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Generalization of the Kolmogorov-Sinai entropy: Logistic- and periodic-like maps at the chaos threshold | | U. Tirnakli
; G.F.J. Garin
; C. Tsallis
; | | Date: |
12 May 2000 | | Subject: | Statistical Mechanics | cond-mat.stat-mech | | Abstract: | We numerically calculate, at the edge of chaos, the time evolution of the nonextensive entropic form $S_q equiv [1-sum_{i=1}^W p_i^q]/[q-1]$ (with $S_1=-sum_{i=1}^Wp_i ln p_i$) for two families of one-dimensional dissipative maps, namely a logistic- and a periodic-like with arbitrary inflexion $z$ at their maximum. At $t=0$ we choose $N$ initial conditions inside one of the $W$ small windows in which the accessible phase space is partitioned; to neutralize large fluctuations we conveniently average over a large amount of initial windows. We verify that one and only one value $q^*<1$ exists such that the $lim_{t oinfty} lim_{W oinfty} lim_{N oinfty} S_q(t)/t$ is {it finite}, {it thus generalizing the (ensemble version of) Kolmogorov-Sinai entropy} (which corresponds to $q^*=1$ in the present formalism). This special, $z$-dependent, value $q^*$ numerically coincides, {it for both families of maps and all $z$}, with the one previously found through two other independent procedures (sensitivity to the initial conditions and multifractal $f(alpha)$ function). | | Source: | arXiv, cond-mat/0005210 | | Services: | Forum | Review | PDF | Favorites | |
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