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Entropy
The concept of entropy arose in the physical sciences during the nineteenth century, particularly in thermodynamics and statistical physics, as a measure of the equilibria and evolution of thermodynamic systems. Two main views developed: the macroscopic view formulated originally by Carnot, Clausius, Gibbs, Planck, and Caratheodory and the microscopic approach associated with Boltzmann and Maxwell. Since then both approaches have made possible deep insights into the nature and behavior of thermodynamic and other microscopically unpredictable processes. However, the mathematical tools used have later developed independently of their original physical background and have led to a plethora of m...
The Mathematics of the Bose Gas and its Condensation
This book contains a unique survey of the mathematically rigorous results about the quantum-mechanical many-body problem that have been obtained by the authors in the past seven years. It addresses a topic that is not only rich mathematically, using a large variety of techniques in mathematical analysis, but is also one with strong ties to current experiments on ultra-cold Bose gases and Bose-Einstein condensation. The book provides a pedagogical entry into an active area of ongoing research for both graduate students and researchers. It is an outgrowth of a course given by the authors for graduate students and post-doctoral researchers at the Oberwolfach Research Institute in 2004. The book also provides a coherent summary of the field and a reference for mathematicians and physicists active in research on quantum mechanics.
Tropical Algebraic Geometry
These notes present a polished introduction to tropical geometry and contain some applications of this rapidly developing and attractive subject. It consists of three chapters which complete each other and give a possibility for non-specialists to make the first steps in the subject which is not yet well represented in the literature. The notes are based on a seminar at the Mathematical Research Center in Oberwolfach in October 2004. The intended audience is graduate, post-graduate, and Ph.D. students as well as established researchers in mathematics.
The Entropy Principle
Entropy – the key concept of thermodynamics, clearly explained and carefully illustrated. This book presents an accurate definition of entropy in classical thermodynamics which does not “put the cart before the horse” and is suitable for basic and advanced university courses in thermodynamics. Entropy is the most important and at the same time the most difficult term of thermodynamics to understand. Many students are discontent with its classical definition since it is either based on “temperature” and “heat” which both cannot be accurately defined without entropy, or since it includes concepts such as “molecular disorder” which does not fit in a macroscopic theory. The phy...
Philosophy of Physics
The ambition of this volume is twofold: to provide a comprehensive overview of the field and to serve as an indispensable reference work for anyone who wants to work in it. For example, any philosopher who hopes to make a contribution to the topic of the classical-quantum correspondence will have to begin by consulting Klaas Landsman's chapter. The organization of this volume, as well as the choice of topics, is based on the conviction that the important problems in the philosophy of physics arise from studying the foundations of the fundamental theories of physics. It follows that there is no sharp line to be drawn between philosophy of physics and physics itself. Some of the best work in t...
Axiomatic Thinking II
In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Göttingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.
Mathematical Problems in Quantum Physics
This volume contains the proceedings of the QMATH13: Mathematical Results in Quantum Physics conference, held from October 8–11, 2016, at the Georgia Institute of Technology, Atlanta, Georgia. In recent years, a number of new frontiers have opened in mathematical physics, such as many-body localization and Schrödinger operators on graphs. There has been progress in developing mathematical techniques as well, notably in renormalization group methods and the use of Lieb–Robinson bounds in various quantum models. The aim of this volume is to provide an overview of some of these developments. Topics include random Schrödinger operators, many-body fermionic systems, atomic systems, effective equations, and applications to quantum field theory. A number of articles are devoted to the very active area of Schrödinger operators on graphs and general spectral theory of Schrödinger operators. Some of the articles are expository and can be read by an advanced graduate student.
Theoretical Fluid Dynamics
This textbook gives an introduction to fluid dynamics based on flows for which analytical solutions exist, like individual vortices, vortex streets, vortex sheets, accretions disks, wakes, jets, cavities, shallow water waves, bores, tides, linear and non-linear free-surface waves, capillary waves, internal gravity waves and shocks. Advanced mathematical techniques ("calculus") are introduced and applied to obtain these solutions, mostly from complex function theory (Schwarz-Christoffel theorem and Wiener-Hopf technique), exterior calculus, singularity theory, asymptotic analysis, the theory of linear and nonlinear integral equations and the theory of characteristics. Many of the derivations, so far contained only in research journals, are made available here to a wider public.
Xivth International Congress On Mathematical Physics
In 2003 the XIV International Congress on Mathematical Physics (ICMP) was held in Lisbon with more than 500 participants. Twelve plenary talks were given in various fields of Mathematical Physics: E Carlen «On the relation between the Master equation and the Boltzmann Equation in Kinetic Theory»; A Chenciner «Symmetries and “simple” solutions of the classical n-body problem»; M J Esteban «Relativistic models in atomic and molecular physics»; K Fredenhagen «Locally covariant quantum field theory»; K Gawedzki «Simple models of turbulent transport»; I Krichever «Algebraic versus Liouville integrability of the soliton systems»; R V Moody «Long-range order and diffraction in math...
Noncommutative Geometry and Physics
This collection of expository articles grew out of the workshop ``Number Theory and Physics'' held in March 2009 at The Erwin Schrodinger International Institute for Mathematical Physics, Vienna. The common theme of the articles is the influence of ideas from noncommutative geometry (NCG) on subjects ranging from number theory to Lie algebras, index theory, and mathematical physics. Matilde Marcolli's article gives a survey of relevant aspects of NCG in number theory, building on an introduction to motives for beginners by Jorge Plazas and Sujatha Ramdorai. A mildly unconventional view of index theory, from the viewpoint of NCG, is described in the article by Alan Carey, John Phillips, and A...
