Non extensive approach to the entropy of symbolic sequences

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/lμ. 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.