Free Energy and Self-Interacting Particles (eBook)

Takashi Suzuki (Autor)

eBook Download: PDF

2008 | 2005
XIII, 370 Seiten
Birkhäuser Boston (Verlag)
978-0-8176-4436-9 (ISBN)

Infos

* Examines a nonlinear system of parabolic PDEs arising in mathematical biology and statistical mechanics

* Describes the whole picture, i.e., the mathematical and physical principles

* Suitable for researchers and grad students in mathematics and applied mathematics who are interested in nonlinear PDEs in stochastic processes, cellular automatons, variational methods, and their applications to physics, chemistry, biology, and engineering

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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.

Inhalt

Contents 7
Preface 9
Acknowledgment 13
1 Summary 14
2 Background 37
3 Fundamental Theorem 47
4 Trudinger–Moser Inequality 71
5 The Green’s Function 90
6 Equilibrium States 115
7 Blowup Analysis for Stationary Solutions 124
8 Multiple Existence 155
9 Dynamical Equivalence 182
10 Formation of Collapses 213
11 Finiteness of Blowup Points 225
12 Concentration Lemma 252
13 Weak Solution 281
14 Hyperparabolicity 296
15 Quantized Blowup Mechanism 310
16 Theory of Dual Variation 326
References 347

Produkt-Details

• Erscheint lt. Verlag: 8.1.2008
• Reihe/Serie: Progress in Nonlinear Differential Equations and Their Applications
• Reihe/Serie: Progress in Nonlinear Differential Equations and Their Applications
• Zusatzinfo: XIII, 370 p.
• Verlagsort: Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Naturwissenschaften ► Biologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Applied mathematics • Calculus • differential equation • Green's Function • Mathematical Biology • Mathematical Physics • Mechanics • Modeling • partial differential equation • Partial differential equations • quantum mechanics • Statistica • thermodynamics
• ISBN-10: 0-8176-4436-9 / 0817644369
• ISBN-13: 978-0-8176-4436-9 / 9780817644369