Microscopic Reversibility and Macroscopic Irreversibility: From the Viewpoint of Algorithmic Randomness

  • Ken HiuraEmail author
  • Shin-ichi Sasa


The emergence of deterministic and irreversible macroscopic behavior from deterministic and reversible microscopic dynamics is understood as a result of the law of large numbers. In this paper, we prove on the basis of the theory of algorithmic randomness that Martin-Löf random initial microstates satisfy an irreversible macroscopic law in the Kac infinite chain model. We find that the time-reversed state of a random state is not random as well as it violates the macroscopic law.


Microscopic reversibility Macroscopic irreversibility Algorithmic randomness Kac model 



The authors thank Naoto Shiraishi and Takahiro Sagawa for their useful comments. The present work was supported by JSPS KAKENHI Grant Number JP17H01148.


  1. 1.
    Lebowitz, J.L.: Boltzmann’s entropy and time’s arrow. Phys. Today 46, 32–38 (1993)CrossRefGoogle Scholar
  2. 2.
    Bricmont, J.: Science of Chaos or Chaos in science? In: Gross, P.R., Levitt, N., Lewis, M.W. (eds.) The Flight from Science and Reason, vol. 775, pp. 131–175. Annals of the New York Academy of Sciences, New York (1996)Google Scholar
  3. 3.
    Lebowitz, J.L., Presutti, E., Spohn, H.: Microscopic models of hydrodynamic behavior. J. Stat. Phys. 51, 841 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Loschmidt, J.: Über den Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft. Sitzungsber. Kais. Akad. Wiss. Wien, Math. Naturwiss. Cl. Abt. II 73, 128 (1876)Google Scholar
  5. 5.
    Zermelo, E.: Über einen Satz der Dynamik und die mechanische Wärmetheorie. Wied. Ann. 57, 485–494 (1896)CrossRefzbMATHGoogle Scholar
  6. 6.
    Li, M., Vitányi, P.M.B.: An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, New York (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Nies, A.: Computability and Randomness. Oxford University Press, Oxford (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Downey, R.G., Hirschfeld, D.R.: Algorithmic Randomness and Complexity. Springer-Verlag, New York (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gács, P.: Lecture notes on descriptional complexity and randomness.
  10. 10.
    Martin-Löf, P.: The definition of random sequences. Inf. Control 9, 602–619 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kac, M.: Probability and Related Topics in Physical Science. Interscience Publishers Inc., New York (1959)zbMATHGoogle Scholar
  12. 12.
    Gottwald, G.A., Oliver, M.: Boltzmann’s Dilemma: an introduction to statistical mechanics via the Kac Ring. SIAM Rev. 51, 613–635 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Maes, C., Netočný, K., Shergelashvili, B.: A selection of nonequilibrium issues. In: Kotecký, R. (ed.) Methods of Contemporary Mathematical Statistical Physics. Lecture Notes in Mathematics, vol. 1970, pp. 247–306. Springer, Berlin (2009)CrossRefGoogle Scholar
  14. 14.
    Sasa, S., Komatsu, T.S.: Thermodynamic irreversibility from high-dimensional Hamiltonian chaos. Prog. Theor. Phys. 103, 1–52 (2000)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Bennett, C.H.: The thermodynamics of computation a review. Int. J. Theor. Phys. 21, 905–940 (1982)CrossRefGoogle Scholar
  16. 16.
    Zurek, W.H.: Thermodynamic cost of computation, algorithmic complexity and the information metric. Nature 341, 119–124 (1989)ADSCrossRefGoogle Scholar
  17. 17.
    Zurek, W.H.: Algorithmic randomness and physical entropy. Phys. Rev. A 40, 4731 (1989)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Caves, C.M.: Information and entropy. Phys. Rev. E. 47, 4010 (1993)ADSCrossRefGoogle Scholar
  19. 19.
    Gács, P.: The Boltzmann Entropy and Randomness Tests. Proc. Workshop on Physics and Computation, IEEE, pp. 209–216 (1994)Google Scholar
  20. 20.
    Cooper, S.B.: Computability Theory. Chapman and Hall/CRC, Boca Raton (2004)zbMATHGoogle Scholar
  21. 21.
    Odifreddi, P.: Classical Recursion Theory, vol. 1. North-Holland Publishing Company, Amsterdam (1990)zbMATHGoogle Scholar
  22. 22.
    Odifreddi, P.: Classical Recursion Theory, vol. 2. North-Holland Publishing Company, Amsterdam (1999)zbMATHGoogle Scholar
  23. 23.
    Ville, J.: Étude Critique de la Notion de Collectif. Monographies des Probabilitités. Calcul des Probabilités et ses Applications. Gauthier-Villars, Paris (1939)Google Scholar
  24. 24.
    Bienvenu, L., Porter, C.: Strong reductions in effective randomness. Theor. Comput. Sci. 459, 55–68 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Solomonoff, R.J.: A formal theory of inductive inference. Part I. Inf. Control 7, 1–22 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Solomonoff, R.J.: A formal theory of inductive inference. Part II. Inf. Control 7, 224–254 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Probl. Inf. Transm. 1, 1–7 (1965)Google Scholar
  28. 28.
    Doob, J.L.: Stochastic Processes. Wiley, New York (1953)zbMATHGoogle Scholar
  29. 29.
    Zubarev, D.N., Morozov, V., Ropke, G.: Statistical Mechanics of Nonequilibrium Processes. Basic Concepts, Kinetic Theory, vol. 1. Wiley, New York (1996)zbMATHGoogle Scholar
  30. 30.
    Zubarev, D.N., Morozov, V., Ropke, G.: Statistical Mechanics of Nonequilibrium Processes. Relaxation and Hydrodynamic Processes, vol. 2. Wiley, New York (1997)zbMATHGoogle Scholar
  31. 31.
    van Lambalgen, M.: Random sequences. Ph.D. Thesis, University of Amsterdam, Amsterdam (1987)Google Scholar
  32. 32.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, New York (2012)zbMATHGoogle Scholar
  33. 33.
    Lefevere, R.: Macroscopic diffusion from a Hamilton-like dynamics. J. Stat. Phys. 151, 861 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lefevere, R.: Fick’s law in a random lattice lorentz gas. Arch. Ration. Mech. Anal. 216, 983 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gács, P.: Uniform test of algorithmic randomness over a general space. Theor. Comput. Sci. 341, 91–137 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hoyrup, M., Rojas, C.: Computability of probability measures and Martin-Löf randomness over metric spaces. Inf. Comput. 207, 830–847 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Galatolo, S., Hoyrup, M., Rojas, C.: Effective symbolic dynamics, random points, statistical behavior, complexity and entropy. Inf. Comput. 208, 23–41 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Gács, P., Hoyrup, M., Rojas, C.: Randomness on computable probability spaces—a dynamical point of view. Theor. Comput. Syst. 48, 465–486 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Gács, P.: Quantum algorithmic entropy. J. Phys. A. 34, 6859–6880 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Vitányi, P.M.: Quantum Kolmogorov complexity based on classical descriptions. IEEE Trans. Inf. Theo. 47, 2464–2479 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Nies, A., Scholz, V.: Martin-Löf quantum states. arXiv:1709.08422
  42. 42.
    Tasaki, H.: Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems. J. Stat. Phys. 163, 937–997 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Goldstein, S., Lebowitz, J.L., Tumulka, R., Zanghí, N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Popescu, S., Short, A.J., Winter, A.: Entanglement and the foundations of statistical mechanics. Nat. Phy. 2, 754 (2006)CrossRefGoogle Scholar
  45. 45.
    Sugita, A.: On the basis of quantum statistical mechanics. Nonlinear Phenom. Complex Syst. 10, 192 (2007)MathSciNetGoogle Scholar
  46. 46.
    Reimann, P.: Typicality for generalized microcanonical ensembles. Phys. Rev. Lett. 99, 160404 (2007)ADSCrossRefGoogle Scholar
  47. 47.
    Rigol, M., Dunjko, V., Olshanii, M.: Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854 (2008)ADSCrossRefGoogle Scholar
  48. 48.
    Iyoda, E., Kaneko, K., Sagawa, T.: Fluctuation theorem for many-body pure quantum states. Phys. Rev. Lett. 119, 100601 (2017)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Kaneko, K., Iyoda, E., Sagawa, T.: Work extraction from a single energy eigenstate. Phys. Rev. E 99, 032128 (2019)ADSCrossRefGoogle Scholar
  50. 50.
    Chetrite, R., Gupta, S.: Two refreshing views of fluctuation theorems through kinematics elements and exponential martingale. J. Stat. Phys. 143, 543 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Neri, I., Rolán, É., Jülicher, F.: Statistics of infima and stopping times of entropy production and applications to active molecular processes. Phys. Rev. X. 7, 011019 (2017)Google Scholar
  52. 52.
    Shafer, G., Vovk, V.: Probability and Finance: It’s Only a Game!. Wiley, New York (2001)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of PhysicsKyoto UniversityKyotoJapan

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