Schrödinger Equations and Diffusion Theory
Schrödinger Equations and Diffusion Theory

Author: Masao Nagasawa

Publisher: Springer Science & Business Media

Published: 2012-12-13

Total Pages: 333

ISBN-13: 3034805608

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Schrödinger Equations and Diffusion Theory addresses the question “What is the Schrödinger equation?” in terms of diffusion processes, and shows that the Schrödinger equation and diffusion equations in duality are equivalent. In turn, Schrödinger’s conjecture of 1931 is solved. The theory of diffusion processes for the Schrödinger equation tells us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrödinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrödinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level. --- This book is a self-contained, very well-organized monograph recommended to researchers and graduate students in the field of probability theory, functional analysis and quantum dynamics. (...) what is written in this book may be regarded as an introduction to the theory of diffusion processes and applications written with the physicists in mind. Interesting topics present themselves as the chapters proceed. (...) this book is an excellent addition to the literature of mathematical sciences with a flavour different from an ordinary textbook in probability theory because of the author’s great contributions in this direction. Readers will certainly enjoy the topics and appreciate the profound mathematical properties of diffusion processes. (Mathematical Reviews)​

Schrödinger Diffusion Processes
Schrödinger Diffusion Processes

Author: Robert Aebi

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 196

ISBN-13: 3034890273

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In 1931 Erwin Schrödinger considered the following problem: A huge cloud of independent and identical particles with known dynamics is supposed to be observed at finite initial and final times. What is the "most probable" state of the cloud at intermediate times? The present book provides a general yet comprehensive discourse on Schrödinger's question. Key roles in this investigation are played by conditional diffusion processes, pairs of non-linear integral equations and interacting particles systems. The introductory first chapter gives some historical background, presents the main ideas in a rather simple discrete setting and reveals the meaning of intermediate prediction to quantum mechanics. In order to answer Schrödinger's question, the book takes three distinct approaches, dealt with in separate chapters: transformation by means of a multiplicative functional, projection by means of relative entropy, and variation of a functional associated to pairs of non-linear integral equations. The book presumes a graduate level of knowledge in mathematics or physics and represents a relevant and demanding application of today's advanced probability theory.

Handbook of Brownian Motion
Handbook of Brownian Motion

Author: Andrei Borodin

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 478

ISBN-13: 3034876521

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There are two parts in this book. The first part is devoted mainly to the proper ties of linear diffusions in general and Brownian motion in particular. The second part consists of tables of distributions of functionals of Brownian motion and re lated processes. The primary aim of this book is to give an easy reference to a large number of facts and formulae associated to Brownian motion. We have tried to do this in a "handbook-style". By this we mean that results are given without proofs but are equipped with a reference where a proof or a derivation can be found. It is our belief and experience that such a material would be very much welcome by students and people working with applications of diffusions and Brownian motion. In discussions with many of our colleagues we have found that they share this point of view. Our original plan included more things than we were able to realize. It turned out very soon when trying to put the plan into practice that the material would be too wide to be published under one cover. Excursion theory, which most of the recent results concerning linear Brownian motion and diffusions can be classified as, is only touched upon slightly here, not to mention Brownian motion in several dimensions which enters only through the discussion of Bessel processes. On the other hand, much attention is given to the theory of local time.

Classes of Linear Operators
Classes of Linear Operators

Author: Israel Gohberg

Publisher: Birkhäuser

Published: 2013-03-09

Total Pages: 563

ISBN-13: 303488558X

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These two volumes constitute texts for graduate courses in linear operator theory. The reader is assumed to have a knowledge of both complex analysis and the first elements of operator theory. The texts are intended to concisely present a variety of classes of linear operators, each with its own character, theory, techniques and tools. For each of the classes, various differential and integral operators motivate or illustrate the main results. Although each class is treated seperately and the first impression may be that of many different theories, interconnections appear frequently and unexpectedly. The result is a beautiful, unified and powerful theory. The classes we have chosen are representatives of the principal important classes of operators, and we believe that these illustrate the richness of operator theory, both in its theoretical developments and in its applicants. Because we wanted the books to be of reasonable size, we were selective in the classes we chose and restricted our attention to the main features of the corresponding theories. However, these theories have been updated and enhanced by new developments, many of which appear here for the first time in an operator-theory text. In the selection of the material the taste and interest of the authors played an important role.

Theoretical Physics 2002
Theoretical Physics 2002

Author: Thomas F. George

Publisher: Nova Publishers

Published: 2002

Total Pages: 266

ISBN-13: 9781590337226

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These twelve chapters, written by scientists from around the world, provide a representative sampling of the latest advances in theoretical physics. The book is divided in five sections, addressing the following topics: optics and quantum mechanics, relativity and cosmology, nuclear physics, thermodynamics and mathematics.

Markov Processes and Quantum Theory
Markov Processes and Quantum Theory

Author: Masao Nagasawa

Publisher: Springer Nature

Published: 2021-06-23

Total Pages: 339

ISBN-13: 3030626881

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This book discusses quantum theory as the theory of random (Brownian) motion of small particles (electrons etc.) under external forces. Implying that the Schrödinger equation is a complex-valued evolution equation and the Schrödinger function is a complex-valued evolution function, important applications are given. Readers will learn about new mathematical methods (theory of stochastic processes) in solving problems of quantum phenomena. Readers will also learn how to handle stochastic processes in analyzing physical phenomena.

Seminar on Stochastic Analysis, Random Fields and Applications
Seminar on Stochastic Analysis, Random Fields and Applications

Author: Erwin Bolthausen

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 392

ISBN-13: 3034870264

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Pure and applied stochastic analysis and random fields form the subject of this book. The collection of articles on these topics represent the state of the art of the research in the field, with particular attention being devoted to stochastic models in finance. Some are review articles, others are original papers; taken together, they will apprise the reader of much of the current activity in the area.

Waves Called Solitons
Waves Called Solitons

Author: Michel Remoissenet

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 248

ISBN-13: 3662030578

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Nonlinearity is a fascinating element of nature whose importance has been appreciated for many years when considering large-amplitude wave motions observed in various fields ranging from fluids and plasmas to solid-state, chemical, biological, and geological systems. Localized large-amplitude waves called solitons, which propagate without spreading and have particle-like properties, represent one of the most striking aspects of nonlinear phenomena. Although a wealth of literature on the subject, including theoretical and numerical studies, is available in good recent books and research journals, very little material has found its way into introductory texbooks and curricula. This is perhaps due to a belief that nonlinear physics is difficult and cannot be taught at an introductory level to undergraduate students and practitioners. Consequently, there is considerable interest in developing practical material suitable for students, at the lowest introductory level. This book is intended to be an elementary introduction to the physics of solitons, for students, physicists, engineers and practitioners. We present the modeling of nonlinear phenomena where soliton-like waves are involved, together with applications to a wide variety of concrete systems and experiments. This book is designed as a book of physical ideas and basic methods and not as an up-to-the minute book concerned with the latest research results. The background in physics and the amount of mathematical knowledge assumed of the reader is within that usually accumulated by junior or senior students in physics.

Computer Vision, Graphics and Image Processing
Computer Vision, Graphics and Image Processing

Author: Prem Kalra

Publisher: Springer

Published: 2007-01-01

Total Pages: 980

ISBN-13: 354068302X

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This book constitutes the refereed proceedings of the Indian Conference on Computer Vision, Graphics and Image Processing, ICVGIP 2006, held in Madurai, India, December 2006. Coverage in this volume includes image restoration and super-resolution, image filtering, visualization, tracking and surveillance, face-, gesture-, and object-recognition, compression, content based image retrieval, stereo/camera calibration, and biometrics.

Stochastic Processes, Physics And Geometry Ii - Proceedings Of The Iii International Conference
Stochastic Processes, Physics And Geometry Ii - Proceedings Of The Iii International Conference

Author: Sergio Albeverio

Publisher: World Scientific

Published: 1995-02-17

Total Pages: 758

ISBN-13: 981454969X

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As was already evident from the previous two meetings, the theory of stochastic processes, the study of geometrical structures, and the investigation of certain physical problems are inter-related. In fact the trend in recent years has been towards stronger interactions between these areas. As a result, a large component of the contributions is concerned with the theory of stochastic processes, quantum theory, and their relations.