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| Author | : Hansjörg Roos |
| Publisher | : |
| Total Pages | : 78 |
| Release | : 1974 |
| Genre | : |
| ISBN | : |
| Author | : |
| Publisher | : European Mathematical Society |
| Total Pages | : 402 |
| Release | : 2009 |
| Genre | : Science |
| ISBN | : 9783037190708 |
Quantum statistical mechanics plays a major role in many fields such as thermodynamics, plasma physics, solid-state physics, and the study of stellar structure. While the theory of quantum harmonic oscillators is relatively simple, the case of anharmonic oscillators, a mathematical model of a localized quantum particle, is more complex and challenging. Moreover, infinite systems of interacting quantum anharmonic oscillators possess interesting ordering properties with respect to quantum stabilization. This book presents a rigorous approach to the statistical mechanics of such systems, in particular with respect to their actions on a crystal lattice. The text is addressed to both mathematicians and physicists, especially those who are concerned with the rigorous mathematical background of their results and the kind of problems that arise in quantum statistical mechanics. The reader will find here a concise collection of facts, concepts, and tools relevant for the application of path integrals and other methods based on measure and integration theory to problems of quantum physics, in particular the latest results in the mathematical theory of quantum anharmonic crystals. The methods developed in the book are also applicable to other problems involving infinitely many variables, for example, in biology and economics.
| Author | : Robert B. Israel |
| Publisher | : Princeton University Press |
| Total Pages | : 257 |
| Release | : 2015-03-08 |
| Genre | : Science |
| ISBN | : 1400868424 |
In this book, Robert Israel considers classical and quantum lattice systems in terms of equilibrium statistical mechanics. He is especially concerned with the characterization of translation-invariant equilibrium states by a variational principle and the use of convexity in studying these states. Arthur Wightman's Introduction gives a general and historical perspective on convexity in statistical mechanics and thermodynamics. Professor Israel then reviews the general framework of the theory of lattice gases. In addition to presenting new and more direct proofs of some known results, he uses a version of a theorem by Bishop and Phelps to obtain existence results for phase transitions. Furthermore, he shows how the Gibbs Phase Rule and the existence of a wide variety of phase transitions follow from the general framework and the theory of convex functions. While the behavior of some of these phase transitions is very "pathological," others exhibit more "reasonable" behavior. As an example, the author considers the isotropic Heisenberg model. Formulating a version of the Gibbs Phase Rule using Hausdorff dimension, he shows that the finite dimensional subspaces satisfying this phase rule are generic. Originally published in 1979. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
| Author | : David Lavis |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 437 |
| Release | : 2013-06-29 |
| Genre | : Science |
| ISBN | : 3662100207 |
Most of the interesting and difficult problems in statistical mechanics arise when the constituent particles of the system interact with each other with pair or multipartiele energies. The types of behaviour which occur in systems because of these interactions are referred to as cooperative phenomena giving rise in many cases to phase transitions. This book and its companion volume (Lavis and Bell 1999, referred to in the text simply as Volume 1) are princi pally concerned with phase transitions in lattice systems. Due mainly to the insights gained from scaling theory and renormalization group methods, this subject has developed very rapidly over the last thirty years. ' In our choice of topics we have tried to present a good range of fundamental theory and of applications, some of which reflect our own interests. A broad division of material can be made between exact results and ap proximation methods. We have found it appropriate to inelude some of our discussion of exact results in this volume and some in Volume 1. Apart from this much of the discussion in Volume 1 is concerned with mean-field theory. Although this is known not to give reliable results elose to a critical region, it often provides a good qualitative picture for phase diagrams as a whole. For complicated systems some kind of mean-field method is often the only tractable method available. In this volume our main concern is with scaling theory, algebraic methods and the renormalization group.
| Author | : Barry Simon |
| Publisher | : Princeton University Press |
| Total Pages | : 534 |
| Release | : 2014-07-14 |
| Genre | : Science |
| ISBN | : 1400863430 |
A state-of-the-art survey of both classical and quantum lattice gas models, this two-volume work will cover the rigorous mathematical studies of such models as the Ising and Heisenberg, an area in which scientists have made enormous strides during the past twenty-five years. This first volume addresses, among many topics, the mathematical background on convexity and Choquet theory, and presents an exhaustive study of the pressure including the Onsager solution of the two-dimensional Ising model, a study of the general theory of states in classical and quantum spin systems, and a study of high and low temperature expansions. The second volume will deal with the Peierls construction, infrared bounds, Lee-Yang theorems, and correlation inequality. This comprehensive work will be a useful reference not only to scientists working in mathematical statistical mechanics but also to those in related disciplines such as probability theory, chemical physics, and quantum field theory. It can also serve as a textbook for advanced graduate students. Originally published in 1993. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
| Author | : D.Y. Petrina |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 460 |
| Release | : 2012-12-06 |
| Genre | : Science |
| ISBN | : 940110185X |
This monograph is devoted to quantum statistical mechanics. It can be regarded as a continuation of the book "Mathematical Foundations of Classical Statistical Mechanics. Continuous Systems" (Gordon & Breach SP, 1989) written together with my colleagues V. I. Gerasimenko and P. V. Malyshev. Taken together, these books give a complete pre sentation of the statistical mechanics of continuous systems, both quantum and classical, from the common point of view. Both books have similar contents. They deal with the investigation of states of in finite systems, which are described by infinite sequences of statistical operators (reduced density matrices) or Green's functions in the quantum case and by infinite sequences of distribution functions in the classical case. The equations of state and their solutions are the main object of investigation in these books. For infinite systems, the solutions of the equations of state are constructed by using the thermodynamic limit procedure, accord ing to which we first find a solution for a system of finitely many particles and then let the number of particles and the volume of a region tend to infinity keeping the density of particles constant. However, the style of presentation in these books is quite different.
| Author | : Jean Ruiz |
| Publisher | : World Scientific |
| Total Pages | : 554 |
| Release | : 1999-05-14 |
| Genre | : |
| ISBN | : 9814543780 |
This invaluable book is a collection of lectures delivered at the Colloquium 'Mathematical Results in Statistical Mechanics' held in Marseilles, France, on July 27-31, 1998, as a satellite colloquium of the Paris conference STATPHYS 20. It covers a large part of the contemporary results in statistical mechanics, from the point of view of mathematical physics, by leading experts in this field. It includes as the main topics, phase transitions, interfaces, disordered systems, Gibbsian and non-Gibbsian states, as well as recent rigorous treatments in quantum statistical mechanics.
| Author | : David Ruelle |
| Publisher | : World Scientific |
| Total Pages | : 236 |
| Release | : 1999-04-14 |
| Genre | : Science |
| ISBN | : 981449500X |
This classic book marks the beginning of an era of vigorous mathematical progress in equilibrium statistical mechanics. Its treatment of the infinite system limit has not been superseded, and the discussion of thermodynamic functions and states remains basic for more recent work. The conceptual foundation provided by the Rigorous Results remains invaluable for the study of the spectacular developments of statistical mechanics in the second half of the 20th century.
| Author | : SERGIO ALBEVERIO; YURI KONDRATIEV; YURI KOZITSKY. |
| Publisher | : |
| Total Pages | : |
| Release | : |
| Genre | : |
| ISBN | : 9783037195703 |
| Author | : Sacha Friedli |
| Publisher | : Cambridge University Press |
| Total Pages | : 643 |
| Release | : 2017-11-23 |
| Genre | : Mathematics |
| ISBN | : 1107184827 |
A self-contained, mathematical introduction to the driving ideas in equilibrium statistical mechanics, studying important models in detail.
