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Mean Field Theories and Dual Variation

  • Takashi Suzuki
Book

Part of the Atlantis Studies in Mathematics for Engineering and Science book series (ASMES, volume 2)

Table of contents

  1. Front Matter
    Pages i-x
  2. Takashi Suzuki
    Pages 1-140
  3. Takashi Suzuki
    Pages 141-284
  4. Back Matter
    Pages 271-288

About this book

Introduction

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.

Keywords

differential equation non-linear differentail equations nonlinear partial differential equation partial differential equation turbulence

Authors and affiliations

  • Takashi Suzuki
    • 1
  1. 1.Division of Mathematical Science Department of Systems InnovationGraduate School of Engineering Science Osaka UniversityOsakaJapan

Bibliographic information