Entropy dimension for deterministic walks in random sceneries

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

<p style='text-indent:20px;'>In this text I study the asymptotics of the complexity function of <i>minimal</i> multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [<xref ref-type="bibr" rid="b12">12</xref>] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the <inline-formula><tex-math id="M1">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M2">\begin{document}$ [0, d] $\end{document}</tex-math></inline-formula>. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all <inline-formula><tex-math id="M3">\begin{document}$ d \ge 2 $\end{document}</tex-math></inline-formula> the possible values for entropy dimensions of <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional SFT are the <inline-formula><tex-math id="M5">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M6">\begin{document}$ [0, d-1] $\end{document}</tex-math></inline-formula>. In the present text I prove formally this result for <inline-formula><tex-math id="M7">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case <inline-formula><tex-math id="M8">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> requires some substantial changes to be made in order to adapt the construction that are not discussed here.</p>

A new criterion for determining the representative elementary volume of translucent porous media and inner contaminant

Representative elementary volume (REV) is essential for measuring and quantifying the effective parameters of a complex heterogeneous medium. To overcome the limitations of the existing REV estimation criteria, a new REV estimation criterion (χi) based on dimensionless range and gradient calculation is proposed in this study to estimate the REV of a translucent material based on light transmission techniques. Three sandbox experiments are performed to estimate REVs of porosity, density, tortuosity, and perchloroethylene (PCE) plume using multiple REV estimation criteria. In comparison with χi, previous REV estimation criteria based on the coefficient of variation (CVi), the entropy dimension (DIi) and the relative gradient error (εgi) are tested in REV quantification of translucent silica and the inner PCE plume to achieve their corresponding effects. Results suggest that the new criterion (χi) can effectively identify the REV in the materials, whereas the coefficient of variation and entropy dimension (F=-2.01×10-12+12π×1.50e-(REV-4.35)22⋅1.502) are not effective. The relative gradient error can make the REV plateau obvious, while random fluctuations make the REV plateau difficult to identify accurately. Therefore, the new criterion is appropriate for REV estimation of the translucent materials and inner contaminant. Models are built based on a Gaussian equation to simulate the distribution of REVs for media properties, whose frequency of REV is dense in the middle and sparse on both sides. REV estimation of the PCE plume indicates that a high level of porosity leads to a large value of mean and standard deviation for REVs of PCE saturation (So) and PCE–water interfacial area (AOW). Fitted equations are derived from distribution of REVs for the PCE plume related to dm (distances from mass center to considered point) and dI (distances from injection position to considered point). Moreover, relationships between REVs of the PCE plume and So are fitted using regression analysis. Results suggest a decreasing trend appears for So-REV when So increases, while AOW-REV increases with increasing So.

A new criterion for determining the representative elementary volume of translucent porous media and inner contaminant

Representative elementary volume (REV) is essential to measure and quantify the effective parameters of a complex heterogeneous medium. Since previous REV estimation criteria having multiple limitations, a new criterion (χi) is proposed to estimate REV of a translucent material based on light transmission techniques. Two sandbox experiments are performed to estimate REVs of porosity, density, tortuosity and perchloroethylene (PCE) plume using multiple REV estimation criteria. In comparison with χi, previous REV estimation criteria based on the coefficient of variation (CVi), the entropy dimension (DIi) and the relative gradient error (εgi) are tested in REV quantification of translucent silica and inner PCE plume to achieve their corresponding effects. Results suggest that new criterion (χi) can effectively identify the REV in the materials, whereas the coefficient of variation (CVi) and entropy dimension (DIi) cannot. The relative gradient error (εgi) can make the REV plateau obvious, while random fluctuations make the REV plateau uneasy to identify accurately. Therefore, the new criterion is appropriate for REV estimation for the translucent materials and inner contaminant. Models are built based on Gaussian equation to simulate the distribution of REVs for media properties, which frequency of REV is dense in the middle and sparse on both sides. REV estimation of PCE plume indicates high level of porosity lead to large value of mean and standard deviation for REVs of PCE saturation (So) and PCE-water interfacial area (AOW). Fitted equations are derived for distribution of REVs for PCE plume related to dm (distances from mass center to considered point) and dI (distances from injection position to considered point). Moreover, relationships between REVs of PCE plume and So are fitted using regression analysis. Results suggest a decreasing trend appears for So-REV when So increases, while Aow-REV increases with increasing of So.