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The book is devoted to the mathematical foundations of nonextensive statistical mechanics. This is the first book containing the systematic presentation of the mathematical theory and concepts related to nonextensive statistical mechanics, a current generalization of Boltzmann-Gibbs statistical mechanics introduced in 1988 by one of the authors and based on a nonadditive entropic functional extending the usual Boltzmann-Gibbs-von Neumann-Shannon entropy. Main mathematical tools like the q-exponential function, q-Gaussian distribution, q-Fourier transform, q-central limit theorems, and other related objects are discussed rigorously with detailed mathematical rational. The book also contains recent results obtained in this direction and challenging open problems. Each chapter is accompanied with additional useful notes including the history of development and related bibliographies for further reading.
The book is devoted to the mathematical foundations of nonextensive statistical mechanics. This is the first book containing the systematic presentation of the mathematical theory and concepts related to nonextensive statistical mechanics, a current generalization of Boltzmann-Gibbs statistical mechanics introduced in 1988 by one of the authors and based on a nonadditive entropic functional extending the usual Boltzmann-Gibbs-von Neumann-Shannon entropy. Main mathematical tools like the q-exponential function, q-Gaussian distribution, q-Fourier transform, q-central limit theorems, and other related objects are discussed rigorously with detailed mathematical rational. The book also contains recent results obtained in this direction and challenging open problems. Each chapter is accompanied with additional useful notes including the history of development and related bibliographies for further reading.
This book focuses on nonextensive statistical mechanics, a current generalization of Boltzmann-Gibbs (BG) statistical mechanics. Conceived nearly 150 years ago by Maxwell, Boltzmann and Gibbs, the BG theory, one of the greatest monuments of contemporary physics, exhibits many impressive successes in physics, chemistry, mathematics, and computational sciences. Presently, several thousands of publications by scientists around the world have been dedicated to its nonextensive generalization. A variety of applications have emerged in complex systems and its mathematical grounding is by now well advanced. Since the first edition release thirteen years ago, there has been a vast amount of new results in the field, all of which have been incorporated in this comprehensive second edition. Heavily revised and updated with new sections and figures, the second edition remains the go-to text on the subject. A pedagogical introduction to the BG theory concepts and their generalizations – nonlinear dynamics, extensivity of the nonadditive entropy, global correlations, generalization of the standard CLT’s, complex networks, among others – is presented in this book, as well as a selection of paradigmatic applications in various sciences together with diversified experimental verifications of some of its predictions. Introduction to Nonextensive Statistical Mechanics is suitable for students and researchers with an interest in complex systems and statistical physics.
This book is a printed edition of the Special Issue "New Trends in Statistical Physics of Complex Systems" that was published in Entropy
Preface, Murray Gell-Mann and Constantino Tsallis. Nonextensive Statistical Mechanics: Construction and Physical Interpretation, Constantino Tsallis. Generalized Nonadditive Information Theory and Quantum Entanglement, Sumiyoshi Abe. Unifying Laws in Multidisciplinary Power-Law Phenomena: Fixed-Point Universality and Nonextensive Entropy, Alberto Robledo. Nonextensive Entropies and Sensitivity to Initial Conditions of Complex Systems, Marcelo L. Lyra. Numerical Analysis of Conservative Maps: A Possible Foundation of Nonextensive Phenomena, Fulvio Baldovin. Nonextensive Effects in Hamiltonian S.
Phase space, ergodic problems, central limit theorem, dispersion and distribution of sum functions. Chapters include Geometry and Kinematics of the Phase Space; Ergodic Problem; Reduction to the Problem of the Theory of Probability; Application of the Central Limit Theorem; Ideal Monatomic Gas; The Foundation of Thermodynamics; and more.
Nonextensive statistical mechanics is now a rapidly growing field and a new stream in the research of the foundations of statistical mechanics. This generalization of the well-known Boltzmann--Gibbs theory enables the study of systems with long-range interactions, long-term memories or multi-fractal structures. This book consists of a set of self-contained lectures and includes additional contributions where some of the latest developments -- ranging from astro- to biophysics -- are covered. Addressing primarily graduate students and lecturers, this book will also be a useful reference for all researchers working in the field.
Written by experts from geophysics, astrophysics and engineering, this unique book on the interdisciplinary aspects of turbulence offers recent advances in the field and covers everything from the very nature of turbulence to some practical applications.
It is widely known that complex systems and complex materials comprise a major interdisciplinary scientific field that draws on mathematics, physics, chemistry, biology, and medicine as well as such social sciences as economics. The role of statistical physics in this new field has been expanding. Statistical physics has shown how phenomena and processes in different research areas that have long been assumed to be unrelated can have a common description. Through the application of statistical physics, methods developed for studying order phenomena in simple systems and processes have been generalized to more complex systems. This volume focuses on recent advances and perspectives in the physics of complex systems and provides both an overview of the field and a more detailed examination of the new ideas and unsolved problems that are currently attracting the attention of researchers. This book should be a useful reference work for anyone interested in this area, whether beginning graduate student or advanced research professional. It provides up-to-date reviews on cutting-edge topics compiled by leading authorities and is designed to both broaden the reader's competence within their own field and encourage the exploration of new problems in related fields.