Free Energy and Self-Interacting Particles

Takashi Suzuki (eds.)0817643028, 9780817643027

This book examines a system of parabolic-elliptic partial differential eq- tions proposed in mathematical biology, statistical mechanics, and chemical kinetics. In the context of biology, this system of equations describes the chemotactic feature of cellular slime molds and also the capillary formation of blood vessels in angiogenesis. There are several methods to derive this system. One is the biased random walk of the individual, and another is the reinforced random walk of one particle modelled on the cellular automaton. In the context of statistical mechanics or chemical kinetics, this system of equations describes the motion of a mean ?eld of many particles, interacting under the gravitational inner force or the chemical reaction, and therefore this system is af?liated with a hierarchy of equations: Langevin, Fokker–Planck, Liouville–Gel’fand, and the gradient ?ow. All of the equations are subject to the second law of thermodynamics — the decrease of free energy. The mat- matical principle of this hierarchy, on the other hand, is referred to as the qu- tized blowup mechanism; the blowup solution of our system develops delta function singularities with the quantized mass.

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Table of contents :
Summary….Pages 1-23
Background….Pages 25-34
Fundamental Theorem….Pages 35-58
Trudinger-Moser Inequality….Pages 59-77
The Green’s Function….Pages 79-103
Equilibrium States….Pages 105-113
Blowup Analysis for Stationary Solutions….Pages 115-145
Multiple Existence….Pages 147-173
Dynamical Equivalence….Pages 175-205
Formation of Collapses….Pages 207-218
Finiteness of Blowup Points….Pages 219-245
Concentration Lemma….Pages 247-275
Weak Solution….Pages 277-291
Hyperparabolicity….Pages 293-306
Quantized Blowup Mechanism….Pages 307-322
Theory of Dual Variation….Pages 323-343