Advertisement

Stable measures and processes in statistical physics

  • Aleksander Weron
  • Karina Weron
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1153)

Abstract

It is shown how α-stable distributions arise in statistical physics. A probabilistic proof of Khalfin’s formula for decaying quantum systems is given. Also ergodic properties of symmetric α-stable flows in classical statistical mechanics are discussed.

Keywords

Stable Distribution Ergodic Property Radioactive Nucleus Classical Statistical Mechanic Decay System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    V. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Benjamin/Cummings Publishing Co. Reading, Mass. 1978.zbMATHGoogle Scholar
  2. [2]
    A. Bellow, Ergodic properties of isometries in Lp spaces, Bull. Amer. Math. Soc. 70 (1966), 366–371.Google Scholar
  3. [3]
    K. Blum, Density Matrix Theory and Applications, Plenum Press, New York 1981.CrossRefzbMATHGoogle Scholar
  4. [4]
    L.A. Bunimovich and Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatteres, Commun. Math. Phys. 78 (1981), 479–497.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    S. Cambanis, Complex symmetric stable variables and processes, Contributions to Statistics: Essays in Honour of Norman L. Johnson, P.K. Sen, Ed., North-Holland, New York 1983, 63–79.Google Scholar
  6. [6]
    S. Cambanis, C.D. Hardin, Jr., and A. Weron, Ergodic properties of stationary stable processes, Center for Stochastic Processes Tech. Rept. No. 59, Univ. of North Carolina, Chapel Hill, 1984.zbMATHGoogle Scholar
  7. [7]
    S. Cambanis and A.R. Soltani, Prediction of stable processes: Spectral and moving average representations. Z. Wahrsch. verw. Geb. 66 (1984), 593–612.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    I.P. Cornfeld, S.V. Fomin, and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, New York 1982.CrossRefzbMATHGoogle Scholar
  9. [9]
    E.B. Davies, Quantum Theory of Open Systems, Academic Press, London, 1976.zbMATHGoogle Scholar
  10. [10]
    J. Doob, The Brownian movement and stochastic equations, Ann. Math. 43 (1942), 351–369.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S.V. Fomin, Normal dynamical systems, (in Russian), Ukr. Mat. J. 2 (1950) no. 2., 25–47.Google Scholar
  12. [12]
    L. Fonda, G.C. Ghirardi, and A. Rimini, Decay theory of unstable quantum systems, Rept. Progr. Phys. 41 (1978) 587–631.CrossRefGoogle Scholar
  13. [13]
    B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Adison-Wesley, Reading, Mass. 1954.zbMATHGoogle Scholar
  14. [14]
    I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York 1980.zbMATHGoogle Scholar
  15. [15]
    U. Grenander, Stochastic processes and statistical inference, Ark. Math. 1 (1950), 195–277.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M.N. Hack, Long tame tails in decay theory, Phys. Lett. 90A (1982), 220–221.CrossRefGoogle Scholar
  17. [17]
    C.D. Hardin, Jr., On the spectral representation of symmetric stable processes, J. Multivariate Anal. 12 (1982) 385–401.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    C.F. Hart and M.D. Girardeau, New variational principle for decaying states, Phys. Rev. Lett. 51 (1983), 1725–1728.CrossRefGoogle Scholar
  19. [19]
    B.D. Hughes, M.F. Shlesinger, and E.W. Montroll, Random walks with self-similar clusters, Proc. Natl. Acad. Sci. USA 78 (1981), 3287–3291.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    G. Jona-Lasinio, The renormalization group: a probabilistic view, II. Nuovo Cim. 26 (1975), 99–137.MathSciNetCrossRefGoogle Scholar
  21. [21]
    L.A. Khalfin, contribution to the decay theory of a quasi stationary state, Zh. Eksp. Teor. Fiz. 33 (1957), 1371–1382.Google Scholar
  22. [22]
    A.I. Khinchin, Mathematical Foundation of Statistical Mechanics, Dover Publ., Inc., New York 1949.zbMATHGoogle Scholar
  23. [23]
    N.S. Krylov and V.A. Fock, The uncertainty relation for energy and time, Zh. Eksp. Teor. Fiz. 17 (1947), 93–107.Google Scholar
  24. [24]
    M.H. Lee, Can the velocity autocorrelation function decay exponentially?, Phys. Rev. Lett. 51 (1983), 1227–1230.CrossRefGoogle Scholar
  25. [25]
    R. LePage, Multidimensional infinitely divisible variables and processes. Part I: Stable case. Tech. Rept. No. 292, Dept. of Statistics, Stanford Univ. 1980.Google Scholar
  26. [26]
    P. Lévy, Theorie des erreurs. La loi de Gauss et les lois exceptionelles, Bull. Soc. Math. France 52 (1924), 49–85.MathSciNetzbMATHGoogle Scholar
  27. [27]
    B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motions, fractional noises and applications., SIAM Review 10 (1968), 422–437.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyushu Univ. A4 (1949), 45–106.MathSciNetzbMATHGoogle Scholar
  29. [29]
    E.W. Montroll and J.T. Bendler, On Lévy (or stable) distributions and the Williams-Watts model of dielectric relaxation, J. Stat. Phys. 34 (1984), 129–162.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    E.W. Montroll and M.F. Shlessinger, On 1/f noise and other distributions with long tails, Proc. Natl. Acad. Sci. USA 79 (1982), 3380–3383.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    E.W. Montroll and B.J. West, On an enriched collection of stochastic processes, in Fluctuation Phenomena, Eds. E.W. Montroll and J.L. Lebowitz, North-Holland, New York 1979, 61–175.CrossRefGoogle Scholar
  32. [32]
    B. Sz.-Nagy and C. Foias, Sur les contractions de l’espace de Hilbert. IV, Acta Sci. Math. 21 (1960), 251–259.MathSciNetzbMATHGoogle Scholar
  33. [33]
    H. Scher and E.W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. 12B (1975), 2455–2477.CrossRefGoogle Scholar
  34. [34]
    J.K.E. Tunaley, Conduction in a random lattice under a potential gradient, J. Appl. Phys. 43 (1972), 4783–4786.CrossRefGoogle Scholar
  35. [35]
    A. Weron, Stable processes and measures: A survey, Lecture Notes in Math. 1080, 306–364, Springer-Verlag 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Weron, A.K. Rajagopal, and K. Weron, Canonical decomposition of dynamical semigroups; Irreversibility and entropy, Preprint 1984.Google Scholar
  37. [37]
    B.J. West and V. Seshadri, Linear systems with Lévy fluctuations, Physics 113A (1982), 203–216.MathSciNetGoogle Scholar
  38. [38]
    V.M. Zolotarev, One-dimensional Stable Distributions, (in Russian), Nauka, Moscow 1983.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Aleksander Weron
    • 1
  • Karina Weron
    • 2
  1. 1.Center for Stochastic Processes Department of StatisticsUniversity of North CarolinaChapel Hill
  2. 2.Department of Physics and AstronomyLouisiana State UniversityBaton Rouge

Personalised recommendations