Turbulent Motion and the Structure of Chaos

analyzing the experimental data and constructing math.ematical models of the processes under study, one has to rely rather on the physical intuition than on the strict calculations. Now let us go one step higher and explain the main title of the book. The concepts of "laminary" and "turbulent" motions were first introduced in hydrodynamics. Since the old days these concepts have considerably broadened; now the laminar and the turbulent motions have been discovered and investigated at all levels of description of nonequilibrium processes in the open systems, from kinetics to reaction diffusion. In any case, one of the principal characteristics of the turbulent motion is the existence of a large number of well-developed macroscopic degrees of freedom. For this reason the turbulent motion is extremely complicated and to a large extent unpredictable. As the laminar and the turbulent flows play an important role in the processes of evolution in the open systems, and in particular, in the processes of self-organization, the need arises for assessing the relative degree of order of laminar and turbulent motions, and also for comparing the degree of order of various turbulent motions. Without being able to make such estimates it will be impossible to determine whether the evolution is going towards higher or towards lower organization when one turbulent state is replaced by another.

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I.1. The Criteria of the Relative Degree of Order of the States of Open Systems.- I.2. Connection Between the Statistical and the Dynamic Descriptions of Motions in Macroscopic Open Systems. The Constructive Role of Dynamic Instability of Motion.- I.3. The Transition from Reversible to Irreversible Equations. The Gibbs Ensemble in the Statistical Theory of Nonequilibrium Processes.- I.4. The Role of Fluctuations at Different Levels of Description. Fluctuation Dissipation Relations.- I.5. Brownian Motion in Open Systems. Molecular and Turbulent Sources of Fluctuations.- I.6. Laminar and Turbulent Motion.- 1. Evolution of Entropy and Entropy Production in Open Systems.- 1.1. Chaos and Order. The Controlling Parameters. Physical Chaos. Evolution and Self-Organization in Open systems.- 1.2. Boltzmann-Shannon-Gibbs Entropy.- 1.3. Entropy Distribution Function.- 1.4. The Gibbs Ensemble. Smoothing over the Physically Infinitesimal Volume. Entropy Including Fluctuations. Space (Time) and Phase Averages. Local Ergodicity Condition.- 1.5. The Kinetic Boltzmann Equation for Statistical and Smoothed Distribuions. Physically Infinitesimal Scales. The Constructive Role of the Dynamic Instability of Motion of Atoms in a Gas.- 1.6. The Role of Nonequilibrium Fluctuations in a Boltzmann Gas. Molecular and Turbulent Sources of Fluctuations.- 1.7. The Kinetic Equations for the N-Particle Distribution Functions. The Leontovich Equation.- 1.8. Boltzmann's H-Theorem for Smoothed (Pulsating) and Deterministic Distributions.- 1.9. Entropy and Entropy Production for Smoothed and Deterministic Distributions.- 1.10. The Gibbs' Theorem.- 1.11. H-Theorem for Open Systems. Kulback Entropy.- 1.12. Evolution in the Space of Controlling Parameters. The S-Theorem.- 1.13. The S-Theorem. LocalFormulation.- 1.14. The Comparison of the Relative Degree of Order of States on the Basis of the S-Theorem Using Experimental Data.- 1.15. Dynamic and Statistical Descriptions of Complex Motions. K-Entropy, Lyapunov Indices. Nonlinear Characteristics of the Trajectory Divergence.- 1.16. Criteria of Dynamic Instability of Motion in Statistical Theory.- 1.17. Entropy as Measure of Diversity in Biological Evolution.- 1.18. The Principle of Minimum Entropy Production in Self-Organization Processes.- 2. Transition From the Reversible Equations of Mechanics to the Irreversible Equations of the Statistical Theory.- 2.1. Two Types of Reversible Processes. Symmetry Properties of Distribution Functions.- 2.2. Liouville Equation and Vlasov Equation. The First Moments and the "Collisionless" Approximations.- 2.3. Reversible Equations in Quantum Statistical Theory.- 2.4. Two Types of Dissipative Kinetic Equations for N-Particle Distributions.- 2.5. Measure of Deficiency (Incompleteness) of the Statistical Description.- 2.6. The Hierarchy of Equations of Fluid Mechanics.- 3. Fluctuation Dissipation Relations.- 3.1. Examples of Fluctuation Dissipation Relations.- 3.2. FDR for N-Particle Distribution Functions.- 3.3. Thermodynamic Form of the FDR. The Callen-Welton Formula.- 3.4. FDR for a Boltzmann gas. The Fluctuative Representation of Boltzmann Collision Integral.- 3.5. FDR for Large-Scale (Kinetic) Fluctuations.- 3.6. Examples of FDR for Large-Scale Fluctuations.- 3.7. System of Quantum Atoms Oscillators.- 3.8. Fluctuation Dissipation Relations in Hydrodynamics.- 3.9. Two Ways of Defining Kinetic Coefficients.- 3.10. The Molecular Langevin Source in the Difffusion Equation.- 3.11. Connection between the Intensities of Langevin Sources and the Correlator of of Phase DensityFluctuations.- 3.12. Natural Flicker Noise ("1/f Noise"). FDR for Flicker Noise.- 3.13. Natural Flicker Noise and Superconductivity.- 4. Brownian Motion.- 4.1. Fokker-Planck and Langevin Equations.- 4.2. Three Definitions of the Langevin and Fokker-Planck Equations.- 4.3. The Fokker-Planck Equations in the Statistical Theory of Nonequilibrium Processes.- 4.4. Transition to the Fokker-Planck Equation from the Smoluchowski equation (the Chapman-Kolmogorov Equation) and from the Master Equation.- 4.5. Langevin.Sources in Kinetic Equations.- 4.6. Langevin Sources in Fokker-Planck and Einstein-Smoluchowski Equations.- 4.7. Turbulent Langevin Sources and Fluctuation Dissipation Relations in Hydrodynamics.- 4.8. Brownian Motion in Systems with a Variable Number of Particles..- 5. The Boltzmann-Gibbs-Shannon Entropy As Measure of the Relative Degree of Order in Open Systems.- 5.1. Van der Pol Generator.- 5.2. Generator with Inertial Nonlinearity.- 5.3. Invariant Measures. Examples of Gibbs Distributions for Open Systems.- 5.4. Generalized Van Der Pol Generators. Bifurcations of the Limiting Cycle Energy and the Period of Oscillations.- 5.5. Dynamic and Statistical Distributions.- 5.6. Comparison Between the Degrees of Order in the Bifurcation Points and in the State of Dynamic Chaos.- 5.7. Evolution of Entropy in Systems with Two Controlling Parameters.- 5.8. A Medium of Linked Generators. The Kinetic Approach in the Theory of Self-Organization.- 5.9. A System of Van der Pol Generators with Common Feedback. Associative Memory and Pattern Recognition.- 5.10. Kinetic Description of Chemically Reacting Systems.- 5.11. A Medium of Bistable Elements. The Kinetic Approach in the Theory of Phase Transitions.- 6. Turbulent Motion. The Structure of Chaos.- 6.1. CharacteristicFeatures of Turbulent Motion. The Main Problems.- 6.2. Incompressible Fluid. Reynolds Equations. Reynolds Stresses.- 6.3. Well-Developed Turbulence. Turbulent Viscosity.- 6.4. Semiempirical Prandtl-Karman Theory of Turbulence.- 6.5. Onset of Turbulence in Steady Couette and Poiseuille Flows.- 6.6. Entropy Production in Laminar and Turbulent Flows.- 6.7. The Principle of Least Dissipation and the Principle of Minimum Entropy Production in Self-Organization Processes.- 6.8. Evolution of Entropy in the Transition from Laminar to Turbulent Flow.- 6.9. Kinetic Description of Hydrodynamic Motion.- Conclusion.- References.