Mean Field Theories And Dual Variation
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Mean Field Theories And Dual Variation Mathematical Structures Of The Mesoscopic Model
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Author : Takashi Suzuki
language : en
Publisher: Springer
Release Date : 2015-11-19
Mean Field Theories And Dual Variation Mathematical Structures Of The Mesoscopic Model written by Takashi Suzuki and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-11-19 with Mathematics categories.
Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.
Mean Field Theories And Dual Variation
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Author : Takashi Suzuki
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-01-01
Mean Field Theories And Dual Variation written by Takashi Suzuki and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-01-01 with Mathematics categories.
A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of “duality” according to the PDE weak solutions and “hierarchy” for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski–Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.
Mean Field Theories And Dual Variation
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Author : Takashi Suzuki
language : en
Publisher: Atlantis Press
Release Date : 2008-09-01
Mean Field Theories And Dual Variation written by Takashi Suzuki and has been published by Atlantis Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-01 with Mathematics categories.
A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of duality according to the PDE weak solutions and hierarchy for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the SmoluchowskiPoisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.
Methods Of Geometry In The Theory Of Partial Differential Equations Principle Of The Cancellation Of Singularities
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Author : Takashi Suzuki
language : en
Publisher: World Scientific
Release Date : 2024-01-22
Methods Of Geometry In The Theory Of Partial Differential Equations Principle Of The Cancellation Of Singularities written by Takashi Suzuki and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-01-22 with Mathematics categories.
Mathematical models are used to describe the essence of the real world, and their analysis induces new predictions filled with unexpected phenomena.In spite of a huge number of insights derived from a variety of scientific fields in these five hundred years of the theory of differential equations, and its extensive developments in these one hundred years, several principles that ensure these successes are discovered very recently.This monograph focuses on one of them: cancellation of singularities derived from interactions of multiple species, which is described by the language of geometry, in particular, that of global analysis.Five objects of inquiry, scattered across different disciplines, are selected in this monograph: evolution of geometric quantities, models of multi-species in biology, interface vanishing of d - δ systems, the fundamental equation of electro-magnetic theory, and free boundaries arising in engineering.The relaxation of internal tensions in these systems, however, is described commonly by differential forms, and the reader will be convinced of further applications of this principle to other areas.
Probabilistic Theory Of Mean Field Games With Applications I
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Author : René Carmona
language : en
Publisher: Springer
Release Date : 2018-03-01
Probabilistic Theory Of Mean Field Games With Applications I written by René Carmona and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-03-01 with Mathematics categories.
This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions. Volume I of the book is entirely devoted to the theory of mean field games without a common noise. The first half of the volume provides a self-contained introduction to mean field games, starting from concrete illustrations of games with a finite number of players, and ending with ready-for-use solvability results. Readers are provided with the tools necessary for the solution of forward-backward stochastic differential equations of the McKean-Vlasov type at the core of the probabilistic approach. The second half of this volume focuses on the main principles of analysis on the Wasserstein space. It includes Lions' approach to the Wasserstein differential calculus, and the applications of its results to the analysis of stochastic mean field control problems. Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. The authors provide a detailed road map through the book allowing different access points for different readers and building up the level of technical detail. The accessible approach and overview will allow interested researchers in the applied sciences to obtain a clear overview of the state of the art in mean field games.
Theory Of Causal Differential Equations
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Author : S. Leela
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-01-01
Theory Of Causal Differential Equations written by S. Leela and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-01-01 with Mathematics categories.
The problems of modern society are both complex and inter-disciplinary. Despite the - parent diversity of problems, however, often tools developed in one context are adaptable to an entirely different situation. For example, consider the well known Lyapunov’s second method. This interesting and fruitful technique has gained increasing signi?cance and has given decisive impetus for modern development of stability theory of discrete and dynamic system. It is now recognized that the concept of Lyapunov function and theory of diff- ential inequalities can be utilized to investigate qualitative and quantitative properties of a variety of nonlinear problems. Lyapunov function serves as a vehicle to transform a given complicated system into a simpler comparison system. Therefore, it is enough to study the properties of the simpler system to analyze the properties of the complicated system via an appropriate Lyapunov function and the comparison principle. It is in this perspective, the present monograph is dedicated to the investigation of the theory of causal differential equations or differential equations with causal operators, which are nonanticipative or abstract Volterra operators. As we shall see in the ?rst chapter, causal differential equations include a variety of dynamic systems and consequently, the theory developed for CDEs (Causal Differential Equations) in general, covers the theory of several dynamic systems in a single framework.
Vortex Dynamics
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Author : İlkay Bakırtaş
language : en
Publisher: BoD – Books on Demand
Release Date : 2022-09-28
Vortex Dynamics written by İlkay Bakırtaş and has been published by BoD – Books on Demand this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-28 with Science categories.
This book discusses vortex dynamics theory from physics, mathematics, and engineering perspectives. It includes nine chapters that cover a variety of research results related to vortex dynamics including nonlinear optics, fluid dynamics, and plasma physics.
Transport In Multilayered Nanostructures The Dynamical Mean Field Theory Approach Second Edition
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Author : James K Freericks
language : en
Publisher: World Scientific
Release Date : 2016-03-15
Transport In Multilayered Nanostructures The Dynamical Mean Field Theory Approach Second Edition written by James K Freericks and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-15 with Science categories.
Over the last 25 years, dynamical mean-field theory (DMFT) has emerged as one of the most powerful new developments in many-body physics. Written by one of the key researchers in the field, this book presents the first comprehensive treatment of this ever-developing topic. Transport in Mutlilayered Nanostructures is varied and modern in its scope, and:A series of over 50 problems help develop the skills to allow readers to reach the level of being able to contribute to research. This book is suitable for an advanced graduate course in DMFT, and for individualized study by graduate students, postdoctoral fellows and advanced researchers wishing to enter the field.
Mean Field Theory
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Author : Vladimir M Kolomietz
language : en
Publisher: World Scientific
Release Date : 2020-05-08
Mean Field Theory written by Vladimir M Kolomietz and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-05-08 with Science categories.
This book describes recent theoretical and experimental developments in the study of static and dynamic properties of atomic nuclei, many-body systems of strongly interacting neutrons and protons. The theoretical approach is based on the concept of the mean field, describing the motion of a nucleon in terms of a self-consistent single-particle potential well which approximates the interactions of a nucleon with all the other nucleons. The theoretical approaches also go beyond the mean-field approximation by including the effects of two-body collisions.The self-consistent mean-field approximation is derived using the effective nucleon-nucleon Skyrme-type interaction. The many-body problem is described next in terms of the Wigner phase space of the one-body density, which provides a basis for semi-classical approximations and leads to kinetic equations. Results of static properties of nuclei and properties associated with small amplitude dynamics are also presented. Relaxation processes, due to nucleon-nucleon collisions, are discussed next, followed by instability and large amplitude motion of excited nuclei. Lastly, the book ends with the dynamics of hot nuclei. The concepts and methods developed in this book can be used for describing properties of other many-body systems.
Free Energy And Self Interacting Particles
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Author : Takashi Suzuki
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-01-08
Free Energy And Self Interacting Particles written by Takashi Suzuki and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-01-08 with Mathematics categories.
* Examines a nonlinear system of parabolic PDEs arising in mathematical biology and statistical mechanics * Describes the whole picture, i.e., the mathematical and physical principles * Suitable for researchers and grad students in mathematics and applied mathematics who are interested in nonlinear PDEs in stochastic processes, cellular automatons, variational methods, and their applications to physics, chemistry, biology, and engineering