Download or read book Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics written by Errico Presutti and published by Springer Science & Business Media. This book was released on 2008-11-01 with total page 478 pages. Available in PDF, EPUB and Kindle. Book excerpt: Collective behavior in systems with many components, blow-ups with emergence of microstructures are proofs of the double, continuum and atomistic, nature of macroscopic systems, an issue which has always intrigued scientists and philosophers. Modern technologies have made the question more actual and concrete with recent, remarkable progresses also from a mathematical point of view. The book focuses on the links connecting statistical and continuum mechanics and, starting from elementary introductions to both theories, it leads to actual research themes. Mathematical techniques and methods from probability, calculus of variations and PDE are discussed at length.
Download or read book Statistical Mechanics of Lattice Systems written by Sacha Friedli and published by Cambridge University Press. This book was released on 2017-11-23 with total page 644 pages. Available in PDF, EPUB and Kindle. Book excerpt: This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kać interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.
Download or read book Making Sense of Statistical Mechanics written by Jean Bricmont and published by Springer Nature. This book was released on 2022-02-11 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many people, including physicists, are confused about what the Second Law of thermodynamics really means, about how it relates to the arrow of time, and about whether it can be derived from classical mechanics. They also wonder what entropy really is: Is it all about information? But, if so, then, what is its relation to fluxes of heat? One might ask similar questions about probabilities: Do they express subjective judgments by us, humans, or do they reflect facts about the world, i.e. frequencies. And what notion of probability is used in the natural sciences, in particular statistical mechanics? This book addresses all of these questions in the clear and pedagogical style for which the author is known. Although valuable as accompaniment to an undergraduate course on statistical mechanics or thermodynamics, it is not a standard course book. Instead it addresses both the essentials and the many subtle questions that are usually brushed under the carpet in such courses. As one of the most lucid accounts of the above questions, it provides enlightening reading for all those seeking answers, including students, lecturers, researchers and philosophers of science.
Download or read book Quantum Mathematics II written by Michele Correggi and published by Springer Nature. This book was released on 2024-01-09 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the second volume that provides an unique overview of the most recent and relevant contributions in the field of mathematical physics with a focus on the mathematical features of quantum mechanics. It is a collection of review papers together with brand new works related to the activities of the INdAM Intensive Period "INdAM Quantum Meetings (IQM22)", which took place at the Politecnico di Milano in Spring 2022 at Politecnico di Milano. The range of topics covered by the book is wide, going ranging from many-body quantum mechanics to quantum field theory and open quantum systems.
Download or read book Theory of Simple Glasses written by Giorgio Parisi and published by Cambridge University Press. This book was released on 2020-01-09 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: This self-contained text describes the modern mean field theory of simple structural glasses using a quantum statistical mechanical approach. Describing the theory in clear and simple terms, this is a valuable resource for graduate students and researchers working in condensed matter physics and statistical mechanics.
Download or read book Physics Avoidance written by Mark Wilson and published by Oxford University Press. This book was released on 2017-10-20 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mark Wilson presents a series of explorations of our strategies for understanding the world. "Physics avoidance" refers to the fact that we frequently cannot reason about nature in the straightforward manner we anticipate, but must seek alternative policies that allow us to address the questions we want answered in a tractable way. Within both science and everyday life, we find ourselves relying upon thought processes that reach useful answers in opaque and roundabout manners. Conceptual innovators are often puzzled by the techniques they develop, when they stumble across reasoning patterns that are easy to implement but difficult to justify. But simple techniques frequently rest upon complex foundations—a young magician learns how to execute a card-guessing trick without understanding how its progressive steps squeeze in on a proper answer. As we collectively improve our inferential skills in this gradually evolving manner, we often wander into unfamiliar explanatory landscapes in which simple words encode physical information in complex and unanticipated ways. Like our juvenile conjurer, we fail to recognize the true strategic rationales underlying our achievements and may turn instead to preposterous rationalizations for our policies. We have learned how to reach better conclusions in a more fruitful way, but we remain baffled by our own successes. At its best, philosophical reflection illuminates the natural developmental processes that generate these confusions and explicates their complexities. But current thinking within philosophy of science and language works to opposite effect by relying upon simplistic conceptions of "cause", "law of nature", "possibility", and "reference" that ignore the strategic complexities in which these concepts become entangled within real life usage. To avoid these distortions, better descriptive tools are required in philosophy. The nine new essays within this volume illustrate this need for finer discriminations through a range of revealing cases, of both historical and contemporary significance.
Download or read book Stochastic Exponential Growth and Lattice Gases written by Dan Pirjol and published by Springer Nature. This book was released on 2022-09-01 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book discusses a class of discrete time stochastic growth processes for which the growth rate is proportional to the exponential of a Gaussian Markov process. These growth processes appear naturally in problems of mathematical finance as discrete time approximations of stochastic volatility models and stochastic interest rates models such as the Black-Derman-Toy and Black-Karasinski models. These processes can be mapped to interacting one-dimensional lattice gases with long-range interactions. The book gives a detailed discussion of these statistical mechanics models, including new results not available in the literature, and their implication for the stochastic growth models. The statistical mechanics analogy is used to understand observed non-analytic dependence of the Lyapunov exponents of the stochastic growth processes considered, which is related to phase transitions in the lattice gas system. The theoretical results are applied to simulations of financial models and are illustrated with Mathematica code. The book includes a general introduction to exponential stochastic growth with examples from biology, population dynamics and finance. The presentation does not assume knowledge of mathematical finance. The new results on lattice gases can be read independently of the rest of the book. The book should be useful to practitioners and academics studying the simulation and application of stochastic growth models.
Download or read book Metastability written by Anton Bovier and published by Springer. This book was released on 2016-02-11 with total page 581 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a concise presentation of a mathematical approach to metastability, a wide-spread phenomenon in the dynamics of non-linear systems - physical, chemical, biological or economic - subject to the action of temporal random forces typically referred to as noise, based on potential theory of reversible Markov processes. The authors shed new light on the metastability phenomenon as a sequence of visits of the path of the process to different metastable sets, and focuses on the precise analysis of the respective hitting probabilities and hitting times of these sets. The theory is illustrated with many examples, ranging from finite-state Markov chains, finite-dimensional diffusions and stochastic partial differential equations, via mean-field dynamics with and without disorder, to stochastic spin-flip and particle-hop dynamics and probabilistic cellular automata, unveiling the common universal features of these systems with respect to their metastable behaviour. The monograph will serve both as comprehensive introduction and as reference for graduate students and researchers interested in metastability.
Download or read book C Algebras and Mathematical Foundations of Quantum Statistical Mechanics written by Jean-Bernard Bru and published by Springer Nature. This book was released on 2023-06-16 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a comprehensive introduction to the mathematical foundations of quantum statistical physics. It presents a conceptually profound yet technically accessible path to the C*-algebraic approach to quantum statistical mechanics, demonstrating how key aspects of thermodynamic equilibrium can be derived as simple corollaries of classical results in convex analysis. Using C*-algebras as examples of ordered vector spaces, this book makes various aspects of C*-algebras and their applications to the mathematical foundations of quantum theory much clearer from both mathematical and physical perspectives. It begins with the simple case of Gibbs states on matrix algebras and gradually progresses to a more general setting that considers the thermodynamic equilibrium of infinitely extended quantum systems. The book also illustrates how first-order phase transitions and spontaneous symmetry breaking can occur, in contrast to the finite-dimensional situation. One of the unique features of this book is its thorough and clear treatment of the theory of equilibrium states of quantum mean-field models. This work is self-contained and requires only a modest background in analysis, topology, and functional analysis from the reader. It is suitable for both mathematicians and physicists with a specific interest in quantum statistical physics.
Download or read book Gibbs Measures and Phase Transitions written by Hans-Otto Georgii and published by Walter de Gruyter. This book was released on 2011 with total page 561 pages. Available in PDF, EPUB and Kindle. Book excerpt: From a review of the first edition: "This book [...] covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics. [...] It is in fact one of the author's stated aims that this comprehensive monograph should serve both as an introductory text and as a reference for the expert." (F. Papangelou
Download or read book Feynman Kac Type Theorems and Gibbs Measures on Path Space written by József Lörinczi and published by Walter de Gruyter. This book was released on 2011-08-29 with total page 521 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph offers a state-of-the-art mathematical account of functional integration methods in the context of self-adjoint operators and semigroups using the concepts and tools of modern stochastic analysis. These ideas are then applied principally to a rigorous treatment of some fundamental models of quantum field theory. In this self-contained presentation of the material both beginners and experts are addressed, while putting emphasis on the interdisciplinary character of the subject.
Download or read book Analysis and Stochastics of Growth Processes and Interface Models written by Peter Mörters and published by OUP Oxford. This book was released on 2008-07-24 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of topical survey articles by leading researchers in the fields of applied analysis and probability theory, working on the mathematical description of growth phenomena. Particular emphasis is on the interplay of the two fields, with articles by analysts being accessible for researchers in probability, and vice versa. Mathematical methods discussed in the book comprise large deviation theory, lace expansion, harmonic multi-scale techniques and homogenisation of partial differential equations. Models based on the physics of individual particles are discussed alongside models based on the continuum description of large collections of particles, and the mathematical theories are used to describe physical phenomena such as droplet formation, Bose-Einstein condensation, Anderson localization, Ostwald ripening, or the formation of the early universe. The combination of articles from the two fields of analysis and probability is highly unusual and makes this book an important resource for researchers working in all areas close to the interface of these fields.
Download or read book Feynman Kac Type Formulae and Gibbs Measures written by József Lörinczi and published by Walter de Gruyter GmbH & Co KG. This book was released on 2020-01-20 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second updated and extended edition of the successful book on Feynman-Kac theory. It offers a state-of-the-art mathematical account of functional integration methods in the context of self-adjoint operators and semigroups using the concepts and tools of modern stochastic analysis. The first volume concentrates on Feynman-Kac-type formulae and Gibbs measures
Download or read book Nonequilibrium and Irreversibility written by Giovanni Gallavotti and published by Springer. This book was released on 2014-06-10 with total page 261 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book concentrates on the properties of the stationary states in chaotic systems of particles or fluids, leaving aside the theory of the way they can be reached. The stationary states of particles or of fluids (understood as probability distributions on microscopic configurations or on the fields describing continua) have received important new ideas and data from numerical simulations and reviews are needed. The starting point is to find out which time invariant distributions come into play in physics. A special feature of this book is the historical approach. To identify the problems the author analyzes the papers of the founding fathers Boltzmann, Clausius and Maxwell including translations of the relevant (parts of) historical documents. He also establishes a close link between treatment of irreversible phenomena in statistical mechanics and the theory of chaotic systems at and beyond the onset of turbulence as developed by Sinai, Ruelle, Bowen (SRB) and others: the author gives arguments intending to support strongly the viewpoint that stationary states in or out of equilibrium can be described in a unified way. In this book it is the "chaotic hypothesis", which can be seen as an extension of the classical ergodic hypothesis to non equilibrium phenomena, that plays the central role. It is shown that SRB - often considered as a kind of mathematical playground with no impact on physical reality - has indeed a sound physical interpretation; an observation which to many might be new and a very welcome insight. Following this, many consequences of the chaotic hypothesis are analyzed in chapter 3 - 4 and in chapter 5 a few applications are proposed. Chapter 6 is historical: carefully analyzing the old literature on the subject, especially ergodic theory and its relevance for statistical mechanics; an approach which gives the book a very personal touch. The book contains an extensive coverage of current research (partly from the authors and his coauthors publications) presented in enough detail so that advanced students may get the flavor of a direction of research in a field which is still very much alive and progressing. Proofs of theorems are usually limited to heuristic sketches privileging the presentation of the ideas and providing references that the reader can follow, so that in this way an overload of this text with technical details could be avoided.
Download or read book Microstructural Randomness and Scaling in Mechanics of Materials written by Martin Ostoja-Starzewski and published by CRC Press. This book was released on 2007-08-13 with total page 500 pages. Available in PDF, EPUB and Kindle. Book excerpt: An area at the intersection of solid mechanics, materials science, and stochastic mathematics, mechanics of materials often necessitates a stochastic approach to grasp the effects of spatial randomness. Using this approach, Microstructural Randomness and Scaling in Mechanics of Materials explores numerous stochastic models and methods used in the m
Download or read book Discrete Variational Problems with Interfaces written by Roberto Alicandro and published by Cambridge University Press. This book was released on 2023-12-31 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many materials can be modeled either as discrete systems or as continua, depending on the scale. At intermediate scales it is necessary to understand the transition from discrete to continuous models and variational methods have proved successful in this task, especially for systems, both stochastic and deterministic, that depend on lattice energies. This is the first systematic and unified presentation of research in the area over the last 20 years. The authors begin with a very general and flexible compactness and representation result, complemented by a thorough exploration of problems for ferromagnetic energies with applications ranging from optimal design to quasicrystals and percolation. This leads to a treatment of frustrated systems, and infinite-dimensional systems with diffuse interfaces. Each topic is presented with examples, proofs and applications. Written by leading experts, it is suitable as a graduate course text as well as being an invaluable reference for researchers.
Download or read book Statistical Continuum Mechanics written by E. Kroner and published by . This book was released on 2014-09-01 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt:
