Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model

Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model

Author: Takashi Suzuki

Publisher: Springer

Published: 2015-11-19

Total Pages: 450

ISBN-13: 9462391548

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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.


Book Synopsis Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model by : Takashi Suzuki

Download or read book Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model written by Takashi Suzuki and published by Springer. This book was released on 2015-11-19 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

MEAN FIELD THEORIES AND DUAL VARIATION

Author: Takashi Suzuki

Publisher: Springer Science & Business Media

Published: 2009-01-01

Total Pages: 299

ISBN-13: 9491216228

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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.


Book Synopsis MEAN FIELD THEORIES AND DUAL VARIATION by : Takashi Suzuki

Download or read book MEAN FIELD THEORIES AND DUAL VARIATION written by Takashi Suzuki and published by Springer Science & Business Media. This book was released on 2009-01-01 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Mean Field Theories and Dual Variation

Author: Takashi Suzuki

Publisher: Atlantis Press

Published: 2008-09-01

Total Pages: 290

ISBN-13: 9789078677093

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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.


Book Synopsis Mean Field Theories and Dual Variation by : Takashi Suzuki

Download or read book Mean Field Theories and Dual Variation written by Takashi Suzuki and published by Atlantis Press. This book was released on 2008-09-01 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.


Mean Field Theories and Dual Variation ; a Mathematical Profile Emerged in the Nonlinear Hierarchy. Atlantis Studies in Mathematics for Engineering and Science

Mean Field Theories and Dual Variation ; a Mathematical Profile Emerged in the Nonlinear Hierarchy. Atlantis Studies in Mathematics for Engineering and Science

Author: T. Suzuki

Publisher:

Published: 2008

Total Pages:

ISBN-13:

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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 Smoluchowskiâ€"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.


Book Synopsis Mean Field Theories and Dual Variation ; a Mathematical Profile Emerged in the Nonlinear Hierarchy. Atlantis Studies in Mathematics for Engineering and Science by : T. Suzuki

Download or read book Mean Field Theories and Dual Variation ; a Mathematical Profile Emerged in the Nonlinear Hierarchy. Atlantis Studies in Mathematics for Engineering and Science written by T. Suzuki and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Smoluchowskiâ€"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 Theory

Mean Field Theory

Author: Vladimir M Kolomietz

Publisher: World Scientific

Published: 2020-05-08

Total Pages: 586

ISBN-13: 9811211795

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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.


Book Synopsis Mean Field Theory by : Vladimir M Kolomietz

Download or read book Mean Field Theory written by Vladimir M Kolomietz and published by World Scientific. This book was released on 2020-05-08 with total page 586 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.


Dynamical Mean Field Theory

Dynamical Mean Field Theory

Author: Jean-Marc Robin

Publisher: Lulu.com

Published: 2010

Total Pages: 166

ISBN-13: 1446638847

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This book is a short introduction to the Dynamical Mean-Field Theory for strongly correlated electrons. Its purpose is to focus on various local decoupling schemes in order to derive a self-consistent approximation and to map the lattice problem onto an impurity problem. Hubbard, Holstein, and Falicov-Kimball models are mainly used to provide examples of calculation. Numerous basic c/c++ programs are given along the book to develop confidence in computing actual numerical results.


Book Synopsis Dynamical Mean Field Theory by : Jean-Marc Robin

Download or read book Dynamical Mean Field Theory written by Jean-Marc Robin and published by Lulu.com. This book was released on 2010 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a short introduction to the Dynamical Mean-Field Theory for strongly correlated electrons. Its purpose is to focus on various local decoupling schemes in order to derive a self-consistent approximation and to map the lattice problem onto an impurity problem. Hubbard, Holstein, and Falicov-Kimball models are mainly used to provide examples of calculation. Numerous basic c/c++ programs are given along the book to develop confidence in computing actual numerical results.


Free Energy and Self-Interacting Particles

Free Energy and Self-Interacting Particles

Author: Takashi Suzuki

Publisher: Springer Science & Business Media

Published: 2008-01-08

Total Pages: 367

ISBN-13: 0817644369

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* 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


Book Synopsis Free Energy and Self-Interacting Particles by : Takashi Suzuki

Download or read book Free Energy and Self-Interacting Particles written by Takashi Suzuki and published by Springer Science & Business Media. This book was released on 2008-01-08 with total page 367 pages. Available in PDF, EPUB and Kindle. Book excerpt: * 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


Non-Local Partial Differential Equations for Engineering and Biology

Non-Local Partial Differential Equations for Engineering and Biology

Author: Nikos I. Kavallaris

Publisher: Springer

Published: 2017-11-28

Total Pages: 300

ISBN-13: 3319679449

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This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grade students in mathematics, engineering, physics, economics, and biology.


Book Synopsis Non-Local Partial Differential Equations for Engineering and Biology by : Nikos I. Kavallaris

Download or read book Non-Local Partial Differential Equations for Engineering and Biology written by Nikos I. Kavallaris and published by Springer. This book was released on 2017-11-28 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grade students in mathematics, engineering, physics, economics, and biology.


Methods Of Geometry In The Theory Of Partial Differential Equations: Principle Of The Cancellation Of Singularities

Methods Of Geometry In The Theory Of Partial Differential Equations: Principle Of The Cancellation Of Singularities

Author: Takashi Suzuki

Publisher: World Scientific

Published: 2024-01-22

Total Pages: 414

ISBN-13: 9811287910

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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.


Book Synopsis Methods Of Geometry In The Theory Of Partial Differential Equations: Principle Of The Cancellation Of Singularities by : Takashi Suzuki

Download or read book Methods Of Geometry In The Theory Of Partial Differential Equations: Principle Of The Cancellation Of Singularities written by Takashi Suzuki and published by World Scientific. This book was released on 2024-01-22 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.


THEORY OF CAUSAL DIFFERENTIAL EQUATIONS

THEORY OF CAUSAL DIFFERENTIAL EQUATIONS

Author: S. Leela

Publisher: Springer Science & Business Media

Published: 2010-01-01

Total Pages: 218

ISBN-13: 9491216252

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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.


Book Synopsis THEORY OF CAUSAL DIFFERENTIAL EQUATIONS by : S. Leela

Download or read book THEORY OF CAUSAL DIFFERENTIAL EQUATIONS written by S. Leela and published by Springer Science & Business Media. This book was released on 2010-01-01 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.