The Almighty Chance

The following sections are included:

• PREFACE

• CONTENTS

The problems of physics of random media are always a source of novel ideas. This field of science is associated in a versatile way with real complexity of random media (as an ensemble of enormous numbers of molecules that form macroscopic bodies, or a plentitude of disordered motions typical of turbulence, etc.). Following the well-known aphorism of David Hilbert, we may only wonder how physicists can deal with a random medium though the subject may be too complicated for them. The simplest fruitful approach consists in the derivation and solution of equations for macroscopic bodies or averaged quantities, e.g., temperature which is essentially the averaged kinetic energy of molecular motion. A problem once proposed by l'Académie Française which considered heat propagation in solids for given boundary conditions has become a basis of classical mathematical physics. J.B. Fourier solved this problem in 1807 and 1811 by separation of variables. This solution brought about such important concepts as the Fourier series and integral, eigenvalues, and eigenfunctions. Later, methods of solution of other equations of mathematical physics were formulated, in particular equations of elastic oscillations and the Maxwell and Schrödinger equations.

The following sections are included:

• CLASSICAL PROBABILITY

• PROBABILITY AS A MEASURE

• THE CENTRAL LIMIT THEOREM

• THE LEAST SQUARES METHOD

• RANDOM FUNCTIONS, FIELDS AND OPERATORS

• ERGODIC PROPERTIES, CHAOS, INTERMITTENCY

• EQUILIBRIUM AND TRANSITION PROBABILITY

• SPATIAL DISTRIBUTION OF PARTICLES IN GASES

The following sections are included:

• THE DIFFUSION COEFFICIENT

• SOLUTIONS OF THE DIFFUSION EQUATION

• DIFFUSION IN INFINITE SPACE AND GREEN'S FUNCTION

• PROBABILISTIC INTERPRETATION OF DIFFUSION

• DIFFUSION IN EXTERNAL FORCE FIELD

• OSMOTIC PRESSURE

• ON REVERSIBILITY OF DIFFUSION IN TIME

The following sections are included:

• EINSTEIN'S DESCRIPTION OF BROWNIAN MOTION

• THE WIENER PATH

• AVERAGING OVER PATHS

• THE WIENER AND FEYNMAN INTEGRALS

• THE WHITE NOISE

• THE ORNSTEIN-UHLENBECK PROCESS

The following sections are included:

• HISTORICAL PERSPECTIVE

• FRACTALS: THICK LINES

• FRACTALS: THICK SURFACES

• FRACTALS: A FOAMY SPACE-TIME

• FRACTALS: CONTOUR LINES

• FRACTALS: DENSE POINT SET

• FRACTALS AND SELF-SIMILARITY

• STRANGE ATTRACTORS

• MEASURING THE ATTRACTOR DIMENSION

The following sections are included:

• CLUSTERS AND PERCOLATION

• INTERMITTENCY AND PERCOLATION

• EIGEN-OSCILLATIONS OF A REGION WITH RANDOM BOUNDARY

• THE ZEROS OF EIGENFUNCTIONS OF FREE AND FORCED OSCILLATIONS

• THE MATHEMATICAL LANGUAGE OF THE PERCOLATION THEORY

The following sections are included:

• THE TURBULENT FLOW WHOSE VELOCITY IS LAMINAR

• THE ROLE OF CHANCE IN TRANSITION TO TURBULENCE

• THE DEVELOPED TURBULENCE

• INTERNAL STRUCTURE AND FRACTAL PROPERTIES OF DEVELOPED TURBULENCE

• TEMPERATURE SPOTS IN RANDOM FLOW

The following sections are included:

• INTERMITTENCY OF A RANDOM QUANTITY

• EVOLUTION OF A RANDOM QUANTITY

• A RANDOM MEDIUM

• THE SHORT-CORRELATED RANDOM MEDIUM

• INTERMITTENCY IN NON-STATIONARY MEDIUM

• STRUCTURE OF THE MOMENTS OF A SCALAR FIELD IN RANDOM MEDIUM

• VORTICITY AND DEFORMATIONS IN A FLOW OF INCOMPRESSIBLE FLUID

• VORTEX FILAMENTS IN HYDRODYNAMICS

• INTERMITTENCY IN NONLINEAR MEDIA

• PROPAGATION OF FIELD FRONTS IN RANDOM MEDIA

• THE INTERMITTENT WORLD

The following sections are included:

• THE KINEMATIC DYNAMO PROBLEM

• THE SHORT-CORRELATED RANDOM FLOW

• GENERATION OF THE MEAN MAGNETIC FIELD

• FLUCTUATIONS OF HELICITY

• HIGHER STATISTICAL MOMENTS AND INTERMITTENCY IN THE FAST DYNAMO

• STRUCTURE OF THE MAGNETIC FIELD

• THE TYPICAL STRUCTURE OF MAGNETIC FIELD

• SELF-EXCITATION OF MAGNETIC FIELDS IN FINITE BODIES

• MAGNETIC FIELD INTERMITTENCY WITH ACCOUNT OF THE LORENTZ FORCE

• THE DYNAMO THEOREM

The following sections are included:

• THE UNIVERSE

• CONSERVATION OF CHARGES

• CONSERVATION OF ENERGY

• THE ASTROPHYSICAL CONSEQUENCES. IS THE PULSATING UNIVERSE INEVITABLE?

• THE SPONTANEOUS FORMATION

The following sections are included:

• EPILOGUE: THE BIRTH OF “DIVINAMICS”

• REFERENCES