MEAN FIELD THEORIES AND DUAL VARIATION [electronic resource] : A Mathematical Profile Emerged in the Nonlinear Hierarchy /
Suzuki, Takashi.author.
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.
Description based upon print version of record.
""Mean Field Theories and Dual Variation""; ""Preface""; ""Contents""; ""Chapter 1 Duality Sealed Variation""; ""1.1. Chemotaxis""; ""1.1.1. Thermal Transport""; ""Heat Equation""; ""1.1.2. Collapse Formation""; ""1.1.3. Mean Field Hierarchy""; ""1.1.4. Stationary State""; ""1.1.5. Localization and Symmetrization""; ""1.1.6. Rescaling""; ""Parabolic Envelope""; ""Generation of theWeak Solution""; ""Second Moment Self-Similarity""; ""1.1.7. Mass Quantization""; ""1.1.8. Blowup Rate""; ""Free Energy Transmission""; ""1.1.9. Blowup in Infinite Time""; ""Formation of the Quantized Collapse""
""Collapse Dynamics""""1.1.10. Time Relaxization""; ""Non-Local Parabolic Equation""; ""Non-Local ODE""; ""Cancer Model""; ""1.1.11. Geometric Flow""; ""Harmonic Heat Flow""; ""Normalized Ricci Flow""; ""1.1.12. Summary""; ""1.2. Toland Duality""; ""1.2.1. Exponential Nonlinearity""; ""1.2.2. Full System of Chemotaxis""; ""1.2.3. Lagrangian""; ""1.2.4. Variational Equivalence""; ""1.2.5. Spectral Equivalence""; ""1.2.6. Stability""; ""1.2.7. Gradient Systems with Duality""; ""1.2.8. Entropy Functional""; ""1.2.9. Zygmund Space""; ""1.2.10. Summary""; ""1.3. Phenomenology""
""1.3.1. Non-Convex Evolution""""Helmholtzâ€? Free Energy""; ""Allen-Cahn Equation""; ""Cahn-Hilliard Equation""; ""Ohta-Kawasakiâ€?s Free Energy""; ""1.3.2. Gradient and Skew-Gradient Systems""; ""Eguchi-Oki-Matsumura Equation""; ""Gierer-Meinhardt Equation""; ""1.3.3. Skew-Gradient Systems with Duality""; ""1.3.4. Semi-Unfolding-Minimality""; ""Semi-Lagrangian""; ""Semi-Skew-Lagrangian""; ""FitzHugh-Nagumo Equation""; ""1.3.5. Kuhn-Tucker Duality""; ""Skew-Lagrangian with Duality""; ""Linear Programing""; ""Stokes System""; ""1.3.6. Summary""; ""1.4. Thermodynamics""
""1.4.1. Stefan Problem""""Level Set Approach""; ""Enthalpy Formulation""; ""Nonlinear Semi-Group Theory""; ""Phase Field Model""; ""1.4.2. Thermal Equilibrium""; ""Quantity of State""; ""Energy Balance""; ""Entropy Increasing""; ""Free Energy""; ""Chemical Potential""; ""Equilibrium""; ""1.4.3. Penrose-Fife Theory""; ""1.4.4. Non-Equilibrium""; ""Balance Laws""; ""Dissipation Function""; ""Entropy Production Rate""; ""Phenomenological Relation""; ""Principle of Detailed Balance""; ""1.4.5. Summary""; ""1.5. Phase Fields""; ""1.5.1. Fix-Caginalp Equation""; ""Thermally Open System""
""Thermal Analysis""""Enthalpy Analysis""; ""1.5.2. Penrose-Fife Equation""; ""1.5.3. Coupled Cahn-Hilliard Equation""; ""1.5.4. Shape Memory Alloys""; ""Three-Dimensional Model""; ""One-Dimensional Case""; ""1.5.5. Time Map""; ""1.5.6. Summary""; ""Chapter 2 Scaling Revealing Hierarchy""; ""2.1. Self-Interacting Continuum""; ""2.1.1. Self-Gravitating Fluids""; ""Euler-Poisson Equation""; ""Conservation Laws""; ""Makinoâ€?s Solution""; ""Second Moment""; ""2.1.2. Equilibrium""; ""Radially Symmetric Equilibrium""; ""Variational Equilibrium""; ""2.1.3. Plasma Confinement""; ""Field Variation""
""Particle Variation""
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.
Differential equations.
Partial differential equations.
Ordinary Differential Equations.
Partial Differential Equations.
Atlantis Studies in Mathematics for Engineering and Science