Acta Applicandae Mathematica

, Volume 22, Issue 2–3, pp 139–282

Ideas in the theory of random media

  • Stanislav A. Molchanov
Article

Abstract

These lectures discuss the ideas of localization, intermittency, and random fluctuations in the theory of random media. These ideas are compared and contrasted with the older approach based on averaging. Within this framework, the topics discussed include: Anderson localization, turbulent diffusion and flows, periodic Schrödinger operators and averaging theory, longwave oscillations of elastic random media, stochastic differential equations, the spectral theory of Hamiltonians with (an infinite sequence of) wells, random Schrödinger operators, electrons in a random homogeneous field, influence of localization effects on the propagation of elastic waves, the Lyapunov spectrum (Lyapunov exponents), the Furstenberg and Oseledec theorems for ann-tuple of identically distributed unimodular matrices and their relation with the spectral theory of random Schrödinger or string operators, Rossby waves, averaging on random Schrödinger operators, percolation mechanisms, the moments method in the theory of sequences of random variables, the evolution of a magnetic field in the turbulent flow of a conducting fluid or plasma (the so-called kinematical dynamo problem), heat transmission in a randomly flowing fluid.

AMS subject classifications (1985)

73B35 81C20 82A55 86A15 76W05 

Key words

Averaging localization intermittency Schrödinger operators magnetic fields random media turbulent diffusion magnetohydrodynamics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    Akanbajev, V.: Some problems of the theory of magnetic fields in random flows (Doctoral thesis) MGU, Moscow (in Russian) (1986)Google Scholar
  2. (2).
    Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev.109, 1492–1503 (1958)Google Scholar
  3. (3).
    Anshelevich, V.V., Khanin, K.M. and Sinai, Ja.G.: Symmetrical random walks in random environments. Comm. Math. Phys.85, 449–470 (1982).Google Scholar
  4. (4).
    Arnold, V.I. and Korkina, E.I.: Growth of magnetic field in three-dimensional incompressible stationary flow. Vestn. MGU, ser. 1, Math., Mech.3, 43–45 (in Russian) (1985)Google Scholar
  5. (5).
    Arnold, L., Papanicolaou, G. and Wihstutz, V.: Asymptotic analysis of the random Lyapunov exponent and rotation number of the random oscillator and applications, SIAM, J. Appl. Math.46 (3) 427–450 (1986)Google Scholar
  6. (6).
    Bakhvalov, N.S. and Panasenko, G.R.: Averaging of Processes in Periodic Media. Moscow:Nauka (in Russian) (1984)Google Scholar
  7. (7).
    Beljaev, M.Ju.: Averaging description of the wave processes in the random media. Appl. Math. Mech.49 (4), 696–700 (1985)Google Scholar
  8. (8).
    Beljaev, A.Ju.: On the Lyapunov exponent of one-dimensional wave equation with random coefficients. Vestn. MGU, ser. 1, Math., Mech.3 17–21 (in Russian) (1987).Google Scholar
  9. (9).
    Berdichevsky, V.L.: Variational Principles of Mechanics of the Continuous Media. Moscow: Nauka (in Russian) (1983)Google Scholar
  10. (10).
    Bogachev, L.V. and Molchanov, S.A.: Models of mean field in the theory of random media. Theor. Math. Phys.81 (2) 281–290 (in Russian) (1989)Google Scholar
  11. (11).
    Braginsky, S.I.: On the theory of a hydromagnetic dynamo. Zh. Eksp. Teor. Fiz.47 2178–2193 (in Russian) (1964)Google Scholar
  12. (12).
    Bulicheva, O.G. and Molchanov, S.A.: Averaging description of the random one-dimensional media. Vestn. MGU, ser. 1, Math., Mech.3 37–46 (in Russian) (1986).Google Scholar
  13. (13).
    Dittrich, P., Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Mean magnetic field in renovating random flow. Astr. Nachr.305 (3) 119–125 (1984)Google Scholar
  14. (14).
    Dichne, A.M.: Conductivity of two-phase system. Zh. Eksp. Teor. Fiz.7 110–116 (in Russian) (1970).Google Scholar
  15. (15).
    Dynamical systems. In: Itogy nauki i techniki, fundamental directions,2 Moscow: VINITI (in Russian) (1985)Google Scholar
  16. (16).
    Feller, W.: An Introduction to Probability Theory and its Applications. Vol. 2, New York, London, Sidney: John Wiley & Sons (1966).Google Scholar
  17. (17).
    Friedman, A.: Partial Differential Equations. New York:Holt, Rinehard and Winston. (1969).Google Scholar
  18. (18).
    Furstenberg, H.: Noncommuting random products. Trans. Amer. Math. Soc.108 (2) 377–428 (1969)Google Scholar
  19. (19).
    Gikhman, I.I. and Skorokhod, A.V.: Theory of Random Processes, Vol. 1–3. Moscow: Nauka (in Russian) (1971–75)Google Scholar
  20. (20).
    Goldsheid, I.Ja., Molchanov, S.A. and Pastur, L.A.: A pure point spectrum of the stochastic one-dimensional Schrödinger equation. Funct. Anal. Appl.11 1–10 (in Russian) (1977)Google Scholar
  21. (21).
    Golytsina, A.G. and Molchanov, S.A.: Multidimensional model of rare scatterers. Dokl. Akad. Nauk S.S.S.R.283 1084–1086 (in Russian) (1985)Google Scholar
  22. (22).
    Gordon, A.Ja.: On the continuous spectrum of one-dimensional Schrödinger operators.13 (3) 77–78 (in Russian) (1979)Google Scholar
  23. (23).
    Gradshtein, I.S. and Ryzhik, I.M.: Tables of Integrals, Sums, Series, Products. Moscow: Fizmatgiz (in Russian) (1963)Google Scholar
  24. (24).
    Grenkova, L.N., Molchanov, S.A. and Sudarev, Ju.N.: On the basic states of one-dimensional disordered structures. Comm. Math. Phys.90 101–123 (1983)Google Scholar
  25. (25).
    Griffits, J.: Systems of interacting cellautomats. Contemp. Math.41 57–64 (1986)Google Scholar
  26. (26).
    Ibragimov, I.A. and Linnik, Ju.V.: Independent and Stationary Connected Random Variables. Moscow: Nauka (in Russian) (1965)Google Scholar
  27. (27).
    Jurinsky, V.V.: On wave propagation in one-dimensional random media. Preprint 9, Mathematical Institute Sib. Dep. Acad. of Sci. U.S.S.R. (in Russian) (1982)Google Scholar
  28. (28).
    Kamenkovich, V.L. and Reznik, G.M.: Rossby waves. In: Physics of the Ocean, Vol. 2. Moscow: Nauka (in Russian) (1978)Google Scholar
  29. (29).
    Kac, M.: Mathematical models of phase transitions. In: Stability and Phase Transitions. Moscow:Mir (in Russian) (1973)Google Scholar
  30. (30).
    Kotani, S.: Lyapunov's exponents and spectra for one-dimensional random Schrödinger operators. Contemp. Math.50 277–286 (1986)Google Scholar
  31. (31).
    Kotani, S.: On a inverse problem for series. Contemp. Math.41 267–280 (1985)Google Scholar
  32. (32).
    Kozlov, S.M.: Averaging method and random walks in nonhomogeneous media. Usp. Math. Nauk40 (2) 61–120 (in Russian) (1985)Google Scholar
  33. (33).
    Kozlov, S.M. and Molchanov, S.A.: On conditions under which central limit theorem is applicable to random walks on lattices. Dokl. Akad. Nauk S.S.S.R.273 410–413 (in Russian) (1984)Google Scholar
  34. (34).
    Kozlov, S.M.: On the value of effective diffusion when the concentration of inclusions is small. First World Congress of the Bernoulli Society, Tashkent, Abstracts of Reports2 656 (in Russian) (1986)Google Scholar
  35. (35).
    Krause, F. and Rädler, K.H.: Meanfield Magnetohydrodynamics and Dynamo Theory. London:Pergamon Press (1980).Google Scholar
  36. (36).
    Larmor, J.: How could a rotating body such as the Sun become a magnet? Rep. Brit. Sci. 159–160 (1919)Google Scholar
  37. (37).
    Lifshitz, I.M., Azbel, M.Ja. and Kaganov, M.I.: Electronic Theory of Metals. Moscow: Nauka (in Russian) (1971)Google Scholar
  38. (38).
    Lifshitz, I.M., Gredeskul, S.A. and Pastur, L.A. Introduction to the Theory of Disordered Media. Moscow:Nauka (in Russian) (1982)Google Scholar
  39. (39).
    Malyshev, V.A. and Milnos, R.A.: Gibbs Random Fields. Moscow:Nauka (in Russian) (1985)Google Scholar
  40. (40).
    Marchenko, V.A. and Khruslov, E.Ja.: The Boundary-Value Problems in a Domain with a Fine-Grain Boundary. Kijev:Naukova Dumka (in Russian) (1974)Google Scholar
  41. (41).
    Martinelli, F. and Scoppola, E.: Introduction to the mathematical theory of Anderson localization. Nota Interna871 Rome.Google Scholar
  42. (42).
    Menaguzzi, M., Frisch, V. and Pouquet, A.: Helical and nonhelical magnetic dynamo. Phys. Rev. Lett.47 1060–1064 (1981)Google Scholar
  43. (43).
    Menshikov, M.V., Molchanov, S.A. and Sidorenko, A.F.: Percolation theory and its application. In: Itogi nauki i techniki, probab. theory, math. stat. and cyb.24 53–110 (in Russian) (1986)Google Scholar
  44. (44).
    Merkurjev, S.P. and Faddajev, L.D.: Quantum theory for several particle systems. Moscow:Nauka (in Russian) (1985)Google Scholar
  45. (45).
    Michailov, A.S. and Uporov, I.V.: Critical effects in media with multiplication, decay and diffusion. Usp. Fiz. Nauk144 79–112 (in Russian) (1984)Google Scholar
  46. (46).
    Moffat, H.K.: Some developments in the theory of turbulance. J. Fluid Mech.106 27–47 (1981)Google Scholar
  47. (47).
    Molchanov, S.A., Piterbarg, L.I. and Sokoloff, D.D.: On generation of the coarse-scale anomalies of ocean surface temperature by short-period atmospheric processes. Izv. Akad. Nauk. S.S.S.R., ser. Physics of Atmosphere and Ocean5 539–545 (in Russian) (1987)Google Scholar
  48. (48).
    Molchanov, S.A. and Piterbarg, L.I.: Averaging in turbulent diffusion problems. In: Probability Theory and Random Processes 35–47 Kijev:Naukova Dumka (in Russian) (1987).Google Scholar
  49. (49).
    Molchanov, S.A. and Piterbarg, L.I.: Turbulent diffusion of gradients of admixtures. Dokl. Akad. Nauk S.S.S.R.293 (5) 1092–1096 (in Russian) (1986).Google Scholar
  50. (50).
    Molchanov, S.A. Piterbarg, L.I., Ruzmaikin, A.A. and Sokoloff, D.D.: Variability of temperature field of the ocean surface. Dokl. Akad. Nauk S.S.S.R.283 1801–1803 (in Russian) (1985)Google Scholar
  51. (51).
    Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Dynamo equations in a random short-term correlated velocity field. Magnitnaja gidrodinamika4 67–73 (in Russian) (1983)Google Scholar
  52. (52).
    Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Dynamo theorem. Geophys. Astrophys. Fluid Dyn.30 242–259 (1984)Google Scholar
  53. (53).
    Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Explicitly solvable model of a hydromagnetic dynamo. Dokl. Akad. Nauk S.S.S.R.295 113–117 (in Russian) (1987)Google Scholar
  54. (54).
    Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Kinematic dynamo in random flow. Usp. Fiz. Nauk145 593–628 (in Russian) (1985)Google Scholar
  55. (55).
    Molchanov, S.A. and Seide, H.: Spectral properties of the general Sturm-Liouville operator with random coefficients. Math. Nachr.109 57–78 (1982)Google Scholar
  56. (56).
    Molchanov, S.A.: The structure of eigenfunctions of one-dimensional disordered structures. Math. U.S.S.R. Izv.12 (1973).Google Scholar
  57. (57).
    Molchanov, S.A., Ruzmaikin, A.A., Sokoloff, D.D., and Zeldovich, Ja.B.: Diffusion and intermittency in non-linear random media. Proc. Acad. Sci. U.S.A.305 1095–1102 (1987)Google Scholar
  58. (58).
    Monin, A.S. and Yaglom, A.M.: Statistical Hydromechanics, Vol. 2. Moscow:Nauka (in Russian) (1967)Google Scholar
  59. (59).
    Mott, N.F.: Electrons in disordered structures. Advances in Physics (Phil. Mag. Suppl.)16 61–79 (1967)Google Scholar
  60. (60).
    Novikov, V.G., Ruzmaikin, A.A. and Sokoloff, D.D.: Fast dynamo in a reflexively invariant random velocity field. Zh. Eksp. Teor. Fiz.85 902–918 (in Russian) (1985)Google Scholar
  61. (61).
    Oseledets, V.I.: Multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc.19 197–231 (in Russian) (1968)Google Scholar
  62. (62).
    Pastur, L.A.: Spectral theory of the random self-adjoint operators. In Itogi nauki i techniki, ser. prob. theory, math. stat. cyb.25 3–67 (in Russian) (1987)Google Scholar
  63. (63).
    Piterbarg, L.I.: Dynamics and forecast of coarse-scale anomalies of the ocean surface. Doctoral thesis, Institute of Oceanology Acad. Sci. U.S.S.R., Moscow (in Russian) (1987)Google Scholar
  64. (64).
    Rayleigh, J.W.: On the influence of obstacles arranged in rectangular order upon the properties of a medium. Scientific Papers, Cambridge Univ. Press4 19–38 (1903)Google Scholar
  65. (65).
    Reed, M. and Simon, B.: Methods of Modern Mathematical Physics. New York: Academic Press (1973–79).Google Scholar
  66. (66).
    Sadovsky, M.A.: On geophysical media and seismical process models. In: Prediction of Earthquakes, Dushanbe-Moscow: Donish 268–273 (in Russian) (1983–84)Google Scholar
  67. (67).
    Semenov, D.V.: Equations of 2nd moment for magnetic field with helicity. Vest. MGU, ser. Phys. 15–21 (in Russian) (1987)Google Scholar
  68. (68).
    Sevastjanov, B.A.: Branching Processes. Moscow:Nauka (in Russian) (1971)Google Scholar
  69. (69).
    Shklovsky, B.N. and Efros, A.L.: Electronic Properties of Legiered Semiconductors. Moscow:Nauka (in Russian) (1979)Google Scholar
  70. (70).
    Simon, B.: Almost periodic Schrödinger operators. Ann. Phys. U.S.A.159 (1) 157–183 (1985)Google Scholar
  71. (71).
    Simon, B., Taylor, M. and Wolff, T.: Some rigorous results for the Anderson model. Phys. Rev. Lett.54 1589–1600 (1985)Google Scholar
  72. (72).
    Simon, B. and Souillard, B.: Franco-American Meeting on the mathematics on random and almost periodic potentials. J. Stat. Phys.36 (1–2) 273–288 (1984)Google Scholar
  73. (73).
    Simon, B. and Wolff, T.: Singular continuous spectrum under rank one perturbation and localization for random hamiltonians. Comm. Pure Appl. Math.39 (1) 75–90 (1986)Google Scholar
  74. (74).
    Sinai, Ya.G.: Limit behaviour of the one-dimensional random walk in random media. Teorija verojatn. i prilozhen.27 (2) 247–258 (in Russian) (1982)Google Scholar
  75. (75).
    Souillard, B.: Mathematical and physical properties of discrete and continuous random Schrödinger operators: a review. Chaotic Behav. Quantum Syst.: Theory and Appl. Proc. NATO Adv. Res. Workshop Quantum Chaos, Como, 20–25 June 1983. New York, London:Plenum 1–10.Google Scholar
  76. (76).
    Stratonovich, R.L.: Conditional Markov Processes. Moscow:Izd. MGU (in Russian) (1986)Google Scholar
  77. (77).
    Taylor, G.I.: Diffusion by continuous movements. Proc. Lond. Math. Soc.A.20 (1921)Google Scholar
  78. (78).
    Tutubalin, V.N.: Central limit theorem for products of random matrices and some of its applications. Sympos. Math.21 101–116 (1977)Google Scholar
  79. (79).
    Virtser, A.D.: On matrix and operator products. Teor. Verojatn. Prilozhen.24 (2) 360–370 (1979)Google Scholar
  80. (80).
    Vishik, M.M.: Periodic dynamo. Vichislytelnaja seimologija Moscow:Nauka19 125–186 (1986)Google Scholar
  81. (81).
    Zeldovich, Ja.B., Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Intermittent passive fields in random media. Zh. Eksp. Teor. Fiz.89 (6) 2061–2072 (in Russian) (1985)Google Scholar
  82. (82).
    Zeldovich, Ja.B., Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Moments and intermittency in random media. Usp. Mat. Nauk41 (4) (in Russian) (1986).Google Scholar
  83. (83).
    Zeldovich, Ja.B., Molchanov, S.A., Ruzmaikin, A.A. and Sokoloff, D.D.: Intermittency in random media. Usp. Fiz. Nauk152 (1) 3–32 (1987)Google Scholar
  84. (84).
    Zeldovich, Ja.B., Molchanov, S.A., Sokoloff, D.D., and Ruzmaikin, A.A.: Generating, diffusion, intermittency in random fields. Soviet Scientific Review in Mathematical Physics. London:Gordon and Breach7 1–120 (1987)Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Stanislav A. Molchanov
    • 1
  1. 1.Moscow State UniversityMoscowU.S.S.R.

Personalised recommendations