The rate of entropy increase at the edge of chaos

Details The rate of entropy increase at the edge of chaos The rate of entropy increase at the edge of chaos

1999-07-27

arxiv.org/abs/cond-mat/9907412v4

Phys.Lett. A273 (2000) 97

Physics

Condensed Matter

Statistical Mechanics

6 pages, Latex, 4 figures included, final version accepted for publication in Physics Letters A

Scientific paper

10.1016/S0375-9601(00)00484-9

Under certain conditions, the rate of increase of the statistical entropy of a simple, fully chaotic, conservative system is known to be given by a single number, characteristic of this system, the Kolmogorov-Sinai entropy rate. This connection is here generalized to a simple dissipative system, the logistic map, and especially to the chaos threshold of the latter, the edge of chaos. It is found that, in the edge-of-chaos case, the usual Boltzmann-Gibbs-Shannon entropy is not appropriate. Instead, the non-extensive entropy $S_q\equiv \frac{1-\sum_{i=1}^W p_i^q}{q-1}$, must be used. The latter contains a parameter q, the entropic index which must be given a special value $q^*\ne 1$ (for q=1 one recovers the usual entropy) characteristic of the edge-of-chaos under consideration. The same q^* enters also in the description of the sensitivity to initial conditions, as well as in that of the multifractal spectrum of the attractor.

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