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This volume describes the current state of knowledge of random spatial processes, particularly those arising in physics. The emphasis is on survey articles which describe areas of current interest to probabilists and physicists working on the probability theory of phase transition. Special attention is given to topics deserving further research. The principal contributions by leading researchers concern the mathematical theory of random walk, interacting particle systems, percolation, Ising and Potts models, spin glasses, cellular automata, quantum spin systems, and metastability. The level of presentation and review is particularly suitable for postgraduate and postdoctoral workers in mathematics and physics, and for advanced specialists in the probability theory of spatial disorder and phase transition.
A collection of survey and research papers that gives a glance of the profound consequences of Molchanov's contributions in stochastic differential equations, spectral theory for deterministic and random operators, localization and intermittency, mathematical physics and optics, and other topics.
This textbook presents an introduction to the use of probability in physics, treating introductory ideas of both statistical physics and of statistical inference, as well the importance of probability in information theory, quantum mechanics, and stochastic processes, in a unified manner. The book also presents a harmonised view of frequentist and Bayesian approaches to inference, emphasising their complementary value. The aim is to steer a middle course between the "cookbook" style and an overly dry mathematical statistics style. The treatment is driven by real physics examples throughout, but developed with a level of mathematical clarity and rigour appropriate to mid-career physics undergraduates. Exercises and solutions are included.
This book is designed as a practical and intuitive introduction to probability, statistics and random quantities for physicists. The book aims at getting to the main points by a clear, hands-on exposition supported by well-illustrated and worked-out examples. A strong focus on applications in physics and other natural sciences is maintained throughout. In addition to basic concepts of random variables, distributions, expected values and statistics, the book discusses the notions of entropy, Markov processes, and fundamentals of random number generation and Monte-Carlo methods.
In this book the author charts the history and development of modern probability theory.
"In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature? This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results"--Publisher's website.
This volume represents the outgrowth of an ongoing workshop on stochastic analysis held in Lisbon. The nine survey articles in the volume extend concepts from classical probability and stochastic processes to a number of areas of mathematical physics. It is a good reference text for researchers and advanced students in the fields of probability, stochastic processes, analysis, geometry, mathematical physics, and physics. Key topics covered include: nonlinear stochastic wave equations, completely positive maps, Mehler-type semigroups on Hilbert spaces, entropic projections, and many others.
The 20 papers contained in this volume span the areas of mathematical physics, dynamical systems, and probability. Yakov Sinai is one of the most important and influential mathematicians of our time, having won the Boltzmann Medal (1986), the Dirac Medal (1992), Dannie Heinemann Prize for Mathematical Physics (1989), Nemmers Prize (2002), and the Wolf Prize in Mathematics (1997). He is well-known as both a mathematician and a physicist, with numerous theorems and proofs bearing his name in both fields, and this book should be of interest to researchers from all fields of the physical sciences.This volume follows Volume I. From the reviews: "The second volume covers statistical mechanics and related topics. It contains 22 papers divided into four groups: Part I: Probability Theory; Part II: Statistical Mechanics; Part III: Mathematical Physics; Part IV: Mathematical Fluid Dynamics. The volume represents Sinai’s work on the above topics spanning almost 40 years: the earliest paper is dated 1972, and the latest 2008. The choice of papers was made by Sinai himself, and he provides commentary for each one. The reader will find a wealth of information and ideas that can still ignite inspiration and motivate students as well as senior researchers. The reader will also have a touch of Sinai’s personality, his taste, enthusiasm, and optimism, which are just as invaluable as his mathematical results." (Nikolai Chernov, Mathematical Reviews 2012e)
For physics students interested in the mathematics they use, and for math students interested in seeing how some of the ideas of their discipline find realization in an applied setting. The presentation strikes a balance between formalism and application, between abstract and concrete. The interconnections among the various topics are clarified both by the use of vector spaces as a central unifying theme, recurring throughout the book, and by putting ideas into their historical context. Enough of the essential formalism is included to make the presentation self-contained.