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Entropy, Volume 19, Issue 3 (March 2017) – 48 articles

Cover Story (view full-size image): Free energy transduction can be obtained when external oscillating fields couple to internal conformational fluctuations contributing to the maintenance of a non-equilibrium state. This would explain the high efficiencies observed in the mechanisms of electrochemical signaling in living cells. The use of structure-based theoretical approaches allows assessing how fluctuation-driven transport occurs in confined systems such as protein channels with unprecedented detail. View this paper
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1619 KiB  
Article
Structure and Dynamics of Water at Carbon-Based Interfaces
by Jordi Martí, Carles Calero and Giancarlo Franzese
Entropy 2017, 19(3), 135; https://doi.org/10.3390/e19030135 - 21 Mar 2017
Cited by 16 | Viewed by 6338
Abstract
Water structure and dynamics are affected by the presence of a nearby interface. Here, first we review recent results by molecular dynamics simulations about the effect of different carbon-based materials, including armchair carbon nanotubes and a variety of graphene sheets—flat and with corrugation—on [...] Read more.
Water structure and dynamics are affected by the presence of a nearby interface. Here, first we review recent results by molecular dynamics simulations about the effect of different carbon-based materials, including armchair carbon nanotubes and a variety of graphene sheets—flat and with corrugation—on water structure and dynamics. We discuss the calculations of binding energies, hydrogen bond distributions, water’s diffusion coefficients and their relation with surface’s geometries at different thermodynamical conditions. Next, we present new results of the crystallization and dynamics of water in a rigid graphene sieve. In particular, we show that the diffusion of water confined between parallel walls depends on the plate distance in a non-monotonic way and is related to the water structuring, crystallization, re-melting and evaporation for decreasing inter-plate distance. Our results could be relevant in those applications where water is in contact with nanostructured carbon materials at ambient or cryogenic temperatures, as in man-made superhydrophobic materials or filtration membranes, or in techniques that take advantage of hydrated graphene interfaces, as in aqueous electron cryomicroscopy for the analysis of proteins adsorbed on graphene. Full article
(This article belongs to the Special Issue Nonequilibrium Phenomena in Confined Systems)
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Figure 1

Figure 1
<p>Water energy density <span class="html-italic">E</span> versus the surface density <span class="html-italic">ρ</span> for three of the four systems considered in this work, the (5,5) and (9,9) CNTs and the graphene sheet. In each panel we show simulation results (squares) and least squares fits to Equation (2) (lines) for <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>323</mn> </mrow> </semantics> </math> K (<b>a</b>), <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>310</mn> </mrow> </semantics> </math> K (<b>b</b>) and <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>298</mn> </mrow> </semantics> </math> K (<b>c</b>).</p> Full article ">Figure 2
<p>Water free energy density <span class="html-italic">F</span> versus the surface per molecule for the (5,5), (9,9) and (12,12) CNTs and the graphene sheet at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>298</mn> </mrow> </semantics> </math> K, as estimated from Equations (1) and (2) using the fitting parameters calculated in <a href="#entropy-19-00135-f001" class="html-fig">Figure 1</a>. For the thinner (5,5), (9,9) CNTs the free energy minimum is around 4.05 Å<sup>2</sup>, while for the (12,12) CNT and the graphene sheet there are no stable minima within the range of densities considered in this work.</p> Full article ">Figure 3
<p>Number of HBs (lines with symbols, scale on the left) and radial oxygen density profiles (lines without symbols, scale on the right) of water adsorbed at the external volume of CNTs as a function of radial distance from the CNTs axis. The number of hydration HBs is normalized with the bulk HB number. Calculations represented with squares or dashed lines, circles or continuous lines, triangles or dotted lines are for (5,5), (9,9), (12,12) CNTs, respectively.</p> Full article ">Figure 4
<p>Orientational order of water outside the CNTs as a function of the distance <span class="html-italic">r</span> from the CNTs surface. The two first Legendre polynomial <math display="inline"> <semantics> <mrow> <mo>〈</mo> <mi>cos</mi> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>〉</mo> </mrow> </semantics> </math> (<b>a</b>) and <math display="inline"> <semantics> <mrow> <mo>〈</mo> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>∗</mo> <mi>cos</mi> <msup> <mi>θ</mi> <mn>2</mn> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>〉</mo> </mrow> </semantics> </math> (<b>b</b>) characterize water dipolar orientations, being <math display="inline"> <semantics> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> the angle between the instantaneous dipole moment of water and the direction normal to the CNT surface. Symbols and lines are as in <a href="#entropy-19-00135-f003" class="html-fig">Figure 3</a>.</p> Full article ">Figure 5
<p>Snapshot of MD simulation at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>275</mn> </mrow> </semantics> </math> K and <math display="inline"> <semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>400</mn> </mrow> </semantics> </math> bar of two fixed parallel graphene plates separated by <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>9.5</mn> </mrow> </semantics> </math> Å and immersed in TIP4P/2005-water (top view). Carbon atoms are in cyan, oxygen in red, hydrogen in white. In our simulations the total number of water molecules, including those outside the confined region, remains constant while the amount of water confined between the graphene sheets depends on the spacing <span class="html-italic">d</span>. Under these conditions the confined water forms a hexagonal ice bilayer.</p> Full article ">Figure 6
<p>Diffusion coefficient <math display="inline"> <semantics> <msub> <mi>D</mi> <mo>∥</mo> </msub> </semantics> </math> of confined TIP4P/2005-water at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>275</mn> </mrow> </semantics> </math> K and <math display="inline"> <semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>400</mn> </mrow> </semantics> </math> bar in a plane parallel to the graphene sheets as a function of inter-plate distance <span class="html-italic">d</span>. The dashed line is a guide to the eyes at <math display="inline"> <semantics> <mrow> <msub> <mi>D</mi> <mo>∥</mo> </msub> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics> </math> nm<sup>2</sup>/ns, a value close to the bulk diffusion coefficient for this water model under these conditions. <math display="inline"> <semantics> <msub> <mi>D</mi> <mo>∥</mo> </msub> </semantics> </math> vanishes for 8.5 Å <math display="inline"> <semantics> <mrow> <mo>&lt;</mo> <mi>d</mi> <mo>&lt;</mo> <mn>10</mn> </mrow> </semantics> </math> Å.</p> Full article ">Figure 7
<p>Rotational relaxation time <math display="inline"> <semantics> <msub> <mi>τ</mi> <mn>1</mn> </msub> </semantics> </math> for confined TIP4P/2005-water at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>275</mn> </mrow> </semantics> </math> K and <math display="inline"> <semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>400</mn> </mrow> </semantics> </math> bar as a function of graphene inter-plate distance <span class="html-italic">d</span>. A large increase of <math display="inline"> <semantics> <msub> <mi>τ</mi> <mn>1</mn> </msub> </semantics> </math>, corresponding to a large slowing-down of the rotational dynamics, occurs for 8.5 Å <math display="inline"> <semantics> <mrow> <mo>&lt;</mo> <mi>d</mi> <mo>&lt;</mo> <mn>10</mn> </mrow> </semantics> </math> Å.</p> Full article ">Figure 8
<p>Density profiles of TIP4P/2005-water at <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>275</mn> </mrow> </semantics> </math> K and <math display="inline"> <semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>400</mn> </mrow> </semantics> </math> bar confined between graphene sheets along the <span class="html-italic">Z</span>-direction perpendicular to the walls at selected inter-plate separations. The occurrence of zeros between the maxima is an evidence of water layering. (<b>a</b>) Two interfacial layers with intermediate liquid water for graphene plates separation <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>17</mn> </mrow> </semantics> </math> Å; (<b>b</b>) two interfacial layers with a slight-deformed intermediate layer for <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics> </math>Å; (<b>c</b>) two layers for <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics> </math> Å; (<b>d</b>) one layer for <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math> Å.</p> Full article ">
2828 KiB  
Article
Permutation Entropy: New Ideas and Challenges
by Karsten Keller, Teresa Mangold, Inga Stolz and Jenna Werner
Entropy 2017, 19(3), 134; https://doi.org/10.3390/e19030134 - 21 Mar 2017
Cited by 64 | Viewed by 11696
Abstract
Over recent years, some new variants of Permutation entropy have been introduced and applied to EEG analysis, including a conditional variant and variants using some additional metric information or being based on entropies that are different from the Shannon entropy. In some situations, [...] Read more.
Over recent years, some new variants of Permutation entropy have been introduced and applied to EEG analysis, including a conditional variant and variants using some additional metric information or being based on entropies that are different from the Shannon entropy. In some situations, it is not completely clear what kind of information the new measures and their algorithmic implementations provide. We discuss the new developments and illustrate them for EEG data. Full article
(This article belongs to the Special Issue Entropy and Electroencephalography II)
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<p>Approximation of Kolmogorov–Sinai (KS) entropy: the direct way.</p> Full article ">Figure 2
<p>Approximation of KS entropy: the conditional way.</p> Full article ">Figure 3
<p>Different estimates of the KS entropy of <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>ω</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>;</mo> <mspace width="0.166667em"/> <mi>ω</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </semantics> </math> for different orbit lengths. On the top: <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mi>…</mi> <mo>,</mo> <mn>7</mn> </mrow> </semantics> </math>; in the middle: <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mi>…</mi> <mo>,</mo> <mn>10</mn> </mrow> </semantics> </math>; on the bottom: <math display="inline"> <semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <mfrac> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </mfrac> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 4
<p>Comparison of empirical Renyi Permutation entropy (eRPE) for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics> </math>, 2, empirical Tsallis Permutation entropy (eTPE) for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> and ePE, computed from EEG recordings before stimulator implantation (data set 1) and after stimulator implantation (data set 2) for 19 channels using a shifted time window, order <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> and delay <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>. Highlighting, in particular, the channels T3 (green line) and P3 (red line) as well as the entropy over all channels by a fat black line. The sampling rate is 256 Hz.</p> Full article ">Figure 5
<p>eRPE for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>, 35 and eTPE for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>35</mn> </mrow> </semantics> </math> computed from data set 1 for 19 channels using a shifted time window, <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math> (cf. <a href="#entropy-19-00134-f004" class="html-fig">Figure 4</a>). T3: green line, P3: red line.</p> Full article ">Figure 6
<p>Entropy of the EEG data sorted by groups for four different entropy measures.</p> Full article ">Figure 7
<p>Boxplots for four different entropy measures sorted by groups.</p> Full article ">Figure 8
<p>One entropy versus another entropy for four entropy combinations.</p> Full article ">Figure 9
<p>Second principal component versus the first one obtained from principal component analysis on three entropy variables.</p> Full article ">
907 KiB  
Article
Spectral Entropy Parameters during Rapid Ventricular Pacing for Transcatheter Aortic Valve Implantation
by Tadeusz Musialowicz, Antti Valtola, Mikko Hippeläinen, Jari Halonen and Pasi Lahtinen
Entropy 2017, 19(3), 133; https://doi.org/10.3390/e19030133 - 20 Mar 2017
Cited by 1 | Viewed by 5339
Abstract
The time-frequency balanced spectral entropy of the EEG is a monitoring technique measuring the level of hypnosis during general anesthesia. Two components of spectral entropy are calculated: state entropy (SE) and response entropy (RE). Transcatheter aortic valve implantation (TAVI) is a less invasive [...] Read more.
The time-frequency balanced spectral entropy of the EEG is a monitoring technique measuring the level of hypnosis during general anesthesia. Two components of spectral entropy are calculated: state entropy (SE) and response entropy (RE). Transcatheter aortic valve implantation (TAVI) is a less invasive treatment for patients suffering from symptomatic aortic stenosis with contraindications for open heart surgery. The goal of hemodynamic management during the procedure is to achieve hemodynamic stability with exact blood pressure control and use of rapid ventricular pacing (RVP) that result in severe hypotension. The objective of this study was to examine how the spectral entropy values respond to RVP and other critical events during the TAVI procedure. Twenty one patients undergoing general anesthesia for TAVI were evaluated. The RVP was used twice during the procedure at a rate of 185 ± 9/min with durations of 16 ± 4 s (range 8–22 s) and 24 ± 6 s (range 18–39 s). The systolic blood pressure during RVP was under 50 ± 5 mmHg. Spectral entropy values SE were significantly declined during the RVP procedure, from 28 ± 13 to 23 ± 13 (p < 0.003) and from 29 ± 12 to 24 ± 10 (p < 0.001). The corresponding values for RE were 29 ± 13 vs. 24 ± 13 (p < 0.006) and 30 ± 12 vs. 25 ± 10 (p < 0.001). Both SE and RE values returned to the pre-RVP values after 1 min. Ultra-short hypotension during RVP changed the spectral entropy parameters, however these indices reverted rapidly to the same value before application of RVP. Full article
(This article belongs to the Special Issue Entropy and Electroencephalography II)
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Figure 1
<p>Block diagram of study protocol and time points of State Entropy, Response Entropy, and hemodynamic variables measurements. RVP—rapid ventricular pacing.</p> Full article ">Figure 2
<p>State entropy (SE) and response entropy (RE) at different time points of general anesthesia during the TAVI procedure. 1: baseline before induction of anesthesia; 2: 1 min after induction of anesthesia; 3: before intubation; 4: after intubation; 5: before skin incision; 6: after skin incision, 7: before dilatation of aortic valve; 8: during dilatation of aortic valve with RVP; 9: 1 min after dilatation of aortic valve; 10: before release of valve prosthesis; 11: during release of valve prosthesis with RVP; 12: 1 min after release of aortic valve stent. Data are mean and ± SD * significant differences between the before and after time point.</p> Full article ">Figure 3
<p>Hemodynamic variables, systolic arterial pressure, mean arterial pressure and heart rate at different time points during the TAVI procedure. For time points, see text in <a href="#entropy-19-00133-f002" class="html-fig">Figure 2</a>.</p> Full article ">
1498 KiB  
Article
Discrepancies between Conventional Multiscale Entropy and Modified Short-Time Multiscale Entropy of Photoplethysmographic Pulse Signals in Middle- and Old- Aged Individuals with or without Diabetes
by Gen-Min Lin, Bagus Haryadi, Chieh-Ming Yang, Shiao-Chiang Chu, Cheng-Chan Yang and Hsien-Tsai Wu
Entropy 2017, 19(3), 132; https://doi.org/10.3390/e19030132 - 18 Mar 2017
Cited by 12 | Viewed by 4607
Abstract
Multiscale entropy (MSE) of physiological signals may reflect cardiovascular health in diabetes. The classic MSE (cMSE) algorithm requires more than 750 signals for the calculations. The modified short-time MSE (sMSE) may have inconsistent outcomes compared with the cMSE at large time scales and [...] Read more.
Multiscale entropy (MSE) of physiological signals may reflect cardiovascular health in diabetes. The classic MSE (cMSE) algorithm requires more than 750 signals for the calculations. The modified short-time MSE (sMSE) may have inconsistent outcomes compared with the cMSE at large time scales and in a disease status. Therefore, we compared the cMSE of 1500 (cMSE1500) consecutive and 1000 photoplethysmographic (PPG) pulse amplitudes with the sMSE of 500 PPG (sMSE500) pulse amplitudes of bilateral fingertips among middle- to old-aged individuals with or without type 2 diabetes. We discovered that cMSE1500 had the smallest value across scale factors 1–10, followed by cMSE1000, and then sMSE500 in both hands. The cMSE1500, cMSE1000 and sMSE500 did not differ at each scale factor in both hands of persons without diabetes and in the dominant hand of those with diabetes. In contrast, the sMSE500 differed at all scales 1–10 in the non-dominant hand with diabetes. In conclusion, autonomic dysfunction, prevalent in the non-dominant hand which had a low local physical activity in the person with diabetes, might be imprecisely evaluated by the sMSE; therefore, using more PPG signal numbers for the cMSE is preferred in such a situation. Full article
(This article belongs to the Special Issue Entropy and Cardiac Physics II)
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<p>PPGA of left hand (PPGA<sub>L</sub>) and right hand (PPGA<sub>R</sub>) were simultaneously acquired from PPGA(1) to PPGA (<span class="html-italic">n</span> = 500, 1000, or 1500).</p> Full article ">Figure 2
<p>Comparisons between sMSE<sub>500</sub> (solid green line), cMSE<sub>1000</sub> (solid blue line), and cMSE<sub>1500</sub> (solid red line) with standard errors (vertical bar) in right hand (<b>a</b>,<b>c</b>) and left hand (<b>b</b>,<b>d</b>) of the unaffected and those with diabetes.</p> Full article ">Figure 3
<p>MSE differences between the unaffected (solid blue line) and the diabetes (solid red line) in dominant right hand (<b>a</b>,<b>c</b>,<b>e</b>) and non-dominant left hand (<b>b</b>,<b>d</b>,<b>f</b>).</p> Full article ">
900 KiB  
Article
Information Submanifold Based on SPD Matrices and Its Applications to Sensor Networks
by Hao Xu, Huafei Sun and Aung Naing Win
Entropy 2017, 19(3), 131; https://doi.org/10.3390/e19030131 - 17 Mar 2017
Cited by 1 | Viewed by 4310
Abstract
In this paper, firstly, manifoldPD(n)consisting of alln×nsymmetric positive-definite matrices is introduced based on matrix information geometry; Secondly, the geometrical structures of information submanifold ofPD(n)are presented including metric, [...] Read more.
In this paper, firstly, manifoldPD(n)consisting of alln×nsymmetric positive-definite matrices is introduced based on matrix information geometry; Secondly, the geometrical structures of information submanifold ofPD(n)are presented including metric, geodesic and geodesic distance; Thirdly, the information resolution with sensor networks is presented by three classical measurement models based on information submanifold; Finally, the bearing-only tracking by single sensor is introduced by the Fisher information matrix. The preliminary analysis results introduced in this paper indicate that information submanifold is able to offer consistent and more comprehensive means to understand and solve sensor network problems for targets resolution and tracking, which are not easily handled by some conventional analysis methods. Full article
(This article belongs to the Special Issue Information Geometry II)
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<p>(<b>a</b>) IFD between two closely spaced targets for range-bearing measurement model; (<b>b</b>) The contour map of <a href="#entropy-19-00131-f001" class="html-fig">Figure 1</a>a.</p> Full article ">Figure 2
<p>(<b>a</b>) <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>F</mi> </msub> </semantics> </math> between two closely spaced targets for range-bearing measurement model; (<b>b</b>) The contour map of <a href="#entropy-19-00131-f002" class="html-fig">Figure 2</a>a.</p> Full article ">Figure 3
<p>(<b>a</b>) IFD between two closely spaced targets for two-bearings measurement model; (<b>b</b>) The contour map of <a href="#entropy-19-00131-f003" class="html-fig">Figure 3</a>a.</p> Full article ">Figure 4
<p>(<b>a</b>) <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>F</mi> </msub> </semantics> </math> between two closely spaced targets for two-bearings measurement model; (<b>b</b>) The contour map of <a href="#entropy-19-00131-f004" class="html-fig">Figure 4</a>a.</p> Full article ">Figure 5
<p>(<b>a</b>) <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>F</mi> </msub> </semantics> </math> between two closely spaced targets for 3D range-bearings measurement sensor network; (<b>b</b>) The contour map of <a href="#entropy-19-00131-f005" class="html-fig">Figure 5</a>a.</p> Full article ">Figure 6
<p>(<b>a</b>) <math display="inline"> <semantics> <msub> <mi>D</mi> <mi>F</mi> </msub> </semantics> </math> between two closely spaced targets for 3D range-bearings measurement sensor network; (<b>b</b>) The contour map of <a href="#entropy-19-00131-f006" class="html-fig">Figure 6</a>a.</p> Full article ">Figure 7
<p>(<b>a</b>) Target information for the sensor network with two-bearings passive sensor networks; (<b>b</b>) The contour map of target information.</p> Full article ">Figure 8
<p>Optimal sensor movement direction for given radius.</p> Full article ">
4315 KiB  
Article
Quantitative EEG Markers of Entropy and Auto Mutual Information in Relation to MMSE Scores of Probable Alzheimer’s Disease Patients
by Carmina Coronel, Heinrich Garn, Markus Waser, Manfred Deistler, Thomas Benke, Peter Dal-Bianco, Gerhard Ransmayr, Stephan Seiler, Dieter Grossegger and Reinhold Schmidt
Entropy 2017, 19(3), 130; https://doi.org/10.3390/e19030130 - 17 Mar 2017
Cited by 35 | Viewed by 6868
Abstract
Analysis of nonlinear quantitative EEG (qEEG) markers describing complexity of signal in relation to severity of Alzheimer’s disease (AD) was the focal point of this study. In this study, 79 patients diagnosed with probable AD were recruited from the multi-centric Prospective Dementia Database [...] Read more.
Analysis of nonlinear quantitative EEG (qEEG) markers describing complexity of signal in relation to severity of Alzheimer’s disease (AD) was the focal point of this study. In this study, 79 patients diagnosed with probable AD were recruited from the multi-centric Prospective Dementia Database Austria (PRODEM). EEG recordings were done with the subjects seated in an upright position in a resting state with their eyes closed. Models of linear regressions explaining disease severity, expressed in Mini Mental State Examination (MMSE) scores, were analyzed by the nonlinear qEEG markers of auto mutual information (AMI), Shannon entropy (ShE), Tsallis entropy (TsE), multiscale entropy (MsE), or spectral entropy (SpE), with age, duration of illness, and years of education as co-predictors. Linear regression models with AMI were significant for all electrode sites and clusters, where R 2 is 0.46 at the electrode site C3, 0.43 at Cz, F3, and central region, and 0.42 at the left region. MsE also had significant models at C3 with R 2 > 0.40 at scales τ = 5 and τ = 6 . ShE and TsE also have significant models at T7 and F7 with R 2 > 0.30 . Reductions in complexity, calculated by AMI, SpE, and MsE, were observed as the MMSE score decreased. Full article
(This article belongs to the Special Issue Entropy and Electroencephalography II)
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<p>Histograms of (<b>a</b>) age; (<b>b</b>) years of education; (<b>c</b>) duration of illness; and (<b>d</b>) MMSE scores of the 79 subjects participating in this study.</p> Full article ">Figure 2
<p>qEEG markers at electrode site T7 or C3. The regression lines are represented by setting co-predictors (age, duration of illness, and years of education) at mean, tabulated in <a href="#entropy-19-00130-t001" class="html-table">Table 1</a>. (<b>a</b>) ShE at T7, (<b>b</b>) TsE at T7, (<b>c</b>) MsE modified at <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math>, &amp; (<b>d</b>) SpE at T7.</p> Full article ">Figure 3
<p>(<b>a</b>) AMI at C3 and (<b>b</b>) central cluster versus MMSE scores. The regression lines of models are represented by setting co-predictors (age, duration of illness, and years of education) at mean, tabulated in <a href="#entropy-19-00130-t001" class="html-table">Table 1</a>.</p> Full article ">Figure 4
<p>The computed R<sup>2</sup> of significant linear regression models (<b>a</b>) ShE, (<b>b</b>) TsE, (<b>c</b>) MsE modified <math display="inline"> <semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>11</mn> </mrow> </semantics> </math>, &amp; (<b>d</b>) AMI at specific electrode, insignificant models are left blank. Electrode sites with R<sup>2</sup> &gt; 0.20 are shaded in light green, R<sup>2</sup> &gt; 0.30 in green, and R<sup>2</sup> &gt; 0.40 in dark green.</p> Full article ">Figure 5
<p>R<sup>2</sup> of regression models at electrode sites with MsE modified, at different scales, as main predictor.</p> Full article ">
889 KiB  
Article
Distance-Based Lempel–Ziv Complexity for the Analysis of Electroencephalograms in Patients with Alzheimer’s Disease
by Samantha Simons and Daniel Abásolo
Entropy 2017, 19(3), 129; https://doi.org/10.3390/e19030129 - 17 Mar 2017
Cited by 29 | Viewed by 5285
Abstract
The analysis of electroencephalograms (EEGs) of patients with Alzheimer’s disease (AD) could contribute to the diagnosis of this dementia. In this study, a new non-linear signal processing metric, distance-based Lempel–Ziv complexity (dLZC), is introduced to characterise changes between pairs of electrodes in EEGs [...] Read more.
The analysis of electroencephalograms (EEGs) of patients with Alzheimer’s disease (AD) could contribute to the diagnosis of this dementia. In this study, a new non-linear signal processing metric, distance-based Lempel–Ziv complexity (dLZC), is introduced to characterise changes between pairs of electrodes in EEGs in AD. When complexity in each signal arises from different sub-sequences, dLZC would be greater than when similar sub-sequences are present in each signal. EEGs from 11 AD patients and 11 age-matched control subjects were analysed. The dLZC values for AD patients were lower than for control subjects for most electrode pairs, with statistically significant differences (p < 0.01, Student’s t-test) in 17 electrode pairs in the distant left, local posterior left, and interhemispheric regions. Maximum diagnostic accuracies with leave-one-out cross-validation were 77.27% for subject-based classification and 78.25% for epoch-based classification. These findings suggest not only that EEGs from AD patients are less complex than those from controls, but also that the richness of the information contained in pairs of EEGs from patients is also lower than in age-matched controls. The analysis of EEGs in AD with dLZC may increase the insight into brain dysfunction, providing complementary information to that obtained with other complexity and synchrony methods. Full article
(This article belongs to the Special Issue Symbolic Entropy Analysis and Its Applications)
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<p>dLZC values for electrode pair P3-O1 for an Alzheimer’s disease (AD) patient with varying epoch sizes. The vertical line corresponds to the epoch size used in this study (1280 data points).</p> Full article ">Figure 2
<p>Average dLZC values for all electrode pairs in the control subjects.</p> Full article ">Figure 3
<p>Average dLZC values for all electrode pairs in the AD patients.</p> Full article ">Figure 4
<p>Statistically significant (<span class="html-italic">p</span> &lt; 0.01, Student’s <span class="html-italic">t</span>-test) electrode pairs for dLZC.</p> Full article ">
890 KiB  
Article
Pairs Generating as a Consequence of the Fractal Entropy: Theory and Applications
by Alexandru Grigorovici, Elena Simona Bacaita, Viorel Puiu Paun, Constantin Grecea, Irina Butuc, Maricel Agop and Ovidiu Popa
Entropy 2017, 19(3), 128; https://doi.org/10.3390/e19030128 - 17 Mar 2017
Cited by 9 | Viewed by 4392
Abstract
In classical concepts, theoretical models are built assuming that the dynamics of the complex system’s stuctural units occur on continuous and differentiable motion variables. In reality, the dynamics of the natural complex systems are much more complicated. These difficulties can be overcome in [...] Read more.
In classical concepts, theoretical models are built assuming that the dynamics of the complex system’s stuctural units occur on continuous and differentiable motion variables. In reality, the dynamics of the natural complex systems are much more complicated. These difficulties can be overcome in a complementary approach, using the fractal concept and the corresponding non-differentiable theoretical model, such as the scale relativity theory or the extended scale relativity theory. Thus, using the last theory, fractal entropy through non-differentiable Lie groups was established and, moreover, the pairs generating mechanisms through fractal entanglement states were explained. Our model has implications in the dynamics of biological structures, in the form of the “chameleon-like” behavior of cholesterol. Full article
(This article belongs to the Special Issue Symbolic Entropy Analysis and Its Applications)
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<p>The states densities <math display="inline"> <semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>Ψ</mi> <mover accent="true"> <mi>Ψ</mi> <mo stretchy="true">¯</mo> </mover> </mrow> </semantics> </math> of the two-dimensional harmonic oscillator: three-dimensional (<b>a</b>); and contour plot (<b>b</b>) dependences, associated with the LDL-HDL pair induced by the quantum numbers <math display="inline"> <semantics> <mrow> <msub> <mi>n</mi> <mi>x</mi> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </semantics> </math>, respectively.</p> Full article ">Figure 2
<p>The informational entropy <math display="inline"> <semantics> <mrow> <mi>I</mi> <mo>=</mo> <mi>ln</mi> <mi>ρ</mi> </mrow> </semantics> </math>: three-dimensional (<b>a</b>); and contour plot dependences (<b>b</b>) associated with the LDL-HDL pair induced by the quantum numbers <math display="inline"> <semantics> <mrow> <msub> <mi>n</mi> <mi>x</mi> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>n</mi> <mi>y</mi> </msub> </mrow> </semantics> </math>, respectively.</p> Full article ">
359 KiB  
Article
Fractional Jensen–Shannon Analysis of the Scientific Output of Researchers in Fractional Calculus
by José A. Tenreiro Machado and António Mendes Lopes
Entropy 2017, 19(3), 127; https://doi.org/10.3390/e19030127 - 17 Mar 2017
Cited by 23 | Viewed by 4791
Abstract
This paper analyses the citation profiles of researchers in fractional calculus. Different metrics are used to quantify the dissimilarities between the data, namely the Canberra distance, and the classical and the generalized (fractional) Jensen–Shannon divergence. The information is then visualized by means of [...] Read more.
This paper analyses the citation profiles of researchers in fractional calculus. Different metrics are used to quantify the dissimilarities between the data, namely the Canberra distance, and the classical and the generalized (fractional) Jensen–Shannon divergence. The information is then visualized by means of multidimensional scaling and hierarchical clustering. The mathematical tools and metrics allow for direct comparison and visualization of researchers based on their relative positioning and on patterns displayed in two- or three-dimensional maps. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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<p>Graphs illustrating the <span class="html-italic">h</span>, <span class="html-italic">g</span>, <math display="inline"> <semantics> <msup> <mi>h</mi> <mn>2</mn> </msup> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>h</mi> <mi>I</mi> <mo>,</mo> <mi>n</mi> <mi>o</mi> <mi>r</mi> <mi>m</mi> </mrow> </semantics> </math> indices for one researcher.</p> Full article ">Figure 2
<p>Graphs illustrating the relationships between the indices <span class="html-italic">h</span>, <span class="html-italic">g</span>, <math display="inline"> <semantics> <msup> <mi>h</mi> <mn>2</mn> </msup> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>h</mi> <mi>I</mi> <mo>,</mo> <mi>n</mi> <mi>o</mi> <mi>r</mi> <mi>m</mi> </mrow> </semantics> </math> for a group of 100 researchers in FC.</p> Full article ">Figure 3
<p>The three-dimensional map generated by the multidimensional scaling (MDS) with <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>D</mi> </mrow> </semantics> </math> for 100 researchers in FC.</p> Full article ">Figure 4
<p>The three-dimensional map generated by the MDS with <math display="inline"> <semantics> <mrow> <mi>J</mi> <mi>S</mi> <mi>D</mi> </mrow> </semantics> </math> for 100 researchers in FC.</p> Full article ">Figure 5
<p>The three-dimensional map generated by the MDS with <math display="inline"> <semantics> <mrow> <mi>J</mi> <mi>S</mi> <msub> <mi>D</mi> <mi>α</mi> </msub> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>7</mn> </mrow> </semantics> </math>, for 100 researchers in FC.</p> Full article ">Figure 6
<p>Hierarchical tree generated by the hierarchical clustering (HC) with <math display="inline"> <semantics> <mrow> <mi>C</mi> <mi>D</mi> </mrow> </semantics> </math> for 100 researchers in FC.</p> Full article ">Figure 7
<p>Hierarchical tree generated by the HC with <math display="inline"> <semantics> <mrow> <mi>J</mi> <mi>S</mi> <mi>D</mi> </mrow> </semantics> </math> for 100 researchers in FC.</p> Full article ">Figure 8
<p>Hierarchical tree generated by the HC with <math display="inline"> <semantics> <mrow> <mi>J</mi> <mi>S</mi> <msub> <mi>D</mi> <mi>α</mi> </msub> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>7</mn> </mrow> </semantics> </math>, for 100 researchers in FC.</p> Full article ">
3239 KiB  
Article
Friction, Free Axes of Rotation and Entropy
by Alexander Kazachkov, Victor Multanen, Viktor Danchuk, Mark Frenkel and Edward Bormashenko
Entropy 2017, 19(3), 123; https://doi.org/10.3390/e19030123 - 17 Mar 2017
Cited by 3 | Viewed by 5121
Abstract
Friction forces acting on rotators may promote their alignment and therefore eliminate degrees of freedom in their movement. The alignment of rotators by friction force was shown by experiments performed with different spinners, demonstrating how friction generates negentropy in a system of rotators. [...] Read more.
Friction forces acting on rotators may promote their alignment and therefore eliminate degrees of freedom in their movement. The alignment of rotators by friction force was shown by experiments performed with different spinners, demonstrating how friction generates negentropy in a system of rotators. A gas of rigid rotators influenced by friction force is considered. The orientational negentropy generated by a friction force was estimated with the Sackur-Tetrode equation. The minimal change in total entropy of a system of rotators, corresponding to their eventual alignment, decreases with temperature. The reported effect may be of primary importance for the phase equilibrium and motion of ubiquitous colloidal and granular systems. Full article
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>(<b>A</b>) Scheme of the hollow top used in the investigation; (<b>B</b>) The top filled with steel balls.</p> Full article ">Figure 2
<p>Thermal image of the spinning top (the mass <math display="inline"> <semantics> <mrow> <msub> <mi>m</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>99.330</mn> <mo> </mo> <mi mathvariant="normal">g</mi> </mrow> </semantics> </math>). Brighter spots correspond to higher temperatures. Bright thermal trace produced by the top on the support is clearly recognized. The scale bar is 10 mm.</p> Full article ">Figure 3
<p>The distribution of temperatures along the axis of rotation (axis <span class="html-italic">Y</span> in <a href="#entropy-19-00123-f002" class="html-fig">Figure 2</a>) observed for the heavy spinning top (<math display="inline"> <semantics> <mrow> <msub> <mi>m</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>99.330</mn> <mo> </mo> <mi mathvariant="normal">g</mi> </mrow> </semantics> </math>).</p> Full article ">Figure 4
<p>Orientation of quasi-steadily rotating transparent tops, filled by agate balls. (<b>A</b>,<b>B</b>) Top containing a single agate ball (<b>A</b>) side view; (<b>B</b>) plan view; (<b>C</b>,<b>D</b>) The same top containing two agate balls; (<b>E</b>) Top containing three agate balls.</p> Full article ">Figure 4 Cont.
<p>Orientation of quasi-steadily rotating transparent tops, filled by agate balls. (<b>A</b>,<b>B</b>) Top containing a single agate ball (<b>A</b>) side view; (<b>B</b>) plan view; (<b>C</b>,<b>D</b>) The same top containing two agate balls; (<b>E</b>) Top containing three agate balls.</p> Full article ">Figure 5
<p>Orientation of quasi-steadily rotating transparent tops, filled by steel balls. (<b>A</b>) Top containing a pair of two steel balls; (<b>B</b>) Top containing a triad of steel balls; (<b>C</b>,<b>D</b>) Rotating top containing two pairs of steel balls.</p> Full article ">Figure 6
<p>Orientation of the rotating coin containing a pair of holes. (<b>A</b>) Orientation of holes corresponding to the maximal momentum of inertia of the coin; (<b>B</b>) Orientation of holes corresponding to the minimal momentum of inertia of the coin, relative to the vertical axis.</p> Full article ">Figure 7
<p>Rectangular parallelepiped falling through air is depicted. If inequality <math display="inline"> <semantics> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>&gt;</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>&gt;</mo> <msub> <mi>I</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> takes place, the falling body will “prefer” rotation around the axis <span class="html-italic">OO</span><sub>1</sub>.</p> Full article ">
282 KiB  
Article
Identity Based Generalized Signcryption Scheme in the Standard Model
by Xiaoqin Shen, Yang Ming and Jie Feng
Entropy 2017, 19(3), 121; https://doi.org/10.3390/e19030121 - 17 Mar 2017
Cited by 11 | Viewed by 3865
Abstract
Generalized signcryption (GSC) can adaptively work as an encryption scheme, a signature scheme or a signcryption scheme with only one algorithm. It is more suitable for the storage constrained setting. In this paper, motivated by Paterson–Schuldt’s scheme, based on bilinear pairing, we first [...] Read more.
Generalized signcryption (GSC) can adaptively work as an encryption scheme, a signature scheme or a signcryption scheme with only one algorithm. It is more suitable for the storage constrained setting. In this paper, motivated by Paterson–Schuldt’s scheme, based on bilinear pairing, we first proposed an identity based generalized signcryption (IDGSC) scheme in the standard model. To the best of our knowledge, it is the first scheme that is proven secure in the standard model. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
244 KiB  
Article
Nonequilibrium Thermodynamics and Scale Invariance
by Leonid M. Martyushev and Vladimir Celezneff
Entropy 2017, 19(3), 126; https://doi.org/10.3390/e19030126 - 16 Mar 2017
Cited by 5 | Viewed by 4326
Abstract
A variant of continuous nonequilibrium thermodynamic theory based on the postulate of the scale invariance of the local relation between generalized fluxes and forces is proposed here. This single postulate replaces the assumptions on local equilibrium and on the known relation between thermodynamic [...] Read more.
A variant of continuous nonequilibrium thermodynamic theory based on the postulate of the scale invariance of the local relation between generalized fluxes and forces is proposed here. This single postulate replaces the assumptions on local equilibrium and on the known relation between thermodynamic fluxes and forces, which are widely used in classical nonequilibrium thermodynamics. It is shown here that such a modification not only makes it possible to deductively obtain the main results of classical linear nonequilibrium thermodynamics, but also provides evidence for a number of statements for a nonlinear case (the maximum entropy production principle, the macroscopic reversibility principle, and generalized reciprocity relations) that are under discussion in the literature. Full article
(This article belongs to the Section Thermodynamics)
446 KiB  
Article
Packer Detection for Multi-Layer Executables Using Entropy Analysis
by Munkhbayar Bat-Erdene, Taebeom Kim, Hyundo Park and Heejo Lee
Entropy 2017, 19(3), 125; https://doi.org/10.3390/e19030125 - 16 Mar 2017
Cited by 23 | Viewed by 8679
Abstract
Packing algorithms are broadly used to avoid anti-malware systems, and the proportion of packed malware has been growing rapidly. However, just a few studies have been conducted on detection various types of packing algorithms in a systemic way. Following this understanding, we elaborate [...] Read more.
Packing algorithms are broadly used to avoid anti-malware systems, and the proportion of packed malware has been growing rapidly. However, just a few studies have been conducted on detection various types of packing algorithms in a systemic way. Following this understanding, we elaborate a method to classify packing algorithms of a given executable into three categories: single-layer packing, re-packing, or multi-layer packing. We convert entropy values of the executable file loaded into memory into symbolic representations, for which we used SAX (Symbolic Aggregate Approximation). Based on experiments of 2196 programs and 19 packing algorithms, we identify that precision (97.7%), accuracy (97.5%), and recall ( 96.8%) of our method are respectively high to confirm that entropy analysis is applicable in identifying packing algorithms. Full article
(This article belongs to the Special Issue Symbolic Entropy Analysis and Its Applications)
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Graphical abstract
Full article ">Figure 1
<p>Structure of single-layer packed, re-packed, and multi-layer packed executables.</p> Full article ">Figure 2
<p>Single-layer packing, re-packing, and multi-layer packing process of a PE file.</p> Full article ">Figure 3
<p>Re-packing and multi-layer packing algorithm detection method.</p> Full article ">Figure 4
<p>Entropy pattern conversion into the symbolic representation. (<b>a</b>) Original entropy pattern; (<b>b</b>) Normalized of entropy pattern; (<b>c</b>) Symbolic representation of normalized entropy pattern.</p> Full article ">Figure 5
<p>Structure of the pattern extraction method.</p> Full article ">Figure 6
<p>Structure of a classifier.</p> Full article ">Figure 7
<p>Entropy patterns converted into SAX using different values of <math display="inline"> <semantics> <mrow> <mi>ϕ</mi> <mo>(</mo> <mi>β</mi> <mo>)</mo> </mrow> </semantics> </math> for the Aspack packers.</p> Full article ">Figure 8
<p>Entropy patterns of single-layer packed and re-packed executable of Notepad.exe when a packer is (<b>a</b>) Alternate_EXE; (<b>b</b>) NsPack; (<b>c</b>) RLPack; (<b>d</b>) nPack; (<b>e</b>) VMProtect. <span class="html-italic">y</span>-axis is entropy values and <span class="html-italic">x</span>-axis is “JMP” instruction numbers.</p> Full article ">Figure 9
<p>Entropy patterns of single-layer packed and multi-layer packed executable of Notepad.exe using two packers. (<b>a</b>) NsPack or Aspack; (<b>b</b>) NsPack and Aspack; (<b>c</b>) NsPack or VMProtect; (<b>d</b>) NsPack and VMProtect; (<b>e</b>) RLPack or VMProtect; (<b>f</b>) RLPack and VMProtect; (<b>g</b>) VMProtect or NsPack; (<b>h</b>) VMProtect and NsPack; (<b>i</b>) VMProtect or RLPack; (<b>j</b>) VMProtect and RLPack. <span class="html-italic">y</span>-axis is entropy values and <span class="html-italic">x</span>-axis is “JMP” instruction numbers.</p> Full article ">
817 KiB  
Article
Witnessing Multipartite Entanglement by Detecting Asymmetry
by Davide Girolami and Benjamin Yadin
Entropy 2017, 19(3), 124; https://doi.org/10.3390/e19030124 - 16 Mar 2017
Cited by 40 | Viewed by 4907
Abstract
The characterization of quantum coherence in the context of quantum information theory and its interplay with quantum correlations is currently subject of intense study. Coherence in a Hamiltonian eigenbasis yields asymmetry, the ability of a quantum system to break a dynamical symmetry generated [...] Read more.
The characterization of quantum coherence in the context of quantum information theory and its interplay with quantum correlations is currently subject of intense study. Coherence in a Hamiltonian eigenbasis yields asymmetry, the ability of a quantum system to break a dynamical symmetry generated by the Hamiltonian. We here propose an experimental strategy to witness multipartite entanglement in many-body systems by evaluating the asymmetry with respect to an additive Hamiltonian. We test our scheme by simulating asymmetry and entanglement detection in a three-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal state. Full article
(This article belongs to the Special Issue Information Geometry II)
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Figure 1

Figure 1
<p>Overlap detection. Two copies <math display="inline"> <semantics> <mrow> <msubsup> <mi>ρ</mi> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> <mi>p</mi> </msubsup> <mo>,</mo> <msubsup> <mi>ρ</mi> <mrow> <msub> <mi>A</mi> <mn>2</mn> </msub> <msub> <mi>B</mi> <mn>2</mn> </msub> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> <mi>p</mi> </msubsup> </mrow> </semantics> </math> of a GHZ-diagonal state are prepared in the state <math display="inline"> <semantics> <mrow> <msub> <mi>ρ</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>p</mi> <msub> <mi>σ</mi> <mi>z</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mo>∀</mo> <mi>i</mi> </mrow> </semantics> </math>. An Hadamard gate <math display="inline"> <semantics> <mrow> <mi>H</mi> <mi>a</mi> <mi>d</mi> </mrow> </semantics> </math> is applied to the qubits <math display="inline"> <semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </semantics> </math>, followed by two CNOT gates on each copy. Then, one evaluates the purity and the overlap terms related to the observables <math display="inline"> <semantics> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </msub> </semantics> </math>, by applying the unitary transformations <math display="inline"> <semantics> <mrow> <msub> <mi>U</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>U</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>A</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mo>⊗</mo> <msub> <mi>U</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>B</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mo>⊗</mo> <msub> <mi>U</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>C</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics> </math> and measuring the ancilla polarisation by means of an interferometric scheme. This consists of an ancilla in the initial state <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <mo>(</mo> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> <mo>)</mo> </mrow> </mrow> </semantics> </math> interacting with the two state copies by a controlled-<span class="html-italic">V</span> gate, being <span class="html-italic">V</span> the swap operator. A second Hadamard gate <span class="html-italic">H</span> is finally applied to the ancilla. The mean value of the ancilla polarisation at the output is <math display="inline"> <semantics> <mrow> <msub> <mrow> <mo>〈</mo> <msub> <mi>σ</mi> <mi>z</mi> </msub> <mo>〉</mo> </mrow> <msup> <mi>α</mi> <mi>out</mi> </msup> </msub> <mo>=</mo> <mi>Tr</mi> <mrow> <mo>[</mo> <mi>V</mi> <mi>ρ</mi> <mo>⊗</mo> <msub> <mi>U</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mi>ρ</mi> <msubsup> <mi>U</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> <mo>†</mo> </msubsup> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mi>Tr</mi> <mrow> <mo>[</mo> <mi>ρ</mi> <msub> <mi>U</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> </msub> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mi>ρ</mi> <msubsup> <mi>U</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> <mo>†</mo> </msubsup> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> <mo>]</mo> </mrow> </mrow> </semantics> </math>, which determines the asymmetry lower bound.</p> Full article ">Figure 2
<p>(Colors Online)—Evaluation of asymmetry in the state <math display="inline"> <semantics> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> </semantics> </math> with respect to the observables <math display="inline"> <semantics> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> </msub> </semantics> </math> (figures (<b>a</b>)–(<b>c</b>) respectively) as a function of the mixing parameter <span class="html-italic">p</span>. The blue dotted line is the quantum Fisher information, here shown for reference; the red dashed line is the bound <math display="inline"> <semantics> <mrow> <mrow/> <msub> <mi>O</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> </msub> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math>; the red continuous line is the approximation <math display="inline"> <semantics> <mrow> <mrow/> <msubsup> <mi>O</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math> obtained by imposing <math display="inline"> <semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics> </math>; and the yellow band is the error region, whose extreme values are <math display="inline"> <semantics> <mrow> <mrow/> <msubsup> <mi>O</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>±</mo> <mo>Δ</mo> <mrow/> <msubsup> <mi>O</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>(Colors Online)—Witnessing entanglement by asymmetry via the inequalities in Equation (<a href="#FD6-entropy-19-00124" class="html-disp-formula">6</a>). (<b>a</b>) Witnessing entanglement in the state <math display="inline"> <semantics> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> </semantics> </math> by computing the quantum Fisher information and the lower bound, as a function of the mixing parameter <span class="html-italic">p</span>. The blue dotted line depicts <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">F</mi> <msub> <mi>J</mi> <mrow> <mi>z</mi> <mo>,</mo> <mn>3</mn> </mrow> </msub> </msub> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>3</mn> </mrow> </semantics> </math>, the red dashed line is <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">O</mi> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>z</mi> </mrow> </msub> </msub> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>3</mn> </mrow> </semantics> </math>, while the red continuous line is <math display="inline"> <semantics> <mrow> <mrow/> <msubsup> <mi>O</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>3</mn> </mrow> </semantics> </math>. Positive values of such quantities signal entanglement. The yellow band is the error region, bounded by the extreme values <math display="inline"> <semantics> <mrow> <mo>(</mo> <msubsup> <mrow> <mi mathvariant="script">O</mi> </mrow> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>±</mo> <mo>Δ</mo> <msubsup> <mrow> <mi mathvariant="script">O</mi> </mrow> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>)</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics> </math>. The <math display="inline"> <semantics> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>x</mi> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </msub> </semantics> </math> cases are not reported as trivially useless, see <a href="#entropy-19-00124-t002" class="html-table">Table 2</a>; (<b>b</b>) Witnessing genuine tripartite entanglement (it is the case <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> of Equation (<a href="#FD6-entropy-19-00124" class="html-disp-formula">6</a>)). The blue dotted line depicts <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">F</mi> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>z</mi> </mrow> </msub> </msub> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>5</mn> </mrow> </semantics> </math>, the red dashed line is <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="script">O</mi> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>z</mi> </mrow> </msub> </msub> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>5</mn> </mrow> </semantics> </math>, while the red continuous line is <math display="inline"> <semantics> <mrow> <mrow/> <msubsup> <mi>O</mi> <msub> <mi>J</mi> <mn>3</mn> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>5</mn> </mrow> </semantics> </math>. The error region (yellow) is bounded by the extreme values of <math display="inline"> <semantics> <mrow> <mo>(</mo> <msubsup> <mrow> <mi mathvariant="script">O</mi> </mrow> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>±</mo> <mo>Δ</mo> <msubsup> <mrow> <mi mathvariant="script">O</mi> </mrow> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>z</mi> </mrow> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>)</mo> <mo>−</mo> <mn>5</mn> </mrow> </semantics> </math>; (<b>c</b>) Witnessing entanglement by computing the average values of the quantum Fisher information and the lower bound over a spin basis <math display="inline"> <semantics> <mrow> <mo>{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>}</mo> </mrow> </semantics> </math>. The blue dotted line is <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi mathvariant="script">F</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics> </math>, the red dashed line is <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi mathvariant="script">O</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics> </math>, while the red continuous line is <math display="inline"> <semantics> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="script">O</mi> <mo>¯</mo> </mover> <msub> <mi>J</mi> <mn>3</mn> </msub> <mi>ap</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics> </math>. The yellow error region is bounded by <math display="inline"> <semantics> <mrow> <mfenced separators="" open="(" close=")"> <msup> <mover accent="true"> <mi mathvariant="script">O</mi> <mo>¯</mo> </mover> <mi>ap</mi> </msup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>±</mo> <msqrt> <mrow> <msub> <mo>∑</mo> <mi>i</mi> </msub> <msup> <mrow> <mo>Δ</mo> <msubsup> <mrow> <mi mathvariant="script">O</mi> </mrow> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>ap</mi> </msubsup> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msubsup> <mi>ρ</mi> <mrow> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mrow> <mi>p</mi> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <msub> <mo>∑</mo> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>Δ</mo> <msubsup> <mrow> <mi mathvariant="script">O</mi> </mrow> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>ap</mi> </msubsup> <mo>Δ</mo> <msubsup> <mrow> <mi mathvariant="script">O</mi> </mrow> <msub> <mi>J</mi> <mrow> <mn>3</mn> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mi>ap</mi> </msubsup> </mrow> </msqrt> </mfenced> <mo>−</mo> <mn>5</mn> </mrow> </semantics> </math>.</p> Full article ">
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Article
On Hölder Projective Divergences
by Frank Nielsen, Ke Sun and Stéphane Marchand-Maillet
Entropy 2017, 19(3), 122; https://doi.org/10.3390/e19030122 - 16 Mar 2017
Cited by 15 | Viewed by 6095
Abstract
We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities and present two novel classes of Hölder divergences and pseudo-divergences [...] Read more.
We describe a framework to build distances by measuring the tightness of inequalities and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities and present two novel classes of Hölder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy–Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians) or affine (e.g., categorical distributions). Those new classes of Hölder distances are invariant to rescaling and thus do not require distributions to be normalized. Finally, we show how to compute statistical Hölder centroids with respect to those divergences and carry out center-based clustering toy experiments on a set of Gaussian distributions which demonstrate empirically that symmetrized Hölder divergences outperform the symmetric Cauchy–Schwarz divergence. Full article
(This article belongs to the Special Issue Information Geometry II)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Hölder proper divergence (bi-parametric) and Hölder improper pseudo-divergence (tri-parametric) encompass Cauchy–Schwarz divergence and skew Bhattacharyya divergence.</p> Full article ">Figure 2
<p>First row: the Hölder pseudo divergence (HPD) <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics> </math>, KL divergence and reverse KL divergence. Remaining rows: the HD <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>γ</mi> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from top to bottom) and <math display="inline"> <semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>{</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from left to right). The reference distribution <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math> is presented as “★”. The minimizer of <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>, if different from <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math>, is presented as “•”. Notice that <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> </mrow> </semantics> </math>. (<b>a</b>) Reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>; (<b>b</b>) reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>.</p> Full article ">Figure 2 Cont.
<p>First row: the Hölder pseudo divergence (HPD) <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics> </math>, KL divergence and reverse KL divergence. Remaining rows: the HD <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>γ</mi> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from top to bottom) and <math display="inline"> <semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>{</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from left to right). The reference distribution <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math> is presented as “★”. The minimizer of <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>, if different from <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math>, is presented as “•”. Notice that <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> </mrow> </semantics> </math>. (<b>a</b>) Reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>; (<b>b</b>) reference categorical distribution <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>/</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>First row: <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math>, where <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>r</mi> </msub> </semantics> </math> is the standard Gaussian distribution and <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>}</mo> </mrow> </semantics> </math> compared to the KL divergence. The rest of the rows: <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>γ</mi> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>r</mi> </msub> <mo>:</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>{</mo> <mn>4</mn> <mo>/</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from top to bottom) and <math display="inline"> <semantics> <mrow> <mi>γ</mi> <mo>∈</mo> <mo>{</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>10</mn> <mo>}</mo> </mrow> </semantics> </math> (from left to right). Notice that <math display="inline"> <semantics> <mrow> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> <mo>=</mo> <msubsup> <mi>D</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi mathvariant="monospace">H</mi> </msubsup> </mrow> </semantics> </math>. The coordinate system is formed by <span class="html-italic">μ</span> (mean) and <span class="html-italic">σ</span> (standard deviation).</p> Full article ">Figure 4
<p>Variational <span class="html-italic">k</span>-means clustering results on a toy dataset consisting of a set of 2D Gaussians organized into two or three clusters. The cluster centroids are represented by contour plots using the same density levels. (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics> </math> (Hölder clustering); (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> (Cauchy–Schwarz clustering); (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>1.1</mn> </mrow> </semantics> </math> (Hölder clustering); (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> (Cauchy–Schwarz clustering).</p> Full article ">
300 KiB  
Article
Variational Principle for Relative Tail Pressure
by Xianfeng Ma and Ercai Chen
Entropy 2017, 19(3), 120; https://doi.org/10.3390/e19030120 - 15 Mar 2017
Cited by 2 | Viewed by 3410
Abstract
We introduce the relative tail pressure to establish a variational principle for continuous bundle random dynamical systems. We also show that the relative tail pressure is conserved by the principal extension. Full article
(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)
2904 KiB  
Article
Thermoeconomic Optimization of an Irreversible Novikov Plant Model under Different Regimes of Performance
by Juan Carlos Pacheco-Paez, Fernando Angulo-Brown and Marco Antonio Barranco-Jiménez
Entropy 2017, 19(3), 118; https://doi.org/10.3390/e19030118 - 15 Mar 2017
Cited by 13 | Viewed by 3955
Abstract
The so-called Novikov power plant model has been widely used to represent some actual power plants, such as nuclear electric power generators. In the present work, a thermo-economic study of a Novikov power plant model is presented under three different regimes of performance: [...] Read more.
The so-called Novikov power plant model has been widely used to represent some actual power plants, such as nuclear electric power generators. In the present work, a thermo-economic study of a Novikov power plant model is presented under three different regimes of performance: maximum power (MP), maximum ecological function (ME) and maximum efficient power (EP). In this study, different heat transfer laws are used: The Newton’s law of cooling, the Stefan–Boltzmann radiation law, the Dulong–Petit’s law and another phenomenological heat transfer law. For the thermoeconomic optimization of power plant models, a benefit function defined as the quotient of an objective function and the total economical costs is commonly employed. Usually, the total costs take into account two contributions: a cost related to the investment and another stemming from the fuel consumption. In this work, a new cost associated to the maintenance of the power plant is also considered. With these new total costs, it is shown that under the maximum ecological function regime the plant improves its economic and energetic performance in comparison with the other two regimes. The methodology used in this paper is within the context of finite-time thermodynamics. Full article
(This article belongs to the Section Thermodynamics)
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Figure 1

Figure 1
<p>Novikov’s model for a thermal power plant. <math display="inline"> <semantics> <mrow> <msub> <mi>Q</mi> <mi>H</mi> </msub> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msub> <mi>Q</mi> <mi>L</mi> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mi>W</mi> </semantics> </math> are quantities per unit time.</p> Full article ">Figure 2
<p>Comparison between the three thermoeconomic objective functions with respect to the internal efficiency for a Newton heat transfer law for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math> (<math display="inline"> <semantics> <mrow> <msup> <mi>F</mi> <mi>N</mi> </msup> </mrow> </semantics> </math> represents the three objective functions with a Newtonian heat transfer law).</p> Full article ">Figure 3
<p>Comparison between the three thermoeconomic objective functions with respect to the internal efficiency, taking into account the maintenance costs of the plant for two values of the parameter <math display="inline"> <semantics> <mrow> <mi>β</mi> <mo>.</mo> </mrow> </semantics> </math> (<math display="inline"> <semantics> <mrow> <msup> <mi>F</mi> <mi>N</mi> </msup> </mrow> </semantics> </math> represents the three objective functions with a Newtonian heat transfer law).</p> Full article ">Figure 4
<p>Comparison between the three thermoeconomic objective functions with respect to the internal efficiency for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>. <math display="inline"> <semantics> <mrow> <msup> <mi>F</mi> <mrow> <mi>P</mi> <mi>h</mi> </mrow> </msup> </mrow> </semantics> </math> represents the three objective functions with a phenomenological heat transfer law.</p> Full article ">Figure 5
<p>Comparison between the three thermoeconomic objective functions with respect to the internal efficiency, taking into account the maintenance costs of the plant for two values of the parameter <math display="inline"> <semantics> <mi>β</mi> </semantics> </math>. <math display="inline"> <semantics> <mrow> <msup> <mi>F</mi> <mrow> <mi>P</mi> <mi>h</mi> </mrow> </msup> </mrow> </semantics> </math> represents the three objective functions with a phenomenological heat transfer law.</p> Full article ">Figure 6
<p>Thermoeconomic objective function under efficient power (EP) conditions with respect to the internal efficiency, taking into account the maintenance costs of the plant for several values of the parameter <math display="inline"> <semantics> <mi>β</mi> </semantics> </math>. We observe how the maxima points tend to the reversible value; that is, the Carnot efficiency (<math display="inline"> <semantics> <mrow> <msub> <mi>η</mi> <mi>C</mi> </msub> </mrow> </semantics> </math>).</p> Full article ">Figure 7
<p>Optimal efficiencies versus fractional fuel cost. <math display="inline"> <semantics> <mrow> <msup> <mi>η</mi> <mi>N</mi> </msup> </mrow> </semantics> </math> represents the three optimal efficiencies with a Newtonian heat transfer law (For <span class="html-italic">R</span> = 1).</p> Full article ">Figure 8
<p>Optimal efficiencies versus fractional fuel cost. <math display="inline"> <semantics> <mrow> <msup> <mi>η</mi> <mi>N</mi> </msup> </mrow> </semantics> </math> represents the three optimal efficiencies with a Newtonian heat transfer law (For <span class="html-italic">R</span> = 0.9).</p> Full article ">Figure 9
<p>Heat rejected for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math> under: (<b>a</b>) maximum power conditions; (<b>b</b>) maximum ecological function conditions; (<b>c</b>) maximum efficient power conditions.</p> Full article ">Figure 10
<p>Ratio of heats rejected between: (<b>a</b>) ecological function conditions and maximum power conditions for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>; (<b>b</b>) efficient power conditions and maximum power conditions for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>.</p> Full article ">Figure 11
<p>Ratio of heat rejected between efficient power and maximum power conditions and ecological function and maximum power conditions for a heat transfer of the Dulong–Petit type and for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>.</p> Full article ">Figure 12
<p>Entropy production under: (<b>a</b>) maximum power conditions for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>; (<b>b</b>) maximum ecological function conditions for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>; (<b>c</b>) maximum efficient power conditions for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>.</p> Full article ">Figure 13
<p>Ratio of entropy production between: (<b>a</b>) ecological function conditions and maximum power conditions for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>; (<b>b</b>) ecological function conditions and maximum power conditions for two values of the parameter <math display="inline"> <semantics> <mi>γ</mi> </semantics> </math>.</p> Full article ">Figure 14
<p>Ratio of entropy production between efficient power and maximum power and ecological function and maximum power conditions, respectively, for a heat transfer of the Dulong–Petit type.</p> Full article ">
353 KiB  
Letter
Specific Emitter Identification Based on the Natural Measure
by Yongqiang Jia, Shengli Zhu and Lu Gan
Entropy 2017, 19(3), 117; https://doi.org/10.3390/e19030117 - 15 Mar 2017
Cited by 15 | Viewed by 4938
Abstract
Specific emitter identification (SEI) techniques are often used in civilian and military spectrum-management operations, and they are also applied to support the security and authentication of wireless communication. In this letter, a new SEI method based on the natural measure of the one-dimensional [...] Read more.
Specific emitter identification (SEI) techniques are often used in civilian and military spectrum-management operations, and they are also applied to support the security and authentication of wireless communication. In this letter, a new SEI method based on the natural measure of the one-dimensional component of the chaotic system is proposed. We find that the natural measures of the one-dimensional components of higher dimensional systems exist and that they are quite diverse for different systems. Based on this principle, the natural measure is used as an RF fingerprint in this letter. The natural measure can solve the problems caused by a small amount of data and a low sample rate. The Kullback–Leibler divergence is used to quantify the difference between the natural measures obtained from diverse emitters and classify them. The data obtained from real application are exploited to test the validity of the proposed method. Experimental results show that the proposed method is not only easy to operate, but also quite effective, even though the amount of data is small and the sample rate is low. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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Figure 1
<p>The histograms obtained from two arbitrary (<b>a</b>) Lorenz; (<b>b</b>) Rösser; and (<b>c</b>) Wiem-Type time series (the <span class="html-italic">x</span>-coordinate); (<b>d</b>) The histograms obtained from the <span class="html-italic">x</span>-coordinates of the Lorenz, Rössler, and Wiem-Type system.</p> Full article ">Figure 2
<p>The histograms obtained from two arbitrary (<b>a</b>) Lorenz; (<b>b</b>) Rösser; and (<b>c</b>) Wiem-Type time series (the <span class="html-italic">y</span>-coordinate). The histograms obtained from two arbitrary (<b>d</b>) Lorenz; (<b>e</b>) Rösser; and (<b>f</b>) Wiem-Type time series (the <span class="html-italic">z</span>-coordinate).</p> Full article ">
3458 KiB  
Article
Fluctuation-Driven Transport in Biological Nanopores. A 3D Poisson–Nernst–Planck Study
by Marcel Aguilella-Arzo, María Queralt-Martín, María-Lidón Lopez and Antonio Alcaraz
Entropy 2017, 19(3), 116; https://doi.org/10.3390/e19030116 - 14 Mar 2017
Cited by 7 | Viewed by 5615
Abstract
Living systems display a variety of situations in which non-equilibrium fluctuations couple to certain protein functions yielding astonishing results. Here we study the bacterial channel OmpF under conditions similar to those met in vivo, where acidic resistance mechanisms are known to yield oscillations [...] Read more.
Living systems display a variety of situations in which non-equilibrium fluctuations couple to certain protein functions yielding astonishing results. Here we study the bacterial channel OmpF under conditions similar to those met in vivo, where acidic resistance mechanisms are known to yield oscillations in the electric potential across the cell membrane. We use a three-dimensional structure-based theoretical approach to assess the possibility of obtaining fluctuation-driven transport. Our calculations show that remarkably high voltages would be necessary to observe the actual transport of ions against their concentration gradient. The reasons behind this are the mild selectivity of this bacterial pore and the relatively low efficiencies of the oscillating signals characteristic of membrane cells (random telegraph noise and thermal noise). Full article
(This article belongs to the Special Issue Nonequilibrium Phenomena in Confined Systems)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>(<b>a</b>) Calculated current-voltage (<span class="html-italic">I-V</span>) curve of OmpF channel in the 3||7 configuration for symmetrical neutral membranes PC||PC; (<b>b</b>) Calculated current-voltage (<span class="html-italic">I-V</span>) curve of OmpF channel in the pH 3||7 configuration for asymmetrical membranes PC||PS. KCl concentration is 25||25 mM. The insets show a representative experimental <span class="html-italic">I-V</span> curve of OmpF single channel at the same conditions as that of the main figure.</p> Full article ">Figure 2
<p>(<b>a</b>) Electrostatic potential in kT/e units across OmpF for an uncharged membrane (PC||PC, left) and an asymmetrically charged membrane (PC||PS, right); (<b>b</b>) 3D ion current paths along the OmpF channel (in grey) for an asymmetrically charged membrane (PC||PS). Ions are represented as streamlines to show the paths followed on average by each ion, not taking into account the flux values. Figures were obtained from PNP-3D calculations and represented using Mayavi, a Python-based program for 3D representation. The conditions used were KCl 25||25 mM, pH 3||7, and no applied electrostatic potential through the system.</p> Full article ">Figure 3
<p>(<b>a</b>) Calculated current-voltage (<span class="html-italic">I-V</span>) curve of OmpF channel in the 3||7 configuration for concentration ratio r = 10 (KCl 250||25 mM) and asymmetrical membranes PC||PS. The inset shows an experimental <span class="html-italic">I-V</span> curve at the same conditions as in the main figure; (<b>b</b>) Concentration profiles for potassium and chloride ions across the OmpF channel (in gray) under an applied potential of +200 mV and −200 mV, as indicated. The conditions used for the calculations are the same as in (<b>a</b>).</p> Full article ">Figure 4
<p>Left panel: Representative calculated traces showing different input signals (constant voltage (<b>a</b>); Gaussian-distributed voltage (<b>b</b>); and random noise (<b>c</b>) and the corresponding output currents; Right panel: Voltage and current (total and K<sup>+</sup> current) distributions obtained from the signals as those shown in the left panel.</p> Full article ">Figure 5
<p>Average current as a function of voltage, calculated from the signals showed in <a href="#entropy-19-00116-f004" class="html-fig">Figure 4</a>.</p> Full article ">Figure 6
<p>(<b>a</b>) Calculated traces showing a triangular input voltage and the corresponding output current. The V and I distributions are almost identical to that obtained for a random distribution (right panel of <a href="#entropy-19-00116-f004" class="html-fig">Figure 4</a>b); (<b>b</b>) Calculated average current as a function of voltage, from the signal in the left panel.</p> Full article ">
5358 KiB  
Article
A Model of Mechanothermodynamic Entropy in Tribology
by Leonid A. Sosnovskiy and Sergei S. Sherbakov
Entropy 2017, 19(3), 115; https://doi.org/10.3390/e19030115 - 14 Mar 2017
Cited by 31 | Viewed by 4994
Abstract
A brief analysis of entropy concepts in continuum mechanics and thermodynamics is presented. The methods of accounting for friction, wear and fatigue processes in the calculation of the thermodynamic entropy are described. It is shown that these and other damage processes of solids [...] Read more.
A brief analysis of entropy concepts in continuum mechanics and thermodynamics is presented. The methods of accounting for friction, wear and fatigue processes in the calculation of the thermodynamic entropy are described. It is shown that these and other damage processes of solids are more adequately described by tribo-fatigue entropy. It was established that mechanothermodynamic entropy calculated as the sum of interacting thermodynamic and tribo-fatigue entropy components has the most general character. Examples of usage (application) of tribo-fatigue and mechanothermodynamic entropies for practical analysis of wear and fatigue processes are given. Full article
(This article belongs to the Special Issue Entropy Application in Tribology)
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<p>Evolution of the thermodynamic (<span class="html-italic">S<sub>T</sub></span>) or the mechanothermodynamic (<span class="html-italic">S<sub>T</sub></span> + <span class="html-italic">S<sub>TF</sub></span>) state of the system (<span class="html-italic">A</span><sub>1</sub>, <span class="html-italic">A</span><sub>2</sub>): (<b>a</b>) oscillatory and asymptotic converging processes; (<b>b</b>) oscillatory and asymptotic diverging processes.</p> Full article ">Figure 2
<p>Sosnovskiy’s generalized rule of interaction of damages.</p> Full article ">Figure 3
<p>Sosnovskiy-Sherbakov rule of the interaction between thermodynamic and tribo-fatigue entropies.</p> Full article ">Figure 4
<p>Dependencies of τ<span class="html-italic"><sub>f</sub></span><sub>σ</sub>(<span class="html-italic">C<sub>T</sub></span>) for Steel 45 (<b>a</b>); silumin (<b>b</b>) and their combined representation (<b>c</b>).</p> Full article ">Figure 5
<p>Dependence of <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mo>−</mo> <mn>1</mn> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math> for steels (<b>a</b>); titanium and its alloys (<b>b</b>).</p> Full article ">Figure 6
<p>Dependence of <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>u</mi> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math> for steels (<b>a</b>); titanium and its alloys (<b>b</b>); aluminum alloys (<b>c</b>) and polymers (<b>d</b>).</p> Full article ">Figure 7
<p>Combined dependencies of <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mo>−</mo> <mn>1</mn> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>u</mi> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mi>T</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math> for steels (<b>a</b>); titanium and its alloys (<b>b</b>).</p> Full article ">
1094 KiB  
Article
Recoverable Random Numbers in an Internet of Things Operating System
by Taeill Yoo, Ju-Sung Kang and Yongjin Yeom
Entropy 2017, 19(3), 113; https://doi.org/10.3390/e19030113 - 13 Mar 2017
Cited by 8 | Viewed by 6285
Abstract
Over the past decade, several security issues with Linux Random Number Generator (LRNG) on PCs and Androids have emerged. The main problem involves the process of entropy harvesting, particularly at boot time. An entropy source in the input pool of LRNG is not [...] Read more.
Over the past decade, several security issues with Linux Random Number Generator (LRNG) on PCs and Androids have emerged. The main problem involves the process of entropy harvesting, particularly at boot time. An entropy source in the input pool of LRNG is not transferred into the non-blocking output pool if the entropy counter of the input pool is less than 192 bits out of 4098 bits. Because the entropy estimation of LRNG is highly conservative, the process may require more than one minute for starting the transfer. Furthermore, the design principle of the estimation algorithm is not only heuristic but also unclear. Recently, Google released an Internet of Things (IoT) operating system called Brillo based on the Linux kernel. We analyze the behavior of the random number generator in Brillo, which inherits that of LRNG. In the results, we identify two features that enable recovery of random numbers. With these features, we demonstrate that random numbers of 700 bytes at boot time can be recovered with the success probability of 90% by using time complexity for 5.20 × 2 40 trials. Therefore, the entropy of random numbers of 700 bytes is merely about 43 bits. Since the initial random numbers are supposed to be used for sensitive security parameters, such as stack canary and key derivation, our observation can be applied to practical attacks against cryptosystem. Full article
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<p>Relations between three entropy pools in LRNG.</p> Full article ">Figure 2
<p><span class="html-italic">Extract unit</span>: Extracting random numbers using get_random_bytes() or /dev/urandom without the entropy transfer.</p> Full article ">Figure 3
<p>Structure of Brillo and three analysis points.</p> Full article ">Figure 4
<p>Flow chart of analysis process.</p> Full article ">Figure 5
<p>Entropy counter of the input pool during boot time in Brillo.</p> Full article ">Figure 6
<p>Code of add_device_randomness().</p> Full article ">Figure 7
<p>Timeline for input and output timing.</p> Full article ">Figure 8
<p>Timeline for identical pattern at boot time in Brillo.</p> Full article ">Figure 9
<p>Timeline for supports of <math display="inline"> <semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi>X</mi> <mn>2</mn> </msub> </semantics> </math>.</p> Full article ">Figure 10
<p>Fitting results for sample values of <math display="inline"> <semantics> <msub> <mi>Y</mi> <mn>1</mn> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi>Y</mi> <mn>2</mn> </msub> </semantics> </math>. (<b>a</b>) Fitting result of <math display="inline"> <semantics> <msub> <mi>Y</mi> <mn>1</mn> </msub> </semantics> </math>; (<b>b</b>) Fitting result of <math display="inline"> <semantics> <msub> <mi>Y</mi> <mn>2</mn> </msub> </semantics> </math>.</p> Full article ">Figure 11
<p>Scenario to recover random numbers during boot time.</p> Full article ">
836 KiB  
Article
Quantum Probabilities as Behavioral Probabilities
by Vyacheslav I. Yukalov and Didier Sornette
Entropy 2017, 19(3), 112; https://doi.org/10.3390/e19030112 - 13 Mar 2017
Cited by 29 | Viewed by 5743
Abstract
We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision [...] Read more.
We demonstrate that behavioral probabilities of human decision makers share many common features with quantum probabilities. This does not imply that humans are some quantum objects, but just shows that the mathematics of quantum theory is applicable to the description of human decision making. The applicability of quantum rules for describing decision making is connected with the nontrivial process of making decisions in the case of composite prospects under uncertainty. Such a process involves deliberations of a decision maker when making a choice. In addition to the evaluation of the utilities of considered prospects, real decision makers also appreciate their respective attractiveness. Therefore, human choice is not based solely on the utility of prospects, but includes the necessity of resolving the utility-attraction duality. In order to justify that human consciousness really functions similarly to the rules of quantum theory, we develop an approach defining human behavioral probabilities as the probabilities determined by quantum rules. We show that quantum behavioral probabilities of humans do not merely explain qualitatively how human decisions are made, but they predict quantitative values of the behavioral probabilities. Analyzing a large set of empirical data, we find good quantitative agreement between theoretical predictions and observed experimental data. Full article
(This article belongs to the Special Issue Foundations of Quantum Mechanics)
342 KiB  
Article
The Two-Time Interpretation and Macroscopic Time-Reversibility
by Yakir Aharonov, Eliahu Cohen and Tomer Landsberger
Entropy 2017, 19(3), 111; https://doi.org/10.3390/e19030111 - 12 Mar 2017
Cited by 35 | Viewed by 7900
Abstract
The two-state vector formalism motivates a time-symmetric interpretation of quantum mechanics that entails a resolution of the measurement problem. We revisit a post-selection-assisted collapse model previously suggested by us, claiming that unlike the thermodynamic arrow of time, it can lead to reversible dynamics [...] Read more.
The two-state vector formalism motivates a time-symmetric interpretation of quantum mechanics that entails a resolution of the measurement problem. We revisit a post-selection-assisted collapse model previously suggested by us, claiming that unlike the thermodynamic arrow of time, it can lead to reversible dynamics at the macroscopic level. In addition, the proposed scheme enables us to characterize the classical-quantum boundary. We discuss the limitations of this approach and its broad implications for other areas of physics. Full article
(This article belongs to the Special Issue Limits to the Second Law of Thermodynamics: Experiment and Theory)
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<p>The universal wavefunction according to (<b>a</b>) the many worlds interpretation (MWI) and (<b>b</b>) the two-state vector-formalism (TSVF).</p> Full article ">
681 KiB  
Article
The Gibbs Paradox, the Landauer Principle and the Irreversibility Associated with Tilted Observers
by Luis Herrera
Entropy 2017, 19(3), 110; https://doi.org/10.3390/e19030110 - 11 Mar 2017
Cited by 35 | Viewed by 4474
Abstract
It is well known that, in the context of General Relativity, some spacetimes, when described by a congruence of comoving observers, may consist of a distribution of a perfect (non–dissipative) fluid, whereas the same spacetime as seen by a “tilted” (Lorentz–boosted) congruence of [...] Read more.
It is well known that, in the context of General Relativity, some spacetimes, when described by a congruence of comoving observers, may consist of a distribution of a perfect (non–dissipative) fluid, whereas the same spacetime as seen by a “tilted” (Lorentz–boosted) congruence of observers may exhibit the presence of dissipative processes. As we shall see, the appearance of entropy-producing processes are related to the high dependence of entropy on the specific congruence of observers. This fact is well illustrated by the Gibbs paradox. The appearance of such dissipative processes, as required by the Landauer principle, are necessary in order to erase the different amount of information stored by comoving observers, with respect to tilted ones. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
2486 KiB  
Article
Entropy Generation Analysis and Performance Evaluation of Turbulent Forced Convective Heat Transfer to Nanofluids
by Yu Ji, Hao-Chun Zhang, Xie Yang and Lei Shi
Entropy 2017, 19(3), 108; https://doi.org/10.3390/e19030108 - 11 Mar 2017
Cited by 55 | Viewed by 7886
Abstract
The entropy generation analysis of fully turbulent convective heat transfer to nanofluids in a circular tube is investigated numerically using the Reynolds Averaged Navier–Stokes (RANS) model. The nanofluids with particle concentration of 0%, 1%, 2%, 4% and 6% are treated as single phases [...] Read more.
The entropy generation analysis of fully turbulent convective heat transfer to nanofluids in a circular tube is investigated numerically using the Reynolds Averaged Navier–Stokes (RANS) model. The nanofluids with particle concentration of 0%, 1%, 2%, 4% and 6% are treated as single phases of effective properties. The uniform heat flux is enforced at the tube wall. To confirm the validity of the numerical approach, the results have been compared with empirical correlations and analytical formula. The self-similarity profiles of local entropy generation are also studied, in which the peak values of entropy generation by direct dissipation, turbulent dissipation, mean temperature gradients and fluctuating temperature gradients for different Reynolds number as well as different particle concentration are observed. In addition, the effects of Reynolds number, volume fraction of nanoparticles and heat flux on total entropy generation and Bejan number are discussed. In the results, the intersection points of total entropy generation for water and four nanofluids are observed, when the entropy generation decrease before the intersection and increase after the intersection as the particle concentration increases. Finally, by definition of Ep, which combines the first law and second law of thermodynamics and attributed to evaluate the real performance of heat transfer processes, the optimal Reynolds number Reop corresponding to the best performance and the advisable Reynolds number Read providing the appropriate Reynolds number range for nanofluids in convective heat transfer can be determined. Full article
(This article belongs to the Special Issue Entropy in Computational Fluid Dynamics)
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<p>Schematic view of circular tube being investigated in this paper.</p> Full article ">Figure 2
<p>Validation of CFD calculated friction factor with Petukhov’s correlation [<a href="#B50-entropy-19-00108" class="html-bibr">50</a>].</p> Full article ">Figure 3
<p>Validation of CFD calculated Nusselt number with Gnielinski’s correlation [<a href="#B50-entropy-19-00108" class="html-bibr">50</a>].</p> Full article ">Figure 4
<p>Validation of present entropy generation model with Bejan’s formula [<a href="#B51-entropy-19-00108" class="html-bibr">51</a>].</p> Full article ">Figure 5
<p>Dimensionless local entropy generation by direct dissipation, turbulence dissipation, mean temperature gradients and fluctuating temperature gradients at different Reynolds number; (<b>a</b>) <span class="html-italic">Re</span> = 5000; (<b>b</b>) <span class="html-italic">Re</span> = 10,000; (<b>c</b>) <span class="html-italic">Re</span> = 20,000; (<b>d</b>) <span class="html-italic">Re</span> = 40,000 (The dimensionless local entropy generation here has been quadrupled).</p> Full article ">Figure 5 Cont.
<p>Dimensionless local entropy generation by direct dissipation, turbulence dissipation, mean temperature gradients and fluctuating temperature gradients at different Reynolds number; (<b>a</b>) <span class="html-italic">Re</span> = 5000; (<b>b</b>) <span class="html-italic">Re</span> = 10,000; (<b>c</b>) <span class="html-italic">Re</span> = 20,000; (<b>d</b>) <span class="html-italic">Re</span> = 40,000 (The dimensionless local entropy generation here has been quadrupled).</p> Full article ">Figure 6
<p>Variation of dimensionless entropy generation rate with Reynolds number and nanoparticle concentration corresponding to different heat flux enforced at the tube wall; (<b>a</b>) <span class="html-italic">q</span> = 50,000 W/m<sup>2</sup>; (<b>b</b>) <span class="html-italic">q</span> = 100,000 W/m<sup>2</sup>; (<b>c</b>) <span class="html-italic">q</span> = 200,000 W/m<sup>2</sup>; (<b>d</b>) <span class="html-italic">q</span> = 500,000 W/m<sup>2</sup>.</p> Full article ">Figure 6 Cont.
<p>Variation of dimensionless entropy generation rate with Reynolds number and nanoparticle concentration corresponding to different heat flux enforced at the tube wall; (<b>a</b>) <span class="html-italic">q</span> = 50,000 W/m<sup>2</sup>; (<b>b</b>) <span class="html-italic">q</span> = 100,000 W/m<sup>2</sup>; (<b>c</b>) <span class="html-italic">q</span> = 200,000 W/m<sup>2</sup>; (<b>d</b>) <span class="html-italic">q</span> = 500,000 W/m<sup>2</sup>.</p> Full article ">Figure 7
<p>Variation of Bejan number with Reynolds number and volume fraction of nanoparticle corresponding to different heat flux enforced at the tube wall; (<b>a</b>) <span class="html-italic">q</span> = 50,000 W/m<sup>2</sup>; (<b>b</b>) <span class="html-italic">q</span> = 100,000 W/m<sup>2</sup>; (<b>c</b>) <span class="html-italic">q</span> = 200,000 W/m<sup>2</sup>; (<b>d</b>) <span class="html-italic">q</span> = 500,000 W/m<sup>2</sup>.</p> Full article ">Figure 8
<p>Performance of the nanofluids as a function of Reynolds number with different heat flux enforced at the wall; (<b>a</b>) <span class="html-italic">q</span> = 50,000 W/m<sup>2</sup>; (<b>b</b>) <span class="html-italic">q</span> = 100,000 W/m<sup>2</sup>; (<b>c</b>) <span class="html-italic">q</span> = 200,000 W/m<sup>2</sup>; (<b>d</b>) <span class="html-italic">q</span> = 500,000 W/m<sup>2</sup>.</p> Full article ">Figure 9
<p>Performance evaluation of the nanofluids as a function of particle concentration with different heat flux enforced at the wall; (<b>a</b>) <span class="html-italic">q</span> = 50,000 W/m<sup>2</sup>; (<b>b</b>) <span class="html-italic">q</span> = 100,000 W/m<sup>2</sup>; (<b>c</b>) <span class="html-italic">q</span> = 200,000 W/m<sup>2</sup>; (<b>d</b>) <span class="html-italic">q</span> = 500,000 W/m<sup>2</sup>.</p> Full article ">
1862 KiB  
Article
Formulation of Exergy Cost Analysis to Graph-Based Thermal Network Models
by Stefano Coss, Elisa Guelpa, Etienne Letournel, Olivier Le-Corre and Vittorio Verda
Entropy 2017, 19(3), 109; https://doi.org/10.3390/e19030109 - 10 Mar 2017
Cited by 9 | Viewed by 4605
Abstract
Information from exergy cost analysis can be effectively used in the design and management of modern district heating networks (DHNs) since it allows to properly account for the irreversibilities in energy conversion and distribution. Nevertheless, this requires the development of suitable graph-based approaches [...] Read more.
Information from exergy cost analysis can be effectively used in the design and management of modern district heating networks (DHNs) since it allows to properly account for the irreversibilities in energy conversion and distribution. Nevertheless, this requires the development of suitable graph-based approaches that are able to effectively consider the network topology and the variations of the physical properties of the heating fluid on a time-dependent basis. In this work, a formulation of exergetic costs suitable for large graph-based networks is proposed, which is consistent with the principles of exergetic costing. In particular, the approach is more compact in comparison to straightforward approaches of exergetic cost formulation available in the literature, especially when applied to fluid networks. Moreover, the proposed formulation is specifically considering transient operating conditions, which is a crucial feature and a necessity for the analysis of future DHNs. Results show that transient effects of the thermodynamic behavior are not negligible for exergy cost analysis, while this work offers a coherent approach to quantify them. Full article
(This article belongs to the Special Issue Thermoeconomics for Energy Efficiency)
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<p>(<b>a</b>) Simple directed graph; (<b>b</b>) thermal graph network.</p> Full article ">Figure 2
<p>Control volumes for (<b>a</b>) exergy and (<b>b</b>) exergy cost analysis.</p> Full article ">Figure 3
<p>Example of thermal network.</p> Full article ">Figure 4
<p>Mass flows and temperature levels for node 5.</p> Full article ">Figure 5
<p><math display="inline"> <semantics> <mrow> <mo>Δ</mo> <msubsup> <mi>k</mi> <mi>n</mi> <mrow> <mo>*</mo> <mo>,</mo> <mi>n</mi> <mi>o</mi> <mi>r</mi> <mi>m</mi> </mrow> </msubsup> </mrow> </semantics> </math>-values for a network in quasi-steady state and transient condition.</p> Full article ">Figure 6
<p>Unit exergy cost evolution of node 7.</p> Full article ">
3660 KiB  
Article
Physical Intelligence and Thermodynamic Computing
by Robert L. Fry
Entropy 2017, 19(3), 107; https://doi.org/10.3390/e19030107 - 9 Mar 2017
Cited by 13 | Viewed by 8620
Abstract
This paper proposes that intelligent processes can be completely explained by thermodynamic principles. They can equally be described by information-theoretic principles that, from the standpoint of the required optimizations, are functionally equivalent. The underlying theory arises from two axioms regarding distinguishability and causality. [...] Read more.
This paper proposes that intelligent processes can be completely explained by thermodynamic principles. They can equally be described by information-theoretic principles that, from the standpoint of the required optimizations, are functionally equivalent. The underlying theory arises from two axioms regarding distinguishability and causality. Their consequence is a theory of computation that applies to the only two kinds of physical processes possible—those that reconstruct the past and those that control the future. Dissipative physical processes fall into the first class, whereas intelligent ones comprise the second. The first kind of process is exothermic and the latter is endothermic. Similarly, the first process dumps entropy and energy to its environment, whereas the second reduces entropy while requiring energy to operate. It is shown that high intelligence efficiency and high energy efficiency are synonymous. The theory suggests the usefulness of developing a new computing paradigm called Thermodynamic Computing to engineer intelligent processes. The described engineering formalism for the design of thermodynamic computers is a hybrid combination of information theory and thermodynamics. Elements of the engineering formalism are introduced in the reverse-engineer of a cortical neuron. The cortical neuron provides perhaps the simplest and most insightful example of a thermodynamic computer possible. It can be seen as a basic building block for constructing more intelligent thermodynamic circuits. Full article
(This article belongs to the Special Issue Maximum Entropy and Its Application II)
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<p>Comparison of dual Venn diagrams (<b>left</b>) and <span class="html-italic">I</span>-diagrams (<b>right</b>).</p> Full article ">Figure 2
<p>Dyadic diagram used for characterizing a binary question.</p> Full article ">Figure 3
<p>Information transmission process depicted as a Carnot cycle on the temperature-entropy plane.</p> Full article ">Figure 4
<p>Carnot process associated with an intelligence process as depicted on the temperature-entropy plane.</p> Full article ">Figure 5
<p>Depiction of delay equalization process relative to neural decisions <span class="html-italic">y</span>(<span class="html-italic">t</span>) made at reference time <span class="html-italic">t</span>.</p> Full article ">Figure 6
<p>Overlay of the logistic function and error functions for <span class="html-italic">T</span> = <span class="html-italic">β</span> = 1 showing they are indistinguishable.</p> Full article ">Figure 7
<p>Sample training codes used against the specified neural optimization objectives.</p> Full article ">Figure 8
<p>Hamming distances between selected training codes.</p> Full article ">Figure 9
<p>(<b>a</b>) Codes that illicit neural firing and (<b>b</b>) the depiction of dendritic field connection strengths.</p> Full article ">Figure 10
<p>Hyper-plane representation for double-matching optimization.</p> Full article ">Figure 11
<p>Colorized map of <span class="html-italic">Z</span>(<span class="html-italic">n</span>, <span class="html-italic">β</span>) showing where phase transition occur. The dark red region is required for cortical operation.</p> Full article ">Figure 12
<p>Depiction of Carnot process underlying the operation of the cortical neuron.</p> Full article ">
233 KiB  
Article
On Quantum Collapse as a Basis for the Second Law of Thermodynamics
by Ruth E. Kastner
Entropy 2017, 19(3), 106; https://doi.org/10.3390/e19030106 - 9 Mar 2017
Cited by 13 | Viewed by 7047
Abstract
It was first suggested by David Z. Albert that the existence of a real, physical non-unitary process (i.e., “collapse”) at the quantum level would yield a complete explanation for the Second Law of Thermodynamics (i.e., the increase in entropy over time). The contribution [...] Read more.
It was first suggested by David Z. Albert that the existence of a real, physical non-unitary process (i.e., “collapse”) at the quantum level would yield a complete explanation for the Second Law of Thermodynamics (i.e., the increase in entropy over time). The contribution of such a process would be to provide a physical basis for the ontological indeterminacy needed to derive the irreversible Second Law against a backdrop of otherwise reversible, deterministic physical laws. An alternative understanding of the source of this possible quantum “collapse” or non-unitarity is presented herein, in terms of the Transactional Interpretation (TI). The present model provides a specific physical justification for Boltzmann’s often-criticized assumption of molecular randomness (Stosszahlansatz), thereby changing its status from an ad hoc postulate to a theoretically grounded result, without requiring any change to the basic quantum theory. In addition, it is argued that TI provides an elegant way of reconciling, via indeterministic collapse, the time-reversible Liouville evolution with the time-irreversible evolution inherent in so-called “master equations” that specify the changes in occupation of the various possible states in terms of the transition rates between them. The present model is contrasted with the Ghirardi–Rimini–Weber (GRW) “spontaneous collapse” theory previously suggested for this purpose by Albert. Full article
(This article belongs to the Special Issue Entropy, Time and Evolution)
912 KiB  
Article
Brownian Dynamics Computational Model of Protein Diffusion in Crowded Media with Dextran Macromolecules as Obstacles
by Pablo M. Blanco, Mireia Via, Josep Lluís Garcés, Sergio Madurga and Francesc Mas
Entropy 2017, 19(3), 105; https://doi.org/10.3390/e19030105 - 9 Mar 2017
Cited by 16 | Viewed by 6178
Abstract
The high concentration of macromolecules (i.e., macromolecular crowding) in cellular environments leads to large quantitative effects on the dynamic and equilibrium biological properties. These effects have been experimentally studied using inert macromolecules to mimic a realistic cellular medium. In this paper, two different [...] Read more.
The high concentration of macromolecules (i.e., macromolecular crowding) in cellular environments leads to large quantitative effects on the dynamic and equilibrium biological properties. These effects have been experimentally studied using inert macromolecules to mimic a realistic cellular medium. In this paper, two different experimental in vitro systems of diffusing proteins which use dextran macromolecules as obstacles are computationally analyzed. A new model for dextran macromolecules based on effective radii accounting for macromolecular compression induced by crowding is proposed. The obtained results for the diffusion coefficient and the anomalous diffusion exponent exhibit good qualitative and generally good quantitative agreement with experiments. Volume fraction and hydrodynamic interactions are found to be crucial to describe the diffusion coefficient decrease in crowded media. However, no significant influence of the hydrodynamic interactions in the anomalous diffusion exponent is found. Full article
(This article belongs to the Special Issue Nonequilibrium Phenomena in Confined Systems)
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<p>Time evolution of the diffusion coefficient obtained in a system with volume fraction of <math display="inline"> <mrow> <mi>ϕ</mi> </mrow> </math> = 0.17. The red line is the result of averaging 400 BD simulations of a tracer particle diffusing among 68 obstacles with radius 4.9 and 1.2 nm respectively. The simulation time is 250 ns and the length of the simulation box is 18 nm.</p> Full article ">Figure 2
<p>Snapshot of one of the performed dynamics. A protein (in red) diffuses among dextran molecules (in yellow) which act as crowding agents.</p> Full article ">Figure 3
<p>Decrease of the normalised diffusion coefficient at long times <math display="inline"> <semantics> <mrow> <mo>(</mo> <msup> <mi>D</mi> <mi>long</mi> </msup> <mo>)</mo> </mrow> </semantics> </math> with the volume fraction at three different values of the stiffness constant <span class="html-italic">k</span> of the interaction potential. BD simulation perfomed: (<b>a</b>) without HI; (<b>b</b>) with HI using the Tokuyama model. The results show that the potential is rigid enough to avoid overlapping. Continuous lines are only to guide the lecturer.</p> Full article ">Figure 4
<p>Comparison of the diffusion coefficient calculated using BD without HI (red triangles), BD with HI using the conventional RPY diffusion tensor (blue squares), BD with HI using the Tokuyama model (cyan circles) and Tokuyama analytical expression (Equation (<a href="#FD8-entropy-19-00105" class="html-disp-formula">8</a>), orange discontinuous line). The results show the important role of short range HI (lubrication forces) in the diffusion reduction of the diffusion coefficient, while the effect of the of long range HI is very low. It is also worth noting the good agreement between the Tokuyama analytical prediction of <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> and the BD simulation with HI using Tokuyama model for <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>short</mi> </msup> </semantics> </math>. The lines are to guide the lecturer.</p> Full article ">Figure 5
<p>Comparison between the normalized <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> obtained using Tokuyama analytic equations (Equations (<a href="#FD6-entropy-19-00105" class="html-disp-formula">6</a>)–(<a href="#FD8-entropy-19-00105" class="html-disp-formula">8</a>)) and the ones resulting from BD simulations at different volume fractions. In (<b>a</b>), the tracer particle (with radius <math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>2</mn> <mo>.</mo> <mn>33</mn> </mrow> </semantics> </math> nm) and the obstacles (<math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>2</mn> <mo>.</mo> <mn>9</mn> </mrow> </semantics> </math> nm) present a similar size. <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> values obtained by Tokuyama method (orange circles) and the ones obtained by BD simulations (red squares) clearly agree. In contrast, in (<b>b</b>), the tracer particle (<math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>4</mn> <mo>.</mo> <mn>90</mn> </mrow> </semantics> </math> nm) is far bigger than the obstacles (<math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>2</mn> </mrow> </semantics> </math> nm). In this case, the Tokuyama method (blue circles) clearly overestimates <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> values obtained by BD simulations (cyan squares).</p> Full article ">Figure 6
<p>Radius of gyration (<math display="inline"> <semantics> <msub> <mi>R</mi> <mi mathvariant="normal">G</mi> </msub> </semantics> </math>, red circles) taken from [<a href="#B44-entropy-19-00105" class="html-bibr">44</a>] and hydrodynamic radius (<math display="inline"> <semantics> <msub> <mi>R</mi> <mi mathvariant="normal">H</mi> </msub> </semantics> </math>, green triangles) taken from [<a href="#B43-entropy-19-00105" class="html-bibr">43</a>] versus molecular weight. The fittings corresponding to the power law (Equation (<a href="#FD9-entropy-19-00105" class="html-disp-formula">9</a>)) are represented with red and green lines. The compact radius (<math display="inline"> <semantics> <msub> <mi>R</mi> <mi mathvariant="normal">c</mi> </msub> </semantics> </math>, blue) and the effective radius (<math display="inline"> <semantics> <msub> <mi>R</mi> <mi>eff</mi> </msub> </semantics> </math>, purple line) versus molecular weight are also depicted. The effective radius for the chosen dextran obstacles (purple squares) in our computations is also plotted. The lines are plotted only to guide the lecturer.</p> Full article ">Figure 7
<p>Decay of <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> corresponding to Streptavidin protein versus dextran concentration. Three different sizes of dextran obstacles: (<b>a</b>) D5; (<b>b</b>) D50; and (<b>c</b>) D400 have been used. The obtained results including the HI with Tokuyama model exhibit better agreement in general with the experimental data showing the relevance of HI in the <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> values in crowded media. The lines are only for guiding the lecturer.</p> Full article ">Figure 8
<p><span class="html-italic">α</span>-Chymiotrypsin protein <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> decay with dextran concentration. Three different sizes of dextran obstacles: D5 (<b>a</b>); D50 (<b>b</b>); and D400 (<b>c</b>) have been used. The obtained results including HI with Tokuyama model exhibit in general better agreement with the experimental data showing the relevance of HI in the <math display="inline"> <semantics> <msup> <mi>D</mi> <mi>long</mi> </msup> </semantics> </math> values in crowded media. Continuous lines are only to guide the lecturer.</p> Full article ">Figure 9
<p>Anomalous diffusion exponent <span class="html-italic">α</span> of Streptavidin versus dextran concentration for three different sizes of dextran obstacles: (<b>a</b>) D5; (<b>b</b>) D50; and (<b>c</b>) D400. The obtained results exhibit good quantitative agreement with the experimental data except for the smallest dextran size D5. In general, no significant influence of HI in the <span class="html-italic">α</span> exponent is obtained. The lines are to guide the lecturer.</p> Full article ">Figure 10
<p><span class="html-italic">α</span>-Chymiotrypsin anomalous diffusion exponent <span class="html-italic">α</span> versus dextran concentration for three different sizes of dextran obstacles: (<b>a</b>) D5; (<b>b</b>) D50; and (<b>c</b>) D400. Computations show good quantitative agreement with experiments except for the smallest dextran size D5. The small difference between the results obtained with and without HI reveals HI are not important in the <span class="html-italic">α</span> exponent value. The lines are only for guiding the lecturer.</p> Full article ">
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Article
Complexity and Vulnerability Analysis of the C. Elegans Gap Junction Connectome
by James M. Kunert-Graf, Nikita A. Sakhanenko and David J. Galas
Entropy 2017, 19(3), 104; https://doi.org/10.3390/e19030104 - 8 Mar 2017
Cited by 4 | Viewed by 5779
Abstract
We apply a network complexity measure to the gap junction network of the somatic nervous system of C. elegans and find that it possesses a much higher complexity than we might expect from its degree distribution alone. This “excess” complexity is seen to [...] Read more.
We apply a network complexity measure to the gap junction network of the somatic nervous system of C. elegans and find that it possesses a much higher complexity than we might expect from its degree distribution alone. This “excess” complexity is seen to be caused by a relatively small set of connections involving command interneurons. We describe a method which progressively deletes these “complexity-causing” connections, and find that when these are eliminated, the network becomes significantly less complex than a random network. Furthermore, this result implicates the previously-identified set of neurons from the synaptic network’s “rich club” as the structural components encoding the network’s excess complexity. This study and our method thus support a view of the gap junction Connectome as consisting of a rather low-complexity network component whose symmetry is broken by the unique connectivities of singularly important rich club neurons, sharply increasing the complexity of the network. Full article
(This article belongs to the Special Issue Complexity, Criticality and Computation (C³))
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Figure 1

Figure 1
<p>(<b>a</b>) To illustrate our method, we generated 10,000 random Erdős-Rényi networks, all with 12 nodes of four degrees. The distribution of our complexity measure <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> is shown, along with the networks having the lowest and highest <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> values; (<b>b</b>) <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mo>Ψ</mo> </mrow> </semantics> </math> of an edge is the amount by which <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> is reduced if that edge is removed. We progressively delete the edges with the highest <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mo>Ψ</mo> </mrow> </semantics> </math> at each step. In this example, most of the complexity in the graph is contained within the upper pentagram-shaped connection structure; the elimination ordering reveals these “complexity-causing” structures.</p> Full article ">Figure 2
<p>The <span class="html-italic">C. elegans</span> gap junction connectome has a complexity score of <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow> <mi>g</mi> <mi>a</mi> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.00143</mn> </mrow> </semantics> </math>. For comparison, we generated 10,000 random networks with the same degree distribution and calculated <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> for each. The distribution of <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math> is approximately normal, with an average score of <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.001173</mn> <mo>±</mo> <mn>0.000015</mn> </mrow> </semantics> </math>. Thus, the actual <span class="html-italic">C. elegans</span> gap junction network has a complexity 16.5 standard deviations above the mean value for its degree distribution.</p> Full article ">Figure 3
<p>Distribution of <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mo>Ψ</mo> </mrow> </semantics> </math> values for the intact <span class="html-italic">C. elegans</span> gap junction connectome. The link between the command interneuron pair AVAL/AVAR is a clear outlier, causing by far the largest drop in network complexity. It is notable that every edge below the red line at <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mo>Ψ</mo> <mo>=</mo> <mo>−</mo> <mn>1.083</mn> </mrow> </semantics> </math> involves at least one interneuron. Another notable feature is that some deletions will actually increase the complexity: deleting the edge between the motor neurons VB03/VA07 causes <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> to increase slightly.</p> Full article ">Figure 4
<p>(<b>a</b>) We iteratively delete connections with the highest <math display="inline"> <semantics> <mrow> <mo>Δ</mo> <mo>Ψ</mo> </mrow> </semantics> </math>, causing the complexity to decay as shown by the blue curve. At each point, we calculate <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math> for 256 random graphs with the same degree distribution. The red line shows the mean <math display="inline"> <semantics> <mrow> <mo>Ψ</mo> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mrow> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math>, with the red band showing the range <math display="inline"> <semantics> <mrow> <mo>±</mo> <mn>2</mn> <mi>σ</mi> </mrow> </semantics> </math>; (<b>b</b>) the same data converted to <span class="html-italic">z</span>-score (i.e., the number of standard deviations by which the actual network differs from random). The <span class="html-italic">C. elegans</span> gap junction network is initially much more complex than randomly expected, but as we successively delete edges, it reveals an underlying network that is much <span class="html-italic">less</span> complex than random.</p> Full article ">Figure 5
<p>(<b>a</b>) As in [<a href="#B24-entropy-19-00104" class="html-bibr">24</a>], the <span class="html-italic">C. elegans</span> synaptic Connectome can be understood to have a “rich club” structure. Edges are classified as “Club” (if between two rich nodes), “Feeder” (if between a rich and poor node), or “Local” (if between two poor nodes); (<b>b</b>) the same curve as <a href="#entropy-19-00104-f004" class="html-fig">Figure 4</a>b, with each deleted edge labeled by class. In the region where the graph is much more complex than average, the procedure disproportionately targets Club and Feeder edges; (<b>c</b>) the fraction of each class which has been deleted at each iteration.</p> Full article ">Figure 6
<p>(<b>a</b>) the initial degree distribution of the graph. Each node has one of 16 unique degree values, which we label by their relative rank from highest to lowest. The first implicated edge <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>i</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>j</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics> </math> connects nodes with degree ranks 1 and 2, such that its “Maximum Degree Rank” is 1; (<b>b</b>) the maximum degree rank of the subsequently targeted edges, plotted in blue. The green line indicates the number of unique degrees, which changes as the degree distribution is altered. The procedure <span class="html-italic">does</span> tend to trim edges to relatively highly connected nodes, but this relationship is not the driving criterion, and it does not simply choose edges based upon connectivity level alone.</p> Full article ">Figure 7
<p>In 50 trials, we deleted all club edges in a random order, then all feeder edges, and then all local edges. The red/green/blue bands show the average resulting complexity curve within one standard deviation. This was repeated for 50 trials in which we instead deleted local edges, then feeder edges, and then club edges. The resulting distribution of complexity curves is indicated by the blue/green/red bands. The edge deletion order prescribed by our algorithm (i.e., the blue curve in <a href="#entropy-19-00104-f004" class="html-fig">Figure 4</a>a) is shown by the black dotted line. A random Club/Feeder/Local deletion order results in a significantly slower complexity decay than our algorithm prescribes, but leads to a much larger decrease in complexity than the Local/Feeder/Club ordering. Thus, the results implicating the rich club are robust to the specific edge deletion order.</p> Full article ">Figure 8
<p>The greedy edge-elimination procedure was repeated for all possible choices of initial edge deletions (i.e., the edge chosen for deletion at the first step). Each row corresponds to a different choice of initial edge, with each column showing the class of the edges subsequently deleted by the iterative procedure. Club and Feeder edges are targeted disproportionately regardless of the initial edge choice.</p> Full article ">
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