# Exploring the Neighborhood of q-Exponentials

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## Abstract

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## 1. Introduction

## 2. Multiple Crossover Statistics

## 3. Linear Combination of Normalized q-Exponentials

## 4. Linear Combination of q-Entropies

## 5. Other Departures—Two-Indices Entropies

#### 5.1. ${S}_{q,\delta}$

#### 5.2. Borges–Roditi Entropy ${S}_{q,{q}^{\prime}}^{BR}$

#### 5.3. ${S}_{q,{q}^{\prime}}$

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**$y\left(x\right)$ (log-log plot). For the case $M=1$ with $(q,a)=(2.7,1)$ (blue curve) and, for the case $M=2$, the crossover between two curves, namely with ${q}_{1}=1$ (black curve) and ${q}_{1}=1.7$ (red curve) respectively, both with $({q}_{2},{\lambda}_{{q}_{2}},{\mu}_{{q}_{1}})=(2.7,1,1\times {10}^{-5})$. For the red curve, we have the crossover characteristic values $({x}_{{c}_{1}},{x}_{{c}_{2}})=(0.588,8.407\times {10}^{8})$, which indicate the passage from one regime to another.

**Figure 2.**Crossovers in $y\left(x\right)$ for $M=3$ (log-log plots) (

**a**) between two curves with $({q}_{1},{q}_{2})=(1,1.7)$ (red curve), $({q}_{1},{q}_{2})=(1.2,1.7)$ (blue curve) respectively, both with $({q}_{3},{a}_{1},{a}_{2},{a}_{3})=(2.7,$$5\times {10}^{-11}$,$1\times {10}^{-4}$,1), and (

**b**) a change was done on the blue curve, with ${q}_{1}=-1$ (black curve); the cutoff occurs at ${x}_{cutoff}\approx 4.48\times {10}^{4}$.

**Figure 3.**$p\left(x\right)$ (log-log plot) of three curves (case $M=2$) with parameters $q=1.11$ and $\beta =0.1$ (blue dashed curve), $\beta =1.5$ (red dashed curve), and their linear combination (black curve) with ${b}_{1}=1\times {10}^{-5}$ and ${b}_{2}=1-{b}_{1}$.

**Figure 4.**$p\left(x\right)$ (log-log plots) of four curves with parameters $q=1.11$, $\beta =1.9$ (blue dashed curve), $\beta =1.5$ (red dashed curve), $\beta =1.2$ (gray dashed curve), and their linear combination (black curve). (

**a**) Four curves with $\beta =1.5$ (blue dashed curve), $\beta =1.1$ (red dashed curve), $\beta =0.1$ (gray dashed curve) and their linear combination (black curve). (

**b**) With ${b}_{1}=1\times {10}^{-5}$, ${b}_{2}=1\times {10}^{-3}$ and ${b}_{3}=1-{b}_{1}-{b}_{2}$, both with $q=1.11$ (case $M=3$).

**Figure 5.**$p\left(x\right)$ (log-log plot) of four curves (case $M=3$) with parameters $\beta =0.1$, $q=1.2$ (blue dashed curve), $q=1.5$ (red dashed curve), $q=1.9$ (gray dashed curve), and their linear combination (black curve) with ${b}_{1}=1\times {10}^{-5}$, ${b}_{2}=1\times {10}^{-3}$ and ${b}_{3}=1-{b}_{1}-{b}_{2}$.

**Figure 6.**Four probability distributions ${p}_{{q}_{1},{q}_{2},{q}_{3}}\left(X\right)$ (M = 3) based on Equation (30) with $({c}_{1},{c}_{2},{c}_{3})=(0.641026,0.006410,0.352564)$. From (31), we respectively obtain the cutoff values ${X}_{c}=1.08$ for $({q}_{1},{q}_{2},{q}_{3})=(1.7,2.1,3.2)$ (blue curve), $1.82$ for $({q}_{1},{q}_{2},{q}_{3})=(1.4,1.9,2.7)$ (black curve), $2.50$ for $({q}_{1},{q}_{2},{q}_{3})=(1.3,1.5,2.0)$ (gray curve) and ${X}_{c}=3.72$ for $({q}_{1},{q}_{2},{q}_{3})=(1.2,1.4,1.7)$.

**Figure 7.**Three probability distributions $p\left(X\right)$ based on Equation (33) with ${c}_{1}=0.3$ and $q=1.01$ hence ${A}_{q}=0.0238786$ (black curve), $q=1.2$ hence ${A}_{q}=0.6798077$ (red curve), and $q=1.5$ hence ${A}_{q}=2.3025270$ (blue curve).

**Figure 8.**Concavity/convexity mapping for (41) with $({c}_{1},{c}_{2})=(0.48,0.52)$, $W=2$ (

**a**) and $W=3$ (

**b**). The green (pink) region represents all points whose entropy (41) is concave (convex). The black region represents all points whose entropy is neither concave nor convex, having two local minima points and a local maximum in between (a global maximum point at $p=0.5$ and divergences at $p=0$ and $p=1$). On the red point is localized the Boltzmann–Gibbs entropy and over the red dashed line cutting the origin, we have all the ${S}_{q}$ entropies. On the concave (convex) region we have ${S}_{q},\phantom{\rule{0.166667em}{0ex}}q>0$ ($q<0$). (

**c**) Four ($W=2$) entropies with ${q}_{2}=1$, and ${q}_{1}=1$ (blue curve), ${q}_{1}=0.2$ (green curve), ${q}_{1}=-0.1$ (black curve) and ${q}_{1}=-1$ (pink curve).

**Figure 9.**Illustrative probability distributions ${p}_{q,\delta}\left(X\right)$. (

**a**) $q=1.2$ and $\delta =0.2$ hence, through (45), ${X}_{c}=1.38$ (gray curve); $\delta =0.3$, hence ${X}_{c}=1.62$ (black curve); $\delta =0.5$ hence ${X}_{c}=2.34$ (red curve) and finally, $\delta =0.9$ hence ${X}_{c}=4.26$ (blue curve); (

**b**) $(q,\delta )=(3.1,0.9)$ hence ${X}_{c}=0.51$ (blue curve); $(q,\delta )=(2.7,0.7)$ hence ${X}_{c}=0.69$ (red curve); $(q,\delta )=(2.5,0.6)$ hence ${X}_{c}=0.78$ (black curve); and $(q,\delta )=(2.1,0.4)$ hence ${X}_{c}=0.96$ (gray curve).

**Figure 10.**Concavity/convexity regions for ${S}_{q,\delta}$ (46) (

**a**) $W=2$. (

**b**) $W=3$. The green (pink) region represents all points whose entropy (41) is concave (convex). The black region represents all points whose entropy is neither concave nor convex, having two local maxima ( inflexion) points and another local minimum (maximum) in between. The points of transition at $\delta =2$ are: ${q}_{c}=1/2$ (both $W=2$ and $W=3$) ($\mathrm{pink}\leftrightarrow \mathrm{black}$); ${q}_{c}=4/3$ ($W=2$) and ${q}_{c}\sim 0.98$ ($W=3$) ($\mathrm{black}\leftrightarrow \mathrm{green}$) and ${q}_{c}=2$ (both cases) ($\mathrm{black}\leftrightarrow \mathrm{purple}$). At $q=1$, we have the transition from non concave to concave at ${\delta}_{c}=1+ln2$ ($W=2$) and for $W=3$, we have ${\delta}_{c}<1+ln3$. The blue dashed horizontal line represents ${S}_{\delta}$, while the red dashed vertical line represents all ${S}_{q}$ entropies, and the red point is the BG entropy. (

**c**) Four cases ($W=2$) for $\delta =2$ with the respective colors: $q=0.4$ and $q=1.8$ (convex and concave regions respectively); $q=0.8$ (black region) and $q=2.5$ (purple region) (non concave and non convex regions).

**Figure 11.**Plot for ${S}_{q,\delta}$ with $W=3$, $q=1$ and $\delta =1+ln3$. We clearly observe that ${\delta}_{c}=1+lnW$ is not valid here, because in this value, the entropy is not concave, much less the values close to this.

**Figure 12.**Plot for $1/lnW\times {\delta}_{c}-lnW$ with ${W}_{max}=9\times {10}^{6}$. Here, ${\delta}_{c}\in (lnW,1+lnW]$. In the inset, we indicate the behavior of that function closer to origin.

**Figure 13.**Plot for $1/lnW\times ({\delta}_{c}-1)/lnW$ with ${W}_{max}=9\times {10}^{6}$. The regression by excluding the $W=2$ and $W=3$ points yields an 8th degree polynomial of $x\equiv 1/lnW$, namely $f\left(x\right)\approx 1-0.794252x-6.20252{x}^{2}+60.9556{x}^{3}-223.39{x}^{4}+466.1{x}^{5}-588.297{x}^{6}+420.626{x}^{7}-130.677{x}^{8}$. It means that, when $W\to \infty $ we have $x\to 0$, thus $\underset{x\to 0}{lim}f\left(x\right)=1$, therefore ${\delta}_{c}\sim 1+lnW$ which diverges at infinity.

**Figure 14.**Eight illustrative Borges–Roditi probability distributions. (

**a**) $(q,{q}^{\prime})=(0.2,0.5)$ (red curve); $(q,{q}^{\prime})=(0.4,0.7)$ (black curve); $(q,{q}^{\prime})=(0.6,0.8)$ (blue curve), and $(q,{q}^{\prime})=(0.8,0.9)$ (gray curve). (

**b**) $(q,{q}^{\prime},{X}_{c})=(1.4,0.9,5.42)$ (red curve), $(q,{q}^{\prime},{X}_{c})=(1.8,0.9,3.03)$ (black curve), $(q,{q}^{\prime},{X}_{c})=(2.8,0.9,1.44)$ (blue curve), and $(q,{q}^{\prime},{X}_{c})=(4.8,0.9,0.7)$ (gray curve).

**Figure 15.**Concavity/convexity for ${S}_{q,{q}^{\prime}}^{BR}$ (49) with (

**a**) $W=2$ and (

**b**) $W=3$. The green (pink) region represents all points whose entropy (49) is concave (convex). The black (purple) region represents all points whose entropy is neither concave nor convex, having two local maxima (inflexion) points and another local minimum (maximum) in between. The red dashed vertical lines represent all ${S}_{q}$ entropies and the red point is the BG entropy, while the light (dark) blue lines represents all Shafee ${S}_{q}^{S}$ (Kaniadakis ${S}_{\kappa}^{K}$) entropies [29,30]. (

**c**) Four illustrative cases ($W=2$) with $q=2$ and its respective colors: ${q}^{\prime}=-0.6$ and ${q}^{\prime}=0.9$ (pink and green regions respectively ); $q=-0.1$ (black region) and $q=2.1$ (purple region).

**Figure 16.**Eight illustrative probability distributions ${p}_{q,{q}^{\prime}}\left(X\right)$. (

**a**) $(q,{q}^{\prime},{X}_{c})=(1.5,1.3,1.5)$ (gray curve), $(q,{q}^{\prime},{X}_{c})=(1.4,1.1,2.21)$ (blue curve), $(q,{q}^{\prime},{X}_{c})=(1.3,0.9,3.96)$ (black curve), and $(q,{q}^{\prime},{X}_{c})=(1.2,0.8,8.59)$ (red curve). (

**b**) $(q,{q}^{\prime},{X}_{c})=(0.8,2.5,0.67)$ (gray curve), $(q,{q}^{\prime},{X}_{c})=(0.7,2.0,1.0)$ (blue curve), $(q,{q}^{\prime},{X}_{c})=(0.5,1.5,2.06)$ (black curve), and $(q,{q}^{\prime},{X}_{c})=(0.3,1.3,3.33)$ (red curve).

**Figure 17.**Concavity/convexity for ${S}_{q,{q}^{\prime}}$ (55) with (

**a**) $W=2$ and (

**b**) $W=3$. The green (pink) region represents all points whose entropy (55) is concave (convex). The black (purple) region represents all points whose entropy is neither concave nor convex, having two local maxima (inflexion) points and another local minimum (maximum) in between. The red dashed vertical line represents all ${S}_{q}$ entropies and the red point is the BG entropy. (

**c**) Four cases ($W=2$) with the respective colors: with $q=0.5$, ${q}^{\prime}=0.5$ and ${q}^{\prime}=1.5$ (pink and green regions) and ${q}^{\prime}=0.87$ (black region), and $({q}^{\prime},q)=(1.9,-3)$ (purple region).

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Santos Lima, H.; Tsallis, C.
Exploring the Neighborhood of *q*-Exponentials. *Entropy* **2020**, *22*, 1402.
https://doi.org/10.3390/e22121402

**AMA Style**

Santos Lima H, Tsallis C.
Exploring the Neighborhood of *q*-Exponentials. *Entropy*. 2020; 22(12):1402.
https://doi.org/10.3390/e22121402

**Chicago/Turabian Style**

Santos Lima, Henrique, and Constantino Tsallis.
2020. "Exploring the Neighborhood of *q*-Exponentials" *Entropy* 22, no. 12: 1402.
https://doi.org/10.3390/e22121402