Symbolic Thermodynamic Formalism And Statistics
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| Author | : Mariusz Urbański |
| Publisher | : Walter de Gruyter GmbH & Co KG |
| Total Pages | : 458 |
| Release | : 2021-11-22 |
| Genre | : Mathematics |
| ISBN | : 3110702681 |
The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen’s formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub’s expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.
| Author | : David Ruelle |
| Publisher | : Cambridge University Press |
| Total Pages | : 198 |
| Release | : 2004-11-25 |
| Genre | : Science |
| ISBN | : 9781139455282 |
Reissued in the Cambridge Mathematical Library this classic book outlines the theory of thermodynamic formalism which was developed to describe the properties of certain physical systems consisting of a large number of subunits. It is aimed at mathematicians interested in ergodic theory, topological dynamics, constructive quantum field theory, the study of certain differentiable dynamical systems, notably Anosov diffeomorphisms and flows. It is also of interest to theoretical physicists concerned with the conceptual basis of equilibrium statistical mechanics. The level of the presentation is generally advanced, the objective being to provide an efficient research tool and a text for use in graduate teaching. Background material on mathematics has been collected in appendices to help the reader. Extra material is given in the form of updates of problems that were open at the original time of writing and as a new preface specially written for this new edition by the author.
| Author | : Luis Barreira |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 300 |
| Release | : 2011-08-24 |
| Genre | : Mathematics |
| ISBN | : 3034802064 |
This self-contained monograph presents a unified exposition of the thermodynamic formalism and some of its main extensions, with emphasis on the relation to dimension theory and multifractal analysis of dynamical systems. In particular, the book considers three different flavors of the thermodynamic formalism, namely nonadditive, subadditive, and almost additive, and provides a detailed discussion of some of the most significant results in the area, some of them quite recent. It also includes a discussion of the most substantial applications of these flavors of the thermodynamic formalism to dimension theory and multifractal analysis of dynamical systems.
| Author | : Mark Pollicott |
| Publisher | : Springer Nature |
| Total Pages | : 536 |
| Release | : 2021-10-01 |
| Genre | : Mathematics |
| ISBN | : 3030748634 |
This volume arose from a semester at CIRM-Luminy on “Thermodynamic Formalism: Applications to Probability, Geometry and Fractals” which brought together leading experts in the area to discuss topical problems and recent progress. It includes a number of surveys intended to make the field more accessible to younger mathematicians and scientists wishing to learn more about the area. Thermodynamic formalism has been a powerful tool in ergodic theory and dynamical system and its applications to other topics, particularly Riemannian geometry (especially in negative curvature), statistical properties of dynamical systems and fractal geometry. This work will be of value both to graduate students and more senior researchers interested in either learning about the main ideas and themes in thermodynamic formalism, and research themes which are at forefront of research in this area.
| Author | : Robert Edward Bowen |
| Publisher | : Springer |
| Total Pages | : 85 |
| Release | : 2008-04-04 |
| Genre | : Mathematics |
| ISBN | : 3540776958 |
For this printing of R. Bowen's book, J.-R. Chazottes has retyped it in TeX for easier reading, thereby correcting typos and bibliographic details. From the Preface by D. Ruelle: "Rufus Bowen has left us a masterpiece of mathematical exposition... Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen's monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems."
| Author | : Marcelo Viana |
| Publisher | : Cambridge University Press |
| Total Pages | : 547 |
| Release | : 2016-02-15 |
| Genre | : Mathematics |
| ISBN | : 1316445429 |
Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. The authors' clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students – invariance, recurrence and ergodicity – as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle and thermodynamical formalism. The 400 exercises increase in difficulty through the text and test the reader's understanding of the whole theory. Hints and solutions are provided at the end of the book.
| Author | : Christian Beck |
| Publisher | : Cambridge University Press |
| Total Pages | : 310 |
| Release | : 1993-07 |
| Genre | : Mathematics |
| ISBN | : 0521433673 |
This book deals with the various thermodynamic concepts used for the analysis of nonlinear dynamical systems. The most important invariants used to characterize chaotic systems are introduced in a way that stresses the interconnections with thermodynamics and statistical mechanics. Among the subjects treated are probabilistic aspects of chaotic dynamics, the symbolic dynamics technique, information measures, the maximum entropy principle, general thermodynamic relations, spin systems, fractals and multifractals, expansion rate and information loss, the topological pressure, transfer operator methods, repellers and escape. The more advanced chapters deal with the thermodynamic formalism for expanding maps, thermodynamic analysis of chaotic systems with several intensive parameters, and phase transitions in nonlinear dynamics.
| Author | : N.W. Tschoegl |
| Publisher | : Elsevier |
| Total Pages | : 279 |
| Release | : 2000-02-14 |
| Genre | : Science |
| ISBN | : 008053211X |
This book summarizes the salient features of both equilibrium and steady-state thermodynamic theory under a uniform postulatory viewpoint. The emphasis is upon the formal aspects and logical structure of thermodynamic theory, allowing it to emerge as a coherent whole, unfettered by much of those details which - albeit indispensable in practical applications - tend to obscure this coherent structure. Largely because of this, statistical mechanics and reference to molecular structure are, barring an occasional allusion, avoided. The treatment is, therefore, 'classical', or - using a perhaps more appropriate word - 'phenomenological'. The volume almost exclusively deals with 'ideal' systems, given that the treatment of 'real' systems properly belongs in the realm of applied, rather than theoretical thermodynamics. For these reasons, only selected ideal systems are covered. Ideal gases are discussed extensively. The ideal solution is treated as an example of a liquid system. The amorphous ideal rubber serves as an example of a solid. The formalism developed in these sections is a model for the treatment of other, more complex systems. This short structural overview is written in the hope that a knowledge of steady-state theory will deepen readers' understanding of thermodynamics as a whole.
| Author | : Zi-Kui Liu |
| Publisher | : Cambridge University Press |
| Total Pages | : 261 |
| Release | : 2016-06-30 |
| Genre | : Science |
| ISBN | : 0521198968 |
Integrates fundamental concepts with experimental data and practical applications, including worked examples and end-of-chapter problems.
| Author | : Normand M. Laurendeau |
| Publisher | : Cambridge University Press |
| Total Pages | : 472 |
| Release | : 2005-11-21 |
| Genre | : Technology & Engineering |
| ISBN | : 1139446185 |
This 2006 textbook discusses the fundamentals and applications of statistical thermodynamics for beginning graduate students in the physical and engineering sciences. Building on the prototypical Maxwell–Boltzmann method and maintaining a step-by-step development of the subject, this book assumes the reader has no previous exposure to statistics, quantum mechanics or spectroscopy. The book begins with the essentials of statistical thermodynamics, pauses to recover needed knowledge from quantum mechanics and spectroscopy, and then moves on to applications involving ideal gases, the solid state and radiation. A full introduction to kinetic theory is provided, including its applications to transport phenomena and chemical kinetics. A highlight of the textbook is its discussion of modern applications, such as laser-based diagnostics. The book concludes with a thorough presentation of the ensemble method, featuring its use for real gases. Numerous examples and prompted homework problems enrich the text.
