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Ideas in the theory of random media. (English) Zbl 0728.73011
Summary: 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 an n-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.

MSC:
74A40 Random materials and composite materials
76W05 Magnetohydrodynamics and electrohydrodynamics
82D20 Statistical mechanical studies of solids
86A15 Seismology (including tsunami modeling), earthquakes
81P20 Stochastic mechanics (including stochastic electrodynamics)
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