#
The Nonadditive Entropy S_{q} and Its Applications in Physics and Elsewhere: Some Remarks

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Additivity versus Extensivity

## 3. Probability Distributions that Are Attractors in the Sense of the Central Limit Theorem

## 4. Remarks on Paradigmatic Low-Dimensional Nonlinear Dynamical Systems Near the Edge of Chaos

#### 4.1. One-Dimensional Dissipative Unimodal Maps

#### 4.2. Two-Dimensional Conservative Maps

## 5. Remarks on Paradigmatic Long-Range-Interacting Many-Body Classical Hamiltonian Systems

## 6. Applications

- (i)
- The velocity distribution of (cells of) Hydra viridissima follows a q-Gaussian probability distribution function (PDF) with $q\simeq 3/2$ [101,102]. Anomalous diffusion has been independently measured as well [101,102], and an exponent $\gamma \simeq 4/3$ has been observed (where the squared space ${x}^{2}$ scales with time t like ${t}^{\gamma}$). Therefore, within the error bars, the prediction $\gamma =\frac{2}{3-q}$ [54] is verified in this system.
- (ii)
- The velocity distribution of (cells of) Dictyostelium discoideum is well fitted by a q-Gaussian PDF with $q\simeq 5/3$ in the vegetative state, and with $q\simeq 2$ in the starved state [103].
- (iii)
- The velocity distribution of the point defects in in defect turbulence, as well as its corresponding anomalous diffusion, have been measured [104]. The results suggest a q-Gaussian PDF with $q\simeq 3/2$, and $\gamma \simeq 4/3$, which constitutes another verification of the prediction $\gamma =\frac{2}{3-q}$ [54].
- (iv)
- The velocity distribution of cold atoms in a dissipative optical lattice was predicted [105] to be a q-Gaussian with $q=1+44\frac{{E}_{r}}{{U}_{0}}$, where ${E}_{r}$ and ${U}_{0}$ are parameters related to the optical lattice potential. This prediction was verified three years later, both in the laboratory and with quantum Monte Carlo techniques [106,107].
- (v)
- (vi)
- The velocity distribution in a driven-dissipative 2D dusty plasma was found to be of the q-Gaussian form, with $q=1.08\pm 0.01$ and $q=1.05\pm 0.01$ at temperatures of $30000\phantom{\rule{0.166667em}{0ex}}K$ and $61000\phantom{\rule{0.166667em}{0ex}}K$ respectively [110].
- (vii)
- The spatial (Monte Carlo) distributions of a trapped ${}^{136}B{a}^{+}$ ion cooled by various classical buffer gases at $300\phantom{\rule{0.166667em}{0ex}}K$ was verified to be of the q-Gaussian form, with q increasing from close to unity to about 1.9 when the mass of the molecules of the buffer increases from that of $He$ to about 200 [111].
- (viii)
- The distributions of price returns and stock volumes at the New York and NASDAQ stock exchanges are well fitted by q-Gaussians and q-exponentials respectively [112,113,114,115]. The volatility smiles that are obtained within this approach also are well fitted. The distributions of interoccurrence times between losses in financial markets are consistently well described by universal q-exponentials, which enables a simple analytical expression for the risk function [116].
- (ix)
- The Bak–Sneppen model of biological evolution exhibits a time-dependence of the spread of damage which is well approached by a q-exponential with $q<1$ [117].
- (x)
- (xi)
- The distributions of returns in the coherent noise model are well fitted with q-Gaussians where q is analytically obtained through $q=\frac{2+\tau}{\tau}$, τ being the exponent associated with the distribution of sizes of the events [120].
- (xii)
- (xiii)
- The distributions of angles in the $HMF$ model approaches as time evolves towards a q-Gaussian form with $q\simeq 1.5$ [123].
- (xiv)
- (xv)
- The relaxation in various paradigmatic spin-glass substances through neutron spin echo experiments is well reproduced by q-exponential forms with $q>1$ [126].
- (xvi)
- The fluctuating time dependence of the width of the ozone layer over Buenos Aires (and, presumably, around the Earth) yields a q-triplet with ${q}_{sen}<1<{q}_{stat\phantom{\rule{0.166667em}{0ex}}state}<{q}_{rel}$ [127].
- (xvii)
- (xviii)
- (xix)
- The tissue radiation response follows a q-exponential form [137].
- (xx)
- (xxi)
- Experimental and simulated molecular spectra due to the rotational population in plasmas are frequently interpreted as two Boltzmann distributions corresponding to two different temperatures. These fittings involve three fitting parameters, namely the two temperatures and the relative proportion of each of the Boltzmann weights. It has been shown [141] that equally good fittings can be obtained with a single q-exponential weight, which has only two fitting parameters, namely q and a single temperature.
- (xxii)
- High energy physics has been since more than one decade handled with q-statistics [142,143,144,145]. During the last decade various phenomena, such as the flux of cosmic rays and others, have been shown to exhibit relevant nonextensive aspects [146,147,148,149]. The distributions of transverse momenta of hadronic jets outcoming from proton-proton collisions (as well as others) have been shown to exhibit q-exponentials with $q\simeq 1.1$. These results have been obtained at the LHC detectors CMS, ATLAS and ALICE [150,151,152,153,154,155,156,157], as well as at SPS and RHIC in Brookhaven [158,159]. Predictions for the rapidities in such experiments have been advanced as well [160].
- (xxiii)
- (xxiv)
- (xxv)
- (xxvi)
- Nonlinear generalizations of the Schroedinger, the Klein–Gordon and the Dirac equations have been implemented which admit q-plane wave solutions as free particles, i.e., solutions of the type ${e}_{q}^{i(kx-\omega t)}$ [194], with the energy given by $E=\hslash \omega $ and the momentum given by $\overrightarrow{p}=\hslash \overrightarrow{k}$, $\forall q$. The nonlinear Schroedinger equation yields $E={p}^{2}/2m$ ($\forall q$), and the nonlinear Klein-Gordon and Dirac equations yield the Einstein relation ${E}^{2}={m}^{2}{c}^{4}+{p}^{2}{c}^{2}$ ($\forall q$).
- (xxvii)

## 7. Final Remarks

## Acknowledgments

## References and Notes

- We should, however, always have in mind that we cannot use the continuous expression (3) for states involving too narrow continuous distributions. Indeed, only expressions such as (1) and (4) can guarantee the non-negativeness of a well defined entropy, whose lowest admissible value is zero (corresponding to the system being at its fundamental state, where all information is available). This feature reflects, of course, the fact that nature ultimately is quantum rather than classical. A well known consequence of this question is the fact that classical specific heats do not vanish in the T → 0 limit, in contrast with the quantum ones, which always do.
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**Figure 1.**Dependence of ${q}_{ent}$ on the central charge c of pure [18] and random [19] one-dimensional magnets undergoing quantum phase transitions at zero temperature, where the entire strongly entangled N-system is in its ground state (hence corresponding to a vanishing entropy since the ground state is a pure state), in contrast with the L-subsystem which is in a mixed state (hence corresponding to a nonvanishing entropy). For this value of q, the block nonadditive entropy ${S}_{q}$ is extensive, whereas its additive BG entropy is nonextensive. Notice that, for the pure magnet, we have that ${q}_{ent}\in [0,1]$, whereas, for the random magnet, we have that ${q}_{ent}\in (-\infty ,1]$. Both cases recover, in the $c\to \infty $ limit, the BG value ${q}_{ent}=1$. 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.

**Figure 2.**Examples of ${q}_{ent}$ and ${q}_{attr}$ in the BG scenario (in blue), in the nonextensive scenario (in red), and in a further generalized scenario (in green). By Rodriguez–Schwammle–Tsallis and Hanel–Thurner–Tsallis we respectively mean [47] and [48]; by Caruso-Tsallis and Saguia–Sarandy we respectively mean [18] and [19] (in this two cases, ${q}_{attr}$ 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 [17], [49], [51], [50]; by Tsallis we mean [10].

**Figure 3.**Examples of algebras connecting the q indices corresponding to different properties. The left figure corresponds to Equation (38); the three black dots correspond to the possible identification proposed in [17] for the q-triplet observed in the solar wind [67], namely $({q}_{sensitivity},{q}_{stationary\phantom{\rule{0.166667em}{0ex}}state,}{q}_{relaxation})=(4,7/4,-1/2)$ (see also [10]). The right figure corresponds to Equation (39).

**Figure 4.**View of the $z=2$ parameter region $1/N$ versus ${(a-{a}_{c})}^{s}$ with $s\simeq 0.9$. Typical values of a are shown, and their conveniently scaled corresponding values of ${N}^{*}$. Any other possible choices for ${N}^{*}$ 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 q-Gaussian. The regions at the left of the largest slope and at the right of the smallest slope are not accessible as far as ${N}^{*}$ values are concerned. Four q-Gaussian examples are presented in Figure 5; the almost vertical line corresponds to a very peaked distribution, and the almost horizontal line corresponds to a Gaussian. See further details in [82].

**Figure 6.**The parameters $(q,\beta )$ corresponding to seven q-Gaussians (four of them are those indicated in Figure 5). These specific seven examples appear to exclude the value $2-{q}_{sen}=1.7555...$, 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 q-Gaussians, assuming that exact q-Gaussians are involved, on top of which a small oscillating component possibly exists. See further details in [82].

**Figure 7.**Fraction of points $n\left(t\right)\equiv N\left(0\right)/N\left(t\right)$ remaining in the $z=2$ system versus time using a trap fraction occupation $\delta =6/7\sim 0.86$; $N\left(0\right)={10}^{6}$ uniformly taken within the interval $[1-{10}^{-10},1]$. The straight line fit shows an escape parameter ${\gamma}_{{q}_{esc}}=0.216...$, numerically consistent with the theoretical one ${\gamma}_{{q}_{esc}}=0.2223...$. See further details in [86].

**Figure 8.**Sensitivity to initial condition versus entropy production, for typical values of $(z,\delta )$. For $(z,\delta )=(2,0)$: ${K}_{{q}_{ent\phantom{\rule{0.166667em}{0ex}}prod}}={\lambda}_{{q}_{sen}}=1.32...$, and ${q}_{ent\phantom{\rule{0.166667em}{0ex}}prod}={q}_{sen}=0.244...$; for $(z,\delta )=(2,6/7)$: ${\gamma}_{{q}_{esc}}=0.222...$, ${K}_{{q}_{ent\phantom{\rule{0.166667em}{0ex}}prod}}=1.1012...$ and ${q}_{ent\phantom{\rule{0.166667em}{0ex}}prod}=0.0919...$. Similar results are obtained for the other values of z. 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 (57): the ordinate corresponds to $({\lambda}_{{q}_{sen}}-{\gamma}_{{q}_{esc}})\phantom{\rule{3.33333pt}{0ex}}t$, and the abscissa corresponds to ${K}_{{q}_{ent\phantom{\rule{0.166667em}{0ex}}prod}}\phantom{\rule{3.33333pt}{0ex}}t$. See further details in [86].

**Figure 9.**Structure of phase space plots of the MacMillan map for $(\mu ,\u03f5)=(1.6,1.2)$, starting from a randomly chosen initial condition in a square $(0,{10}^{-6})\times (0,{10}^{-6})$, and for N iterates. See further details in [90].

**Figure 10.**Probability distributions of the rescaled sums of iterates of the MacMillan map for $(\mu ,\u03f5)=(1.6,1.2)$ are seen to converge to a $(q=1.6)$-Gaussian. This is shown in the left panel for the central part of the distribution (for $N<{2}^{18}$), and in the right panel for the tail part ($N>{2}^{18}$). The number of initial conditions that have been randomly chosen from a square $(0,{10}^{-6})\times (0,{10}^{-6})$ are indicated as well. See further details in [90].

**Figure 11.**Illustrative classical N-body d-dimensional models, with a two-body interaction which asymptotically decays as a $-1/{r}^{\alpha}$ attractive potential ($\alpha \ge 0$). The total energy is extensive (i.e., it grows like N) if $\alpha >d$ (short-range interactions), and nonextensive (i.e., it grows faster than N) if $0\le \alpha \le d$ (long-range interactions). The HMF model corresponds to the α-XY inertial ferromagnetic model [24] with $\alpha =0$. The dotted line $\alpha =(d-2)$ corresponds to the d-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 $d=1$α-XY model. Although not indicated in this figure, the Lennard–Jones model for fluids corresponds to the extensive region since $d=3$ and $\alpha =6$. It is necessary to have $\alpha /d>1$ for the BG partition function to be finite. 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.

**Figure 12.**An example of robust q-Gaussian momenta distribution ($P\left(p\right)\propto {e}_{q}^{-\beta {p}^{2}}$ with $q\simeq 1.6$) associated with a single initial condition [94]. 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 β can change, but not the value of q (within the error bar).

**Figure 13.**Time-averaged (centered and rescaled) distribution of velocities in the Fermi–Pasta–Ulam β model in the neighborhood of the π-mode for $N=128$, ϵ characterizing the distance of a single trajectory to the π-mode. Top panel: For $\u03f5=1$ and $\u03f5=5$ the largest Lyapunov exponent is relatively large, and the corresponding distributions are Gaussians (continuous lines). Bottom panel:: For $\u03f5=0.006$, the largest Lyapunov exponent is close to zero, and the corresponding distribution approaches a ${q}_{attr}$-Gaussian with ${q}_{attr}\simeq 1.5$ (to enable a comparison, a Gaussian distribution is indicated as well). Further details in [95].

**Figure 14.**Distributions of velocities for Kuramoto model with $N=20000$ oscillators. Left panels: When the model parameter K equals 2.53, the LLE is close to 0.06, and the distribution approaches a Gaussian. Right panels: When the model parameter K equals 0.6, the LLE nearly vanishes, and the distribution approaches a q-Gaussian with $q=1.7$. Further details in [97].

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Tsallis, C.
The Nonadditive Entropy S_{q} and Its Applications in Physics and Elsewhere: Some Remarks. *Entropy* **2011**, *13*, 1765-1804.
https://doi.org/10.3390/e13101765

**AMA Style**

Tsallis C.
The Nonadditive Entropy S_{q} and Its Applications in Physics and Elsewhere: Some Remarks. *Entropy*. 2011; 13(10):1765-1804.
https://doi.org/10.3390/e13101765

**Chicago/Turabian Style**

Tsallis, Constantino.
2011. "The Nonadditive Entropy S_{q} and Its Applications in Physics and Elsewhere: Some Remarks" *Entropy* 13, no. 10: 1765-1804.
https://doi.org/10.3390/e13101765