R.G. Zaripov. Self-organization аnd irreversibility in nonextensive systems. Kazan: Rep. Tat. Асаd. Sci. Publishing House "Fen" 2002, 251 р., ISBN 5-7544-0196-5
This book is devoted to new results of statistic theory of nonextensive systems concerned, in the first place, theoretic and informational approachs to investigation of self-organization and irreversibility. On the basis of half-norm definition in q-spase of probability, known and new measures of order and disorder are derived by statistic, group and variation approachs. General S-theorem and I-theorem of there measures on evolution in time and in space of control parameters are started and proved. New ways of solution of irreversibility problems are proposed, techniques of kinetic equations building are developed and statistical criteria of equilibrium stability and of time evolution of systems are derived. Measures in the multifractal theory are considered from unified position.
For researchers, under- and postgraduate students learning irreversible processes in open systems.
THE AUTHOR: Rinat G. Zaripov (b. 1947), graduated from Kazan State University in 1970. Doctor of Sciences (Phys.-Math.), Professor at Kazan А.N.Tupolev State Technical University, Deputy Director and Laboratory Head of Continuum Mechanics. Institute of Mechanics & Engineering the Russian Academy of Sciences, Kazan Science Center, Russian Асаd. of Sci., 2/31, Lоbachevsky Str., Kazan, 420111, Russia.
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Chapter 1. Gibbs statistical model
Hamilton motion equations. Evolution of mechanical system.
Lagrangian and principle of minimum of action function
Motion equation of phase volume. Theorem of Liuvillian measure conservation.
Probabilities and mean values by measure
Kullback discrimination information
Liuvillian kinetic equation. Evolution of statistical system
Neumann quantum equation. Evolution of quantum system
Physical statistics. Nonequilibrium entropy and discrimination information.
Chapter 2. Disorder-order transitions
Gibbsian measure for equilibrium states.
Uniform family of measures in space of intensive parameters.
Principle of Boltzmann-Gibbs-Shannon entropy maximum.
Uncertainty relation for equilibrium state. Bloch equation
Multimeasure equilibrium distribution. Stability condition.
Principle of Kullback discrimination information minimum.
Spontaneous transitions and thermodinamics of information processes in open systems.
Gibbs theorem and H-theorem. Self decay
Forced transitions and self-organization. I-theorem.
I-theorem for quantum systems.
Self-organization of Fermi- and Bose gases.
Boundaries of variation of degree of order. h-theorem.
Nonparametric estimate of degree of order
Instability of disorder and local I-theorem
Chapter 3. Irreversible kinetic equations
Kinetic equation with source. Statistical criterion of evolution
Nonequilibrium statistical principle
Nonequilibrium variational principle
Irreversibility and evolution in open systems.
Quantum kinetic equation with source
Chapter 4. Statistical model of nonextensive systems.
Probability q-space. Half-norms and statistical characteristics
Half-norms of distributions.
Golder inequality. Tsallis entropy and discrimination information
Principle of half-norm minimum
Types of q-entropies and q-discrimination informations.
Parametrizated distribution. Renyi entropy.
Extremum of Renyi entropy. Tsallis definition
Generalized parametrizated distribution.
Renyi discrimination information
Extremum of Renyi discrimination information. Tsallis definition.
Fluctuations of nonequilibrium microscopic entropy and discrimination information
Half-norms and multifractal measures
Two-parametrical discrimination information
Chapter 5. Self-decay and self-organization
Tsallis measures for equilibrium states.
Low boundary for half-norm.
Half-norm minimum and equilibrium q-distribution
Parametrizated distribution and equilibrium states.
Equilibrium microscopic q-entropies and their fluctuations.
Equations for equilibrium q-distribution
Spontaneous transitions and generalized Gibbs theorem. H-theorem.
S-theorem and I-theorem for classic nonextensive systems
Generalized statistical criterion of stability
Thermal stability of equilibrium.
Self-organization in chaotic dynamics
Chapter 6. Evolution and irreversibility.
Microscopic q-entropy production. Generalized kinetic equation with source.
Generalized statistical criterion of evolution
Schwartz inequality in q-space and kinetic equation.
Direction of nonextensive systems evolution.
Entropy and irreversibility
Evolution in diffusion processes.
Nonequilibrium principle of minimum of variance
of microscopic entropy production.
Chapter 7. Some mathematical problems of statistical models.
Equilibrum state with effective Hamiltonian
Equation for distribution with effective Hamiltonian.
Group of macroscopic q-entropy and its notation
Isomorphism of group of matrixes.
Group of microscopic q-entropy and its notation.
Groups of macroscopic and microscopic energies.
Fluctuations and superoperators.