Free Energy And Self-interacting Particles

This book examines a nonlinear system of parabolic partial differential equations 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 w