TITLE:
Tree Network Formation in Poisson Equation Models and the Implications for the Maximum Entropy Production Principle
AUTHORS:
Hiroshi Serizawa, Takashi Amemiya, Kiminori Itoh
KEYWORDS:
Dissipative Structure, Far from Equilibrium, Fractal, Poisson Equation, Maximum Entropy Production (MEP) Principle, Minimum Entropy Production (MinEP) Principle, Tree Network
JOURNAL NAME:
Natural Science,
Vol.6 No.7,
April
25,
2014
ABSTRACT: This paper presents
not only practical but also instructive mathematical models to simulate tree
network formation using the Poisson equation and the Finite Difference Method
(FDM). Then, the implications for entropic theories are discussed from the viewpoint
of Maximum Entropy Production (MEP). According to the MEP principle, open
systems existing in the state far from equilibrium are stabilized when entropy
production is maximized, creating dissipative structures with low entropy
such as the tree-shaped network. We prepare two simulation models: one is the
Poisson equation model that simulates the state far from equilibrium, and the
other is the Laplace equation model that simulates the isolated state or the
state near thermodynamic equilibrium. The output of these equations is
considered to be positively correlated to entropy production of the system.
Setting the Poisson equation model so that entropy production is maximized,
tree network formation is advanced. We suppose that this is due to the invocation
of the MEP principle, that is, entropy of the system is lowered by emitting
maximal entropy out of the system. On the other hand, tree network formation is
not observed in the Laplace equation model. Our simulation results will offer
the persuasive evidence that certifies the effect of the MEP principle.