Free Energy and Self-Interacting Particles

  • Takashi Suzuki

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 62)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Pages 1-23
  3. Pages 25-34
  4. Pages 35-58
  5. Pages 79-103
  6. H Tanaka
    Pages 105-113
  7. Pages 147-173
  8. Pages 175-205
  9. Pages 207-218
  10. Pages 247-275
  11. Pages 277-291
  12. Pages 293-306
  13. Pages 323-343
  14. Back Matter
    Pages 345-366

About this book


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.


Applied Mathematics Green's function Mathematical Biology Mathematical Physics Partial Differential Equations STATISTICA calculus differential equation mechanics modeling partial differential equation quantum mechanics thermodynamics

Editors and affiliations

  • Takashi Suzuki
    • 1
  1. 1.Department of System Innovation Division of Mathematical ScienceOsaka University Graduate School of Engineering ScienceOsakaJapan

Bibliographic information