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Entropy, Volume 13, Issue 10 (October 2011) – 6 articles , Pages 1746-1903

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581 KiB  
Article
On the Growth Rate of Non-Enzymatic Molecular Replicators
by Harold Fellermann and Steen Rasmussen
Entropy 2011, 13(10), 1882-1903; https://doi.org/10.3390/e13101882 - 21 Oct 2011
Cited by 6 | Viewed by 6803
Abstract
It is well known that non-enzymatic template directed molecular replicators X + nO -> 2X exhibit parabolic growth d[X]/dt -> k[X]1/2. Here, we analyze the dependence of the effective replication rate constant k on [...] Read more.
It is well known that non-enzymatic template directed molecular replicators X + nO -> 2X exhibit parabolic growth d[X]/dt -> k[X]1/2. Here, we analyze the dependence of the effective replication rate constant k on hybridization energies, temperature, strand length, and sequence composition. First we derive analytical criteria for the replication rate k based on simple thermodynamic arguments. Second we present a Brownian dynamics model for oligonucleotides that allows us to simulate their diffusion and hybridization behavior. The simulation is used to generate and analyze the effect of strand length, temperature, and to some extent sequence composition, on the hybridization rates and the resulting optimal overall rate constant k. Combining the two approaches allows us to semi-analytically depict a replication rate landscape for template directed replicators. The results indicate a clear replication advantage for longer strands at lower temperatures in the regime where the ligation rate is rate limiting. Further the results indicate the existence of an optimal replication rate at the boundary between the two regimes where the ligation rate and the dehybridization rates are rate limiting. Full article
(This article belongs to the Special Issue Emergence in Chemical Systems)
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Figure 1

Figure 1
<p>Minimal template directed replicator: two complementary oligomers hybridize to a template strand (upper part). An irreversible ligation reaction transforms the oligomers into the complementary copy of the template. The newly obtained double strand can dehybridize (lower part) thus allowing for iteration of the process. We assume that ligation is rate limiting, which implies that hybridization and dehybridization are in local equilibrium.</p> Full article ">Figure 2
<p>Effective replication rate <span class="html-italic">k</span> (given by Equation <a href="#FD15-entropy-13-01882" class="html-disp-formula">15</a>) as a function of strand length and temperature. For strands below a critical length <math display="inline"> <msup> <mi>N</mi> <mo>*</mo> </msup> </math> (here 10) the rate increases with temperature, for strands longer than <math display="inline"> <msup> <mi>N</mi> <mo>*</mo> </msup> </math>, the replication rate grows with decreasing temperature. The value of <math display="inline"> <msup> <mi>N</mi> <mo>*</mo> </msup> </math> is determined through Equation (<a href="#FD16-entropy-13-01882" class="html-disp-formula">16</a>). Note the saddle point of the surface where <math display="inline"> <msup> <mi>T</mi> <mo>*</mo> </msup> </math> and <math display="inline"> <msup> <mi>N</mi> <mo>*</mo> </msup> </math> intersect (Equation (<a href="#FD12-entropy-13-01882" class="html-disp-formula">12</a>) (<math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>H</mi> <mtext>base</mtext> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mo>−</mo> <mn>1</mn> <mo>.</mo> <mn>5</mn> <msub> <mi>k</mi> <mtext>B</mtext> </msub> <msup> <mi>T</mi> <mo>′</mo> </msup> </mrow> </math>, <math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>S</mi> <mtext>base</mtext> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mo>−</mo> <mn>1</mn> <msub> <mi>k</mi> <mtext>B</mtext> </msub> </mrow> </math>, <math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>H</mi> <mtext>init</mtext> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>0</mn> <mo>.</mo> <mn>50</mn> <msub> <mi>k</mi> <mtext>B</mtext> </msub> <msup> <mi>T</mi> <mo>′</mo> </msup> </mrow> </math>, <math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>S</mi> <mtext>init</mtext> </msub> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>1</mn> <mo>.</mo> <mn>25</mn> <msub> <mi>k</mi> <mtext>B</mtext> </msub> </mrow> </math>, <math display="inline"> <mrow> <mo>Δ</mo> <msubsup> <mi>H</mi> <mtext>L</mtext> <mo>‡</mo> </msubsup> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>5</mn> <mo>.</mo> <mn>25</mn> <msub> <mi>k</mi> <mtext>B</mtext> </msub> <msup> <mi>T</mi> <mo>′</mo> </msup> </mrow> </math>, <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </math>).</p> Full article ">Figure 3
<p>Geometry of the nucleotide strands. The figure shows the angles that define inner- and intermolecular interactions for one nucleobase (shaded in grey).</p> Full article ">Figure 4
<p>Diffusion coefficients measured for different strand lengths and temperatures (symbols) fitted to the prediction of the Einstein-Stokes relation (solid lines). For each parameter pair, 40 simulation runs over <math display="inline"> <mrow> <mn>1000</mn> <mi>τ</mi> </mrow> </math> have been averaged.</p> Full article ">Figure 5
<p>Radius of gyration measured for different strand lengths and bending potentials (symbols) fitted to the prediction of the Flory mean field theory (solid lines). For each parameter pair, 40 simulation runs over <math display="inline"> <mrow> <mn>400</mn> <mi>τ</mi> </mrow> </math> have been averaged. The upper panel shows results for homopolymers (e.g., poly-C), the lower panel compares those to radii of self-complementary strands. The plots also show the boundaries for maximally stretched chains (<math display="inline"> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>—upper dotted line) and the expectation value of an ideal chain (<math display="inline"> <mrow> <mi>ν</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>3</mn> <mo>/</mo> <mn>5</mn> </mrow> </math>—lower dotted line).</p> Full article ">Figure 5 Cont.
<p>Radius of gyration measured for different strand lengths and bending potentials (symbols) fitted to the prediction of the Flory mean field theory (solid lines). For each parameter pair, 40 simulation runs over <math display="inline"> <mrow> <mn>400</mn> <mi>τ</mi> </mrow> </math> have been averaged. The upper panel shows results for homopolymers (e.g., poly-C), the lower panel compares those to radii of self-complementary strands. The plots also show the boundaries for maximally stretched chains (<math display="inline"> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>—upper dotted line) and the expectation value of an ideal chain (<math display="inline"> <mrow> <mi>ν</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>3</mn> <mo>/</mo> <mn>5</mn> </mrow> </math>—lower dotted line).</p> Full article ">Figure 6
<p>Systems of size <math display="inline"> <msup> <mn>10</mn> <mn>3</mn> </msup> </math> are initialized with two complementary strands of length <span class="html-italic">N</span>. The sequence information is taken from the <span class="html-italic">N</span> central nucleotides of the master sequence denoted in each panel (e.g., <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>6</mn> </mrow> </math> implies sequence CABACD in the first panel). Each system is simulated over <math display="inline"> <mrow> <mn>50000</mn> <mi>τ</mi> </mrow> </math>, and the average fraction <span class="html-italic">χ</span> of hybridized nucleobases is determined. Error bars show the average and standard deviation of 40 measurements. Solid lines show the theoretical prediction <math display="inline"> <mrow> <mi>χ</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mfenced separators="" open="(" close=")"> <mn>1</mn> <mo>+</mo> <msup> <mi>e</mi> <mfrac> <mrow> <mo>Δ</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mo>Δ</mo> <mi>S</mi> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </msup> </mfenced> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> fitted individually to each data set via <math display="inline"> <mrow> <mo>Δ</mo> <mi>S</mi> </mrow> </math> and <math display="inline"> <mrow> <mo>Δ</mo> <mi>H</mi> </mrow> </math>. Melting temperatures <math display="inline"> <msub> <mi>T</mi> <mtext>m</mtext> </msub> </math> are obtained from the relation <math display="inline"> <mrow> <mi>χ</mi> <mo>(</mo> <msub> <mi>T</mi> <mtext>m</mtext> </msub> <mo>)</mo> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>, and their scaling as a function of strand length is depicted in the inlays for the cases where enough melting points had been observed.</p> Full article ">Figure 7
<p>Melting curves for an oligomer that hybridizes to the left hand side of the master sequence in the presence of the right hand side oligomer. Data is obtained with the procedure described in <a href="#entropy-13-01882-f006" class="html-fig">Figure 6</a>. For the analyzed master sequence, the results are comparable to those of two complementary strands of length <math display="inline"> <mrow> <mi>N</mi> <mo>/</mo> <mn>2</mn> </mrow> </math> (dotted lines).</p> Full article ">Figure 8
<p>Hybridization energy changes <math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>G</mi> <mtext>T</mtext> </msub> </mrow> </math> obtained from the measurements of <a href="#sec4dot3-entropy-13-01882" class="html-sec">Section 4.3</a>, sequence ACDCABACDCABACDCABAC (symbols), fitted to the analytical model of Equation (<a href="#FD13-entropy-13-01882" class="html-disp-formula">13</a>) via the parameters <math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>H</mi> <mtext>base</mtext> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>.</mo> <mn>81</mn> </mrow> </math>, <math display="inline"> <mrow> <mo>Δ</mo> <mi>S</mi> <mo>=</mo> <mo>−</mo> <mn>0</mn> <mo>.</mo> <mn>756</mn> </mrow> </math>, <math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>H</mi> <mtext>init</mtext> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>470</mn> </mrow> </math>, and <math display="inline"> <mrow> <mo>Δ</mo> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mo>.</mo> <mn>58</mn> </mrow> </math>. Since <math display="inline"> <mrow> <msub> <mrow> <mo>[</mo> <mi>X</mi> <mo>]</mo> </mrow> <mtext>total</mtext> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>001</mn> </mrow> </math>, we can estimate <math display="inline"> <mrow> <mo>Δ</mo> <msub> <mi>S</mi> <mtext>init</mtext> </msub> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>33</mn> </mrow> </math>.</p> Full article ">Figure 9
<p>Effective equilibrium constant <math display="inline"> <mrow> <msubsup> <mi>K</mi> <mtext>O</mtext> <mn>2</mn> </msubsup> <mo>/</mo> <msqrt> <mrow> <mn>2</mn> <msub> <mi>K</mi> <mtext>T</mtext> </msub> </mrow> </msqrt> </mrow> </math>, obtained from the measurements of <a href="#entropy-13-01882-f008" class="html-fig">Figure 8</a> (colored) compared to the theoretical prediction of Equation (<a href="#FD13-entropy-13-01882" class="html-disp-formula">13</a>) (mesh). Data shaded in gray is extrapolated from dehybridization experiments.</p> Full article ">Figure 10
<p>Final replication rate <span class="html-italic">k</span> as a function of template length and temperature. The figure is produced by superposing the data from <a href="#entropy-13-01882-f009" class="html-fig">Figure 9</a> with the Arrhenius equation for the ligation reaction following Equation (<a href="#FD15-entropy-13-01882" class="html-disp-formula">15</a>) with <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> <mo>,</mo> <mo>Δ</mo> <msubsup> <mi>H</mi> <mtext>L</mtext> <mo>‡</mo> </msubsup> <mo>=</mo> <mn>6</mn> <mo>.</mo> <mn>52</mn> <msub> <mi>k</mi> <mtext>B</mtext> </msub> <msup> <mi>T</mi> <mo>′</mo> </msup> </mrow> </math>. For this parametrization, the critical strand length <math display="inline"> <msup> <mi>N</mi> <mo>*</mo> </msup> </math> above which the temperature dependence of the reaction inverts is 8.</p> Full article ">
443 KiB  
Article
Quantifying Dynamical Complexity of Magnetic Storms and Solar Flares via Nonextensive Tsallis Entropy
by Georgios Balasis, Ioannis A. Daglis, Constantinos Papadimitriou, Anastasios Anastasiadis, Ingmar Sandberg and Konstantinos Eftaxias
Entropy 2011, 13(10), 1865-1881; https://doi.org/10.3390/e13101865 - 14 Oct 2011
Cited by 29 | Viewed by 7387
Abstract
Over the last couple of decades nonextensive Tsallis entropy has shown remarkable applicability to describe nonequilibrium physical systems with large variability and multifractal structure. Herein, we review recent results from the application of Tsallis statistical mechanics to the detection of dynamical changes related [...] Read more.
Over the last couple of decades nonextensive Tsallis entropy has shown remarkable applicability to describe nonequilibrium physical systems with large variability and multifractal structure. Herein, we review recent results from the application of Tsallis statistical mechanics to the detection of dynamical changes related with the occurrence of magnetic storms. We extend our review to describe attempts to approach the dynamics of magnetic storms and solar flares by means of universality through Tsallis statistics. We also include a discussion of possible implications on space weather forecasting efforts arising from the verification of Tsallis entropy in the complex system of the magnetosphere. Full article
(This article belongs to the Special Issue Tsallis Entropy)
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Figure 1

Figure 1
<p>From top to bottom are shown <math display="inline"> <msub> <mi>D</mi> <mrow> <mi>s</mi> <mi>t</mi> </mrow> </msub> </math> time series along with time variations of Shannon entropies and Tsallis entropies, <math display="inline"> <msub> <mi>S</mi> <mi>q</mi> </msub> </math>. Tsallis entropies were derived using a <span class="html-italic">q</span> parameter value of 1.84. The 31 March and 6 November 2001 magnetic storms are marked with red. The triangles denote the time intervals corresponding to the five time windows used for the calculations presented in <a href="#entropy-13-01865-f002" class="html-fig">Figure 2</a>. The red dashed line in the <math display="inline"> <msub> <mi>S</mi> <mi>q</mi> </msub> </math> plot marks a possible boundary value (0.7) for the transition of the system to the lower complexity, which is characteristic of the different state of the magnetosphere.</p> Full article ">Figure 2
<p>The normalized Tsallis entropies <math display="inline"> <msub> <mi>S</mi> <mi>q</mi> </msub> </math> calculated at the five time windows, derived after the initial <math display="inline"> <msub> <mi>D</mi> <mrow> <mi>s</mi> <mi>t</mi> </mrow> </msub> </math> time series was divided into five shorter time intervals as shown in <a href="#entropy-13-01865-f001" class="html-fig">Figure 1</a>, for ten different values of the entropic index <span class="html-italic">q</span> (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>.</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>.</mo> <mn>5</mn> <mo>,</mo> <mn>1</mn> <mo>.</mo> <mn>76</mn> <mo>,</mo> <mn>1</mn> <mo>.</mo> <mn>84</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>.</mo> <mn>5</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </math>). On abscissa it is noted the central day of each time window. The arrows point to the Tsallis entropies corresponding to <math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>84</mn> </mrow> </math>.</p> Full article ">Figure 3
<p>We use the Gutenberg-Richter (G-R) type law for the nonextensive Tsallis statistics (Equation 6) to calculate the relative cumulative number of <math display="inline"> <msub> <mi>D</mi> <mrow> <mi>s</mi> <mi>t</mi> </mrow> </msub> </math> data, <math display="inline"> <mrow> <mi>N</mi> <mo>(</mo> <mo>&gt;</mo> <mi>m</mi> <mo>)</mo> <mo>/</mo> <mi>N</mi> </mrow> </math> (upper panel). There is an excellent agreement of the aforementioned formula with the <math display="inline"> <msub> <mi>D</mi> <mrow> <mi>s</mi> <mi>t</mi> </mrow> </msub> </math> time series. The threshold is <math display="inline"> <mrow> <mo>−</mo> <mn>30</mn> </mrow> </math> nT which results in 164 events, and the associated Tsallis entropic parameter is <math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>84</mn> </mrow> </math>.</p> Full article ">Figure 4
<p>GOES-12 5-minute averages X-ray flux, <math display="inline"> <msub> <mi>X</mi> <mi>l</mi> </msub> </math> (1–-8 Angstrom) time series (upper panel). The 20 January 2005 solar flare is marked with red. We use the Gutenberg-Richter (G-R) type law for the nonextensive Tsallis statistics (Equation 6) to calculate the relative cumulative number of X-ray flux data, <math display="inline"> <mrow> <mi>N</mi> <mo>(</mo> <mo>&gt;</mo> <mi>m</mi> <mo>)</mo> <mo>/</mo> <mi>N</mi> </mrow> </math> (lower panel). There is an excellent agreement of the aforementioned formula with the X-ray flux time series. The threshold is 10<math display="inline"> <msup> <mrow/> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </math> W/m<math display="inline"> <msup> <mrow/> <mn>2</mn> </msup> </math> which results in 141 events, and the associated parameter is <math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>82</mn> </mrow> </math>.</p> Full article ">
297 KiB  
Article
Entropy Generation Analysis of Desalination Technologies
by Karan H. Mistry, Ronan K. McGovern, Gregory P. Thiel, Edward K. Summers, Syed M. Zubair and John H. Lienhard V
Entropy 2011, 13(10), 1829-1864; https://doi.org/10.3390/e13101829 - 30 Sep 2011
Cited by 248 | Viewed by 19633
Abstract
Increasing global demand for fresh water is driving the development and implementation of a wide variety of seawater desalination technologies. Entropy generation analysis, and specifically, Second Law efficiency, is an important tool for illustrating the influence of irreversibilities within a system on the [...] Read more.
Increasing global demand for fresh water is driving the development and implementation of a wide variety of seawater desalination technologies. Entropy generation analysis, and specifically, Second Law efficiency, is an important tool for illustrating the influence of irreversibilities within a system on the required energy input. When defining Second Law efficiency, the useful exergy output of the system must be properly defined. For desalination systems, this is the minimum least work of separation required to extract a unit of water from a feed stream of a given salinity. In order to evaluate the Second Law efficiency, entropy generation mechanisms present in a wide range of desalination processes are analyzed. In particular, entropy generated in the run down to equilibrium of discharge streams must be considered. Physical models are applied to estimate the magnitude of entropy generation by component and individual processes. These formulations are applied to calculate the total entropy generation in several desalination systems including multiple effect distillation, multistage flash, membrane distillation, mechanical vapor compression, reverse osmosis, and humidification-dehumidification. Within each technology, the relative importance of each source of entropy generation is discussed in order to determine which should be the target of entropy generation minimization. As given here, the correct application of Second Law efficiency shows which systems operate closest to the reversible limit and helps to indicate which systems have the greatest potential for improvement. Full article
(This article belongs to the Special Issue Entropy Generation Minimization)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>When the control volume is selected suitably far away from the physical system, all inlet and outlet streams are at ambient temperature and pressure. The temperature of the streams inside the control volume, denoted by <math display="inline"> <semantics> <msubsup> <mi>T</mi> <mi>i</mi> <mo>′</mo> </msubsup> </semantics> </math>, might not be at <math display="inline"> <semantics> <msub> <mi>T</mi> <mn>0</mn> </msub> </semantics> </math>.</p> Full article ">Figure 2
<p>Addition of a high temperature reservoir and a Carnot engine to the control volume model shown in <a href="#entropy-13-01829-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 3
<p>Entropy is generated in the process of a stream reaching thermal equilibrium with the environment.</p> Full article ">Figure 4
<p>A typical flow path for a forward feed multiple effect distillation system.</p> Full article ">Figure 5
<p>Entropy production in the various components of a 6 effect forward feed multiple effect distillation system.</p> Full article ">Figure 6
<p>Relative contribution of sources of entropy generation in a forward feed multiple effect distillation system. Irreversibilities in the effects dominate. Total specific entropy generation is 196 J/kg-K.</p> Full article ">Figure 7
<p>A typical flow path for a once-through multistage flash system.</p> Full article ">Figure 8
<p>Sources of entropy generation in a 24 stage once through multistage flash system.</p> Full article ">Figure 9
<p>Relative contribution of sources of entropy generation in a once-through multistage flash system. Irreversibilities in the feed heaters dominate. Total specific entropy generation is 423 J/kg-K.</p> Full article ">Figure 10
<p>Flow path for a basic direct contact membrane distillation system.</p> Full article ">Figure 11
<p>Relative contribution of sources of entropy generation in a direct contact membrane distillation system. Total specific entropy generation is 925.4 J/kg-K.</p> Full article ">Figure 12
<p>Single effect mechanical vapor compression process.</p> Full article ">Figure 13
<p>Relative contribution of sources of entropy generation in a mechanical vapor compression system. Total specific entropy generation is 98.0 J/kg-K. Contributions of the temperature disequilibrium of the distillate and brine streams are 0.5% and 0.2%, respectively.</p> Full article ">Figure 14
<p>A typical flow path for a single stage reverse osmosis system.</p> Full article ">Figure 15
<p>Relative contribution of sources to entropy generation in the reverse osmosis system. Irreversibilities associated with product flow through the membrane dominates. Total specific entropy generation is 19.4 J/kg-K.</p> Full article ">Figure 16
<p>A schematic diagram of a closed air open water, water heated humidification-dehumidification desalination cycle.</p> Full article ">Figure 17
<p>Relative contribution of sources to entropy generation in the closed air open water, water heated humidification-dehumidification system. Irreversibilities in the dehumidifier dominate. Total specific entropy generation is 370 J/kg-K.</p> Full article ">Figure 18
<p>GOR versus Second Law efficiency for closed air open water humidification-dehumidification cycle configurations analyzed by Mistry <span class="html-italic">et al.</span> [<a href="#B6-entropy-13-01829" class="html-bibr">6</a>]. The original data, <a href="#entropy-13-01829-f018" class="html-fig">Figure 18</a>a ([<a href="#B6-entropy-13-01829" class="html-bibr">6</a>], <a href="#entropy-13-01829-f010" class="html-fig">Figure 10</a>), shows no correlation between GOR and the old definition of <math display="inline"> <semantics> <msub> <mi>η</mi> <mi mathvariant="italic">II</mi> </msub> </semantics> </math>. <a href="#entropy-13-01829-f018" class="html-fig">Figure 18</a>b shows that using a minimum least work of separation based definition for Second Law efficiency results in a positive correlation between the energetic performance (GOR) and Second Law performance (<math display="inline"> <semantics> <msub> <mi>η</mi> <mi mathvariant="italic">II</mi> </msub> </semantics> </math>) of the cycles.</p> Full article ">Figure 19
<p>Second Law efficiencies calculated for the systems modeled in this paper. Reverse osmosis has a substantially higher Second Law efficiency than the other desalination processes considered in this paper.</p> Full article ">Figure 20
<p>The least work of separation is minimized when the recovery ratio approaches zero.</p> Full article ">
10102 KiB  
Article
Tsallis Entropy for Geometry Simplification
by Pascual Castelló, Carlos González, Miguel Chover, Mateu Sbert and Miquel Feixas
Entropy 2011, 13(10), 1805-1828; https://doi.org/10.3390/e13101805 - 29 Sep 2011
Cited by 3 | Viewed by 7528
Abstract
This paper presents a study and a comparison of the use of different information-theoretic measures for polygonal mesh simplification. Generalized measures from Information Theory such as Havrda–Charvát–Tsallis entropy and mutual information have been applied. These measures have been used in the error metric [...] Read more.
This paper presents a study and a comparison of the use of different information-theoretic measures for polygonal mesh simplification. Generalized measures from Information Theory such as Havrda–Charvát–Tsallis entropy and mutual information have been applied. These measures have been used in the error metric of a surfaces implification algorithm. We demonstrate that these measures are useful for simplifying three-dimensional polygonal meshes. We have also compared these metrics with the error metrics used in a geometry-based method and in an image-driven method. Quantitative results are presented in the comparison using the root-mean-square error (RMSE). Full article
(This article belongs to the Special Issue Tsallis Entropy)
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Figure 1

Figure 1
<p>Simplification example. (a) Original Model. 49694 triangles. (b) Model simplified to 12% with our algorithm using TVMI (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>). 6285 triangles.</p> Full article ">Figure 2
<p>The two possible half edge collapses for the edge highlighted with a thicker line. Triangles in grey will be removed.</p> Full article ">Figure 3
<p>Galo model. (a) Original model. T = 6592. (b) QSlim. T = 600. (c) IDS. T = 600. (d) TVE(<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>). T = 600. (e) TVMI(<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>). T = 600. T indicates the number of triangles. Different approximations of Galo model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a). TVE and TVMI preserve the comb and tail better than QSlim and IDS.</p> Full article ">Figure 4
<p>RMSE for Galo model. (a) RMSE <span class="html-italic">vs.</span> different alpha values. T = 600. (b) Decimation %. High percentage values indicate that the model has been simplified slightly. Low values correspond to a very coarse model.</p> Full article ">Figure 5
<p>Skull model. (a) Original model. T = 9934. (b) QSlim. T = 1784. (c) IDS. T = 1784. (d) TVE (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>). T = 1783. (e) TVMI (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>). T = 1784. Different approximations of Skull model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a).</p> Full article ">Figure 6
<p>Close-ups of Skull model. (a) Original model. (b) QSlim. T = 1784. RMSE = 47.58. (c) IDS. T = 1784. RMSE = 42.05. (d) TVE (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>). T = 1783. RMSE = 42.53. (e) TVMI (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math>). T = 1784. RMSE = 40.24. These images show that the region around the mouth, especially the teeth in the lower junk, is preserved better in TVMI, IDS and TVE than in QSlim. In the bottom row, difference images are shown. These difference images were produced by superimposing the simplified image over the original image. Here black signifies no difference, while red corresponds to maximum difference.</p> Full article ">Figure 7
<p>RMSE for Skull model. (a) RMSE <span class="html-italic">vs.</span> different alpha values. T = 1784. (b) Decimation %.</p> Full article ">Figure 7 Cont.
<p>RMSE for Skull model. (a) RMSE <span class="html-italic">vs.</span> different alpha values. T = 1784. (b) Decimation %.</p> Full article ">Figure 8
<p>Brush model. (a) Original model. T = 20698. (b) QSlim. T = 1200. (c) IDS. T = 1199. (d) TVE (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>). T = 1199. (e) TVMI (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math>). T = 1199. Different approximations of Brush model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a). The results show that our measures (TVE and TVMI) are capable of retaining more polygons in the brush pins than IDS and QSlim.</p> Full article ">Figure 8 Cont.
<p>Brush model. (a) Original model. T = 20698. (b) QSlim. T = 1200. (c) IDS. T = 1199. (d) TVE (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>). T = 1199. (e) TVMI (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math>). T = 1199. Different approximations of Brush model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a). The results show that our measures (TVE and TVMI) are capable of retaining more polygons in the brush pins than IDS and QSlim.</p> Full article ">Figure 9
<p>RMSE for Brush model. (a) RMSE <span class="html-italic">vs.</span> different alpha values. T = 1200. (b) Decimation %.</p> Full article ">Figure 9 Cont.
<p>RMSE for Brush model. (a) RMSE <span class="html-italic">vs.</span> different alpha values. T = 1200. (b) Decimation %.</p> Full article ">Figure 10
<p>Junk model. (a) Original model. T = 61242. (b) QSlim. T = 6212. (c) IDS. T = 6211. (d) TVE (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>). T = 6218. (e) TVMI (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math>). T = 6219. Different approximations of Junk model obtained with QSlim, IDS, TVE and TVMI. The original model is shown in (a). All the visual simplifications (IDS, TVE and TVMI) preserve the ropes better than the purely geometric simplification (QSlim). The silhouette of the model is retained better with TVE and TVMI than with IDS, see for example the sail at the ship’s stern in (c).</p> Full article ">Figure 11
<p>Close-ups of Junk model. (a) Original model. (b) QSlim. T = 6212. RMSE = 30.15. (c) IDS. T = 6211. RMSE = 28.18. (d) TVE (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>). T = 6218. RMSE = 27.46. (e) TVMI (<math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math>). T = 6219. RMSE = 26.59. The ropes, some masts and poles are retained better in TVE, TVMI and IDS than in QSlim. TVE (see (e)) achieves an improvement about 6% over IDS (see (c)) and over 14% over QSlim (see (b)).</p> Full article ">Figure 12
<p>RMSE for Junk model. (a) RMSE <span class="html-italic">vs.</span> different alpha values. T = 6212. (b) Decimation %.</p> Full article ">Figure 12 Cont.
<p>RMSE for Junk model. (a) RMSE <span class="html-italic">vs.</span> different alpha values. T = 6212. (b) Decimation %.</p> Full article ">
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Review
The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks
by Constantino Tsallis
Entropy 2011, 13(10), 1765-1804; https://doi.org/10.3390/e13101765 - 28 Sep 2011
Cited by 161 | Viewed by 12536
Abstract
The nonadditive entropy Sq has been introduced in 1988 focusing on a generalization of Boltzmann–Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the [...] Read more.
The nonadditive entropy Sq has been introduced in 1988 focusing on a generalization of Boltzmann–Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that Sq has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, we exhibit recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results. Full article
(This article belongs to the Special Issue Tsallis Entropy)
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<p>Dependence of <math display="inline"> <semantics> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </semantics> </math> on the central charge <span class="html-italic">c</span> of pure [<a href="#B18-entropy-13-01765" class="html-bibr">18</a>] and random [<a href="#B19-entropy-13-01765" class="html-bibr">19</a>] one-dimensional magnets undergoing quantum phase transitions at zero temperature, where the entire strongly entangled <span class="html-italic">N</span>-system is in its ground state (hence corresponding to a vanishing entropy since the ground state is a <span class="html-italic">pure</span> state), in contrast with the <span class="html-italic">L</span>-subsystem which is in a <span class="html-italic">mixed</span> state (hence corresponding to a nonvanishing entropy). For this value of <span class="html-italic">q</span>, the block <span class="html-italic">nonadditive</span> entropy <math display="inline"> <semantics> <msub> <mi>S</mi> <mi>q</mi> </msub> </semantics> </math> is <span class="html-italic">extensive</span>, whereas its <span class="html-italic">additive</span> BG entropy is <span class="html-italic">nonextensive</span>. Notice that, for the pure magnet, we have that <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics> </math>, whereas, for the random magnet, we have that <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mo>-</mo> <mi>∞</mi> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </mrow> </semantics> </math>. Both cases recover, in the <math display="inline"> <semantics> <mrow> <mi>c</mi> <mo>→</mo> <mi>∞</mi> </mrow> </semantics> </math> limit, the BG value <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>. These examples definitively clarify that additivity and extensivity are different properties. The only reason for which they have been confused (and still are confused in the mind of not few scientists!) is the fact that, during 140 years, the systems that have been addressed are simple, and not complex, thermodynamically speaking. For such non-pathological systems, the additive BG entropy happens to be extensive, and is naturally the one that should be used.</p> Full article ">Figure 2
<p>Examples of <math display="inline"> <semantics> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mi>q</mi> <mrow> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> </semantics> </math> in the BG scenario (in <span class="html-italic">blue</span>), in the nonextensive scenario (in <span class="html-italic">red</span>), and in a further generalized scenario (in <span class="html-italic">green</span>). By Rodriguez–Schwammle–Tsallis and Hanel–Thurner–Tsallis we respectively mean [<a href="#B47-entropy-13-01765" class="html-bibr">47</a>] and [<a href="#B48-entropy-13-01765" class="html-bibr">48</a>]; by Caruso-Tsallis and Saguia–Sarandy we respectively mean [<a href="#B18-entropy-13-01765" class="html-bibr">18</a>] and [<a href="#B19-entropy-13-01765" class="html-bibr">19</a>] (in this two cases, <math display="inline"> <semantics> <msub> <mi>q</mi> <mrow> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> </semantics> </math> refers to the distributions of energies rather than to distributions of momenta); by Tsallis–Gell–Mann–Sato, Moyano–Tsallis–Gell–Mann, Marsh–Fuentes–Moyano–Tsallis, Thistleton–Marsh–Nelson–Tsallis we mean [<a href="#B17-entropy-13-01765" class="html-bibr">17</a>], [<a href="#B49-entropy-13-01765" class="html-bibr">49</a>], [<a href="#B51-entropy-13-01765" class="html-bibr">51</a>], [<a href="#B50-entropy-13-01765" class="html-bibr">50</a>]; by Tsallis we mean [<a href="#B10-entropy-13-01765" class="html-bibr">10</a>].</p> Full article ">Figure 3
<p>Examples of algebras connecting the <span class="html-italic">q</span> indices corresponding to different properties. The <span class="html-italic">left</span> figure corresponds to Equation (<a href="#FD38-entropy-13-01765" class="html-disp-formula">38</a>); the three black dots correspond to the possible identification proposed in [<a href="#B17-entropy-13-01765" class="html-bibr">17</a>] for the <span class="html-italic">q</span>-triplet observed in the solar wind [<a href="#B67-entropy-13-01765" class="html-bibr">67</a>], namely <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi>q</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>n</mi> <mi>s</mi> <mi>i</mi> <mi>t</mi> <mi>i</mi> <mi>v</mi> <mi>i</mi> <mi>t</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow> <mi>s</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mi>a</mi> <mi>r</mi> <mi>y</mi> <mspace width="0.166667em"/> <mi>s</mi> <mi>t</mi> <mi>a</mi> <mi>t</mi> <mi>e</mi> <mo>,</mo> </mrow> </msub> <msub> <mi>q</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> <mi>a</mi> <mi>x</mi> <mi>a</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> <mo>/</mo> <mn>4</mn> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> (see also [<a href="#B10-entropy-13-01765" class="html-bibr">10</a>]). The <span class="html-italic">right</span> figure corresponds to Equation (<a href="#FD39-entropy-13-01765" class="html-disp-formula">39</a>).</p> Full article ">Figure 4
<p>View of the <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> parameter region <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <mi>N</mi> </mrow> </semantics> </math> versus <math display="inline"> <semantics> <msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>-</mo> <msub> <mi>a</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> <mi>s</mi> </msup> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>≃</mo> <mn>0</mn> <mo>.</mo> <mn>9</mn> </mrow> </semantics> </math>. Typical values of <span class="html-italic">a</span> are shown, and their conveniently scaled corresponding values of <math display="inline"> <semantics> <msup> <mi>N</mi> <mo>*</mo> </msup> </semantics> </math>. Any other possible choices for <math display="inline"> <semantics> <msup> <mi>N</mi> <mo>*</mo> </msup> </semantics> </math> yield lines that remain between the lines with the largest and the smallest slopes shown in the figure. If one approaches the critical point (origin) along any of these lines in this region, the probability distribution function appears to gradually approach, excepting for a small oscillating contribution, a <span class="html-italic">q</span>-Gaussian. The regions at the left of the largest slope and at the right of the smallest slope are not accessible as far as <math display="inline"> <semantics> <msup> <mi>N</mi> <mo>*</mo> </msup> </semantics> </math> values are concerned. Four <span class="html-italic">q</span>-Gaussian examples are presented in <a href="#entropy-13-01765-f005" class="html-fig">Figure 5</a>; the almost vertical line corresponds to a very peaked distribution, and the almost horizontal line corresponds to a Gaussian. See further details in [<a href="#B82-entropy-13-01765" class="html-bibr">82</a>].</p> Full article ">Figure 5
<p>Data collapse of distribution probabilities of sums for the cases <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mi>N</mi> <mo>*</mo> </msup> <mo>=</mo> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> </semantics> </math> for four representative examples. The <span class="html-italic">q</span>-Gaussians appear to be approached through finite-<span class="html-italic">N</span> effects (see also <a href="#entropy-13-01765-f004" class="html-fig">Figure 4</a>). See further details in [<a href="#B82-entropy-13-01765" class="html-bibr">82</a>].</p> Full article ">Figure 5 Cont.
<p>Data collapse of distribution probabilities of sums for the cases <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <msup> <mi>N</mi> <mo>*</mo> </msup> <mo>=</mo> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> </semantics> </math> for four representative examples. The <span class="html-italic">q</span>-Gaussians appear to be approached through finite-<span class="html-italic">N</span> effects (see also <a href="#entropy-13-01765-f004" class="html-fig">Figure 4</a>). See further details in [<a href="#B82-entropy-13-01765" class="html-bibr">82</a>].</p> Full article ">Figure 6
<p>The parameters <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>β</mi> <mo>)</mo> </mrow> </semantics> </math> corresponding to seven <span class="html-italic">q</span>-Gaussians (four of them are those indicated in <a href="#entropy-13-01765-f005" class="html-fig">Figure 5</a>). These specific seven examples appear to exclude the value <math display="inline"> <semantics> <mrow> <mn>2</mn> <mo>-</mo> <msub> <mi>q</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>7555</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math>, which could have been a plausible result. At the present numerical precision, even if quite high, it is not possible to infer whether the analytical result corresponding to the present observations would be only one or a set of <span class="html-italic">q</span>-Gaussians, assuming that exact <span class="html-italic">q</span>-Gaussians are involved, on top of which a small oscillating component possibly exists. See further details in [<a href="#B82-entropy-13-01765" class="html-bibr">82</a>].</p> Full article ">Figure 7
<p>Fraction of points <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>≡</mo> <mi>N</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>/</mo> <mi>N</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics> </math> remaining in the <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> system versus time using a trap fraction occupation <math display="inline"> <semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>6</mn> <mo>/</mo> <mn>7</mn> <mo>∼</mo> <mn>0</mn> <mo>.</mo> <mn>86</mn> </mrow> </semantics> </math>; <math display="inline"> <semantics> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </semantics> </math> uniformly taken within the interval <math display="inline"> <semantics> <mrow> <mo>[</mo> <mn>1</mn> <mo>-</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math>. The straight line fit shows an escape parameter <math display="inline"> <semantics> <mrow> <msub> <mi>γ</mi> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>s</mi> <mi>c</mi> </mrow> </msub> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>216</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math>, numerically consistent with the theoretical one <math display="inline"> <semantics> <mrow> <msub> <mi>γ</mi> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>s</mi> <mi>c</mi> </mrow> </msub> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>2223</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math>. See further details in [<a href="#B86-entropy-13-01765" class="html-bibr">86</a>].</p> Full article ">Figure 8
<p>Sensitivity to initial condition <span class="html-italic">versus</span> entropy production, for typical values of <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> </mrow> </semantics> </math>. For <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics> </math>: <math display="inline"> <semantics> <mrow> <msub> <mi>K</mi> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mspace width="0.166667em"/> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>d</mi> </mrow> </msub> </msub> <mo>=</mo> <msub> <mi>λ</mi> <msub> <mi>q</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>n</mi> </mrow> </msub> </msub> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>32</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math>, and <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mspace width="0.166667em"/> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>q</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>244</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math>; for <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>z</mi> <mo>,</mo> <mi>δ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>6</mn> <mo>/</mo> <mn>7</mn> <mo>)</mo> </mrow> </semantics> </math>: <math display="inline"> <semantics> <mrow> <msub> <mi>γ</mi> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>s</mi> <mi>c</mi> </mrow> </msub> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>222</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msub> <mi>K</mi> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mspace width="0.166667em"/> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>d</mi> </mrow> </msub> </msub> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>1012</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mspace width="0.166667em"/> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>0919</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> </semantics> </math>. Similar results are obtained for the other values of <span class="html-italic">z</span>. The continuous line corresponds to a fit with a slope 1.004..., numerically very close to unity, as expected. These examples neatly illustrate the validity of Equation (<a href="#FD57-entropy-13-01765" class="html-disp-formula">57</a>): the ordinate corresponds to <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <msub> <mi>q</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>n</mi> </mrow> </msub> </msub> <mo>-</mo> <msub> <mi>γ</mi> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>s</mi> <mi>c</mi> </mrow> </msub> </msub> <mo>)</mo> <mspace width="3.33333pt"/> <mi>t</mi> </mrow> </semantics> </math>, and the abscissa corresponds to <math display="inline"> <semantics> <mrow> <msub> <mi>K</mi> <msub> <mi>q</mi> <mrow> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mspace width="0.166667em"/> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>d</mi> </mrow> </msub> </msub> <mspace width="3.33333pt"/> <mi>t</mi> </mrow> </semantics> </math>. See further details in [<a href="#B86-entropy-13-01765" class="html-bibr">86</a>].</p> Full article ">Figure 9
<p>Structure of phase space plots of the MacMillan map for <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>μ</mi> <mo>,</mo> <mi>ϵ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>1</mn> <mo>.</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>.</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math>, starting from a randomly chosen initial condition in a square <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math>, and for <span class="html-italic">N</span> iterates. See further details in [<a href="#B90-entropy-13-01765" class="html-bibr">90</a>].</p> Full article ">Figure 10
<p>Probability distributions of the rescaled sums of iterates of the MacMillan map for <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>μ</mi> <mo>,</mo> <mi>ϵ</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>1</mn> <mo>.</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>.</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> are seen to converge to a <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>6</mn> <mo>)</mo> </mrow> </semantics> </math>-Gaussian. This is shown in the left panel for the central part of the distribution (for <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>&lt;</mo> <msup> <mn>2</mn> <mn>18</mn> </msup> </mrow> </semantics> </math>), and in the right panel for the tail part (<math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>&gt;</mo> <msup> <mn>2</mn> <mn>18</mn> </msup> </mrow> </semantics> </math>). The number of initial conditions that have been randomly chosen from a square <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> are indicated as well. See further details in [<a href="#B90-entropy-13-01765" class="html-bibr">90</a>].</p> Full article ">Figure 11
<p>Illustrative classical <span class="html-italic">N</span>-body <span class="html-italic">d</span>-dimensional models, with a two-body interaction which asymptotically decays as a <math display="inline"> <semantics> <mrow> <mo>-</mo> <mn>1</mn> <mo>/</mo> <msup> <mi>r</mi> <mi>α</mi> </msup> </mrow> </semantics> </math> attractive potential (<math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math>). The total energy is <span class="html-italic">extensive</span> (<span class="html-italic">i.e.</span>, it grows like <span class="html-italic">N</span>) if <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mi>d</mi> </mrow> </semantics> </math> (<span class="html-italic">short-range</span> interactions), and <span class="html-italic">nonextensive</span> (<span class="html-italic">i.e.</span>, it grows faster than <span class="html-italic">N</span>) if <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mi>d</mi> </mrow> </semantics> </math> (<span class="html-italic">long-range</span> interactions). The HMF model corresponds to the <span class="html-italic">α</span>-XY inertial ferromagnetic model [<a href="#B24-entropy-13-01765" class="html-bibr">24</a>] with <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>. The dotted line <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> corresponds to the <span class="html-italic">d</span>-dimensional classical gravitation (assuming a physical cut-off at very short distances, in order to avoid further nonintegrability, coming from the short-distance limit). The dashed vertical line corresponds to the <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math><span class="html-italic">α</span>-XY model. Although not indicated in this figure, the Lennard–Jones model for fluids corresponds to the extensive region since <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics> </math>. It is necessary to have <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>/</mo> <mi>d</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </semantics> </math> for the BG partition function to be <span class="html-italic">finite</span>. This is however not sufficient in general for the preferred collective stationary state to be BG thermal equilibrium. Indeed, ergodicity in the entire Γ phase-space (or in a symmetry-broken part of it) is necessary as well.</p> Full article ">Figure 12
<p>An example of robust <span class="html-italic">q</span>-Gaussian momenta distribution (<math display="inline"> <semantics> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> <mo>∝</mo> <msubsup> <mi>e</mi> <mi>q</mi> <mrow> <mrow> <mo>-</mo> <mi>β</mi> </mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </msubsup> </mrow> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>≃</mo> <mn>1</mn> <mo>.</mo> <mn>6</mn> </mrow> </semantics> </math>) associated with a single initial condition [<a href="#B94-entropy-13-01765" class="html-bibr">94</a>]. As time flows after the so-called quasi-stationary state (QSS) is leaved towards the regime whose temperature is that of the analytical BG result, the value of <span class="html-italic">β</span> can change, but not the value of <span class="html-italic">q</span> (within the error bar).</p> Full article ">Figure 13
<p>Time-averaged (centered and rescaled) distribution of velocities in the Fermi–Pasta–Ulam <span class="html-italic">β</span> model in the neighborhood of the <span class="html-italic">π</span>-mode for <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>128</mn> </mrow> </semantics> </math>, <span class="html-italic">ϵ</span> characterizing the distance of a single trajectory to the <span class="html-italic">π</span>-mode. <span class="html-italic">Top panel:</span> For <math display="inline"> <semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math> the largest Lyapunov exponent is relatively large, and the corresponding distributions are Gaussians (continuous lines). <span class="html-italic">Bottom panel:</span>: For <math display="inline"> <semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>006</mn> </mrow> </semantics> </math>, the largest Lyapunov exponent is close to zero, and the corresponding distribution approaches a <math display="inline"> <semantics> <msub> <mi>q</mi> <mrow> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> </semantics> </math>-Gaussian with <math display="inline"> <semantics> <mrow> <msub> <mi>q</mi> <mrow> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>r</mi> </mrow> </msub> <mo>≃</mo> <mn>1</mn> <mo>.</mo> <mn>5</mn> </mrow> </semantics> </math> (to enable a comparison, a Gaussian distribution is indicated as well). Further details in [<a href="#B95-entropy-13-01765" class="html-bibr">95</a>].</p> Full article ">Figure 14
<p>Distributions of velocities for Kuramoto model with <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>20000</mn> </mrow> </semantics> </math> oscillators. <span class="html-italic">Left panels:</span> When the model parameter <span class="html-italic">K</span> equals 2.53, the LLE is close to 0.06, and the distribution approaches a Gaussian. <span class="html-italic">Right panels:</span> When the model parameter <span class="html-italic">K</span> equals 0.6, the LLE nearly vanishes, and the distribution approaches a <span class="html-italic">q</span>-Gaussian with <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>7</mn> </mrow> </semantics> </math>. Further details in [<a href="#B97-entropy-13-01765" class="html-bibr">97</a>].</p> Full article ">
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Article
Projective Power Entropy and Maximum Tsallis Entropy Distributions
by Shinto Eguchi, Osamu Komori and Shogo Kato
Entropy 2011, 13(10), 1746-1764; https://doi.org/10.3390/e13101746 - 26 Sep 2011
Cited by 22 | Viewed by 9144
Abstract
We discuss a one-parameter family of generalized cross entropy between two distributions with the power index, called the projective power entropy. The cross entropy is essentially reduced to the Tsallis entropy if two distributions are taken to be equal. Statistical and probabilistic properties [...] Read more.
We discuss a one-parameter family of generalized cross entropy between two distributions with the power index, called the projective power entropy. The cross entropy is essentially reduced to the Tsallis entropy if two distributions are taken to be equal. Statistical and probabilistic properties associated with the projective power entropy are extensively investigated including a characterization problem of which conditions uniquely determine the projective power entropy up to the power index. A close relation of the entropy with the Lebesgue space Lp and the dual Lq is explored, in which the escort distribution associates with an interesting property. When we consider maximum Tsallis entropy distributions under the constraints of the mean vector and variance matrix, the model becomes a multivariate q-Gaussian model with elliptical contours, including a Gaussian and t-distribution model. We discuss the statistical estimation by minimization of the empirical loss associated with the projective power entropy. It is shown that the minimum loss estimator for the mean vector and variance matrix under the maximum entropy model are the sample mean vector and the sample variance matrix. The escort distribution of the maximum entropy distribution plays the key role for the derivation. Full article
(This article belongs to the Special Issue Tsallis Entropy)
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Figure 1
<p>t-distribution <math display="inline"> <mrow> <mo>(</mo> <mi>γ</mi> <mo>=</mo> <mo>-</mo> <mn>0</mn> <mo>.</mo> <mn>4</mn> <mo>)</mo> </mrow> </math>, Gaussian <math display="inline"> <mrow> <mo>(</mo> <mi>γ</mi> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </math> and Wigner <math display="inline"> <mrow> <mo>(</mo> <mi>γ</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </math> distributions.</p> Full article ">Figure 1 Cont.
<p>t-distribution <math display="inline"> <mrow> <mo>(</mo> <mi>γ</mi> <mo>=</mo> <mo>-</mo> <mn>0</mn> <mo>.</mo> <mn>4</mn> <mo>)</mo> </mrow> </math>, Gaussian <math display="inline"> <mrow> <mo>(</mo> <mi>γ</mi> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </math> and Wigner <math display="inline"> <mrow> <mo>(</mo> <mi>γ</mi> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </math> distributions.</p> Full article ">
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