*2.3. Tsallis Entropy*

In a vast variety of systems that exhibit long-range interactions or long-term memory or being of a multifractal nature, they have been found to be better described by a generalized statistical-mechanical formalism proposed by Tsallis [56,57]. Tsallis, inspired by multifractal concepts, introduced an entropic expression characterized by an index, *qTS*, which leads to non-extensive statistics [56,57]:

$$S\_{\mathbb{P}\_{\rm TS}} = k \frac{1}{q\_{\rm TS} - 1} \left( 1 - \sum\_{i=1}^{W} p\_i^{q\_{\rm TS}} \right) \tag{15}$$

where *qTS* is a real number, *k* is the Boltzmann's constant from statistical thermodynamics, *pi* are probabilities associated with the microscopic configurations, and *W* is their total number. It is important to note that there is a remarkable conceptual similarity between Tsallis' entropy definition and the notion of Rényi entropies.

The entropic index, *qTS*, describes the deviation of Tsallis entropy from the standard Boltzmann–Gibbs entropy. Indeed, using *p* (*qTS*−1) *<sup>i</sup>* <sup>=</sup> *<sup>e</sup>*(*qTS*−1)ln (*pi*) <sup>∼</sup> <sup>1</sup> <sup>+</sup> (*qTS* <sup>−</sup> <sup>1</sup>)ln(*pi*) in the limit *qTS* → 1, we recover the Boltzmann–Gibbs entropy

$$S\_1 = -k \sum\_{i=1}^{W} p\_i \ln(p\_i)\_\prime \tag{16}$$

as the thermodynamic analog of the information-theoretic Shannon entropy. From this point and for the rest of this article, we will refer to the entropy calculated by Equation (16) as the Shannon entropy.

For *qTS* = 1, the entropic index, *qTS*, characterizes the degree of non-extensivity reflected in the following pseudo-additivity rule:

$$\frac{S\_{q\_{TS}}(A+B)}{k} = \frac{S\_{q\_{TS}}(A)}{k} + \frac{S\_{q\_{TS}}(B)}{k} + (q\_{TS}-1)\frac{S\_{q\_{TS}}(A)}{k}\frac{S\_{q\_{TS}}(B)}{k},\tag{17}$$

where *A* and *B* are two subsystems. In case these subsystems have special probability correlations, extensivity does not hold for *qTS* = 1 (*S*<sup>1</sup> = *S*1(*A*) + *S*1(*B*)), but may occur for *SqTS* , with a particular value of the index, *qTS* = 1. Such systems are called nonextensive [56]. The cases *qTS* > 1 and *qTS* < 1 correspond to sub-additivity or superadditivity, respectively. As in the case of Rényi entropies, we may think of *qTS* as a bias parameter: *qTS* < 1 privileges rare events, while *qTS* > 1 highlights prominent events [58].

It is noted that the parameter, *qTS*, itself is not a measure of the complexity of the system but measures the degree of the non-extensivity of the system. In turn, the temporal variations of the Tsallis entropy, *SqTS* , for some *qTS*, quantify the dynamical changes of the complexity of the system. In particular, lower *SqTS* values characterize the portions of the signal with lower complexity [55].
