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, Fokkera
"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.
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