Mean Field Theories and Dual Variation by Takashi Suzuki (Osaka University, Japan) Series: Atlantis Studies in Mathematics for Engineering and Science, Volume 2. 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 socalled 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 multiscale mathematical explanations of the Smoluchowski–Poisson model in nonequilibrium thermodynamics, twodimensional turbulence theory, selfdual gauge theory, and so forth. This book shows how and why many different nonlinear problems are interconnected in terms of the properties of duality and scaling, and the way to analyze them mathematically.
 ScalingRevealing Hierarchy:
HigherDimensional Blowup Publishing Date: January 2009 
