A non extensive approach to the entropy of symbolic sequences

Abstract

Symbolic sequences with long-range correlations are expected to result in a slow regression to a steady state of entropy increase. However, we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained, provided that the extensive entropy of Tsallis with entropic index q is adopted, thereby resulting in a new form of entropy that we shall refer to as Kolmogorov-Sinai-Tsallis (KST) entropy. We assume that the same symbols, either 1 or −1, are repeated in strings of length l, with the probability distribution p(l) ∝ 1 l µ . The numerical evaluation of the KST entropy suggests that at the value µ = 2 a sort of abrupt transition might occur. For the values of µ in the range 1 < µ < 2 the entropic index q is expected to vanish, as a consequence of the fact that in this case the average length < l > diverges, thereby breaking the balance between determinism and randomness in favor of determinism. In the region µ ≥ 2 the entropic index q seems to depend on µ through the power law expression q = (µ − 2) α with α ≈ 0.13 (q = 1 with µ > 3). It is argued that this phase-transition like property signals the onset of the thermodynamical regime at µ = 2.

Symbolic sequences and Tsallis entropy

“…More precisely, we investigate one dimensional symbolic sequences with long-range correlations which are generated by using the numerical experiment presented in Ref. [22]. The procedure uses two random numbers to obtain a lattice with N sites which represent the symbolic sequence.…”

The Thermodynamics of Social Processes: The Teen Birth Phenomenon

“…[10] applied the non-extensive entropy to the analysis of a symbolic sequence with long-range correlation. The sequence examined in Ref.…”

“…The sequence examined in Ref. [10] is equivalent to the CMM model with a vanishing weight for the random component. In this case it is shown that the entropy undergoes a regime of linear increase in time (the time t rather than the logarithmic time τ ) if an entropic index Q = 1 is adopted.…”

“…The traditional KS entropy serves the basic purpose of establishing in an objective way if a process is random. To detect randomness in a sequence characterized by long-range correlation it has been recently proposed [10] to replace the Shannon indicator with the non-extensive Tsallis indicator [11]. However, the computational detection of the Kolmogorov-Sinai-Tsallis (KST) entropy meets the same computational difficulties as that of the ordinary KS entropy.…”

“…The same approach has to be adopted with the KST entropy [10], the only significant difference being that the Shannon entropy is replaced by the Tsallis entropy. It is evident that this way of proceeding is limited to windows of small size, since the number of symbol combinations to take into account grows quickly with the window size and rapidly reaches the maximum size compatible with the current generation of computers.…”