Time evolution of thermodynamic entropy for conservative and dissipative chaotic maps

Abstract We consider several low-dimensional chaotic maps started in far-from-equilibrium initial conditions and we study the process of relaxation to equilibrium. In the case of conservative maps the Boltzmann–Gibbs entropy S ( t ) increases linearly in time with a slope equal to the Kolmogorov–Sinai entropy rate. The same result is obtained also for a simple case of dissipative system, the logistic map, when considered in the chaotic regime. A very interesting results is found at the chaos threshold. In this case, the usual Boltzmann–Gibbs is not appropriate and in order to have a linear increase, as for the chaotic case, we need to use the generalized q -dependent Tsallis entropy S q ( t ) with a particular value of a q different from 1 (when q =1 the generalized entropy reduces to the Boltzmann–Gibbs). The entropic index q appears to be characteristic of the dynamical system.