, Thermodynamic limits to information harvesting by sensory systems. J. Stat. Mech. P01014 (2015)

    Google Scholar 

  64. S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, M. Sano, Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys. 6, 988 (2010)

    Article  Google Scholar 

  65. A. Berut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, E. Lutz, Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483, 187 (2012)

    Article  ADS  Google Scholar 

  66. J.V. Koski, V.F. Maisi, T. Sagawa, J.P. Pekola, 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 

  67. H.S. Leff, A.F. Rex (eds.), Maxwell’s demon 2: Entropy, Classical and Quantum Information, Computing (Princeton University Press, New Jersey, 2003)

    Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. L. Szilard, Über die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Z. Phys. 53, 840 (1929)

    Article  ADS  MATH  Google Scholar 

  70. D.J. Evans, D.J. Searles, The fluctuation theorem. Adv. Phys. 51, 1529 (2002)

    Article  ADS  Google Scholar 

  71. D.J. Evans, G.P. Morriss, Statistical mechanics of nonequilibrium liquids, 2nd edn. (Cambridge Univ. Press, Cambridge, 2008)

    Book  MATH  Google Scholar 

  72. K. Sekimoto, Stochastic Energetics (Springer, New York, 2010)

    Book  MATH  Google Scholar 

  73. U. Seifert, Stochastic thermodynamics: principles and perspectives. Eur. Phys. J. B 64, 423 (2008)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  75. D.J. Evans, E.G.D. Cohen, G.P. Morriss, Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401 (1993)

    Article  ADS  MATH  Google Scholar 

  76. G. Gallavotti, E.G.D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694 (1995)

    Article  ADS  Google Scholar 

  77. J. Kurchan, Fluctuation theorem for stochastic dynamics. J. Phys. A: Math. Gen. 31, 3719 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  78. J.L. Lebowitz, H. Spohn, A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  79. D.J. Searles, D.J. Evans, Fluctuation theorem for stochastic systems. Phys. Rev. E 60, 159 (1999)

    Article  ADS  Google Scholar 

  80. C. Jarzynski, Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690 (1997)

    Article  ADS  Google Scholar 

  81. C. Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: a master-equation approach. Phys. Rev. E 56, 5018 (1997)

    Article  ADS  Google Scholar 

  82. G.E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721 (1999)

    Article  ADS  Google Scholar 

  83. C. Jarzynski, Hamiltonian derivation of a detailed fluctuation theorem. J. Stat. Phys. 98, 77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  84. K. Sekimoto, Kinetic characterization of heat bath and the energetics of thermal ratchet models. J. Phys. Soc. Jpn. 66, 1234 (1997)

    Article  ADS  Google Scholar 

  85. K. Sekimoto, Langevin equation and thermodynamics. Prog. Theor. Phys. Supp. 130, 17 (1998)

    Article  ADS  Google Scholar 

  86. T. Hatano, S.-I. Sasa, Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463 (2001)

    Article  ADS  Google Scholar 

  87. G.E. Crooks, Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90, 1481 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  88. G. Hummer, A. Szabo, Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proc. Natl. Acad. Sci. 98, 3658 (2001)

    Article  ADS  Google Scholar 

  89. J. Liphardt, S. Dumont, S.B. Smith, I. Tinoco, C. Bustamante, Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality. Science 296, 1832 (2002)

    Article  ADS  Google Scholar 

  90. E.H. Trepagnier, C. Jarzynski, F. Ritort, G.E. Crooks, C.J. Bustamante, J. Liphardt, Experimental test of Hatano and Sasa’s nonequilibrium steady-state equality. Proc. Natl. Acad. Sci. USA 101, 15038 (2004)

    Article  ADS  Google Scholar 

  91. R. Van Zon, S. Ciliberto, E.G.D. Cohen, Power and heat fluctuation theorems for electric circuits. Phys. Rev. Lett. 92, 130601 (2004)

    Article  Google Scholar 

  92. D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, C. Bustamante, Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 437, 231 (2005)

    Article  ADS  Google Scholar 

  93. S. Schuler, T. Speck, C. Tietz, J. Wrachtrup, U. Seifert, Experimental test of the fluctuation theorem for a driven two-level system with time-dependent rates. Phys. Rev. Lett. 94, 180602 (2005)

    Article  ADS  Google Scholar 

  94. C. Tietz, S. Schuler, T. Speck, U. Seifert, J. Wrachtrup, Measurement of stochastic entropy production. Phys. Rev. Lett. 97, 050602 (2006)

    Article  ADS  Google Scholar 

  95. D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, A. Petrosyan, Entropy production and time asymmetry in nonequilibrium fluctuations. Phys. Rev. Lett. 98, 150601 (2007)

    Article  ADS  Google Scholar 

  96. S. Toyabe, T. Okamoto, T. Watanabe-Nakayama, H. Taketani, S. Kudo, E. Muneyuki, Nonequilibrium energetics of a single F1-ATPase molecule. Phys. Rev. Lett. 104, 198103 (2010)

    Article  ADS  Google Scholar 

  97. K. Hayashi, H. Ueno, R. Iino, H. Noji, Fluctuation theorem applied to F1-ATPase. Phys. Rev. Lett. 104, 218103 (2010)

    Article  ADS  Google Scholar 

  98. J. Mehl, B. Lander, C. Bechinger, V. Blickle, U. Seifert, Role of hidden slow degrees of freedom in the fluctuation theorem. Phys. Rev. Lett. 108, 220601 (2012)

    Article  ADS  Google Scholar 

  99. M. Minsky, Steps toward artificial intelligence. Comput. Thought 406, 450 (1963)

    Google Scholar 

  100. J. Pearl, Fusion, propagation, and structuring in belief networks. Artif. Intell. 29, 241 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  101. C.M. Bishop, Pattern Recognition and Machine Learning (Springer, New York, 2006)

    MATH  Google Scholar 

  102. J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (Morgan Kaufmann, San Mateo, 1988)

    MATH  Google Scholar 

  103. J. Pearl, Causality: Models, Reasoning and Inference (MIT press, Cambridge, 2000)

    MATH  Google Scholar 

  104. F.V. Jensen, T.D. Nielsen, Bayesian Networks and Decision Graphs (Springer, New York, 2009)

    MATH  Google Scholar 

  105. T. Schreiber, Measuring information transfer. Phys. Rev. Lett. 85, 461 (2000)

    Article  ADS  Google Scholar 

  106. L. Barnett, A.B. Barrett, A.K. Seth, Granger causality and transfer entropy are equivalent for Gaussian variables. Phys. Rev. Lett. 103, 238701 (2009)

    Article  ADS  Google Scholar 

  107. C.W.J. Granger, Investigating causal relations by econometric and cross-spectral methods. Econometrica 37, 424438 (1969)

    Google Scholar 

  108. A. Hlavackova-Schindler, M. Palus, M. Vejmelka, Causality detection based on information-theoretic approaches in time series analysis. Phys. Rep. 441, 1–46 (2007)

    Article  ADS  Google Scholar 

  109. M. Staniek, K. Lehnertz, Symbolic transfer entropy. Phys. Rev. Lett. 100, 158101 (2008)

    Article  ADS  Google Scholar 

  110. Y. Oono, M. Paniconi, Steady state thermodynamics. Prog. Theor. Phys. Suppl. 130, 29–44 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  111. S.-I. Sasa, H. Tasaki, Steady state thermodynamics. J. Stat. Phys. 125, 125 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  112. T.S. Komatsu, N. Nakagawa, Expression for the stationary distribution in nonequilibrium steady states. Phys. Rev. Lett. 100, 030601 (2008)

    Article  ADS  Google Scholar 

  113. K.H. Kim, H. Qian, Fluctuation theorems for a molecular refrigerator. Phys. Rev. E 75, 022102 (2007)

    Article  ADS  Google Scholar 

  114. D. Kleckner, D. Bouwmeester, Sub-kelvin optical cooling of a micromechanical resonator. Nature 444, 75 (2006)

    Article  ADS  Google Scholar 

  115. M. Poggio, C.L. Degen, H.J. Mamin, D. Rugar, Feedback cooling of a cantilever’s fundamental mode below 5 mK. Phys. Rev. Lett. 99, 017201 (2007)

    Article  ADS  Google Scholar 

  116. T.J. Kippenberg, K.J. Vahala, Cavity optomechanics: back-action at the mesoscale. Science 321, 1172 (2008)

    Article  ADS  Google Scholar 

  117. L. Tongcang, S. Kheifets, M.G. Raizen, Millikelvin cooling of an optically trapped microsphere in vacuum. Nat. Phys. 7, 527 (2011)

    Article  Google Scholar 

  118. J. Gieseler, B. Deutsch, R. Quidant, L. Novotny, Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 109, 103603 (2012)

    Article  ADS  Google Scholar 

  119. N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 1992)

    MATH  Google Scholar 

  120. J. Wilks, The Third Law of Thermodynamics (Oxford University Press, Oxford, 1961)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, The University of Tokyo, Tokyo, Japan

    Sosuke Ito

Authors
  1. Sosuke Ito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sosuke Ito .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Singapore

About this chapter

Cite this chapter

Ito, S. (2016). Introduction to Information Thermodynamics on Causal Networks. In: Information Thermodynamics on Causal Networks and its Application to Biochemical Signal Transduction. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-10-1664-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation