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Entropy 2016, 18(4), 155; doi:10.3390/e18040155

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

1
LAMIH UMR CNRS 8201, Université de Valenciennes, Valenciennes 59313, France
2
Laboratoire MSMP, Arts et Métiers ParisTech, Lille 59046, France
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: Kevin H. Knuth
Received: 26 October 2015 / Revised: 15 March 2016 / Accepted: 12 April 2016 / Published: 21 April 2016
(This article belongs to the Special Issue Entropy Generation in Thermal Systems and Processes 2015)
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Abstract

In a previous investigation (Bigerelle and Iost, 2004), the authors have proposed a physical interpretation of the instability λ = Δtx2 > 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. View Full-Text
Keywords: instability; entropy; parabolic partial differential equations; Monte Carlo simulations; data compression; information theory instability; entropy; parabolic partial differential equations; Monte Carlo simulations; data compression; information theory
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Bigerelle, M.; Naceur, H.; Iost, A. Analyses of the Instabilities in the Discretized Diffusion Equations via Information Theory. Entropy 2016, 18, 155.

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