Skip to main content
Top

2019 | OriginalPaper | Chapter

Further Developments of Sinai’s Ideas: The Boltzmann–Sinai Hypothesis

Author : Nándor Simányi

Published in: The Abel Prize 2013-2017

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter we present a brief survey of the rich and manifold developments of Sinai’s ideas, dating back to 1963, concerning his exact mathematical formulation of Boltzmann’s original ergodic hypothesis. These developments eventually lead to the 2013 proof of the so called “Boltzmann-Sinai Ergodic Hypothesis”.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
2.
4.
go back to reference E. Hopf. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91:261–304, 1939.MathSciNetMATH E. Hopf. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig, 91:261–304, 1939.MathSciNetMATH
5.
go back to reference A. Katok, J.-M. Strelcyn, F. Ledrappier, and F. Przytycki. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, volume 1222 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986.CrossRef A. Katok, J.-M. Strelcyn, F. Ledrappier, and F. Przytycki. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, volume 1222 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986.CrossRef
6.
go back to reference A. Krámli, N. Simányi, and D. Szász. Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3D torus. Nonlinearity, 2(2):311–326, 1989.MathSciNetMATHCrossRef A. Krámli, N. Simányi, and D. Szász. Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3D torus. Nonlinearity, 2(2):311–326, 1989.MathSciNetMATHCrossRef
7.
go back to reference A. Krámli, N. Simányi, and D. Szász. A “transversal” fundamental theorem for semi-dispersing billiards. Comm. Math. Phys., 129(3):535–560, 1990.MathSciNetMATHCrossRef A. Krámli, N. Simányi, and D. Szász. A “transversal” fundamental theorem for semi-dispersing billiards. Comm. Math. Phys., 129(3):535–560, 1990.MathSciNetMATHCrossRef
10.
go back to reference N.S. Krylov. Works on the Foundations of Statistical Physics. Princeton University Press, Princeton, NJ, 1979. N.S. Krylov. Works on the Foundations of Statistical Physics. Princeton University Press, Princeton, NJ, 1979.
11.
go back to reference D. Ornstein and B. Weiss. On the Bernoulli nature of systems with some hyperbolic structure. Ergodic Theory Dynam. Systems, 18(2):441–456, 1998.MathSciNetMATHCrossRef D. Ornstein and B. Weiss. On the Bernoulli nature of systems with some hyperbolic structure. Ergodic Theory Dynam. Systems, 18(2):441–456, 1998.MathSciNetMATHCrossRef
12.
go back to reference Ya.B. Pesin. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv., 32(4):55–114, 1977.MATHCrossRef Ya.B. Pesin. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv., 32(4):55–114, 1977.MATHCrossRef
13.
go back to reference N. Simányi. The K-property of N billiard balls. I. Invent. Math., 108(3):521–548, 1992. N. Simányi. The K-property of N billiard balls. I. Invent. Math., 108(3):521–548, 1992.
14.
go back to reference N. Simányi. The K-property of N billiard balls. II. Computation of neutral linear spaces. Invent. Math., 110(1):151–172, 1992. N. Simányi. The K-property of N billiard balls. II. Computation of neutral linear spaces. Invent. Math., 110(1):151–172, 1992.
16.
go back to reference N. Simányi. Proof of the Boltzmann–Sinai ergodic hypothesis for typical hard disk systems. Invent. Math., 154(1):123–178, 2003.MathSciNetMATHCrossRef N. Simányi. Proof of the Boltzmann–Sinai ergodic hypothesis for typical hard disk systems. Invent. Math., 154(1):123–178, 2003.MathSciNetMATHCrossRef
17.
20.
go back to reference N. Simányi and D. Szász. The K-property of 4D billiards with nonorthogonal cylindric scatterers. J. Statist. Phys., 76(1–2):587–604, 1994.MathSciNetMATHCrossRef N. Simányi and D. Szász. The K-property of 4D billiards with nonorthogonal cylindric scatterers. J. Statist. Phys., 76(1–2):587–604, 1994.MathSciNetMATHCrossRef
22.
go back to reference N. Simányi and D. Szász. Non-integrability of cylindric billiards and transitive Lie group actions. Ergodic Theory Dynam. Systems, 20(2):593–610, 2000.MathSciNetMATHCrossRef N. Simányi and D. Szász. Non-integrability of cylindric billiards and transitive Lie group actions. Ergodic Theory Dynam. Systems, 20(2):593–610, 2000.MathSciNetMATHCrossRef
23.
go back to reference Ya.G. Sinai. On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Soviet Math. Dokl., 4:1818–1822, 1963. Ya.G. Sinai. On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Soviet Math. Dokl., 4:1818–1822, 1963.
24.
go back to reference Ya.G. Sinai. Dynamical systems with elastic reflections. Russ. Math. Surv., 25(2):137–189, 1970.MATHCrossRef Ya.G. Sinai. Dynamical systems with elastic reflections. Russ. Math. Surv., 25(2):137–189, 1970.MATHCrossRef
25.
go back to reference Ya.G. Sinai. Development of Krylov’s ideas. Afterword to N.S. Krylov, Works on the Foundations of Statistical Physic, pages 239–281. Princeton University Press, Princeton, NJ,1979. Ya.G. Sinai. Development of Krylov’s ideas. Afterword to N.S. Krylov, Works on the Foundations of Statistical Physic, pages 239–281. Princeton University Press, Princeton, NJ,1979.
26.
go back to reference Ya.G. Sinai and N.I. Chernov. Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Russ. Math. Surv., 42(3):181–207, 1987.MATHCrossRef Ya.G. Sinai and N.I. Chernov. Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Russ. Math. Surv., 42(3):181–207, 1987.MATHCrossRef
29.
30.
go back to reference M. Wojtkowski. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys., 105(3):391–414, 1986.MathSciNetMATHCrossRef M. Wojtkowski. Principles for the design of billiards with nonvanishing Lyapunov exponents. Comm. Math. Phys., 105(3):391–414, 1986.MathSciNetMATHCrossRef
Metadata
Title
Further Developments of Sinai’s Ideas: The Boltzmann–Sinai Hypothesis
Author
Nándor Simányi
Copyright Year
2019
DOI
https://doi.org/10.1007/978-3-319-99028-6_12

Premium Partner