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

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.

Clarifying common mathematical structures used in wide area of sciences; physics, chemistry, biology Striking mathematical phenomena emerged from these structures, recognized only by these new points of view; quantized blowup mechanism, spatially homogenization, interface regularity, and so forth Linking of physical principles and mathematical analysis to nonlinear non-stationary models, that is the nonlinear spectral mechanics, especially valid in the region of near from equilibrium Includes supplementary material: sn.pub/extras

Inhalt
Chemotaxis.- Time Relaxization.- Toland Duality.- Phenomenology.- Phase Transition.- Critical Phenomena of Isolated Systems.- Self-Interacting Fluids.- Magnetic Fields.- Boltzmann-Poisson Equation.- Particle Kinetics.- Parabolic Equations.- Gauge Fields.- Higher- Dimensional Blowup.