The maximum entropy principle for non-equilibrium phase transitions: Determination of order parameters, slaved modes, and emerging patterns

1986 ◽  
Vol 63 (4) ◽  
pp. 487-491 ◽  
Author(s):  
H. Haken
Keyword(s):  
Phase Transitions ◽  
Maximum Entropy ◽  
Equilibrium Phase ◽  
Maximum Entropy Principle ◽  
Order Parameters ◽  
Emerging Patterns ◽  
Entropy Principle ◽  
Non Equilibrium

On Singular Closures for the 5-Moment System in Kinetic Gas Theory

2015 ◽  
Vol 17 (2) ◽  
pp. 371-400 ◽  
Author(s):  
Roman Pascal Schaerer ◽  
Manuel Torrilhon
Keyword(s):  
Maximum Entropy ◽  
Maximum Entropy Principle ◽  
Second Order ◽  
Gas Flows ◽  
Moment Equations ◽  
Moment System ◽  
Entropy Principle ◽  
Kinetic Gas Theory ◽  
Closure Theory ◽  
Non Equilibrium

AbstractMoment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which suffer not only from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. Amore recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

Natural partitioning of orientation elements and determination of the ODF from individual orientations according to the maximum-entropy principle

1999 ◽  
Vol 32 (3) ◽  
pp. 404-408
Author(s):  
Y. D. Wang ◽  
A. Vadon ◽  
J. Bessières ◽  
J. J. Heizmann
Keyword(s):  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Maximum Entropy Principle ◽  
Orientation Distribution ◽  
Entropy Method ◽  
Euler Space ◽  
Entropy Principle ◽  
Orientation Density ◽  
Preliminary Orientation

The orientation distribution function (ODF) of a polycrystalline material is usually constructed from individual orientations by the harmonic method on the assumption of a certain function distribution in the Euler space around each orientation. In the present paper, a new method is developed to determine the ODF from individual orientations. A natural partitioning of the orientation elements in the Euler space around some clustered orientations is proposed. Thus, the preliminary values of orientation density in the elements are directly estimated by the volumes of the orientation elements and the number of grains (or measured points) in each orientation element. Then, the texture vector is further refined using the maximum-entropy method with the preliminary orientation densities as constraints. The validity of this method is exemplified by the texture analysis of a cubic material from individual orientations modelled by Gaussian distribution.

An entropy concentration theorem: applications in artificial intelligence and descriptive statistics

1990 ◽  
Vol 27 (2) ◽  
pp. 303-313 ◽  
Author(s):  
Claudine Robert
Keyword(s):  
Artificial Intelligence ◽  
Maximum Entropy ◽  
Large Deviation ◽  
Maximum Entropy Principle ◽  
Statistical Modelling ◽  
Linear Constraints ◽  
Descriptive Statistics ◽  
Entropy Principle ◽  
Maximum Entropy Distribution ◽  
Mathematical Justification

The maximum entropy principle is used to model uncertainty by a maximum entropy distribution, subject to some appropriate linear constraints. We give an entropy concentration theorem (whose demonstration is based on large deviation techniques) which is a mathematical justification of this statistical modelling principle. Then we indicate how it can be used in artificial intelligence, and how relevant prior knowledge is provided by some classical descriptive statistical methods. It appears furthermore that the maximum entropy principle yields to a natural binding between descriptive methods and some statistical structures.

SINE ENTROPY FOR UNCERTAIN VARIABLES

2013 ◽  
Vol 21 (05) ◽  
pp. 743-753 ◽  
Author(s):  
KAI YAO ◽  
JINWU GAO ◽  
WEI DAI
Keyword(s):  
Maximum Entropy ◽  
Uncertainty Theory ◽  
Maximum Entropy Principle ◽  
Expected Value ◽  
Uncertain Variables ◽  
Entropy Principle

Entropy is a measure of the uncertainty associated with a variable whose value cannot be exactly predicated. In uncertainty theory, it has been quantified so far by logarithmic entropy. However, logarithmic entropy sometimes fails to measure the uncertainty. This paper will propose another type of entropy named sine entropy as a supplement, and explore its properties. After that, the maximum entropy principle will be introduced, and the arc-cosine distributed variables will be proved to have the maximum sine entropy with given expected value and variance.

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