Tree Network Formation in Poisson Equation Models and the Implications for the Maximum Entropy Production Principle

Affiliation(s)

Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, Japan.

Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, Japan.

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.

KEYWORDS

Dissipative Structure, Far from Equilibrium, Fractal, Poisson Equation, Maximum Entropy Production (MEP) Principle, Minimum Entropy Production (MinEP) Principle, Tree Network

Dissipative Structure, Far from Equilibrium, Fractal, Poisson Equation, Maximum Entropy Production (MEP) Principle, Minimum Entropy Production (MinEP) Principle, Tree Network

Cite this paper

Serizawa, H. , Amemiya, T. and Itoh, K. (2014) Tree Network Formation in Poisson Equation Models and the Implications for the Maximum Entropy Production Principle.*Natural Science*, **6**, 514-527. doi: 10.4236/ns.2014.67050.

Serizawa, H. , Amemiya, T. and Itoh, K. (2014) Tree Network Formation in Poisson Equation Models and the Implications for the Maximum Entropy Production Principle.

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[2] Rodriguez-Iturbe, I. and Rinaldo, A. (1997) Fractal River Basins: Chance and Self-Organization. Cambridge University Press, Cambridge.

[3] Shimokawa, M. and Ohta, S. (2012) Annihilative Fractals Formed in Rayleigh-Taylor instability. Fractals, 20, 97-104.

http://dx.doi.org/10.1142/S0218348X12500090

[4] Prigogine, I. (1962) Introduction to Non-Equilibrium Thermodynamics. Wiley-Interscience, New York.

[5] Prigogine, I. and Stengers, I. (1984) Order out of Chaos: Man’s New Dialogue with Nature. Bantam Books, New York.

[6] Kondepudi, D. and Prigogine, I. (1998) Modern Thermodynamics: From Heat Engines to Dissipative Structures. John Wiley & Sons, New York.

[7] Kleidon, A. and Lorenz, R.D. (2004) Entropy Production by Earth System Processes. In: Kleidon, A. and Lorenz, R.D., Eds., Non-Equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond, Springer-Verlag, Berlin, 1-20.

[8] Martyushev, L.M. and Seleznev, V.D. (2006) Maximum Entropy Production Principle in Physics, Chemistry and Biology. Physics Reports, 426, 1-45.

http://dx.doi.org/10.1016/j.physrep.2005.12.001

[9] Schrodinger, E. (1944) What Is Life? The Physical Aspect of the Living Cell. Cambridge University Press, Cambridge.

[10] Shimokawa, S. and Ozawa, H. (2002) On the Thermodynamics of the Oceanic General Circulation: Irreversible Transition to a State with Higher Rate of Entropy Production. Quarterly Journal of the Royal Meteorological Society, 128, 2115-2128.

http://dx.doi.org/10.1256/003590002320603566

[11] Meysman, F.J.R. and Bruers, S. (2010) Ecosystem Functioning and Maximum Entropy Production: A Quantitative Test of Hypotheses. Philosophical Transactions of the Royal Society B-Biological Science, 365, 1405-1416.

http://dx.doi.org/10.1098/rstb.2009.0300

[12] Bejan, A. and Lorente, S. (2006) Constructal Theory of Generation of Configuration in Nature and Engineering. Journal of Applied Physics, 100, Article ID: 041301.

http://dx.doi.org/10.1063/1.2221896

[13] Bejan, A. (2007) Constructal Theory of Pattern Formation. Hydrology and Earth System Sciences, 11, 753-768.

http://dx.doi.org/10.5194/hess-11-753-2007

[14] Kleidon, A. (2010) Life, Hierarchy, and the Thermodynamic Machinery of Planet Earth. Physics of Life Reviews, 7, 424-460. http://dx.doi.org/10.1016/j.plrev.2010.10.002

[15] Bejan, A. (2010) Design in Nature, Thermodynamics, and the Constructal Law. Comment on “Life, Hierarchy, and the Thermodynamic Machinery of Planet Earth” by Kleidon. Physics of Life Reviews, 7, 467-470.

http://dx.doi.org/10.1016/j.plrev.2010.10.008

[16] Kleidon, A. (2010) Life as the Major Driver of Planetary Geochemical Disequilibrium. Reply to Comments on “Life, Hierarchy, and the Thermodynamic Machinery of Planet Earth”. Physics of Life Reviews, 7, 473-476.

http://dx.doi.org/10.1016/j.plrev.2010.11.001

[17] Errera, M.R. and Bejan, A. (1998) Deterministic Tree Networks for River Drainage Basins. Fractals, 6, 245-261.

http://dx.doi.org/10.1142/S0218348X98000298

[18] Marin, C.A. and Errera, M.R. (2009) A Comparison between Random and Deterministic Dynamics of River Drainage Basins Formation. Engenharia Térmica (Thermal Engineering), 8, 65-71.

[19] Miyamoto, H., Baker, V.R. and Lorenz, R.D. (2004) Entropy and the Shaping and the Landscape by Water. In: Kleidon, A. and Lorenz R.D., Eds., Non-Equilibrium Thermodynamics and the Production of Entropy: Life, Earth, and Beyond, Springer-Verlag, Berlin, 135-146.