Time evolution of thermodynamic entropy for conservative and dissipative chaotic maps

We consider several low-dimensional chaotic maps started in far-from-equili brium initial conditions and we study the process of relaxation to equilibr ium. In the case of conservative maps the Boltzmann-Gibbs entropy S(t) incr eases linearly in time with a slope equal to the Kolmogorov-Sinai entropy r ate. The same result is obtained also for a simple case of dissipative syst em, the logistic map, when considered in the chaotic regime. A very interes ting results is found at the chaos threshold. In this case, the usual Boltz mann-Gibbs is not appropriate and in order to have a linear increase, as fo r the chaotic case, we need to use the generalized q-dependent Tsallis entr opy S-q(t) with a particular value of a q different from 1 (when q = I the generalized entropy reduces to the Boltzmann-Gibbs). The entropic index q a ppears to be characteristic of the dynamical system. (C) 2001 Elsevier Scie nce Ltd. All rights reserved.