# Entropy

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^{4}

## Definition

**:**

## 1. Introduction

## 2. Basics

#### 2.1. Definitions and Properties of Entropy

#### 2.2. Additivity versus Extensivity

- Exponential class $W\left(N\right)\sim A{\mu}^{N}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(A>0;\phantom{\rule{0.166667em}{0ex}}\mu >1)$:This is the typical case within the BG theory. We have ${S}_{BG}\left(N\right)=klnW\left(N\right)\sim Nln\mu +lnA\propto N$, therefore ${S}_{BG}$ is extensive, as thermodynamically required.
- Power-law class $W\left(N\right)\sim B{N}^{\rho}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(B>0;\phantom{\rule{0.166667em}{0ex}}\rho >0)$:We should not use ${S}_{BG}$ since it implies ${S}_{BG}\left(N\right)=klnW\left(N\right)\sim \rho lnN+lnB\propto lnN$, thus violating thermodynamics. We verify instead that ${S}_{q=1-1/\rho}\left(N\right)=k{ln}_{q=1-1/\rho}W\left(N\right)\propto N$, as thermodynamically required.
- Stretched exponential class $W\left(N\right)\sim C{\nu}^{{N}^{\gamma}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(C>0;\phantom{\rule{0.166667em}{0ex}}\nu >1;\phantom{\rule{0.166667em}{0ex}}0<\gamma <1)$:In this instance, no value of q exists which would imply an extensive entropy ${S}_{q}$. We can instead used ${S}_{\delta}$ with $\delta =1/\gamma $. Indeed, ${S}_{\delta =1/\gamma}\left(N\right)=k{[lnW\left(N\right)]}^{\delta}\propto N$, as thermodynamically required.
- Logarithmic class $W\left(N\right)\sim DlnN\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}(D>0)$:In this case, no values of $(q,\delta )$ exist which imply an extensive entropy ${S}_{q,\delta}$. Instead, we can use the Curado entropy [33] ${S}_{\lambda}^{C}\left(N\right)=k\left(\right)open="["\; close="]">{e}^{\lambda \phantom{\rule{0.166667em}{0ex}}W\left(N\right)}-{e}^{\lambda}$ with $\lambda =1/D$. Indeed, we can verify that ${S}_{\lambda =1/D}^{C}\left(N\right)\propto N$, as thermodynamically required.

#### 2.3. Range of Interactions

#### 2.4. Thermodynamics and Legendre Transformations

#### 2.5. Classification of Entropic Functionals

#### 2.6. Boltzmann–Gibbs and Nonextensive Statistical Mechanics

#### 2.6.1. Boltzmann–Gibbs Statistical Mechanics

#### 2.6.2. q-Generalization of the Boltzmann–Gibbs Theory

## 3. Results and Applications

#### 3.1. In Physics

#### 3.1.1. Nonlinear Dynamical Systems

#### 3.1.2. First-Principle Calculation of q for a Quantum Hamiltonian System

#### 3.1.3. Long-Range Interactions

#### 3.1.4. Overdamped Many-Body Systems

#### 3.1.5. Low Energy Physics

#### 3.1.6. High Energy Physics

#### 3.2. Beyond Physics

#### 3.2.1. Mathematics

#### 3.2.2. Chemistry

#### 3.2.3. Economics

#### 3.2.4. Biology and Life Sciences

#### 3.2.5. Computer Sciences

#### 3.2.6. Random Networks

#### 3.2.7. Image and Signal Processing

#### 3.2.8. Engineering

## 4. Final Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## Entry Link on the Encyclopedia Platform

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**Figure 1.**Typical behaviors of $W\left(N\right)$ (number of nonzero-probability states of a system with N random variables) in the $N\to \infty $ limit and entropic functionals which, under the assumption of equal probabilities for all states with nonzero probability, yield extensive entropies for specific values of the corresponding (nonadditive) entropic indices. In what concerns the exponential class $W\left(N\right)\sim A{\mu}^{N}$, ${S}_{BG}$ is not the unique entropy that yields entropic extensivity; the (additive) Renyi entropic functional ${S}_{q}^{R}$ also is extensive for all values of q. Analogously, in what concerns the stretched-exponential class $W\left(N\right)\sim C{\nu}^{{N}^{\gamma}}$, the (nonadditive) entropic functional ${S}_{\delta}$ is not unique. All the entropic families illustrated in this table contain ${S}_{BG}$ as a particular case, except ${S}_{\lambda}^{C}$, which is appropriate for the logarithmic class $W\left(N\right)\sim DlnN$. In the limit $N\to \infty $, the inequalities ${\mu}^{N}\gg {\nu}^{{N}^{\gamma}}\gg {N}^{\rho}\gg lnN\gg 1$ are satisfied, hence ${lim}_{N\to \infty}{\nu}^{{N}^{\gamma}}/{\mu}^{N}={lim}_{N\to \infty}{N}^{\rho}/{\mu}^{N}={lim}_{N\to \infty}lnN/{\mu}^{N}=0$. This exhibits that, in all these nonadditive cases, the occupancy of the full phase space corresponds essentially to zero Lebesgue measure, similarly to a whole class of (multi) fractals. If the equal probabilities hypothesis is not satisfied, specific analysis becomes necessary and the results might be different.

**Figure 2.**The classical systems for which $\alpha /d>1$ correspond to an extensive total energy and typically involve absolutely convergent series, whereas the so-called nonextensive systems ($0\le \alpha /d<1$ for the classical ones) correspond to a superextensive total energy and typically involve divergent series. The marginal systems ($\alpha /d=1$ here) typically involve conditionally convergent series, which therefore depend on the boundary conditions, i.e., typically on the external shape of the system. Capacitors constitute a notorious example of the $\alpha /d=1$ case. The model usually referred to in the literature as the Hamiltonian-Mean-Field (HMF) one [35] lies on the $\alpha =0$ axis (for all values of $d>0$). The models usually referred to as the d-dimensional $\alpha $-XY [36], $\alpha $-Heisenberg [37,38,39] and $\alpha $-Fermi–Pasta–Ulam (or $\alpha $-Fermi–Pasta–Ulam–Tsingou problem [40,41]) [42,43,44] models lie parallel to the vertical axis at abscissa d (for all values of $\alpha \ge 0$). The standard Lennard–Jones gas is located at $(d,\alpha )=(3,6)$. From [33].

**Figure 3.**Schematic representation of the different scaling regimes for classical d-dimensional systems. In the case of attractive long-ranged interactions (i.e., $0\le \alpha /d\le 1$, $\alpha $ characterizing the interaction range in a potential with the form $1/{r}^{\alpha}$; for example, Newtonian gravitation corresponds to $(d,\alpha )=(3,1)$), we may distinguish three classes of thermodynamic variables, namely, those scaling with ${L}^{\theta}$, named pseudo-intensive (L is a characteristic linear length, $\theta $ is a system-dependent parameter), those scaling with ${L}^{d+\theta}$ with $\theta =d-\alpha $, the pseudo-extensive ones (the energies), and those scaling with ${L}^{d}$ (which are always extensive). In the case of short-ranged interactions (i.e., $\alpha >d$), we have $\theta =0$ and the energies recover their standard ${L}^{d}$ extensive scaling, falling within the same class of S, N, V, M, etc., whereas the previous pseudo-intensive variables become truly intensive ones (independent of L); this is the region, with only two classes of variables that is covered by the traditional textbooks of thermodynamics. From [26,30,33,47,48,49].

**Figure 4.**It has been proved [54] that ${S}_{q}$ is the unique entropic form which simultaneously is trace-form, composable, and recovers ${S}_{BG}$ as a particular instance. ${S}_{q}$ (hence ${S}_{BG}$), the Renyi entropy ${S}_{q}^{R}$ [14], the Tempesta $(a,b,\alpha )$-entropy ${S}_{a,b,\alpha}^{T}$ (Equation (9.1) in [57]), the Jensen–Pazuki–Pruessner–Tempesta entropy ${S}_{\gamma ,\alpha}^{JPPT}$ [58] and many others belong to the class of group entropies and are therefore composable. To facilitate the identification, we are here using the following notations: Sharma–Mittal entropy ${S}_{q,r}^{SM}$ [20,21,22], Landsberg–Vedral–Rajagopal–Abe entropy ${S}_{q}^{LVRA}$ [59,60,61], Tsallis–Mendes–Plastino entropy ${S}_{q}^{TMP}$, Arimoto entropy ${S}_{q}^{Ar}$ [62], Curado–Tempesta–Tsallis entropy ${S}_{a,b,r}^{CTT}$ [63], Borges–Roditi entropy ${S}_{q,{q}^{\prime}}^{BR}$ [64], Abe entropy ${S}_{q}^{Ab}$ [65], Kaniadakis entropy ${S}_{\kappa}^{K}$ [66,67,68], Kaniadakis–Lissia–Scarfone entropy ${S}_{\kappa ,r}^{KLS}$ [69], Anteneodo–Plastino entropy ${S}_{\eta}^{AP}$ [70], Hanel–Thurner entropy ${S}_{c,d}^{HT}$ [71,72], ${S}_{q,\delta}$ [30], Schwammle–Tsallis entropy ${S}_{q,{q}^{\prime}}^{ST}$ [73], the Tempesta $(\alpha ,\beta ,q)$-entropy ${S}_{\alpha ,\beta ,q}^{T}$ [74], the Curado b-entropy ${S}_{b}^{C}$ [75,76], the Curado $\lambda $-entropy ${S}_{\lambda}^{C}$ [33], and the exponential c-entropy ${S}_{c}^{E}$ [77]; one more exponential form is, in fact, available in the literature, namely the Pal and Pal non-composable trace-form entropy ${S}^{PP}$ [78]. The entropic form ${S}_{\lambda}^{C}$ is a rare case which does not include ${S}_{BG}$ and is neither trace-form nor composable. From [33].

**Figure 5.**Time evolution of ${S}_{q}$ for $a=2$. The interval $[-1,1]$ is partitioned into W equal cells. The initial distribution consists of $M={10}^{6}$ points placed at random inside a randomly picked cell. We indicate three typical values of q and the two cases $W={10}^{4}$ and $W={10}^{5}$. Results are averages over 100 runs. From [95].

**Figure 6.**Numerical confirmation (full circles) of the q-generalized Pesin-like identity ${K}_{q}^{\left(k\right)}={\lambda}_{q}^{\left(k\right)}$ at the logistic-map edge of chaos. On the ordinate, we plot the q-logarithm of ${\xi}_{{t}_{k}}$ (equal to ${\lambda}_{q}^{\left(k\right)}\phantom{\rule{0.166667em}{0ex}}t$), and, on the abscissa, ${S}_{q}$ (equal to ${K}_{q}^{\left(q\right)}\phantom{\rule{0.166667em}{0ex}}t$), both for $q=0.2445...$ The dashed line is a linear fit. Inset: The full lines are from the analytic result. From [96].

**Figure 7.**Data collapse of probability density functions (in log-linear plots) for $T={2}^{2n}$, where $2n$ is an odd number (

**top**) or an even number (

**bottom**). As n increases, good fits with ${q}_{attractor}$-Gaussian $\propto {e}_{{q}_{attractor}}^{-\beta {y}^{2}}$ (with $({q}_{attractor},\beta )=(1.63,6.2)$ (

**top**), and $({q}_{stat},\beta )=(1.70,6.2)$ (

**bottom**)) are obtained for increasingly large regions. Insets: Linear-linear plots of the data for a better visualization of the central part. From [97].

**Figure 8.**Phase portrait of the standard map for representative values of K. In each case, black dots represent the region of chaotic sea in the available phase space and all other colors represent different stability islands. From [100].

**Figure 9.**Lyapunov exponent results of the phase portrait of the standard map. The same representative K values are used. For each case, Lyapunov exponents are calculated for 200,000 initial conditions. In the calculation, each initial condition is iterated 107 times. From [100].

**Figure 12.**The index ${q}_{entropy}$ as a function of the inverse central charge $1/c$. The universality classes of some specific models are indicated, see [94]. The BG value ${q}_{entropy}=1$ is recovered in the $c\to \infty $ limit.

**Figure 13.**A typical single-initial-condition one-momentum distribution $P\left(p\right)$ for $N={10}^{6}$, $u=0.69$, $\tau =1$ (corresponding to 5 molecular-dynamics steps), calculated in the region $[{t}_{min};{t}_{max}]=[200,000;500,000]$ for $\alpha =0.9$ (top plot), and $\alpha =2.0$ (bottom plot). The upper temperature indicated in the $\alpha =0.9$ inset coincides with that analytically calculated within BG statistical mechanics, namely ${T}_{kin}\equiv 2K\left(t\right)/N\simeq 0.475$. The horizontal line of the $\alpha =2.0$ inset corresponds to the numerically calculated time average; indeed, analytical solutions are only available for $\alpha <1$ and in the $\alpha \to \infty $ limit. The continuous curves correspond to $P\left(\tilde{p}\right)/{P}_{0}={e}_{{q}_{n}}^{-{\beta}_{{q}_{n}}^{\left({P}_{0}\right)}\left[\tilde{p}{P}_{0}\right){]}^{2}/2}$ with $({q}_{n},{\beta}_{{q}_{n}}^{\left({P}_{0}\right)})=(1.58,11.2)$ for $\alpha =0.9$ and (1.0,6.4) for $\alpha =2.0$. Notice that, for $\alpha =0.9$, $1/{\beta}_{{q}_{n}}^{\left({P}_{0}\right)}\ne T$. Each distribution has been rescaled with its own ${P}_{0}$. From [102].

**Figure 14.**Distributions of the time-averages of the momenta ${\overline{p}}_{i}$ and of the energies ${\overline{E}}_{i}$ (with $\tau =1$) for $\alpha /d=0.9$, in $d=1$, 2 and 3 dimensions. The simulations were done for the energy per particle $u=0.69$ and total number of rotators $N={10}^{6}$.

**Top**: Distribution $P\left({\overline{p}}_{i}\right)$ is shown (${P}_{0}\equiv P({\overline{p}}_{i}=0)$), the full line being a q-Gaussian with ${q}_{p}=1.59$ and ${\beta}_{p}=5.6$, and the dashed line being a Gaussian $(q=1)$. The left inset shows the same data in a q-logarithm versus squared-momentum representation; a straight line is obtained as expected (since ${ln}_{q}\left({e}_{q}^{x}\right)=x$).

**Bottom**: The full line represents the q-exponential $P\left({\overline{E}}_{i}\right)=P\left(\mu \right){exp}_{{q}_{E}}[-{\beta}_{E}({\overline{E}}_{i}-\mu )]$, with ${q}_{E}=1.31$ (${\beta}_{E}=48.0$, $\mu =0.69$, and $P\left(\mu \right)=12$); the corresponding exponential (dashed line) is also shown for comparison. Since the density of states is necessary to reproduce the entire range of data, a chemical potential $\mu $ was introduced in the fitting. The bottom inset shows a straight line by using the q-logarithm in the ordinate. The kinetic temperature $T\left(t\right)\equiv 2K\left(t\right)/N$ and time window $\Delta t$, along which the time averages were calculated, coincide in both cases (shown as insets). One notices that, in all plots, the collapse of all dimensions occurs with nearly the same value of q. From [103].

**Figure 15.**Quantum Monte Carlo verification (

**left panels**) [(

**a**) Analytic and numerical distributions; (

**b**) Analytical and numerical functions $q({U}_{0}/{E}_{r})$], and laboratory verification with ${C}_{s}$ atoms (

**right panels**) [(

**Top a**) Analytical and experimental distributions; (

**Bottom b**) Experimental frequency dependence of q; (

**Left a**) Linear-linear representation of the experimental distribution of momenta, the black curve corresponding to the present q-Gaussian, the red curve corresponding to a Maxwellian distribution; (

**Right b**) The same in log-log representation.] of the 2003 Lutz prediction. From [106].

**Figure 16.**Probability density functions of the horizontal components of the randomly fluctuating displacements tracked during two typical increments of shear strain ($\Delta \gamma =7.3\times {10}^{-4}$ and $\Delta \gamma ={10}^{-1}$). The scatters correspond to experimental data, and the solid lines correspond to q-Gaussian fittings. From [110].

**Figure 17.**Evolution of the measured value q as a function of the squared inverse of the strain increment for both the experiments and simulations. The dashed line corresponds to a regression using the function $q(1/\sqrt{\Delta \gamma})=1+atanh(b/\sqrt{\Delta \gamma})$, with $(a,b)=(0.521,0.096)$. Inset: The same plot for data from a simulation that highlights the limit $q=1$ when $\Delta \gamma \to \infty $. The fitted parameters for simulations were $(a,b)=(0.387,0.057)$. From [110].

**Figure 18.**Verification of the scaling law $\alpha =2/(3-q)$ [108] for several regimes of diffusion.

**Top**: Evolution of the measured diffusion exponent $\alpha $ as a function of $1/\sqrt{\Delta \gamma}$; the dashed line is a direct application of the scaling law from the fit of the values shown in Figure 17, $\alpha (1/\sqrt{\Delta \gamma})=2/[3-q(1/\sqrt{\Delta \gamma})]$.

**Inset**: A typical diffusion curve showing the mean square displacement fluctuations $\langle {x}^{2}\rangle $ as a function of the shear strain $\gamma $; this allows the assessment of the diffusion exponent $\alpha \equiv \mu $ for each strain window tested. In the case shown, it corresponds to the smallest strain window, the rightmost point in the curve in the main panel. Note that for a constant strain rate, $\gamma $ is proportional to time.

**Bottom**: Measure of the deviation of the data relative to the scaling law prediction ${\alpha}_{P}=2/(3-q)$, as a function of $1/\sqrt{\Delta \gamma}$, showing a remarkable agreement of the order of $\pm 2\%$. From [110].

**Figure 19.**Experimental transverse momentum distribution of hadrons in $pp$ collisions at central rapidity y compared with theoretical q-exponentials with $q\simeq 1.13\pm 0.02$ and $T\simeq (0.14\pm 0.01)\phantom{\rule{0.166667em}{0ex}}GeV$. The corresponding Boltzmann–Gibbs (purely exponential) fit is illustrated as the dashed curve. The data and the analytical curves have been divided by a constant factor as indicated, for a better visualization. The ratios data/fit are shown at the bottom, where a nearly log-periodic behavior is observed on top of the q-exponential one. See [111] for details.

**Figure 20.**The measured AMS-02 data are very well fitted by linear combination of escort and non-escort distributions (solid lines); ${q}_{1}=13/11$ and ${q}_{2}=1/(2-{q}_{1})=11/9$. From [112].

**Figure 22.**

**Left panel**: Cumulative distributions of absolute normalized returns corresponding to different time scales $\Delta t$ for the 100 American companies with the highest market capitalization (points), and the fitted cumulative q-Gaussian distributions (lines). In order to better visualize the associated results, each q-Gaussian CDF and the respective experimental data have been multiplied by a positive factor $c\ne 1$.

**Right panel**: Dependence of the index q on the time scale $\Delta t$, for the estimated q-Gaussian pdfs of normalized absolute returns in the left panel. Inset: log-log representation exhibiting a power-law dependence of the type $q-1\propto {\left(\Delta t\right)}^{-\tau}$ , with $\tau =0.081\pm 0.004$. From [124].

**Figure 23.**PDFs of the inter-nucleotide intervals A-A, T-T (open symbols); G-G, C-C (full symbols) in the DNA sequences from Homo Sapiens and Bacteria full genomes (in scaled form). Dashed lines show the best fits by a q-exponential distribution $A=1/{[1+(q-1)\beta (l/L)]}^{\frac{1}{q-1}}$. While in Bacteria the approximation by a single q-exponential with $q\sim 1.1$ and $\beta \sim 1.5$ is possible, in H. Sapiens, a sum of two q-exponentials with $q\sim 1.11$ and $\beta \sim 1.5$ and 0.1 makes the best fit. To avoid overlapping, the PDFs for Bacteria are shifted downwards by two decades. For comparison, dotted lines show corresponding exponential PDFs. From [125].

**Figure 24.**Sample of a network with 100 sites for the following model parameters $(d,{\alpha}_{A},{\alpha}_{G},\eta ,{w}_{0})=(2,1,5,1,1)$, where ${\alpha}_{A}$ and ${\alpha}_{G}$ respectively characterize the attachment and geographical ranges; $\eta $ and ${w}_{0}$ are parameters of the distribution of widths. As illustrated for this choice of parameters, hubs (highly connected nodes) naturally emerge in the network. Each link has a specific width ${w}_{ij}$ and the total energy ${\epsilon}_{i}$ associated with the site i will be given by half of the sum over all link widths connected to the site i (see zoom of site i). From [136].

**Figure 25.**In all cases, the energy distribution is well fitted with the form ${p}_{q}\propto {e}_{q}^{-{\beta}_{q}\epsilon}$. (

**a**) q as a function of ${\alpha}_{A}/d$ (black solid line); $q=4/3$ for $0\le {\alpha}_{A}/d\le 1$ and decreases exponentially with ${\alpha}_{A}/d$ for ${\alpha}_{A}/d>1$, down to ${q}_{\infty}=1$. (

**b**) ${\beta}_{q}$ as a function of ${\alpha}_{A}/d$ for $\eta =1,2,3$ and ${w}_{0}=1,5,10$, for typical values of ${\alpha}_{A}/d$; ${\beta}_{q}$ increases with $\eta $ and decreases with ${w}_{0}$ and ${\alpha}_{A}/d$. From [136].

**Figure 26.**Without q-entropy enhancement with $q\ne 1$, detection of microcalcifications is meager: 80.21% Tps (true positives) with 8.1 Fps (false positives), whereas upon introduction of the q-entropy, the results surge to 96.55% Tps with 0.4 Fps. Detection results from the experiment: (

**a**) mdb236, (

**b**) output with the Mcs enhanced, (

**c**) output with the Mcs extracted, (

**d**) mdb216, (

**e**) output with the Mcs enhanced, (

**f**) output with the Mcs extracted. From [137].

**Figure 27.**Sample scans from the dataset before and after enhancement showing infected lungs. (

**a**) Original Computer Tomography scans, with red circles highlighting some regions where fibrosis can be seen; (

**b**) enhanced Computer Tomography scans using $q=0.5$. From [138].

**Figure 28.**Cumulative pdfs of the inter-event times of acoustic emission for the last three loadings of the specimens.

**Left**: C1, C2 (concrete) and

**Right**: B1, B2 (basalt). The q-exponential fittings are also shown. From [139].

**Figure 29.**The values of the entropic index of the q-exponential fits, reproducing the complementary cumulative pdfs obtained from experimental data about the AE inter-event time series for both the concrete (C1, C2—full circles) and basalt (B1, B2—full squares) specimens, are reported as function of $1/{\beta}_{q}$. Linear fits of the reported values are also shown. The macroscopic failure occurs when $1/{\beta}_{q}$ vanishes. The values for q and $1/{\beta}_{q}$ extracted from analogous failure tests with cement mortar specimens in [140] are also reported (triangles). From [139].

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Tsallis, C.
Entropy. *Encyclopedia* **2022**, *2*, 264-300.
https://doi.org/10.3390/encyclopedia2010018

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Tsallis C.
Entropy. *Encyclopedia*. 2022; 2(1):264-300.
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