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Analyses of the Instabilities in the Discretized Diffusion Equations via Information Theory

Article dans une revue avec comité de lecture
Author
BIGERELLE, Maxence
NACEUR, Hakim
1303 Laboratoire d'Automatique, de Mécanique et d'Informatique industrielles et Humaines - UMR 8201 [LAMIH]
IOST, Alain
211915 Mechanics surfaces and materials processing [MSMP]

URI
http://hdl.handle.net/10985/10901
DOI
10.3390/e18040155
Date
2016
Journal
Entropy

Abstract

In a previous investigation (Bigerelle and Iost, 2004), the authors have proposed a physical interpretation of the instability λ = Δt/Δx2 > 1/2 of the parabolic partial differential equations when solved by finite differences. However, our results were obtained using integration techniques based on erf functions meaning that no statistical fluctuation was introduced in the mathematical background. In this paper, we showed that the diffusive system can be divided into sub-systems onto which a Brownian motion is applied. Monte Carlo simulations are carried out to reproduce the macroscopic diffusive system. It is shown that the amount of information characterized by the compression ratio of information of the system is pertinent to quantify the entropy of the system according to some concepts introduced by the authors (Bigerelle and Iost, 2007). Thanks to this mesoscopic discretization, it is proved that information on each sub-cell of the diffusion map decreases with time before the unstable equality λ = 1/2 and increases after this threshold involving an increase in negentropy, i.e., a decrease in entropy contrarily to the second principle of thermodynamics.

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