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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Chemotaxis

Abstract
Chemotaxis is a feature that living things are attracted by special chemical source, which results in, for example, the formation of spores in the case of cellular slime molds.
Takashi Suzuki

Chapter 2. Time Relaxization

Abstract
Different models can share common stationary states. In the previous chapter we described the quantized blowup mechanics of Smoluchowski-Poisson equation in two-space dimensions. This chapter is devoted to the full system of chemotaxis and its relatives.
Takashi Suzuki

Chapter 3. Toland Duality

Abstract
The collapse mass quantization, (1.​26) to (1.​8), has its origin in a stationary state (1.​38) which provides a variational structure of duality between the particle density u and the field distribution v.
Takashi Suzuki

Chapter 4. Phenomenology

Abstract
An important feature of dual variation is the unfolding-minimality concerning the stationary state. There are, however, systems provided only with the semi-duality, that is semi-unfolding and minimality.
Takashi Suzuki

Chapter 5. Phase Transition

Abstract
Having model (A) and model (B) equations, we devote this chapter to the physical principles that derive model (C) equations consistent with the non-equilibrium thermodynamics.
Takashi Suzuki

Chapter 6. Critical Phenomena of Isolated Systems

Abstract
Regarding the order parameter as the “field component,” we see the semi-dual variational structure in phase field equations associated with several critical phenomena. Particularly, the nonlinear eigenvalue problem with non-local term arises as the stationary state of the closed system. Recognizing the profile of the total set of stationary solutions and distinguishing their stability and instability are the first step to clarify the process of self-organization. In this chapter we study these variational structures sealed in several mathematical models of self-assembly.
Takashi Suzuki

Chapter 7. Self-interacting Fluids

Abstract
Macroscopic state of particles that constitute self-interacting fluid is formulated by a system of equations provided with self-duality between the particle density and the field distribution.
Takashi Suzuki

Chapter 8. Magnetic Fields

Abstract
The electro-magnetic fields are described by the Maxwell equation. Concerning the static case, first, we have the regularity in a specific components of the solution across the interface.
Takashi Suzuki

Chapter 9. Boltzmann-Poisson Equation

Abstract
The equilibrium statistical mechanics provides the derivation of the mean field limit of many self-interacting particles in the equilibrium state.
Takashi Suzuki

Chapter 10. Particle Kinetics

Abstract
This chapter is devoted to several mathematical modelings and analysis used in non-equilibrium statistical mechanics. The first two sections are concerned with the macroscopic description of the non-stationary mean field equation based on the microscopic and mezoscopic overviews, respectively.
Takashi Suzuki

Chapter 11. Parabolic Equations

Abstract
A degenerate parabolic equation arises in the context of astrophysics whereby the critical exponent is detected from the scaling invariance of the model compatible to the total mass conservation.
Takashi Suzuki

Chapter 12. Gauge Fields

Abstract
In the first quantization the movement of the particle density is described by a transformation of the Hamiltonian of classical particles. We take this process under the Maxwell gauge, using the Bogomol’nyi structure and the self-duality, to obtain again the exponential nonlinearity competing to the two-dimensional diffusion. Then we can show the quantized blowup mechanism for solutions to this Boltzmann-Poisson equation in two-space dimension. We develop the blowup analysis based on the scaling invariance of the problem and reveals the quantized blowup mechanism through several hierarchical arguments. This process provides with the fundamental motivation of the study on the system of chemotaxis in Chap. 1, that is the nonlinear spectral mechanics.
Takashi Suzuki

Chapter 13. Higher-Dimensional Blowup

Abstract
Mass and energy quantizations are observed even in higher-space dimension.
Takashi Suzuki

Backmatter

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