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

2015-11-19 Mathematics
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Release : 2015-11-19
Genre : Mathematics
Kind : eBook
Book Rating : 548/5 ( reviews)

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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. This book was released on 2015-11-19. 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

2008-09-01 Mathematics
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Release : 2008-09-01
Genre : Mathematics
Kind : eBook
Book Rating : 093/5 ( reviews)

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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. This book was released on 2008-09-01. 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.

Semilinear Elliptic Equations

2020-10-12 Mathematics
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Release : 2020-10-12
Genre : Mathematics
Kind : eBook
Book Rating : 286/5 ( reviews)

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Book Synopsis Semilinear Elliptic Equations by : Takashi Suzuki

Download or read book Semilinear Elliptic Equations written by Takashi Suzuki. This book was released on 2020-10-12. Available in PDF, EPUB and Kindle. Book excerpt: This authoritative monograph presents in detail classical and modern methods for the study of semilinear elliptic equations, that is, methods to study the qualitative properties of solutions using variational techniques, the maximum principle, blowup analysis, spectral theory, topological methods, etc. The book is self-contained and is addressed to experienced and beginning researchers alike.

Vortex Dynamics

2022-09-28 Science
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Release : 2022-09-28
Genre : Science
Kind : eBook
Book Rating : 244/5 ( reviews)

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Book Synopsis Vortex Dynamics by : İlkay Bakırtaş

Download or read book Vortex Dynamics written by İlkay Bakırtaş. This book was released on 2022-09-28. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Non-Local Partial Differential Equations for Engineering and Biology

2017-11-28 Technology & Engineering
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Release : 2017-11-28
Genre : Technology & Engineering
Kind : eBook
Book Rating : 449/5 ( reviews)

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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. This book was released on 2017-11-28. 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.

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