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.