Free Energy and Self-Interacting Particles
Suzuki, Takashi.
Free Energy and Self-Interacting Particles [electronic resource] / edited by Takashi Suzuki. - XIII, 366 p. 7 illus. online resource. - Progress in Nonlinear Differential Equations and Their Applications ; 62 . - Progress in Nonlinear Differential Equations and Their Applications ; 62 .
Summary -- Background -- Fundamental Theorem -- Trudinger-Moser Inequality -- The Green's Function -- Equilibrium States -- Blowup Analysis for Stationary Solutions -- Multiple Existence -- Dynamical Equivalence -- Formation of Collapses -- Finiteness of Blowup Points -- Concentration Lemma -- Weak Solution -- Hyperparabolicity -- Quantized Blowup Mechanism -- Theory of Dual Variation.
This book examines a nonlinear system of parabolic partial differential equations (PDEs) arising in mathematical biology and statistical mechanics. In the context of biology, the system typically describes the chemotactic feature of cellular slime molds. One way of deriving these equations is via the random motion of a particle in a cellular automaton. In statistical mechanics the system is associated with the motion of the mean field of self-interacting particles under gravitational force. Physically, such a system is related to Langevin, Fokker-Planck, Liouville and gradient flow equations. Mathematically, the mechanism can be referred to as a quantized blowup. This book describes the whole picture, i.e., the mathematical and physical principles: derivation of a series of equations, biological modeling based on biased random walks, the study of equilibrium states via the variational structure derived from the free energy, and the quantized blowup mechanism based on several PDE techniques.
9780817644369 99780817644369
10.1007/0-8176-4436-9 doi
MATHEMATICS.
CHEMISTRY--MATHEMATICS.
DIFFERENTIAL EQUATIONS, PARTIAL.
BIOLOGY--MATHEMATICS.
MATHEMATICAL PHYSICS.
ENGINEERING MATHEMATICS.
MATHEMATICS.
PARTIAL DIFFERENTIAL EQUATIONS.
APPLICATIONS OF MATHEMATICS.
MATHEMATICAL METHODS IN PHYSICS.
MATHEMATICAL BIOLOGY IN GENERAL.
APPL.MATHEMATICS/COMPUTATIONAL METHODS OF ENGINEERING.
MATH. APPLICATIONS IN CHEMISTRY.
515.353
Free Energy and Self-Interacting Particles [electronic resource] / edited by Takashi Suzuki. - XIII, 366 p. 7 illus. online resource. - Progress in Nonlinear Differential Equations and Their Applications ; 62 . - Progress in Nonlinear Differential Equations and Their Applications ; 62 .
Summary -- Background -- Fundamental Theorem -- Trudinger-Moser Inequality -- The Green's Function -- Equilibrium States -- Blowup Analysis for Stationary Solutions -- Multiple Existence -- Dynamical Equivalence -- Formation of Collapses -- Finiteness of Blowup Points -- Concentration Lemma -- Weak Solution -- Hyperparabolicity -- Quantized Blowup Mechanism -- Theory of Dual Variation.
This book examines a nonlinear system of parabolic partial differential equations (PDEs) arising in mathematical biology and statistical mechanics. In the context of biology, the system typically describes the chemotactic feature of cellular slime molds. One way of deriving these equations is via the random motion of a particle in a cellular automaton. In statistical mechanics the system is associated with the motion of the mean field of self-interacting particles under gravitational force. Physically, such a system is related to Langevin, Fokker-Planck, Liouville and gradient flow equations. Mathematically, the mechanism can be referred to as a quantized blowup. This book describes the whole picture, i.e., the mathematical and physical principles: derivation of a series of equations, biological modeling based on biased random walks, the study of equilibrium states via the variational structure derived from the free energy, and the quantized blowup mechanism based on several PDE techniques.
9780817644369 99780817644369
10.1007/0-8176-4436-9 doi
MATHEMATICS.
CHEMISTRY--MATHEMATICS.
DIFFERENTIAL EQUATIONS, PARTIAL.
BIOLOGY--MATHEMATICS.
MATHEMATICAL PHYSICS.
ENGINEERING MATHEMATICS.
MATHEMATICS.
PARTIAL DIFFERENTIAL EQUATIONS.
APPLICATIONS OF MATHEMATICS.
MATHEMATICAL METHODS IN PHYSICS.
MATHEMATICAL BIOLOGY IN GENERAL.
APPL.MATHEMATICS/COMPUTATIONAL METHODS OF ENGINEERING.
MATH. APPLICATIONS IN CHEMISTRY.
515.353