Some Properties of Fractal Tsallis Entropy

Abstract

We introduce fractal Tsallis entropy and show that it satisfies Shannon–Khinchin axioms. Analogously to Tsallis divergence (or Tsallis relative entropy, according to some authors), fractal Tsallis divergence is defined and some properties of it are studied. Within this framework, Lesche stability is verified and an example concerning the microcanonical ensemble is given. We generalize the LMC complexity measure (LMC is Lopez-Ruiz, Mancini and Calbert), apply it to a two-level system and define the statistical complexity by using the Euclidean and Wootters’ distance measures in order to analyze it for two-level systems.

1. Introduction

In recent years, the concept of entropy has been intensively studied and multiple generalizations have appeared, i.e., Tsallis entropy, Varma entropy, Rényi entropy, Kaniadakis entropy, fractional entropy, fractal entropy, natural time entropy, etc., with applications in many areas such as earthquakes [1,2,3,4,5,6], stock exchanges [7,8], plasma [9,10,11], signal processing [12], Markov chains [13,14,15], astrophysics [16,17], model selection [18,19], finance [20,21,22,23] and Lie symmetries [24,25].

There are also many applications in medicine, more exactly the following: electroencephalographic (EEG) records [26,27,28], medical images [29,30,31], electromyogram (EMG) records [32,33], electrocardiograms (ECG) ([34] and references therein) and photoplethysmography (PPG) ([35] and references therein).

Deng et al. [26] proposed, for detection at reset state of Alzheimer’s disease complex abnormality, MMSWEP (multivariate multi-scale weighted permutation entropy). This method takes into consideration permutation patterns, that incorporate the correlations between different brain regions, different temporal scales and the amplitude information. In the beginning, the synthetic data are used in order to validate MMSWEP performance; then, the complexity features of Alzheimer’s disease patients are extracted. Moreover, the normal controls in different EEG frequency bands are obtained using the MMSWEP method.

Tylová et al. [27] used the permutation entropy of equidistantly sampled data. This method leads to bias reduction of permutation entropy estimates, time complexities and memory decrease of permutation analysis; hence, there are no limitations for the permutation sample and EEG signal lengths.

Yin et al. [28] introduced MPEr (multiscale permutation Rényi entropy), a new complexity algorithm which is applied to epileptic EEG signals. Its performance is compared to MPE—multiscale permutation entropy; PEr—permutation Rényi entrop; and other complexity measures algorithms. Moreover, the complexity of different epileptic EEG signals and the statistical analysis is performed using MPEr algorithm.

Chen and Ramabadran [29] implemented the entropy-coded DPCM (Differential Pulse Code Modulation) method using two different quantizers, each having a large number of quantization levels. The authors evaluated this method from the point of view of quality of reconstructed images and compression performance using some MR (magnetic resonance) and US (ultrasound) images. They also show that the entropy-coded DPCM method can meaningfully outperform the JPEG (Joint Photographics Experts Group) standard.

Rodrigues and Giraldi [30] proposed a generalization of the pseudo-additive property of Tsallis entropy. The image segmentation is presented as an application of this entropy. As in other papers, we can see that precisely the nonadditiveness of Tsallis entropy helps us to obtain results which are applied in many areas.

Studholme et al. [31] studied the development of entropy-based registration criteria for automatic alignment of multi-modality 3D medical images. The authors introduced a normalized measure. This new measure is defined as the ratio of the sum of the marginal entropies and the joint entropy. Using the experiments on clinical data and a simple image model, the new normalized measure and the effect of changing overlap are compared. The normalized entropy measure proves significantly improved behaviour.

Cuesta-Frau [32] defined SlopEn (Slope Entropy). This concept operates in a different way, by using a novel encoding method based on the slope generated by two consecutive data samples, disregarding time series amplitude information and keeping the symbolic representation of subsequences.

Zhang and Su [33] proposed an improved particle swarm optimization algorithm. The experimental results show that the combined sample entropy algorithm has strong robustness, real-time capability and anti-interference in the signal recognition of dual-channel EMGs.

In order to define most of the entropies mentioned above, a deformed logarithm was used. The fractal and fractional entropies were defined using the classical logarithm. Wang [36] introduced the fractal entropy in order to describe complex systems which exhibit fractal or chaotic phase space. Ubriaco [37] introduced the fractional entropy and used it to study anomalous diffusion in [38]. Radhakrishnan et al. [39] defined a two-parameter generalized entropy which can be reduced, in some particular cases, to these entropies. Using these ideas, we introduce, in this paper, fractal Tsallis entropy. More exactly, the Tsallis logarithm is used instead of the natural logarithm. We prove Shannon–Khinchin axioms corresponding to this entropy. We introduce the fractal Tsallis divergence and study some properties of it. As a thermodynamic property, we check Lesche stability for the aforementioned divergence and give an example concerning microcanonical ensemble. We define a generalization of the LMC (Lopez-Ruiz, Mancini and Calbet) complexity measure (see [40]), by using the fractal Tsallis entropy in order to apply it to a two-level system. We construct complexity measures for disequilibrium distances as Euclidean distance and Wootters’ distance.

2. Fractal Tsallis Entropy

Definition 1. 

For any $\alpha \in {\mathbb{R}}^{*}$ and any $x\in [0,\infty )$, we define the Tsallis logarithm via

$${log}_{\alpha }^{T}x=\left\{\begin{array}{cc}{\displaystyle \frac{x^{\alpha }-1}{\alpha }}\hfill & \mathrm{if}x>0\hfill \\ 0\hfill & \mathrm{if}x=0.\hfill \end{array}\right.$$

For more informations on the properties of the functions defined by Tsallis, we recommend [41]. In this paper, we denote $\alpha =1-q$, where q is the Tsallis entropic index.

We consider $\alpha \in (0,\infty )$, $\beta \in (0,1]$, $n\in {\mathbb{N}}^{*}=\{1,2,\dots \}$ and $p=(p_1,\dots ,p_n)$ a probability distribution, i.e., $p_1,\dots ,p_n\in [0,1]$ such that ${\displaystyle \sum _{i=1}^n{p}_i=1}$.

Definition 2. 

We define the fractal Tsallis entropy by

$${\displaystyle S_{\alpha ,\beta }^T\left(p\right)=S_{\alpha ,\beta }^T(p_1,\dots ,p_n)=-\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left(p_i\right)=\sum _{i=1}^n\frac{p_i^{\beta }-p_i^{\alpha +\beta }}{\alpha }.}$$

Remark 1. 

If $\beta =1$, we obtain Tsallis entropy (see [42,43]) in the previous definition.

Theorem 1. 

We have the following properties for fractal Tsallis entropy:

$1.$$S_{\alpha ,\beta }^T\left(p\right)\ge 0$.

$2.$ The function $S_{\alpha ,\beta }^T$ is continuous in each variable.

$3.$$S_{\alpha ,\beta }^T(p_1,\dots ,p_n,0)=S_{\alpha ,\beta }^T(p_1,\dots ,p_n)$.

$4.$$S_{\alpha ,\beta }^T(p_1,\dots ,p_n)\le S_{\alpha ,\beta }^T\left({\displaystyle \frac{1}{n},\dots ,\frac{1}{n}}\right)$.

Proof. 

$1.$ Because $\alpha +\beta >\beta$ and $p_i\in [0,1]$ for any $i\in \{1,\dots ,n\}$, we have $p_i^{\alpha +\beta }\le p_i^{\beta }$; hence, $S_{\alpha ,\beta }^T\left(p\right)\ge 0$.

$2.$ Because in the expression of $S_{\alpha ,\beta }^T\left(p\right)$ only operations with continuous functions appear, we obtain the conclusion.

$3.$${\displaystyle S_{\alpha ,\beta }^T(p_1,\dots ,p_n,0)=-\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left(p_i\right)-0^{\beta }{log}_{\alpha }^T\left(0\right)=-\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left(p_i\right)=S_{\alpha ,\beta }^T(p_1,\dots ,p_n)}$.

$4.$ Taking into account the continuity (see $2.$), there exists $(p_1^{*},\dots ,p_n^{*})\in {[0,1]}^n$ such that ${\displaystyle \sum _{i=1}^n{p}_i^{*}=1}$ and $S_{\alpha ,\beta }^T(p_1^{*},\dots ,p_n^{*})=\underset{(p_1,\dots ,p_n)\in {[0,1]}^n}{max}{S}_{\alpha ,\beta }^T(p_1,\dots ,p_n)$.

If there is $i\in \{1,\dots ,n\}$ such that $p_i^{*}=1$, then $p_j^{*}=0$ for any $j\in \{1,\dots ,n\}$, $j\ne i$ and $S_{\alpha ,\beta }^T(p_1^{*},\dots ,p_n^{*})=0$, which is minimum, not maximum.

Hence $(p_1^{*},\dots ,p_n^{*})\in {(0,1)}^n$ and $(p_1^{*},\dots ,p_n^{*})$ is a stationary point of $S_{\alpha ,\beta }^T$ conditioned by ${\displaystyle \sum _{i=1}^n{p}_i=1}$. Applying Lagrange’s Multipliers Theorem, we obtain that, for any $i,j\in \{1,\dots ,n\}$, the following relationships hold:

$$\begin{array}{c}\beta {\left(p_i^{*}\right)}^{\beta -1}-(\alpha +\beta ){\left(p_i^{*}\right)}^{\alpha +\beta -1}=\beta {\left(p_j^{*}\right)}^{\beta -1}-(\alpha +\beta ){\left(p_j^{*}\right)}^{\alpha +\beta -1}⟺\\ {\left(p_i^{*}\right)}^{\beta -1}\left(\beta -(\alpha +\beta ){\left(p_i^{*}\right)}^{\alpha }\right)={\left(p_j^{*}\right)}^{\beta -1}\left(\beta -(\alpha +\beta ){\left(p_j^{*}\right)}^{\alpha }\right).\end{array}$$

We shall prove that $p_i^{*}=p_j^{*}$.

If $p_i^{*}<p_j^{*}$, then $\beta {\left(p_i^{*}\right)}^{\beta -1}\ge \beta {\left(p_j^{*}\right)}^{\beta -1}$ and $\beta -(\alpha +\beta ){\left(p_i^{*}\right)}^{\alpha }>\beta -(\alpha +\beta ){\left(p_j^{*}\right)}^{\alpha }$; hence

${\left(p_i^{*}\right)}^{\beta -1}\left(\beta -(\alpha +\beta ){\left(p_i^{*}\right)}^{\alpha }\right)>{\left(p_j^{*}\right)}^{\beta -1}\left(\beta -(\alpha +\beta ){\left(p_j^{*}\right)}^{\alpha }\right)$, which is a contradiction.

If $p_i^{*}>p_j^{*}$, then $\beta {\left(p_i^{*}\right)}^{\beta -1}\le \beta {\left(p_j^{*}\right)}^{\beta -1}$ and $\beta -(\alpha +\beta ){\left(p_i^{*}\right)}^{\alpha }<\beta -(\alpha +\beta ){\left(p_j^{*}\right)}^{\alpha }$; hence

${\left(p_i^{*}\right)}^{\beta -1}\left(\beta -(\alpha +\beta ){\left(p_i^{*}\right)}^{\alpha }\right)<{\left(p_j^{*}\right)}^{\beta -1}\left(\beta -(\alpha +\beta ){\left(p_j^{*}\right)}^{\alpha }\right)$, which is a contradiction.

So $p_i^{*}=p_j^{*}$ for any $i,j\in \{1,\dots ,n\}$ and, because ${\displaystyle \sum _{i=1}^n{p}_i^{*}=1}$, it follows that ${\displaystyle p_i^{*}=\frac{1}{n}}$ for any $i\in \{1,\dots ,n\}$. □

Notation. We denote

$${\displaystyle S_{\alpha ,\beta }^{T,max}=S_{\alpha ,\beta }^T\left({\displaystyle \frac{1}{n},\dots ,\frac{1}{n}}\right)=\frac{n\left({\displaystyle \frac{1}{n^{\beta }}-\frac{1}{n^{\alpha +\beta }}}\right)}{\alpha }=\frac{n^{\alpha }-1}{\alpha n^{\alpha +\beta -1}}.}$$

Lemma 1 

(see [43]). We have the following pseudo-additivity property concerning the Tsallis logarithm:

$${log}_{\alpha }^T\left(xy\right)={log}_{\alpha }^T\left(x\right)+{log}_{\alpha }^T\left(y\right)+\alpha {log}_{\alpha }^T\left(x\right){log}_{\alpha }^T\left(y\right),$$

valid for any $x,y\in (0,\infty )$.

Theorem 2. 

Let $m,n\in {\mathbb{N}}^{*}$. We take, for any $i\in \{1,\dots ,m\}$ and any $j\in \{1,\dots ,n\}$, the probability of occurrence of a joint event, denoted by $p_{ij}$, in which $p_i$ and $q_j$ are probabilities of occurrence of the individual events. We assume that the two events are independent, i.e., $p_{ij}=p_i{q}_j$ for any $i\in \{1,\dots ,m\}$ and $j\in \{1,\dots ,n\}$.

Let $P=(p_{11},\dots ,p_{1n},p_{21},\dots ,p_{mn})$.

Then we have the following pseudo-additivity property for fractal Tsallis entropy:

$$S_{\alpha ,\beta }^T\left(P\right)=\left({\displaystyle \sum _{j=1}^n{q}_j^{\beta }}\right)S_{\alpha ,\beta }^T(p_1,\dots ,p_m)+\left({\displaystyle \sum _{i=1}^m{p}_i^{\beta }}\right)S_{\alpha ,\beta }^T(q_1,\dots ,q_n)+\alpha S_{\alpha ,\beta }^T(p_1,\dots ,p_m)S_{\alpha ,\beta }^T(q_1,\dots ,q_n).$$

Proof. 

Using Lemma 1, we obtain:

$$\begin{array}{c}{\displaystyle S_{\alpha ,\beta }^T\left(P\right)=\sum _{i=1}^m\sum _{j=1}^n{p}_{ij}^{\beta }{log}_{\alpha }^T\left(p_{ij}\right)=\sum _{i=1}^m\sum _{j=1}^n{\left(p_i{q}_j\right)}^{\beta }{log}_{\alpha }^T\left(p_i{q}_j\right)=}\\ {\displaystyle \sum _{i=1}^m\sum _{j=1}^n{p}_i^{\beta }{q}_j^{\beta }\left({log}_{\alpha }^T\left(p_i\right)+{log}_{\alpha }^T\left(q_j\right)+\alpha {log}_{\alpha }^T\left(p_i\right){log}_{\alpha }^T\left(q_j\right)\right)=}\\ \left({\displaystyle \sum _{j=1}^n{q}_j^{\beta }}\right)S_{\alpha ,\beta }^T(p_1,\dots ,p_m)+\left({\displaystyle \sum _{i=1}^m{p}_i^{\beta }}\right)S_{\alpha ,\beta }^T(q_1,\dots ,q_n)+\alpha S_{\alpha ,\beta }^T(p_1,\dots ,p_m)S_{\alpha ,\beta }^T(q_1,\dots ,q_n).\end{array}$$

□

3. Fractal Tsallis Divergence

Let $\alpha \in (0,\infty )$, $\beta \in (0,1]$, $n\in {\mathbb{N}}^{*}$ and $p=(p_1,\dots ,p_n)$, $q=(q_1,\dots ,q_n)$ be two probability distributions. We use the convention ${\displaystyle \frac{0}{0}=0}$ and assume that the following implication is true: $q_i=0⟹p_i=0$.

Definition 3. 

The fractal Tsallis divergence is given by

$${\displaystyle D_{\alpha ,\beta }^T\left(p\right|\left|q\right)=\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i}{q_i}}\right).}$$

Remark 2. 

If $\beta =1$, we obtain Tsallis divergence (see [44,45,46]) in the last definition.

Lemma 2 

(see [47]). For any $x\in (0,\infty )\setminus \left\{1\right\}$, let ${\phi }_x:\mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$,

$${\displaystyle {\phi }_x\left(t\right)=\frac{x^t-1}{t}.}$$

The function ${\phi }_x$ is strictly increasing.

Theorem 3. 

We have the following properties for the fractal Tsallis divergence:

$1.$ If ${\displaystyle \sum _{i=1}^n{p}_i^{\beta }\ge \sum _{i=1}^n{q}_i^{\beta }}$, then $D_{\alpha ,\beta }^T\left(p\right|\left|q\right)\ge 0$.

$2.$ Assume that ${\displaystyle \sum _{i=1}^n{p}_i^{\beta }\ge \sum _{i=1}^n{q}_i^{\beta }}$. The next assertions are equivalent:

$\left(i\right)$$D_{\alpha ,\beta }^T\left(p\right|\left|q\right)=0$.

$\left(ii\right)$$p=q$ (i.e., $p_i=q_i$ for any $i\in \{1,\dots ,n\}$).

$3.$$D_{\alpha ,\beta }^T\left(p\right|\left|q\right)$ is a continuous function in each variable.

$4.$$$\begin{gathered} D_{\alpha ,\beta }^T((p_1,\dots ,p_i,\dots ,p_j,\dots ,p_n)\left|\right|(q_1,\dots ,q_i,\dots ,q_j,\dots ,q_n)) = \\ {D}_{\alpha ,\beta }^T((p_1,\dots ,p_j,\dots ,p_i,\dots ,p_n)\left|\right|(q_1,\dots ,q_j,\dots ,q_i,\dots ,q_n)) \end{gathered}$$.

$5.$$D_{\alpha ,\beta }^T((p_1,\dots ,p_n,0)\left|\right|(q_1,\dots ,q_n,0))=D_{\alpha ,\beta }^T((p_1,\dots ,p_n)\left|\right|(q_1,\dots ,q_n))$.

Proof. 

$1.$${\displaystyle D_{\alpha ,\beta }^T\left(p\right|\left|q\right)=\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i}{q_i}}\right)=\sum _{i=1}^n{p}_i^{\beta }\frac{{\left({\displaystyle \frac{p_i}{q_i}}\right)}^{\alpha }-1}{\alpha }}$.

Using Lemma 2, we obtain that ${\displaystyle \frac{{\left({\displaystyle \frac{p_i}{q_i}}\right)}^{\alpha }-1}{\alpha }\ge \frac{{\left({\displaystyle \frac{p_i}{q_i}}\right)}^{-\beta }-1}{-\beta }}$.

Hence ${\displaystyle D_{\alpha ,\beta }^T\left(p\right|\left|q\right)\ge \sum _{i=1}^n{p}_i^{\beta }\frac{{\left({\displaystyle \frac{p_i}{q_i}}\right)}^{-\beta }-1}{-\beta }=\frac{{\displaystyle \sum _{i=1}^n{p}_i^{\beta }-\sum _{i=1}^n{q}_i^{\beta }}}{\beta }\ge 0}$.

$2.$$\left(i\right)⟹\left(ii\right)$

If there exists $i_0\in \{1,\dots ,n\}$ such that $p_{i_0}\ne q_{i_0}$, then

$${\displaystyle \frac{{\left({\displaystyle \frac{p_{i_0}}{q_{i_0}}}\right)}^{\alpha }-1}{\alpha }>\frac{{\left({\displaystyle \frac{p_{i_0}}{q_{i_0}}}\right)}^{-\beta }-1}{-\beta };}$$

hence

$${\displaystyle \sum _{i=1}^n{p}_i^{\beta }\frac{{\left({\displaystyle \frac{p_i}{q_i}}\right)}^{\alpha }-1}{\alpha }>\sum _{i=1}^n{p}_i^{\beta }\frac{{\left({\displaystyle \frac{p_i}{q_i}}\right)}^{-\beta }-1}{-\beta }=\sum _{i=1}^n\frac{p_i^{\beta }-q_i^{\beta }}{\beta }\ge 0,}$$

which is a contradiction.

So $p_i=q_i$ for any $i\in \{1,\dots ,n\}$.

$\left(ii\right)⟹\left(i\right)$.

If $p_i=q_i$ for any $i\in \{1,\dots ,n\}$, then

$${\displaystyle D_{\alpha ,\beta }^T\left(p\right|\left|q\right)=\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left(1\right)=\sum _{i=1}^n{p}_i^{\beta }·0=0.}$$

$3.$ Because in the expression of $D_{\alpha ,\beta }^T\left(p\right|\left|q\right)$ only operations with continuous functions appear, we obtain the conclusion.

$4.$$$\begin{gathered} D_{\alpha ,\beta }^T((p_1,\dots ,p_i,\dots ,p_j,\dots ,p_n)\left|\right|(q_1,\dots ,q_i,\dots ,q_j,\dots ,q_n)) =p_1^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_1}{q_1}}\right)+ \\ \dots +p_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i}{q_i}}\right)+···+p_j^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_j}{q_j}}\right)+···+p_n^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_n}{q_n}}\right) = \\ {p}_1^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_1}{q_1}}\right)+\dots +p_j^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_j}{q_j}}\right)+···+p_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i}{q_i}}\right)+···+p_n^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_n}{q_n}}\right)= \\ {D}_{\alpha ,\beta }^T((p_1,\dots ,p_j,\dots ,p_i,\dots ,p_n)\left|\right|(q_1,\dots ,q_j,\dots ,q_i,\dots ,q_n)) \end{gathered}$$.

$5.$ Using the convention ${\displaystyle \frac{0}{0}=0}$, we have

$$\begin{array}{c}{\displaystyle D_{\alpha ,\beta }^T((p_1,\dots ,p_n,0)\left|\right|(q_1,\dots ,q_n,0))=\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i}{q_i}}\right)+0^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{0}{0}}\right)=}\\ {\displaystyle \sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i}{q_i}}\right)=D_{\alpha ,\beta }^T((p_1,\dots ,p_n)\left|\right|(q_1,\dots ,q_n)).}\end{array}$$

□

The following example shows that, if we drop the condition ${\displaystyle \sum _{i=1}^n{p}_i^{\beta }\ge \sum _{i=1}^n{q}_i^{\beta }}$ in Theorem 3, we can have $D_{\alpha ,\beta }^T\left(p\right|\left|q\right)<0$.

Example 1. 

Let $n=3$, ${\displaystyle \alpha =\frac{1}{2}}$, ${\displaystyle \beta =\frac{1}{2}}$, $p=(p_1,p_2,p_3)=\left({\displaystyle \frac{4^2}{{100}^2},\frac{{25}^2}{{100}^2},\frac{9^2}{{100}^2}}\right)$ and $q=(q_1,q_2,q_3)=\left({\displaystyle \frac{1}{{100}^2},\frac{{50}^2}{{100}^2},\frac{9^2}{{100}^2}}\right)$.

We remark that

$${\displaystyle \sum _{i=1}^3{p}_i^{\beta }=\frac{38}{100}<\frac{60}{100}=\sum _{i=1}^3{q}_i^{\beta }.}$$

We also have

$${\displaystyle D_{\alpha ,\beta }^T\left(p\right|\left|q\right)=\sum _{i=1}^3\left({\displaystyle \frac{p_i}{q_i^{\frac{1}{2}}}-p_i^{\frac{1}{2}}}\right)=\frac{16}{100}+\frac{25}{2·100}+\frac{9}{100}-\frac{38}{100}<0.}$$

Theorem 4. 

Let $m,n\in {\mathbb{N}}^{*}$, $p^{\left(1\right)}=(p_1^{\left(1\right)},\dots ,p_m^{\left(1\right)})$, $q^{\left(1\right)}=(q_1^{\left(1\right)},\dots ,q_n^{\left(1\right)})$, $p^{\left(2\right)}=(p_1^{\left(2\right)},\dots ,$$p_m^{\left(2\right)})$, $q^{\left(2\right)}=(q_1^{\left(2\right)},\dots ,q_n^{\left(2\right)})$ be probability distributions and $P^{\left(1\right)}=(p_{11}^{\left(1\right)},\dots ,p_{1n}^{\left(1\right)},p_{21}^{\left(1\right)},\dots ,p_{mn}^{\left(1\right)})$, $P^{\left(2\right)}=(p_{11}^{\left(2\right)},\dots ,p_{1n}^{\left(2\right)},p_{21}^{\left(2\right)},\dots ,p_{mn}^{\left(2\right)})$, where $p_{ij}^{\left(1\right)}=p_i^{\left(1\right)}{q}_j^{\left(1\right)}$, $p_{ij}^{\left(2\right)}=p_i^{\left(2\right)}{q}_j^{\left(2\right)}$ for any $i\in \{1,\dots ,m\}$ and $j\in \{1,\dots ,n\}$. Hence

$$\begin{array}{c}{D}_{\alpha ,\beta }^T(P^{\left(1\right)}\left|\right|P^{\left(2\right)})=\left({\displaystyle \sum _{j=1}^n{\left(q_j^{\left(1\right)}\right)}^{\beta }}\right)D_{\alpha ,\beta }^T(p^{\left(1\right)}\left|\right|p^{\left(2\right)})+\left({\displaystyle \sum _{i=1}^m{\left(p_i^{\left(1\right)}\right)}^{\beta }}\right)D_{\alpha ,\beta }^T(q^{\left(1\right)}\left|\right|q^{\left(2\right)})+\\ \alpha D_{\alpha ,\beta }^T(p^{\left(1\right)}\left|\right|p^{\left(2\right)})D_{\alpha ,\beta }^T(q^{\left(1\right)}\left|\right|q^{\left(2\right)}).\end{array}$$

Proof. 

We have:

$$\begin{array}{c}{\displaystyle D_{\alpha ,\beta }^T(P^{\left(1\right)}\left|\right|P^{\left(2\right)})=\sum _{i=1}^m\sum _{j=1}^n{\left(p_{ij}^{\left(1\right)}\right)}^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_{ij}^{\left(1\right)}}{p_{ij}^{\left(2\right)}}}\right)=\sum _{i=1}^m\sum _{j=1}^n{\left(p_i^{\left(1\right)}{q}_j^{\left(1\right)}\right)}^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i^{\left(1\right)}{q}_j^{\left(1\right)}}{p_i^{\left(2\right)}{q}_j^{\left(2\right)}}}\right)=}\\ {\displaystyle \sum _{i=1}^m\sum _{j=1}^n{\left(p_i^{\left(1\right)}\right)}^{\beta }{\left(q_j^{\left(1\right)}\right)}^{\beta }\left({log}_{\alpha }^T\left({\displaystyle \frac{p_i^{\left(1\right)}}{p_i^{\left(2\right)}}}\right)+{log}_{\alpha }^T\left({\displaystyle \frac{q_j^{\left(1\right)}}{q_j^{\left(2\right)}}}\right)+\alpha {log}_{\alpha }^T\left({\displaystyle \frac{p_i^{\left(1\right)}}{p_i^{\left(2\right)}}}\right){log}_{\alpha }^T\left({\displaystyle \frac{q_j^{\left(1\right)}}{q_j^{\left(2\right)}}}\right)\right)=}\\ {\displaystyle \left({\displaystyle \sum _{j=1}^n{\left(q_j^{\left(1\right)}\right)}^{\beta }}\right)\sum _{i=1}^m{\left(p_i^{\left(1\right)}\right)}^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i^{\left(1\right)}}{p_i^{\left(2\right)}}}\right)+\left({\displaystyle \sum _{i=1}^m{\left(p_i^{\left(1\right)}\right)}^{\beta }}\right)\sum _{j=1}^n{\left(q_j^{\left(1\right)}\right)}^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{q_j^{\left(1\right)}}{q_j^{\left(2\right)}}}\right)+}\\ \alpha \left({\displaystyle \sum _{i=1}^m{\left(p_i^{\left(1\right)}\right)}^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{p_i^{\left(1\right)}}{p_i^{\left(2\right)}}}\right)}\right)\left({\displaystyle \sum _{j=1}^n{\left(q_j^{\left(1\right)}\right)}^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{q_j^{\left(1\right)}}{q_j^{\left(2\right)}}}\right)}\right)=\\ \left({\displaystyle \sum _{j=1}^n{\left(q_j^{\left(1\right)}\right)}^{\beta }}\right)D_{\alpha ,\beta }^T(p^{\left(1\right)}\left|\right|p^{\left(2\right)})+\left({\displaystyle \sum _{i=1}^m{\left(p_i^{\left(1\right)}\right)}^{\beta }}\right)D_{\alpha ,\beta }^T(q^{\left(1\right)}\left|\right|q^{\left(2\right)}) +\\ \alpha D_{\alpha ,\beta }^T(p^{\left(1\right)}\left|\right|p^{\left(2\right)})D_{\alpha ,\beta }^T(q^{\left(1\right)}\left|\right|q^{\left(2\right)}).\end{array}$$

□

Because the fractal Tsallis divergence is not generally symmetric in p and q, we define the symmetric measure

$${\displaystyle J_{\alpha ,\beta }^T(p,q)=\frac{1}{2}\left(D_{\alpha ,\beta }^T\left(p\right|\left|q\right)+D_{\alpha ,\beta }^T\left(q\right|\left|p\right)\right),}$$

which has the following properties:

$1.$$J_{\alpha ,\beta }^T(p,q)=J_{\alpha ,\beta }^T(q,p)$.

$2.$ If ${\displaystyle \sum _{i=1}^n{p}_i^{\beta }=\sum _{i=1}^n{q}_i^{\beta }}$, then $J_{\alpha ,\beta }^T(p,q)\ge 0$.

$3.$ Assume that ${\displaystyle \sum _{i=1}^n{p}_i^{\beta }=\sum _{i=1}^n{q}_i^{\beta }}$. The next assertions are equivalent:

$\left(i\right)$$J_{\alpha ,\beta }^T(p,q)=0$

$\left(ii\right)$$p=q$.

In the sequel we withdraw the assumption $q_i=0⟹p_i=0$. The mathematical expression corresponding to the modified fractal Tsallis divergence is

$${\displaystyle {\mathcal{D}}_{\alpha ,\beta }^T(p,q)=D_{\alpha ,\beta }^T\left({\displaystyle p\left|\right|\frac{p+q}{2}}\right)=\sum _{i=1}^n{p}_i^{\beta }\left({log}_{\alpha }^T\left({\displaystyle \frac{p_i}{{\displaystyle \frac{p_i+q_i}{2}}}}\right)\right)=\sum _{i=1}^n{p}_i^{\beta }\left({log}_{\alpha }^T\left({\displaystyle \frac{2p_i}{p_i+q_i}}\right)\right).}$$

This measure satisfies all the aforementioned properties of the fractal Tsallis divergence, but it is not symmetric. As above, we consider

$${\displaystyle {\mathcal{J}}_{\alpha ,\beta }^T(p,q)=\frac{1}{2}\left({\mathcal{D}}_{\alpha ,\beta }^T(p,q)+{\mathcal{D}}_{\alpha ,\beta }^T(q,p)\right).}$$

We also have:

$1.$${\mathcal{J}}_{\alpha ,\beta }^T(p,q)={\mathcal{J}}_{\alpha ,\beta }^T(q,p)$.

$2.$ If ${\displaystyle \sum _{i=1}^n{p}_i^{\beta }\ge \sum _{i=1}^n{\left({\displaystyle \frac{p_i+q_i}{2}}\right)}^{\beta }}$ and ${\displaystyle \sum _{i=1}^n{q}_i^{\beta }\ge \sum _{i=1}^n{\left({\displaystyle \frac{p_i+q_i}{2}}\right)}^{\beta }}$, then ${\mathcal{J}}_{\alpha ,\beta }^T(p,q)\ge 0$.

$3.$ Assume that ${\displaystyle \sum _{i=1}^n{p}_i^{\beta }\ge \sum _{i=1}^n{\left({\displaystyle \frac{p_i+q_i}{2}}\right)}^{\beta }}$ and ${\displaystyle \sum _{i=1}^n{q}_i^{\beta }\ge \sum _{i=1}^n{\left({\displaystyle \frac{p_i+q_i}{2}}\right)}^{\beta }}$. The next assertions are equivalent:

$\left(i\right)$${\mathcal{J}}_{\alpha ,\beta }^T(p,q)=0$.

$\left(ii\right)$$p=q$.

The next proposition gives a relationship between the last two symmetric measures.

Proposition 1. 

We have

$${\displaystyle {\mathcal{J}}_{\alpha ,\beta }^T(p,q)\le \frac{1}{2}{J}_{\frac{\alpha }{2},\beta }^T(p,q).}$$

Proof. 

Because, for any $i\in \{1,\dots ,n\}$,

$${\displaystyle \frac{2p_i}{p_i+q_i}\le \frac{\sqrt{p_i}}{\sqrt{q_i}},}$$

we obtain

$${\displaystyle {log}_{\alpha }^T\left({\displaystyle \frac{2p_i}{p_i+q_i}}\right)\le \frac{1}{2}{log}_{\frac{\alpha }{2}}^T\left({\displaystyle \frac{p_i}{q_i}}\right).}$$

Hence

$${\displaystyle \sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{2p_i}{p_i+q_i}}\right)\le \frac{1}{2}\sum _{i=1}^n{p}_i^{\beta }{log}_{\frac{\alpha }{2}}^T\left({\displaystyle \frac{p_i}{q_i}}\right).}$$

In the same way, it can be seen that

$${\displaystyle \sum _{i=1}^n{q}_i^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{2q_i}{p_i+q_i}}\right)\le \frac{1}{2}\sum _{i=1}^n{q}_i^{\beta }{log}_{\frac{\alpha }{2}}^T\left({\displaystyle \frac{q_i}{p_i}}\right).}$$

We conclude that

$${\displaystyle {\mathcal{J}}_{\alpha ,\beta }^T(p,q)\le \frac{1}{2}{J}_{\frac{\alpha }{2},\beta }^T(p,q).}$$

□

4. Thermodynamic Properties

It is known that, by optimizing the entropy subject to the norm constraint and the energy constraint, we can obtain the canonical probability distribution $p_i$. Using an analogous procedure for the fractal Tsallis entropy, we construct the functional

$$\begin{array}{c}{\displaystyle L_{\alpha ,\beta }^T(p_1,\dots ,p_n)=-\sum _{i=1}^n{p}_i^{\beta }{log}_{\alpha }^T\left(p_i\right)-A\left({\displaystyle \sum _{i=1}^n{p}_i-1}\right)-B\left({\displaystyle \sum _{i=1}^n{p}_i{\epsilon }_i-E}\right)=}\\ {\displaystyle \sum _{i=1}^n\frac{p_i^{\beta }-p_i^{\alpha +\beta }}{\alpha }-A\left({\displaystyle \sum _{i=1}^n{p}_i-1}\right)-B\left({\displaystyle \sum _{i=1}^n{p}_i{\epsilon }_i-E}\right),}\end{array}$$

where A and B are Lagrange’s multipliers, ${\epsilon }_i$ is the energy eigenvalue and E is the internal energy.

Differentiating the functional $L_{\alpha ,\beta }^T$, we obtain

$${\displaystyle \frac{\partial L_{\alpha ,\beta }^T}{\partial p_i}=\frac{\beta p_i^{\beta -1}-(\alpha +\beta )p_i^{\alpha +\beta -1}}{\alpha }-(A+B{\epsilon }_i).}$$

When the functional $L_{\alpha ,\beta }^T$ attains a maximum, its variation with respect to $p_i$ is null; hence

$${\displaystyle \frac{\beta p_i^{\beta -1}-(\alpha +\beta )p_i^{\alpha +\beta -1}}{\alpha }=A+B{\epsilon }_i.}$$

(1)

Integrating the last relationship with respect to $p_i$, we obtain

$${\displaystyle \frac{p_i^{\beta }-p_i^{\alpha +\beta }}{\alpha }=(A+B{\epsilon }_i)p_i+\mathcal{C},}$$

where $\mathcal{C}$ is a real constant.

Now we can find the $p_i$’s for given $\alpha$ and $\beta$ either analytically or numerically. The specific values of $\alpha$ and $\beta$ come from real physical systems.

4.1. Lesche Stability

Lesche introduced in [48,49] a stability criterion to study the stabilities of Rényi and Boltzmann–Gibbs entropies. This stability criterion is concerned with the fact that an infinitesimal change in the probabilities will produce equally infinitesimal changes in the observable; i.e., for any $\epsilon >0$, there exists $\delta >0$ such that for any $n\in {\mathbb{N}}^{*}$ and for any $p=(p_1,\dots ,p_n)$ and $q=(q_1,\dots ,q_n)$, two probability distributions, the following implication is true:

$${\displaystyle \sum _{i=1}^n\left|p_i-q_i\right|<\delta ⟹\frac{\left|S_{\alpha ,\beta }^T\left(p\right)-S_{\alpha ,\beta }^T\left(q\right)\right|}{S_{\alpha ,\beta }^{T,max}}<\epsilon .}$$

In order to show that fractal Tsallis entropy satisfies Lesche stability, we will use the stability criterion from [50]. More exactly, it is sufficient to prove that $\underset{{∥p-q∥}_1\to 0}{lim}\underset{n\to \infty }{lim}B\left({∥p-q∥}_1,n\right)=0$, where

$${\displaystyle {∥p-q∥}_1=\sum _{i=1}^n\left|p_i-q_i\right|}$$

and

$${\displaystyle B\left({∥p-q∥}_1,n\right)=\frac{{\displaystyle {\int }_0^{\frac{{∥p-q∥}_1}{n}}{f}^{-1}\left(t\right)dt+\frac{{∥p-q∥}_1}{n}}}{{\displaystyle {\int }_0^{\frac{1}{n}}{f}^{-1}\left(t\right)dt-\frac{1}{n}{\int }_0^1{f}^{-1}\left(t\right)dt}},}$$

the function $f^{-1}\left(p\right)$ being the inverse probability distribution found in (1).

More exactly, in the last formula, we used Equation (18) from [50], having ${\displaystyle f^{-1}\left(p\right)=\frac{\beta p^{\beta -1}-(\alpha +\beta )p^{\alpha +\beta -1}}{\alpha }}$ and $t_{min}=-1$.

Hence

$$\begin{array}{c}{\displaystyle B\left({∥p-q∥}_1,n\right)=\frac{{\displaystyle {\left({\displaystyle \frac{{∥p-q∥}_1}{n}}\right)}^{\beta }-{\left({\displaystyle \frac{{∥p-q∥}_1}{n}}\right)}^{\alpha +\beta }+\frac{\alpha {∥p-q∥}_1}{n}}}{{\displaystyle \frac{1}{n^{\beta }}-\frac{1}{n^{\alpha +\beta }}}}=}\\ {\displaystyle \left({\displaystyle {\left({\displaystyle \frac{{∥p-q∥}_1}{n}}\right)}^{\beta }-{\left({\displaystyle \frac{{∥p-q∥}_1}{n}}\right)}^{\alpha +\beta }+\frac{\alpha {∥p-q∥}_1}{n}}\right)·\frac{n^{\alpha +\beta }}{n^{\alpha }-1}=}\\ {\displaystyle {\left({∥p-q∥}_1\right)}^{\beta }·\frac{n^{\alpha }}{n^{\alpha }-1}-\frac{{\left({∥p-q∥}_1\right)}^{\alpha +\beta }}{n^{\alpha }-1}+\alpha {∥p-q∥}_1·\frac{n^{\alpha +\beta -1}}{n^{\alpha }-1}.}\end{array}$$

Now, we can easily see that $\underset{{∥p-q∥}_1\to 0}{lim}\underset{n\to \infty }{lim}B\left({∥p-q∥}_1,n\right)=0$; i.e., the Lesche stability criterion is satisfied.

4.2. Generic Example in the Microcanonical Ensemble

The microcanonical ensemble can be used to describe an isolated system in the thermodynamic equilibrium. All the microstates are equally probable in a microcanonical picture. Hence, taking into account the equiprobability, i.e., ${\displaystyle p_i=\frac{1}{W}}$ for any $i\in \{1,\dots ,W\}$, the fractal Tsallis entropy becomes

$$\begin{array}{c}{\displaystyle S_{\alpha ,\beta }^T\left(p\right)=-\sum _{i=1}^W{\left({\displaystyle \frac{1}{W}}\right)}^{\beta }{log}_{\alpha }^T\left({\displaystyle \frac{1}{W}}\right)=W·\frac{1}{W^{\beta }}·\frac{{\displaystyle 1-\frac{1}{W^{\alpha }}}}{\alpha }=W·\frac{{\displaystyle \frac{1}{W^{\beta }}-\frac{1}{W^{\beta +\alpha }}}}{\alpha }=}\\ {\displaystyle \frac{W^{1-\beta }-W^{1-\alpha -\beta }}{\alpha },}\end{array}$$

where W is the total number of microstates. In this case, we call $S_{\alpha ,\beta }^T$ the microcanonical fractal Tsallis entropy derived from the fractal Tsallis entropy.

For this entropy, the temperature is given via

$${\displaystyle \frac{1}{T}=\frac{\partial S_{\alpha ,\beta }^T}{\partial E}=\frac{(1-\beta )W^{-\beta }-(1-\alpha -\beta )W^{-\alpha -\beta }}{\alpha }·\frac{\partial W}{\partial E}.}$$

If $W=C{E}^f$, the temperature is given by

$$\begin{array}{c}{\displaystyle \frac{1}{T}=\frac{(1-\beta ){\left(C{E}^f\right)}^{-\beta }-(1-\alpha -\beta ){\left(C{E}^f\right)}^{-\alpha -\beta }}{\alpha }·fC{E}^{f-1}=}\\ {\displaystyle \frac{(1-\beta )C^{1-\beta }{E}^{(1-\beta )f-1}-(1-\alpha -\beta )C^{1-\alpha -\beta }{E}^{(1-\alpha -\beta )f-1}}{\alpha };}\end{array}$$

i.e.,

$${\displaystyle T=\frac{\alpha }{(1-\beta )C^{1-\beta }{E}^{(1-\beta )f-1}-(1-\alpha -\beta )C^{1-\alpha -\beta }{E}^{(1-\alpha -\beta )f-1}}.}$$

(2)

Although Formula (2) is given only for the microcanonical ensemble, a direct extension of this method to include other kinds of adiabatic ensembles can be obtained.

5. Complexity Measures

For a system consisting of N accessible states with a set of probabilities $P=(p_1,\dots ,p_n)$ such that ${\displaystyle \sum _{i=1}^N{p}_i=1}$, the complexity measure is given as $C\left(P\right)=S\left(P\right)D\left(P\right)$, where ${\displaystyle D\left(P\right)=\sum _{i=1}^N{\left({\displaystyle p_i-\frac{1}{N}}\right)}^2}$.

The LMC (López-Ruiz, Mancini and Calbet) measure of complexity is considered to be a nonextensive quantity. Yamano [51] proposed a generalized measure of complexity based on Tsallis entropy with a view to absorbing the nonadditive features of the entropy. Using these ideas, we define a statistical measure corresponding to the fractal Tsallis entropy via

$$C_{\alpha ,\beta }^T\left(P\right)=S_{\alpha ,\beta }^T\left(P\right)D\left(P\right)=-\left({\displaystyle \sum _{i=1}^N{p}_i^{\beta }{log}_{\alpha }^T\left(p_i\right)}\right)\left({\displaystyle \sum _{i=1}^N{\left({\displaystyle p_i-\frac{1}{N}}\right)}^2}\right).$$

In cases in which we work with a two-level system with probabilities $P=(p,1-p)$, the expressions for the entropy and the disequilibrium measure are as follows:

$${\displaystyle S_{\alpha ,\beta }^T\left(P\right)=-p^{\beta }{log}_{\alpha }^T\left(p\right)-{(1-p)}^{\beta }{log}_{\alpha }^T(1-p)=\frac{p^{\beta }-p^{\alpha +\beta }+{(1-p)}^{\beta }-{(1-p)}^{\alpha +\beta }}{\alpha },}$$

and

$$D\left(P\right)=2{\left({\displaystyle p-\frac{1}{2}}\right)}^2,$$

respectively.

Having this framework, the statistical complexity is

$${\displaystyle C_{\alpha ,\beta }^T\left(P\right)=\frac{2\left[p^{\beta }-p^{\alpha +\beta }+{(1-p)}^{\beta }-{(1-p)}^{\alpha +\beta }\right]}{\alpha }·{\left({\displaystyle p-\frac{1}{2}}\right)}^2.}$$

The natural step is to generalize this measure to the continuous case, considering the probability distribution function $p\left(x\right)$ with the normalization condition ${\displaystyle {\int }_{-\infty }^{\infty }p\left(x\right)dx=1}$. Fractal Tsallis entropy and the disequilibrium measure are, in this case,

$${\displaystyle S_{\alpha ,\beta }^T\left(P\right)=-{\int }_{-\infty }^{\infty }{\left(p\left(x\right)\right)}^{\beta }{log}_{\alpha }^T\left(p\left(x\right)\right)dx,}$$

$${\displaystyle D\left(P\right)={\int }_{-\infty }^{\infty }{\left(p\left(x\right)\right)}^{2}dx,}$$

respectively.

With this framework, the fractal Tsallis generalization of the LMC complexity measure is

$$C_{\alpha ,\beta }^T\left(P\right)=\left({\displaystyle -{\int }_{-\infty }^{\infty }{\left(p\left(x\right)\right)}^{\beta }{log}_{\alpha }^T\left(p\left(x\right)\right)dx}\right)\left({\displaystyle {\int }_{-\infty }^{\infty }{\left(p\left(x\right)\right)}^{2}dx}\right).$$

Martin et al. [52] introduced a modified definition in order to scale the complexity to lie in the interval $[0,1]$. The modified statistical measure of complexity corresponding to fractal Tsallis entropy is given as follows:

$${\displaystyle {\mathcal{C}}_{\alpha ,\beta }^T\left(P\right)=\frac{S_{\alpha ,\beta }^T\left(P\right)}{S_{\alpha ,\beta }^{T,max}}·{\mathcal{D}}_x\left(P\right),}$$

where ${\mathcal{D}}_x$ is the normalized disequilibrium measure based on a particular distance measure x.

In the sequel we give some examples of the disequilibrium based on different distance measures.

5.1. Disequilibrium Measure Based on Euclidean Distance

In this case, the disequilibrium ${\mathcal{D}}_E$ is given by

$${\displaystyle {\mathcal{D}}_E\left(P\right)={\mathcal{N}}_E\sum _{i=1}^N{∥(p_1,\dots ,p_N)-\left({\displaystyle \frac{1}{N},\dots ,\frac{1}{N}}\right)∥}_E,}$$

where N is the number of accessible states and ${\displaystyle {\mathcal{N}}_E=\frac{N}{N-1}}$.

In the case of a two-level system with probabilities $P=(p,1-p)$, we have:

$${\displaystyle S_{\alpha ,\beta }^T\left(P\right)=-p^{\beta }{log}_{\alpha }^T\left(p\right)-{(1-p)}^{\beta }{log}_{\alpha }^T(1-p)=\frac{p^{\beta }-p^{\alpha +\beta }+{(1-p)}^{\beta }-{(1-p)}^{\alpha +\beta }}{\alpha },}$$

$${\displaystyle S_{\alpha ,\beta }^{T,max}=\frac{2^{1-\beta }-2^{1-\alpha -\beta }}{\alpha }}$$

and

$${\mathcal{D}}_E\left(P\right)={\left({\displaystyle p-\frac{1}{2}}\right)}^2+{\left({\displaystyle 1-p-\frac{1}{2}}\right)}^2=2{\left({\displaystyle p-\frac{1}{2}}\right)}^2.$$

Hence, in this case,

$$\begin{array}{c}{\displaystyle {\mathcal{C}}_{\alpha ,\beta }^T\left(P\right)=\frac{{\displaystyle \frac{p^{\beta }-p^{\alpha +\beta }+{(1-p)}^{\beta }-{(1-p)}^{\alpha +\beta }}{\alpha }}}{{\displaystyle \frac{2^{1-\beta }-2^{1-\alpha -\beta }}{\alpha }}}·2{\left({\displaystyle p-\frac{1}{2}}\right)}^2=}\\ {\displaystyle \frac{2\left[p^{\beta }-p^{\alpha +\beta }+{(1-p)}^{\beta }-{(1-p)}^{\alpha +\beta }\right]}{2^{1-\beta }-2^{1-\alpha -\beta }}·{\left({\displaystyle p-\frac{1}{2}}\right)}^2.}\end{array}$$

However, the Euclidean distance used above do not take into account the stochastic nature of the probabilities. In order to avoid this, the statistical distance measure introduced by Wootters (see [53]) can be used to redefine the disequilibrium.

5.2. Disequilibrium Measure Based on Wootters’ Distance

Wootters [53] introduced a new statistical measure in which the distance between two quantities is determined by the size of statistical fluctuations in a measurement. Let $P=(p_1,\dots ,p_N)$ and $Q=(q_1,\dots ,q_N)$ be two probability distributions. Assume that they are indistinguishable and the difference between them is smaller than the size of the typical fluctuation. Then the statistical distance is the shortest curve connecting two points in the probability space. This shortest distance is equal to the angle between the two probability vectors and is given by

$$D_W(P,Q)={∥P-Q∥}_W={cos}^{-1}\left({\displaystyle \sum _{i=1}^N\sqrt{p_i}\sqrt{q_i}}\right).$$

In the case $Q=\left({\displaystyle \frac{1}{N},\frac{1}{N},\dots ,\frac{1}{N}}\right)$, we denote

$$D_W\left(P\right)={∥(p_1,\dots ,p_N)-\left({\displaystyle \frac{1}{N},\dots ,\frac{1}{N}}\right)∥}_W={cos}^{-1}\left({\displaystyle \sum _{i=1}^N\sqrt{{\displaystyle \frac{p_i}{N}}}}\right).$$

In the case the two probability distributions coincide with each other, this measure vanishes. It attains the maximum value in cases in which each outcome has a positive probability according to one distribution and zero probability according to the other distribution. Using Wootters’ distance, Martin et al. [52] defined the disequilibrium measure for the extensive Boltzmann–Gibbs entropy. We continue this work, dealing with fractal Tsallis entropy. The disequilibrium defined using a normalized version of the statistical distance is given via

$${\mathcal{D}}_W(P,Q)={\mathcal{N}}_W{cos}^{-1}\left({\displaystyle \sum _{i=1}^N\sqrt{p_i}\sqrt{q_i}}\right),$$

where N is the number of accessible states and ${\displaystyle {\mathcal{N}}_W=\frac{1}{{cos}^{-1}\left({\displaystyle \frac{1}{\sqrt{N}}}\right)}}$ is the corresponding normalization constant.

As above, in the case $Q=\left({\displaystyle \frac{1}{N},\dots ,\frac{1}{N}}\right)$, we denote

$${\mathcal{D}}_W\left(P\right)={\mathcal{N}}_W{cos}^{-1}\left({\displaystyle \sum _{i=1}^N\sqrt{{\displaystyle \frac{p_i}{N}}}}\right).$$

The statistical measure of complexity of a two-level system with probabilities $P=(p,1-p)$, computed using this disequilibrium distance, is

$${\displaystyle {\mathcal{C}}_{\alpha ,\beta }^T\left(P\right)=\frac{p^{\beta }-p^{\alpha +\beta }+{(1-p)}^{\beta }-{(1-p)}^{\alpha +\beta }}{2^{1-\beta }-2^{1-\alpha -\beta }}·\frac{{cos}^{-1}\left(\sqrt{{\displaystyle \frac{p}{2}}}+\sqrt{{\displaystyle \frac{1-p}{2}}}\right)}{{cos}^{-1}\left({\displaystyle \frac{1}{\sqrt{2}}}\right)}.}$$

Remark 3. 

We use the hypothesis $\beta \in (0,1]$ only at Theorem 1 point 3 and when we show that Lesche stability is valid for fractal Tsallis entropy. The rest of the proofs are valid without this assumption. Of course, in the absence of this hypothesis, we do not know the value of $S_{\alpha ,\beta }^{T,max}$.

6. Conclusions

We introduced fractal Tsallis entropy and fractal Tsallis divergence, which generalize Tsallis entropy and Tsallis divergence, respectively. The standard properties, like Shannon–Khinchin axioms, are verified. We also showed that, in some cases, Lesche stability is preserved. A generalization of the LMC complexity measure is introduced, which is applied to measure the complexity of a two-level system. We analyzed the complexity of a given system by using a normalized form of the entropy and distance measure. The complexity is defined with the help of two different distance measures, namely the normalized form of the Euclidean distance and the normalized Wootters’ distance; then, the complexity of a two-level system is thoroughly analyzed. Because the Wootters’ distance for the disequilibrium measure captures the stochastic effects of the probability distribution, we remark that it is more appropriate to use it. Motivated by the applications of entropy in medical image processing, we hope that this new entropy is a step forward in this framework.