| Free Energy and Self-Interacting Particles 2005 Edition Contributor(s): Suzuki, Takashi (Author) |
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ISBN: 0817643028 ISBN-13: 9780817643027 Publisher: Birkhauser OUR PRICE: $104.49 Product Type: Hardcover - Other Formats Published: May 2005 Annotation: 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, on the other hand, 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, which involve the issues of free energy and the second law of thermodynamics. Mathematically, the mechanism can be referred to as a quantized blowup. Actually, it is regarded as a nonlinear theory of quantum mechanics, and it comes from the mass and location quantization of the singular limit for the associated nonlinear eigenvalue problems. 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. Free Energy and Self-Interacting Particles is suitable for researchers and graduate students of mathematics and applied mathematics who are interested in nonlinear PDEs in stochastic processes, cellular automatons, variational methods, and their applications to natural sciences. It is also suitable for researchers in other fields such as physics, chemistry, biology, and engineering. |
| Additional Information |
| BISAC Categories: - Mathematics | Differential Equations - General - Mathematics | Applied - Science | Physics - Mathematical & Computational |
| Dewey: 515.353 |
| LCCN: 2004062340 |
| Series: Progress in Nonlinear Differential Equations and Their Appli |
| Physical Information: 0.91" H x 6.22" W x 9.52" (1.48 lbs) 370 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: 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. |