Synopses & Reviews
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.
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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.
Synopsis
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.
Synopsis
Self-contained and structurally coherent, this introduction to the theory and applications of diffusion processes deploys them to analyze Schrödinger's equations, using relative entropy and the theory of transformations to forge apparently identical processes.
About the Author
Masao Nagasawa is professor of mathematics at the University of Zurich, Switzerland.
Table of Contents
Preface.- I Introduction and Motivation.- II Diffusion Processes and their Transformations.- III Duality and Time Reversal of Diffusion Processes.- IV Equivalence of Diffusion and Schrödinger Equations.- V Variational Principle.- VI Diffusion Processes in q-Representation.- VII Segregation of a Population.- VIII The Schrödinger Equation can be a Boltzmann Equation.- IX Applications of the Statistical Model for Schrödinger Equations.- X Relative Entropy and Csiszar's Projection.- XI Large Deviations.- XII Non-Linearity Induced by the Branching Property.- Appendix.- References.- Index.