Exploring the Neighborhood of q-Exponentials

Abstract

The q-exponential form $e_q^x\equiv {[1+(1-q)x]}^{1/(1-q)}(e_1^x=e^x)$ is obtained by optimizing the nonadditive entropy $S_q\equiv k\frac{1-{\sum }_i{p}_i^q}{q-1}$ (with $S_1=S_{BG}\equiv -k{\sum }_i{p}_{i}ln{p}_i$, where BG stands for Boltzmann–Gibbs) under simple constraints, and emerges in wide classes of natural, artificial and social complex systems. However, in experiments, observations and numerical calculations, it rarely appears in its pure mathematical form. It appears instead exhibiting crossovers to, or mixed with, other similar forms. We first discuss departures from q-exponentials within crossover statistics, or by linearly combining them, or by linearly combining the corresponding q-entropies. Then, we discuss departures originated by double-index nonadditive entropies containing $S_q$ as particular case.

1. Introduction

Nonadditive entropies have been used as a basis to explain a diversity of phenomena, from astrophysics to the oscillatory behavior of El Niño [1,2,3], from DNA to financial markets [4,5] from high-energy physics of collisions to granular matter and cold atoms [6,7,8], among many others. It turns out that wide classes of complex systems can be satisfactorily handled within a generalization of Boltzmann–Gibbs (BG) statistical mechanics based on the nonadditive entropy

$$S_q\equiv k\frac{1-{\sum }_{i=1}^W{p}_i^q}{q-1}=k\sum _{i=1}^W{p}_i{ln}_q\frac{1}{p_i}(q\in \mathbb{R};S_1=S_{BG}\equiv -k\sum _{i=1}^W{p}_{i}ln{p}_i;\sum _{i=1}^W{p}_i=1),$$

(1)

where W is the total number of microstates and k is a conventional positive constant (usually $k=k_B$ in physics, and $k=1$ in computational sciences), the q-logarithmic function being defined as ${ln}_{q}z\equiv \frac{z^{1-q}-1}{1-q}$ (${ln}_{1}z=lnz$). This theory is currently referred to as nonextensive statistical mechanics, or q-statistics for short [9,10,11]. The optimization of $S_q$ with simple constraints yields

$$p_i=\frac{e_q^{-{\beta }_q{E}_i}}{{\sum }_{j=1}^W{e}_q^{-{\beta }_q{E}_j}},$$

(2)

where $\left\{E_i\right\}$ are the energy eigenvalues, and the q-exponential function (inverse of the q-logarithmic function) is defined as follows:

$$e_q^x\equiv {[1+(1-q)x]}_{+}^{\frac{1}{1-q}}(q\in \mathbb{R};e_1^x=e^x),$$

(3)

where ${\left[z\right]}_{+}=z$ if $z>0$ and zero if $z\le 0$; notice that this definition implies that, for $q<1$, there is a cutoff at $x_{cutoff}=-1/(1-q)<0$ [9]. In the limit $q\to 1$, Equation (2) recovers the celebrated BG weight.

The aim of the present article is to discuss in detail some departures from a pure q-exponential function which frequently emerge in real situations. Such variations are used in the statistics of nucleotides in full genomes [4], the re-association of folded proteins [12], standard map for intermediate values of the control parameter [13], to mention but a few. We focus on crossover statistics (Section 2), linear combinations of q-exponential functions (Section 3), linear combinations of q-entropies (Section 4), and some two-indices entropies, namely $S_{q,\delta }$[14], $S_{q,q^{\prime }}^{BR}$[15] and $S_{q,q^{\prime }}$ [16] (Section 5).

2. Multiple Crossover Statistics

Crossover statistics is often useful whenever the phenomenon which is focused on exhibits a q-exponential behavior within a range of the relevant variables, and then makes a crossover to another q-exponential function with a different index q. Although rare, it can, in principle, happen that several crossovers successively occur one after the other. We will refer to it as multiple crossover statistics.

Illustrations of such crossovers can be found in [12,17,18,19,20,21].

Let us consider the following ordinary differential equation:

$$\frac{dy}{dx}=-a{y}^q(y\left(0\right)=1;a\in \mathbb{R}).$$

(4)

Its solution is given by

$$y\left(x\right)=e_q^{-ax}.$$

(5)

Multiple crossovers emerge from the following nonlinear ordinary differential equation:

$$\frac{dy}{dx}=-\sum _{k=1}^M{a}_k{y}^{q_k}(q_1<q_2<\cdots <q_M),$$

(6)

with $y\left(0\right)=1$, and $0\le a_1\le a_2\le \cdots \le a_M$, where the right-hand term is constituted by a linear combination of nonlinear terms. Consequently

$$x={\int }_y^1\frac{dz}{{\sum }_{k=1}^M{a}_k{z}^{q_k}}.$$

(7)

We know that Equation (7) has analytical solutions for $M=1$ and $M=2$ (Figure 1). For $M>2$, we need to solve this equation numerically.

Particularly for crossover between two curves ($M=2$) with $q_1$ and $q_2$, we have:

$$\frac{dy}{dx}=-a_1{y}^{q_1}-a_2{y}^{q_2}=-{\mu }_{q_1}{y}^{q_1}-({\lambda }_{q_2}-{\mu }_{q_1})y^{q_2}(y\left(0\right)=1),$$

(8)

where we have identified $(a_1,a_2)\equiv ({\mu }_{q_1},{\lambda }_{q_2}-{\mu }_{q_1})$ in order to facilitate the connection with the notation used in [12]. Let us incidentally mention that this equation enabled the study of the anomalous behavior of folded proteins.

To solve Equation (8), we use Equation (7), which yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x& =\frac{1}{{\mu }_{q_1}}\left\{\frac{y^{1-q_1}-1}{q_1-1}-\frac{\left(\frac{{\lambda }_{q_2}}{{\mu }_{q_1}}\right)-1}{1+q_2-2q_1}\right.\hfill \\ \hfill & \times \left[H\left(1;q_2-2q_1,q_2-q_1,\left(\frac{{\lambda }_{q_2}}{{\mu }_{q_1}}\right)-1\right)\right.\hfill \\ \hfill & \left.-H\left.\left(y;q_2-2q_1,q_2-q_1,\left(\frac{{\lambda }_{q_2}}{{\mu }_{q_1}}\right)-1\right)\right]\right\}\hfill \end{array}\end{array}$$

(9)

with

$$H(y;a,b,c)=y^{1+a}{}_2{F}_1\left(\frac{1+a}{b},1;\frac{1+a+b}{c};-y^{b}c\right),$$

(10)

where ${}_2{F}_1$ is a hypergeometric function.

For the particular case $q_1=1$, we obtain

$$y=\frac{1}{{\left[1-\frac{{\lambda }_{q_2}}{{\mu }_1}+\frac{{\lambda }_{q_2}}{{\mu }_1}{e}^{(q_2-1){\mu }_{1}x}\right]}^{\frac{1}{q_2-1}}}.$$

(11)

It is certainly worth mentioning that its $q_2=2$ instance yields $y={\left[1-\frac{{\lambda }_2}{{\mu }_1}+\frac{{\lambda }_2}{{\mu }_1}{e}^{{\mu }_{1}x}\right]}^{-1}$, whose ${\lambda }_2/{\mu }_1>>1$ asymptotic behavior becomes $y\propto 1/[e^{{\mu }_{1}x}-1]$. It is precisely through this ordinary-differential path that Planck found, in his historical 19 October 1900 paper, the thermostatistical factor which eventually led to his celebrated law for the black-body radiation with the ultimate identification ${\mu }_{1}x\to h\nu /k_{B}T$ [22,23].

For the case $M=3$, we have

$$\frac{dy}{dx}=-a_1{y}^{q_1}-a_3{y}^{q_2}-a_3{y}^{q_3}$$

(12)

whose analytical solution is intractable. Therefore, we use numerical methods to solve it. In contrast, the characteristic values $(x_{c_1},x_{c_2},x_{c_3})$ where changes of behavior of the curve occur are analytically accessible. Those values are obtained through the following considerations. For the characteristic value $x_{c_1}$, we have

$$y\left(x_{c_1}\right)\sim {\left[(q_3-1)a_3{x}_{c_1}\right]}^{-\frac{1}{q_3-1}}\sim 1.$$

(13)

Consequently

$$x_{c_1}=\frac{1}{\left[(q_3-1)a_3\right]}.$$

(14)

For $x_{c_2}$ we have

$$y\left(x_{c_2}\right)\sim {\left[(q_2-1)a_2{x}_{c_2}\right]}^{-\frac{1}{q_2-1}}\sim {\left[(q_3-1)a_3{x}_{c_2}\right]}^{-\frac{1}{q_3-1}},$$

(15)

hence

$$x_{c_2}=\frac{{\left[(q_3-1)a_3\right]}^{\frac{q_2-1}{q_3-q_2}}}{{\left[(q_2-1)a_2\right]}^{\frac{q_3-1}{q_3-q_2}}}.$$

(16)

Similarly, we have

$$x_{c_3}=\frac{{\left[(q_2-1)a_2\right]}^{\frac{q_1-1}{q_2-q_1}}}{{\left[(q_1-1)a_1\right]}^{\frac{q_2-1}{q_2-q_1}}}.$$

(17)

Therefore, for the $M=3$ particular case whose parameter values are $a_1=5\times {10}^{-11},a_2=1\times {10}^{-4}$ and $a_3=1$, with $q_1=1.2,q_2=1.7$ and $q_3=2.7$, we have $x_{c_1}\approx 0.59$, $x_{c_2}\approx 1.68\times {10}^7$ and $x_{c_3}\approx 5.47\times {10}^{13}$, as shown in Figure 2a,b. It is similarly possible to study multiple crossovers for the case $M>3$.

3. Linear Combination of Normalized q-Exponentials

For a linear combination of normalized q-exponentials, we consider a probability distribution function $P=P\left(x\right)$, $x\in X\subset {\mathbb{R}}^{+}$ such that:

$$P\left(x\right)=\sum _{k=1}^M{b}_k{p}_k\left(x\right)=\sum _{k=1}^M{b}_k\frac{e_{q_k}^{-{\beta }_{q_k}x}}{Z_{q_k}}(q_1\le q_2\le \cdots \le q_M<2;{\beta }_{q_k}>0,\forall k),$$

(18)

with ${\sum }_{k=1}^M{b}_k=1(b_k\ge 0)$, $\left\{Z_{q_k}\right\}$ being normalization factors (the upper limit $q<2$ emerges in order to $\left\{Z_{q_k}\right\}$ being finite). Those quantities are determined by imposing, for all $k\in \{1,...,M\}$,

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }dx{p}_k\left(x\right)& =& 1if{q}_k\ge 1,\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {\int }_0^{\frac{1}{{\beta }_{q_k}(1-q_k)}}dx{p}_k\left(x\right)& =& 1if{q}_k<1.\hfill \end{array}$$

(20)

It follows

$$Z_{q_k}=\frac{1}{{\beta }_{q_k}(2-q_k)},\forall q_k<2.$$

(21)

Let us focus on two specific particular cases, namely $M=2$ with $q_1=q_2\equiv q$, and $M=3$ with $q_1=q_2=q_3\equiv q$; ${\beta }_{q_1}\equiv {\beta }_1$, ${\beta }_{q_2}\equiv {\beta }_2$, ${\beta }_{q_3}\equiv {\beta }_3$, and $Z_{q_k}\equiv Z_k$. It follows that

$$p\left(x\right)=b_1\frac{e_q^{-{\beta }_{1}x}}{Z_1}+b_2\frac{e_q^{-{\beta }_{2}x}}{Z_2}$$

(22)

with $b_2=1-b_1$, $1/Z_1={\beta }_1(2-q)$, and $1/Z_2={\beta }_2(2-q)$, and

$$p\left(x\right)=b_1\frac{e_q^{-{\beta }_{1}x}}{Z_1}+b_2\frac{e_q^{-{\beta }_{2}x}}{Z_2}+b_3\frac{e_q^{-{\beta }_{3}x}}{Z_3}$$

(23)

with $b_3=1-b_1-b_2$, $1/Z_1={\beta }_1(2-q)$, $1/Z_2={\beta }_2(2-q)$ and $1/Z_3={\beta }_3(2-q)$. See Figure 3 and Figure 4.

In Figure 4 ($M=3$), we fix the value $q_k=1.11$ for $k=1,2,3$. Another illustration of the linear combination consists of fixing the value ${\beta }_{q_k}=\beta$ for $k=1,2,3$ and using three different values for $q_k$. In the case illustrated in Figure 5, the linear combination remains close to the curve corresponding to $(q,\beta )=(1.2,0.1)$.

$$p\left(x\right)=b_1\frac{e_{q_1}^{-\beta x}}{Z_{q_1}}+b_2\frac{e_{q_2}^{-\beta x}}{Z_{q_2}}+b_3\frac{e_{q_3}^{-\beta x}}{Z_{q_3}}$$

(24)

with $b_3=1-b_1-b_2$, $1/Z_{q_1}=\beta (2-q_1)$, $1/Z_{q_2}=\beta (2-q_2)$ and $1/Z_{q_3}=\beta (2-q_3)$.

Linear combinations of this kind (either of q-exponentials, or of q-Gaussians) have been fruitfully used in [4,13,24,25].

4. Linear Combination of q-Entropies

A linear combination of q-entropies can be written as follows:

$$S\left(\left\{p_i\right\}\right)=\sum _{k=1}^M{c}_k{S}_{q_k}\left(\left\{p_i\right\}\right)(q_1<q_2<\cdots <q_M)(c_k\ge 0).$$

(25)

This expression is generically not normalized. If we happen to prefer normalization for some specific reason, it is enough to divide Equation (25) by ${\sum }_{k=1}^M{c}_k$.

With the constraints ${\sum }_i{p}_i-1=0$ and ${\sum }_i{p}_i{E}_i-U=0$, where U is the internal energy of the system and $\left\{E_i\right\}$ are the energy eigenvalues, we define the functional $f({\alpha }_1,{\alpha }_2,\left\{p_i\right\})$ as follows:

$$f({\alpha }_1,{\alpha }_2,\left\{p_i\right\})\equiv \sum _{k=1}^M{c}_k{S}_{q_k}\left(\left\{p_i\right\}\right)+{\alpha }_1\left(1-\sum _i{p}_i\right)+{\alpha }_2\left(U-\sum _i{p}_i{E}_i\right).$$

(26)

Then, through extremization, we obtain

$$\frac{\partial }{\partial p_j}f=0=\sum _k{c}_k\left\{{ln}_{q_k}\frac{1}{p_j}-{\left(\frac{1}{p_j}\right)}^{1-q_k}\right\}-{\alpha }_1-{\alpha }_2{E}_j$$

(27)

hence

$$E\left(p_j\right)=-\frac{{\alpha }_1}{{\alpha }_2}+\frac{1}{{\alpha }_2}\sum _k{c}_k\left\{{ln}_{q_k}\frac{1}{p_j}-{\left(\frac{1}{p_j}\right)}^{1-q_k}\right\}.$$

(28)

We introduce convenient new variables, namely

$${\alpha }_1\equiv -{\alpha }_2\mu ,{\alpha }_2\equiv \beta .$$

(29)

This enables us to express $X_j\equiv \beta (E_j-\mu )$ as an explicit function of $p_j$, namely

$$X_j=\sum _k{c}_k\left\{{ln}_{q_k}\frac{1}{p_j}-{\left(\frac{1}{p_j}\right)}^{1-q_k}\right\}.$$

(30)

The cutoff condition, whenever present, is given by ${lim}_{p_j\to 0}X(p_j,q_1,q_2,\cdots ,q_M)\equiv X_c(q_1,q_2,\cdots ,q_M)$. For instance, for $M=3$, we have (see Figure 6)

$$X_c(q_1,q_2,q_3)=\frac{c_1}{q_1-1}+\frac{c_2}{q_2-1}+\frac{c_3}{q_3-1},(1<q_1\le q_2\le q_3).$$

(31)

The $M=2$ particular case of (25) has been focused on in [24]:

$$S\left(\left\{p_i\right\}\right)=c_1{S}_{BG}\left(\left\{p_i\right\}\right)+c_2{S}_q\left(\left\{p_i\right\}\right)$$

(32)

where one of the entropies is the BG entropy (i.e., $q_1=1$), and the other one $S_q\left(\left\{p_i\right\}\right)$ corresponds to $q_2\equiv q\ne 1$. Then, we have (see Figure 7)

$$p_j={\left\{aW\left(A_q{e}^{-(q-1)X_j}\right)\right\}}^{\frac{1}{q-1}}$$

(33)

where $W\left(z\right)$ is the Lambert function, implicitly defined by $W{e}^W=z$ (see, for instance, [26]), $A_q\equiv \frac{1}{a}{e}^{-(q-1)\left(1-\frac{c_2}{c_1(q-1)}\right)}$, ${\alpha }_1\equiv -\mu {\alpha }_2$, $\beta \equiv \frac{{\alpha }_2}{c_1}$ and $X_j\equiv \beta (E_j-\mu )$ (this definition of $\beta$ differs from that in Equation (29)), with $a\equiv \frac{c_1}{c_{2}q}=\frac{c_1}{(1-c_1)q}$. $A_q$ is determined via the normalization of the probabilities $\left\{p_j\right\}$, i.e.,

$$\sum _j{p}_j=\sum _j{\left\{aW\left(A_q{e}^{-(q-1)X_j}\right)\right\}}^{\frac{1}{q-1}}=1.$$

(34)

In other words, $A_q$ implicitly depends on $(q,c_1)$. Whenever appropriate, we may go to the continuum limit. If it is allowed to consider $X\ge 0$, we have

$${\int }_0^{\infty }{\left\{aW\left(A_q{e}^{-(q-1)X}\right)\right\}}^{\frac{1}{q-1}}dX=1,$$

(35)

hence

$$q{a}^{-\frac{1}{q-1}}=W{\left(A_q\right)}^{\frac{1}{q-1}}\left[q+W\left(A_q\right)\right].$$

(36)

This expression determines a as an explicit function of $(q,A_q)$.

It is known that, in nonextensive statistical mechanics [27], the constraints under which the entropy is optimized might be chosen with escort distributions, namely, ${\sum }_i{p}_i-1=0$ and $\frac{{\sum }_i{p}_i^q{E}_i}{{\sum }_i{p}_i^q}-U_q=0$. We then have

$$\tilde{f}({\alpha }_1,{\alpha }_2,\left\{p_i\right\})\equiv c_1{S}_{BG}\left(\left\{p_i\right\}\right)+c_2{S}_q\left(\left\{p_i\right\}\right)+{\alpha }_1\left(1-\sum _i{p}_i\right)+{\alpha }_2\left[U_q-\frac{{\sum }_i{p}_i^q{E}_i}{{\sum }_i{p}_i^q}\right]$$

(37)

hence

$$p_j={\left\{a{e}_q^{(q-1)}W\left(B_q{e}_q^{-(q-1)X_j}\right)\right\}}^{\frac{1}{q-1}},$$

(38)

where $X_j\equiv {\beta }^{\prime }(E_j-\mu )$ with ${\beta }^{\prime }$ defined as

$${\beta }^{\prime }\equiv \frac{\beta }{{\sum }_j{p}_j^q+(1-q)\beta U_q}$$

(39)

with $\beta \equiv \frac{{\alpha }_2}{c_1}$. Clearly, $B_q$ is determined by

$$\sum _j{p}_j=\sum _k{\left\{a{e}_q^{(q-1)}W\left(B_q{e}_q^{-(q-1)X_j}\right)\right\}}^{\frac{1}{q-1}}=1.$$

(40)

Let us remind at this point that extremizing $S_q$ with standard constraints is equivalent to extremizing $S_{2-q}$ with escort constraints. The equivalence implies in doing the transformation $q\to 2-q$ [27,28].

Let us address now the concavity/convexity of $S\left\{p_i\right\}$. We illustrate with the linear combination of two ($M=2$) q-entropies with $q_1$ and $q_2$, assuming $p_1\equiv p_2\equiv \cdots \equiv p_{(W-1)}\equiv p$ and $p_W=1-(W-1)p$. In other words, we consider

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_{q_1,q_2}\left(p\right)=& c_1\left[(W-1)p{ln}_{q_1}\left(\frac{1}{p}\right)+(1-(W-1)p){ln}_{q_1}\left(\frac{1}{1-(W-1)p}\right)\right]+\hfill \\ \hfill & c_2\left[(W-1)p{ln}_{q_2}\left(\frac{1}{p}\right)+(1-(W-1)p){ln}_{q_2}\left(\frac{1}{1-(W-1)p}\right)\right].\hfill \end{array}\end{array}$$

(41)

The study of concavity of (41) can be done in the $(q_1,q_2)$ space, taking also into consideration the regions of non admissibility in which the entropy is neither concave nor convex.

We clearly note that when $W=3$ (see Figure 8b), the black region is reduced compared to the $W=2$ case (Figure 8a). This result tends to suggest that the black region tends to disappear at $W\to \infty$, while the pink (convex) region predominates.

5. Other Departures—Two-Indices Entropies

We focus here on other type of departures from pure q-exponentials, originated now from two-indices nonadditive entropies which recover $S_q$ as particular instances.

5.1. $S_{q,\delta }$

From [14], we have

$$S_{q,\delta }\equiv \sum _{i=1}^W{p}_i{\left[{ln}_q\frac{1}{p_i}\right]}^{\delta }(q\in \mathbb{R};\delta >0).$$

(42)

We verify that $S_{q,1}=S_q$. Extremization of $S_{q,\delta }$ under usual constraints yields

$$E\left(p_j\right)=-\frac{{\alpha }_1}{{\alpha }_2}+\frac{1}{{\alpha }_2}\left\{{\left[{ln}_q\frac{1}{p_j}\right]}^{\delta }-\delta {\left(\frac{1}{p_j}\right)}^{1-q}{\left[{ln}_q\frac{1}{p_j}\right]}^{\delta -1}\right\}.$$

(43)

Through (29), we have

$$X_j=\left\{{\left[{ln}_q\frac{1}{p_j}\right]}^{\delta }-\delta {\left(\frac{1}{p_j}\right)}^{1-q}{\left[{ln}_q\frac{1}{p_j}\right]}^{\delta -1}\right\}.$$

(44)

Taking into account the transformation $q\to 2-q$ mentioned below Equation (40), the cutoff occurs for $q>1$, and $X_c(q,\delta )$ is given by (see Figure 9)

$$X_c(q,\delta )={(q-1)}^{-\delta }(q>1).$$

(45)

We verify that $p_{q,\delta }\left(X\right)$ is single-valued for $q\ge \delta$ and multi-valued otherwise.

Let us now consider the case $p_1\equiv p_2\equiv \cdots \equiv p_{(W-1)}\equiv p$ and $p_W=1-(W-1)p$ hence

$$S_{q,\delta }\left(p\right)=(W-1)p{\left[{ln}_q\left(\frac{1}{p}\right)\right]}^{\delta }+(1-(W-1)p){\left[{ln}_q\left(\frac{1}{1-(W-1)p}\right)\right]}^{\delta },$$

(46)

where $p\in \left[0,\frac{1}{W-1}\right]$. This expression will help us to study the concavity/convexity of the entropy for increasing values of W. See Figure 10 and Figure 11.

The black region is clearly reduced for $W=3$ (see Figure 10b), but the purple region at, for example, $\delta =3.8$ and $q=2.15$, invades the concave region. It is not excluded that the purple region gradually expands with W in such way that it approaches the black region.

We noticed that an inadvertence occurred in [14]. Indeed, it was therein indicated that, for all entropies $S_{\delta }$, it would be ${\delta }_c\left(W\right)=1+lnW$, but this is not exactly so in some cases. As we verify in what follows, we always have ${\delta }_c\in (lnW,1+lnW]$. Therefore, the formula in [14] constitutes an upper bound of ${\delta }_c$.

The probability is limited by $p\ge \frac{1}{W-1}$. Numerically, we analyze the plot $1/lnW\times {\delta }_c-lnW$. If it was ${\delta }_c=1+lnW$ for all entropies $S_{\delta }$, we should obtain ${\delta }_c-lnW=1$ for all values of W, which is not the case.

The interpretation of ${\delta }_c$ is given by the transition green ↔ black; no transition black ↔ pink appears to exist.

We notice in Figure 10, Figure 11, Figure 12 and Figure 13 that the divergence of ${\delta }_c$ in the limit $W\to \infty$ means that $S_{\delta }$ is concave in the thermodynamic limit for any positive $\delta$.

5.2. Borges–Roditi Entropy $S_{q,q^{\prime }}^{BR}$

Borges and Roditi [15] extended the entropy $S_q$ as follows:

$$S_{q,q^{\prime }}^{BR}=\frac{{\sum }_{i=1}^W{p}_i^q-{\sum }_{i=1}^W{p}_i^{q^{\prime }}}{q^{\prime }-q},((q,q^{\prime })\in {\mathbb{R}}^2),$$

(47)

with $S_{q,1}^{BR}=S_{1,q}^{BR}=S_q$, where $BR$ stands for Borges–Roditi; notice that $S_{q,q^{\prime }}^{BR}=S_{q^{\prime },q}^{BR}$.

Extremization with usual constraints, and using (29), we have:

$$X_j=\frac{1}{q^{\prime }-q}\left(q{p}_j^{q-1}-q^{\prime }{p}_j^{q^{\prime }-1}\right).$$

(48)

For $q,q^{\prime }<1$, p monotonically decreases to zero when X increases to infinity. For $q,q^{\prime }>1$, p is multivalued, hence physically inadmissible. For $q<1,q^{\prime }>1$ (hence, for $q>1,q^{\prime }<1$ ), p is single-valued and exhibits a cutoff at $X_c$. See Figure 14 for typical examples.

Let us focus now on the concavity of $S_{q,q^{\prime }}^{BR}$. By considering the same case that led to Equation (46), we obtain here

$$S_{q,q^{\prime }}\left(p\right)=\frac{1}{q^{\prime }-q}\left[(W-1)p^q+{(1-(W-1)p)}^q-(W-1)p^{q^{\prime }}-{(1-(W-1)p)}^{q^{\prime }}\right].$$

(49)

The purple region undergoes a slight change whether we compare the Figure 15a ($W=2$) and Figure 15b ($W=3$), although it appears that the rectangular purple region at $W=3$ does not increase for $W>3$. Indeed, if it did that, it would affect the BG and $S_q$ entropies whose convexity/concavity are known. With respect to the black region, the fact of that region shrinks from $W=2$ to $W=3$ suggests that it possibly disappears in $W\to \infty$.

5.3. $S_{q,q^{\prime }}$

On the basis of some algebraic properties, $S_q$ has been generalized in [16,31,32]:

$$S_{q,q^{\prime }}=\sum _{i=1}^W{p}_i{ln}_{q,q^{\prime }}\frac{1}{p_i}$$

(50)

with

$${ln}_{q,q^{\prime }}z\equiv \frac{1}{1-q^{\prime }}\left[exp\left(\frac{1-q^{\prime }}{1-q}(z^{1-q}-1)\right)-1\right].$$

(51)

We verify that ${ln}_{q,1}={ln}_{1,q}={ln}_q$, hence $S_{q,1}=S_{1,q}=S_q$. with $S_{q,1}=S_{1,q}=S_q$. Clearly, we can reformulate (51) in terms of ${ln}_q$ such that

$${ln}_{q,q^{\prime }}z=\frac{1}{1-q^{\prime }}\left[exp\left((1-q^{\prime }){ln}_{q}z\right)-1\right].$$

(52)

The reformulated version of the extremized entropy $S_{q,q^{\prime }}$ is written as

$$X_j=exp\left((1-q^{\prime }){ln}_q\frac{1}{p_j}\right)\left[\frac{1}{1-q^{\prime }}-{\left(\frac{1}{p_j}\right)}^{1-q}\right]-\frac{1}{1-q^{\prime }}$$

(53)

The cutoff equation $X_c(q,q^{\prime })$ is given by

$$X_c(q,q^{\prime })=\frac{1}{1-q^{\prime }}\left[e^{-\frac{1-q^{\prime }}{1-q}}-1\right],q>1.$$

(54)

For $q>1$ and $0<q^{\prime }<1$, p is single-valued and exhibits a cutoff at $X_c$ (see Figure 16). For $q,q^{\prime }<1$, p is multi-valued, hence, it is inadequate for physical purposes. For $0<q<1$ and $q^{\prime }>1$, p exhibits clearly a cutoff.

Analogously to (46), we write the Equation (50) as

$$S_{q,q^{\prime }}\left(p\right)=(W-1)p{ln}_{q,q^{\prime }}\left(\frac{1}{p}\right)+(1-(W-1)p){ln}_{q,q^{\prime }}\left(\frac{1}{1-(W-1)p}\right).$$

(55)

In Figure 17a,b, we observe that the purple region appears to remain the same for all $W\ge 2$. In contrast, the black region for $W=3$ is slightly smaller than that for $W=2$, which suggests that, in $W\to \infty$, such a region might disappear. We checked for large values of W, and this scenario is confirmed. This happens in two different ways: the black region close to the BG point gradually disappears, being replaced by the pink (convex) region, and the black region in the negative part of $q^{\prime }$ also disappears, being replaced by the green (concave) region.

6. Conclusions

In summary, we have explored here various mathematical properties related to extensions of q-exponentials and q-entropies, including some double-index nonadditive entropies.

In the case of crossover statistics (Equation (7)), there are multiple changes in the slopes of the corresponding log-log plots. The values of the abscissa at which the relevant quantities make crossovers between two successive regimes are characterized by $x_c$, analytically calculated in all cases, as illustrated in Figure 1 and Figure 2.

When we consider linear combinations of normalized q-exponentials, we may focus on the influence of the $q_k$’s and of the ${\beta }_k$’s in Equation (18). For a single value of ${\beta }_k$ and various values for the $q_k$’s, the result might be close to one of the q-exponentials, whereas if we adopt a single value of $q_k$ and various values for the ${\beta }_k$’s, the outcome might be sensibly different from all the q-exponentials, as illustrated in Figure 3, Figure 4 and Figure 5.

With respect to the linear combination of q-entropies, it is generically impossible to have the probability distribution $p_j$ in Equation (30) as an explicit function of $X_j$. Notice, however, that we do have $X_j$ as an explicit function of $p_j$. This is in contrast with the case where we have linear combinations of the normalized q-exponentials. The final results for these two types of linear combinations clearly differ, as first shown in [24]. Let us emphasize that, consistently, the operations of linearly combining and entropic extremization do not commute.

In addition to that, for the linear combination of two nonadditive entropies (case M = 2), as well as for the three double-index nonadditive entropies (namely, $S_{q,\delta }$, $S_{q,q^{\prime }}^{BR}$ and $S_{q,q^{\prime }}$), we have studied their convexity/concavity in the indices-space. The results depend naturally on the total number of states (W). The limit $W\to \infty$ is particularly interesting, since it corresponds to the thermodynamical limit. We verify that, in the case of a linear combination of two q-entropies ($M=2$), the concave region remains one and the same for all values of W. Indeed, the value of W only affects the size of the convex region, as illustrated in Figure 8. It seems plausible that, in the $W\to \infty$ limit, the only possibilities which remain are either concave or convex. In what concerns $S_{q,\delta }$, $S_{q,q^{\prime }}^{BR}$ and $S_{q,q^{\prime }}$, regions in the indices-space exist, for a given value of W, where the entropy is concave, or convex, or none of them, as illustrated in Figure 10, Figure 15 and Figure 17. For all these three entropies, the region which is neither concave nor convex does not disappear even for $W\to \infty$. In particular, we have studied in detail the case of $S_{\delta }$ ($q=1$ and $\delta >0$), and have obtained that convexity never emerges, $\forall \delta ,\forall W$. A critical value ${\delta }_c\left(W\right)$ exists such that $S_{\delta }$ is concave for $\delta <{\delta }_c\left(W\right)$ and neither concave nor convex for $\delta >{\delta }_c\left(W\right)$; moreover, in the $W\to \infty$ limit, we verify that ${\delta }_c\left(W\right)\sim lnW$. The results displayed in the present paper could hopefully guide the use of entropies differing from $S_q$ for large classes of natural, artificial and social complex systems.